irreversibility and universality in quantum computation · 2016-01-22 · university of cagliari...

University of Cagliari

Irreversibility and Universality in

Quantum Computation

by

Giuseppe Sergioli

S.S.D.: M/Fil 02

Dipartimento di Scienze Pedagogiche e Filosofiche

Dottorato in Storia, Filosofia e Didattica delle Scienze, XXI ciclo

Supervisor: Professor Roberto Giuntini

http://www.unica.it

http://www.scform.unica.it/pub/netdocs/11/index.php?module=documents&JAS_DocumentManager_op=viewDocument&JAS_Document_id=805

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To my father

About this thesis

At the end of the XIX century, classical physics was in a position to solve a very large

number of problems [33]: Maxwell’s equations and classical statistical mechanics were

able to provide a good description of electromagnetic and thermodynamic phenomena,

but were considered only as a pragmatic approximation to the true laws of physics.

For instance, classical statistical mechanics could not explain how the walls of an empty

cavity would ever reach equilibrium with the electromagnetic radiation enclosed in that

cavity: the walls of the cavity are made of atoms, which can absorb or emit radiation.

The number of these atoms is about 1025 and the walls have a finite number of degrees of

freedom. The radiation field, on the other hand, can be Fourier analysed in orthogonal

modes, and its energy is distributed among these modes. In each one of the modes,

the field oscillates with a fixed frequency, like a harmonic oscillator. Thus, the radia-

tion is dynamically equivalent to an infinite set of harmonic oscillators. Under these

circumstances, the law of equipartition of energy (E = kT per harmonic oscillator, on

the average) can never be satisfied: the vacuum in the cavity, having an infinite heat

capacity, would absorb all the thermal energy of the walls. Agreement with experimen-

tal data could be obtained only by modifying, ad hoc, some laws of physics. Planck

assumed that energy exchanges between an atom and a radiation mode of frequency υ

could occur only in integral multiples of hυ (the so called quantum of energy), where h

is the so called Planck constant. Soon afterwards, Einstein sharpened Planck’s hypoth-

esis in order to explain the photoelectric effect -the ejection of electrons from materials

irradiated by light. Einstein did not go as far as to explicitly write that light consisted

of particles, but this was strongly suggested by this work.

The notion of physical reality acquires a new meaning with quantum phenomena, dif-

ferent from its meaning in classical physics. The meaning of position, energy, and time

in the quantum context are very different from the classical ones. So a new language to

describe the quantum world is in order.

This language develops out of two primitive notions: preparations and tests.

A preparation is an experimental procedure that is completely specified, but it may

involve also stochastic processes, e.g. thermal fluctuations, provided that the statistical

properties of the stochastic process are known, or at least reproducible.

A test, like a preparation, is an experimental procedure that is completely specified, but

it also includes a final step, where the information on the state of the physical system

is supplied to the observer. In a strict sense, quantum theory is a set of rules allowing

the computation of probabilities for the outcomes of tests under specified preparations.

At the very beginning of the XX century, quantum mechanics and computation theory

were two fundamental theories studied in completely separated ways. Subsequently, the

increasing miniaturisation of the hardware parts of computing devices and the strenu-

ous attempts to increase computational efficiency demanded a new idea of computation.

The first author who envisaged an application of quantum mechanics to computation

theory was Richard Feynman. He demonstrated [20] that no Turing machine could ever

simulate some physical systems without incurring into an exponential performance slow-

down, while an universal quantum simulator would perform far more efficiently. After

the seminal work of David Deutsch, who provided in 1985 [18] the first mathematical

framework for the so called universal quantum Turing machine, the literature concern-

ing what we can call the quantum approach to computation has enormously increased,

bringing to what, nowadays, is a science of its own: quantum computation.

This thesis consists of tree chapters and five appendices: in the first chapter we introduce

some basic notions of quantum computation and we try to focus on the differences

between classical and quantum computation. More precisely, we stress the fact that if on

the one hand classical computation is essentially irreversible, on the other hand quantum

computation is not. The aim of the second chapter is to show how the behaviour of

certain irreversible transformations can be exactly or approximately simulated via some

particular quantum operations, termed polynomial quantum operations. In the third

chapter we deal with a fundamental issue of quantum computation: universality. We

consider the universal set whose members are the Toffoli gate and Hadamard gate [3],

and we propose an algebraic investigation of its main features.

Contents

About this thesis ii

List of Figures vi

1 Reversibility and irreversibility in quantum computation 1

1.1 Basic notions of quantum computation . . . . . . . . . . . . . . . . . . . . 2

1.1.1 Qubits and superposition states . . . . . . . . . . . . . . . . . . . . 2

1.1.2 Tensor spaces, factorised states, quregisters . . . . . . . . . . . . . 4

1.1.3 Quantum gates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.1.4 Semiclassical and genuinely quantum gates . . . . . . . . . . . . . 8

1.1.5 Genuine entanglement gates . . . . . . . . . . . . . . . . . . . . . . 11

1.2 From pure to mixed states . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.3 Reversible and irreversible transformations . . . . . . . . . . . . . . . . . . 18

2 Irreversible quantum tranformations 21

2.1 Quantum gates from statistical operators . . . . . . . . . . . . . . . . . . 21

2.1.1 The Lukasiewicz truncated sum and its properties . . . . . . . . . 23

2.1.2 Fuzzy irreducibility of genuine quantum gates . . . . . . . . . . . . 23

2.2 Representing same irreversible transformation as quantum operation . . . 29

2.2.1 Polynomial quantum operations . . . . . . . . . . . . . . . . . . . . 29

2.2.2 Representing IAND and Lukasiewicz transformations as quantumoperations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3 An approximately universal system of quantum computational gates 38

3.1 The Toffoli and the Hadamard gates . . . . . . . . . . . . . . . . . . . . . 40

3.2 Reversible and irreversible quantum computational structures . . . . . . . 44

3.3 The complex quantum computational algebra . . . . . . . . . . . . . . . . 46

A Preliminaries 51

B Basic of linear algebra and functional analysis 55

B.1 Hilbert spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

B.1.1 Operators of a Hilbert space . . . . . . . . . . . . . . . . . . . . . 57

C The von Neumann’s axiomatization of quantum theory 61

C.1 Quantum computational background . . . . . . . . . . . . . . . . . . . . . 63

iv

C.1.1 Qubits and superposition states . . . . . . . . . . . . . . . . . . . . 63

D What is a quantum computer? 66

E Bibliographical remarks 69

Acknowledgements 70

Bibliography 71

List of Figures

1.1 The Bloch sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 The Mach-Zehnder interferometer . . . . . . . . . . . . . . . . . . . . . . . 10

1.3 quregisters Vs. qumixes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.4 The square . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.5 quregisters Vs. qumixes II . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.1 The Lukasiewicz function . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.2 The approximating polynomial . . . . . . . . . . . . . . . . . . . . . . . . 36

2.3 A comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

vi

Chapter 1

Reversibility and irreversibility in

quantum computation

Classical circuit theory is basically irreversible in the sense that Boolean functions (gates)

are generally described as many-to-one: the same output-bits may correspond to dif-

ferent input-bits. We know, however, that every Boolean gate has its own reversible

counterpart, as shown by Toffoli [39]. The main idea is to consider the input-bits of

a reversible gate as composed by two parts: a control -component which carries over

the “actual” input-value and a target-component (ancilla), whose final value (after the

application of the gate) represents the “actual” output. The price to pay is the increase

of the computational space, due to the number of extra ancilla-bits needed to make

the circuit reversible. Unlike the classical circuit model, quantum computation “orig-

inates” in a naturally reversible way, since quantum gates are interpreted as unitary

operators acting on pure states (qubits or quregisters) of the Hilbert space associated

with the quantum circuit at issue. Being unitary, quantum gates represent reversible

time-evolution of the circuit in question.

What we mentioned so far is formulated in the usual approach to quantum computation,

which is essentially based on quregisters and unitary operators of convenient Hilbert

spaces. However, such a representation is unduly restrictive, since it does not encompass

open systems, where interactions with the environment and some measurement-processes

may occur. In this case, the time-evolution of the system is no longer reversible.

To deal with these phenomena a more general model is required, and it can be obtained

by replacing quregisters and unitary operators by density operators (qumixes) and by

unitary quantum operations, respectively (see [4] and [24]).

1

Reversibility and Irreversibility 2

From a physical point of view, using qumixes instead of quregisters has plenty of ad-

vantages. In fact, physical systems are always somehow interacting with the environ-

ment. Hence, quantum states are better represented by qumixes (mixed states) instead

of quregisters (pure states). Moreover, as shown by Aharonov, Kitaev and Nisan [4],

taking into account quantum circuits with qumixes allows us to treat some critical prob-

lems, such as measurements in the middle of a computation, decoherence, noise, and so

on, which cannot be adequately modelled in the usual approach to quantum comput-

ing. Also, the Aharonov-Kitaev-Nisan model and the standard model are polynomially

equivalent in computational power [4].

1.1 Basic notions of quantum computation

1.1.1 Qubits and superposition states

In quantum mechanics a physical system is mathematically interpreted as a Hilbert

space. We say that a state, i.e. a unit vector in such a Hilbert space[10], is pure if and

only if it represents a maximal information quantity, i.e. an information on the physical

system that could not be consistently augmented by any further observation.

Consider a two-dimensional Hilbert space H, and let {|0〉 , |1〉} be its canonical orthonor-

mal basis. The quantum computational counterpart of the bit -the basic information

quantity of classical information theory- is the quantum bit (qubit), i.e. any unit vector

|ψ〉 in C2. The general form of a qubit is:

|ψ〉 = a0 |0〉+ a1 |1〉 ,

where a0, a1 are complex numbers such that |a0|2+|a1|2 = 1, as required by the unitarity

hypothesis.1

Qubits, therefore, correspond to pure states: in fact, as dictated by the Born rule,

• |a0|2 yields the probability of the information described by the pure state |0〉,which, from a logical viewpoint, corresponds to falsity;

• |a1|2 yields the probability of the information described by the pure state |1〉,corresponding to truth.

Therefore, |0〉 and |1〉 represent maximal and certain pieces of information, while a

superposition |ψ〉 corresponds to a maximal but uncertain piece of information. A

1For an exposition of the basics of linear algebra and functional analysis see Appendix A.


superposition |ψ〉 of the states |0〉 and |1〉 is a new state absolutely distinct from both

|0〉 and |1〉; this typically holistic phenomenon is known as the superposition principle [5].

For example, consider an idealised atom with a single electron and two energy levels:

a ground state (identified with |0〉), which we suppose to be the current state of the

electron, and an excited state (identified with |1〉). By shining a light pulse of half the

duration as the one needed to perform a change of the energy level from |0〉 to |1〉, we

can effect a “half-flip” between the two logical states. The ensuing state of the atom

is neither |0〉 nor |1〉, but rather a superposition of both states: |ψ〉 = a0 |0〉 + a1 |1〉.The electron is neither in the ground state, nor in the excited state, but “halfway in

between”.

Suppose, now, that we measure the energy of such an electron. The measurement process

will not admit an uncertain result: the electron must be detected in either one of the

two levels. The respective probabilities that the electron will be detected in the ground

or in the excited level will be |a0|2 and |a1|2. The electron has changed again its energy

level, since the measurement procedure “has forced” |ψ〉 to collapse into only one of the

two possible states. In some sense (see e.g. [13]) the measurement procedure tells us

nothing about how was |ψ〉 before the measurement, but it causes an irreversible change

of the initial state |ψ〉.

Let |ψ〉 , |φ〉 ∈ H be two vectors differing only by a complex number c, i.e. |ψ〉 =

c |φ〉. Since |ψ〉 and |φ〉 represent the same physical state, we will say that |ψ〉 and

|φ〉 are equivalent and we will write |ψ〉 ' |φ〉. Consequently, the proper phase space

of a quantum system is not the original Hilbert space H, but rather the space of rays

associated to it, called projective Hilbert space:

P (H) = H/ ' .

Points in P (H) are 1-dimensional rays in H or, equivalently, 1-dimensional projectors

|ψ〉 → P|ψ〉.

Clearly, |ψ〉 ' |φ〉 iff P|ψ〉 = P|φ〉.

It can also be seen that normalized vectors in Cn+1 determine a 2n + 1-dimensional

sphere

S2n+1 ={|ψ〉 ∈ Cn+1 : 〈ψ |ψ〉 = 1

}.

Given two points |ψ〉 and |φ〉 in S2n+1, they define the same quantum state iff |ψ〉 =

eiα |φ〉. The associated projective Hilbert subspace

PS(Cn+1

)= S2n+1/ '


is usually called the complex projective space.

We can suggest as an example PS(C2)

= S3/ '= S2, the space of quantum states

of a 2-level system, i.e. a qubit. This 2-dimensional sphere is commonly called the

Bloch-Poincare sphere (ball).

Figure 1.1: The Bloch sphere

1.1.2 Tensor spaces, factorised states, quregisters

Any pair of Hilbert spaces H1,H2 gives rise to a new Hilbert space H1⊗H2: the tensor

product of H1 and H2.

Tensor products play a key role in the mathematical representation of compound quan-

tum systems.

Definition 1.1. Let H1,H2 be two Hilbert spaces over the same field D, either of the

reals or the complex numbers. A Hilbert space H is the tensor product of H1 and H2

iff the following conditions are satisfied:

1. there exists a map ⊗ (the tensor product) from the Cartesian product of H1×H2

into H that satisfies the following conditions:

(a) the tensor product ⊗ is linear in each component, in other words, ∀ |ψ〉 , |ϕ〉 ∈H1, ∀ |χ〉 , |δ〉 ∈ H2,∀a, b ∈ D:

i. (a |ψ〉+ b |ϕ〉)⊗ |χ〉 = (a |ψ〉 ⊗ |χ〉) + (b |ϕ〉 ⊗ |χ〉),

ii. |ψ〉 ⊗ (a |χ〉+ b |δ〉) = (|ψ〉 ⊗ a |χ〉) + (|ψ〉 ⊗ b |δ〉);

(b) the external product with a scalar carries over to the tensor product, in other

words ∀ |ψ〉 ∈ H1, ∀ |ϕ〉 ∈ H2, ∀a ∈ D:

a (|ψ〉 ⊗ |ϕ〉) = (a |ψ〉)⊗ |ϕ〉 = |ψ〉 ⊗ (a |ϕ〉) ;


2. every vector can be expressed as a linear combination of the vectors of the set

{|ψ〉 ⊗ |ϕ〉 : |ψ〉 ∈ H1, |ϕ〉 ∈ H2} .

As required by condition 2., every vector in H1 ⊗ H2 can be expressed as a linear

combination of vectors of the form |ψ〉 ⊗ |ϕ〉.

If {|ψ〉i}i∈I and{|ϕ〉j

}j∈J

are two orthonormal bases for H1 and H2 respectively, then

the set{|ψ〉i ⊗ |ϕ〉j : i ∈ I, j ∈ J

}is an orthonormal basis for H1 ⊗ H2. Moreover, if

{|ψ〉1 , ..., |ψ〉n} and {|ϕ〉1 , ..., |ϕ〉m}, are orthonormal bases of the finite dimensional

Hilbert spaces H1 and H2, then every vector can be written as

|ψ〉 =n∑i=1

m∑j=1

aij |ψ〉i ⊗ |ϕ〉j .

Suppose we have to deal with a physical system S composed by n other systems, say

S1, ..., Sn. Let HSi be the Hilbert spaces associated to Si, for 1 ≤ i ≤ n. The space Hassociated to S will be the tensor product (see Definition 1.1) HS1 ⊗... ⊗ HSn of the

spaces associated to S1, ..., Sn. If Si = Sj for every i, j, we resort to the notation ⊗nHSi

in place of HSi ⊗...⊗HSi . Once again, the space H will be “something different” from

the spaces HS1 , ...,HSn . Given m vector spaces HS1 , ...,HSm and a state |ψ〉 ∈ HS1

⊗...⊗HSm, we call |ψ〉 a factorised state iff |ψ〉 = |ψ1〉 ⊗ ...⊗ |ψm〉, for |ψj〉 ∈ HSj and

1 ≤ j ≤ m (see e.g. [10]). In general, it is not the case that every vector in a tensor

product space is amenable to factorisation; entangled states, in fact, are nonfactorisable

states, i.e. there is no way to express them as tensor products of pure states in Hilbert

spaces with a lower dimension (see below).2

As we have seen, qubits “live” in the space C2. Quregisters are the tensor product

analogues of qubits: by quregister, in fact, we mean any unit vector in ⊗nC2. As an

example, let us consider the space ⊗3C2, whose canonical basis is

{|000〉 , |001〉 , |010〉 , |011〉 , |100〉 , |101〉 , |110〉 , |111〉} .

A quregister will be a vector

|φ〉 = a0 |000〉+ a1 |001〉+ a2 |010〉+ a3 |011〉+ a4 |100〉+ a5 |101〉+ a6 |110〉+ a7 |111〉 ,2For the sake of simplicity we will sometimes denote |ψ1〉 ⊗ ...⊗ |ψm〉 by |ψ1, ..., ψm〉.


where the ai’s are complex numbers such that7∑i=0|ai|2 = 1.

We will call any factorized unit vector |φ〉 = |x1, ..., xn〉 of ⊗nC2, where x1, ..., xn are

variables ranging over the set {0, 1}, an n-configuration of ⊗nC2. 3

It is not hard to see that one can identify each n-configuration with a natural number

i ∈ [0, 2n−1], for i = 2n−1x1 + 2n−2x2 + ...+xn; intuitively, any n−configuration can be

read as a natural number in its binary codification. In other words, one can concisely

express a quregister |φ〉 as

|φ〉 =2n−1∑j=0

cj || j〉〉 ,

where cj is a complex number, || j〉〉 is the n-configuration corresponding to the number

j, and

2n−1∑j=0|cj |2 = 1. Let R

(⊗nC2

)be the set of all quregisters of ⊗nC2. We denote by

R :=

∞⋃n=1

(R(⊗nC2

))the set of all quregisters in C2 or in a tensor power of C2.

1.1.3 Quantum gates

Just like classical gates can be identified with functions on {0, 1}, quantum computa-

tional gates are, mathematically speaking, special operators whose arguments may be

either qubits, or quregisters. Additional features that are sometimes required in the

definition of quantum gate are linearity and unitarity [31]. However, neither the first

[13] nor the second property [4] is universally acknowledged as a necessary condition for

a quantum transformation.

If we consider the classical truth table of the conjunction, it is immediate to see that

it represents a typical many-to-one irreversible transformation, meaning that several

inputs correspond to the same output, so it is impossible to retrieve the initial state of

3The set of all n-configurations B(n) =

{|x1, ..., xn〉 : xi ∈ {0, 1} and

n∑1

xi = 1

}is an orthonormal

basis for ⊗nC2. We call B(n) the computational basis.


the transformation from its final state:

(0, 0) → 0

(0, 1) → 0

(1, 0) → 0

(1, 1) → 1

Instead, we will see that the quantum computational conjunction does not share this

feature. We will introduce this operation in a number of successive steps.

For any n,m ≥ 1, the Petri-Toffoli gate is the unitary operator T (n,m,1) such that,

for every element |x1, ..., xn〉 ⊗ |y1, ..., ym〉 ⊗ |z〉 of the computational basis B(n+m+1)

(shortened as |x〉 ⊗ |y〉 ⊗ |z〉),

T (n,m,1)(|x〉 ⊗ |y〉 ⊗ |z〉) = |x〉 ⊗ |y〉 ⊗∣∣xnym+z

⟩,

where + represents the sum modulo 2. For instance, T (1,1,1) trasforms any factorised

vector |x〉 ⊗ |y〉 ⊗ |z〉 into the vector obtained by leaving the first two factors (referred

to as the control bits) unchanged, while replacing |z〉 (the target bit) by∣∣xy+z

⟩. This

yields the following “table”:

|000〉 → |000〉

|001〉 → |001〉

|010〉 → |010〉

|011〉 → |011〉

|100〉 → |100〉

|101〉 → |101〉

|110〉 → |111〉

|111〉 → |110〉 .

T (1,1,1) behaves like the identity matrix on the first six basis elements, while interchang-

ing the last two basis elements.

Let us stress the fact that the Toffoli gate is a unitary operator, that is the Toffoli

gate operator is necessarily reversible (for it must admit of an inverse). The matrix


representation of T (1,1,1) relative to the computational basis is the following:

1 0 0 0 0 0 0 0

0 1 0 0 0 0 0 0

0 0 1 0 0 0 0 0

0 0 0 1 0 0 0 0

0 0 0 0 1 0 0 0

0 0 0 0 0 1 0 0

0 0 0 0 0 0 0 1

0 0 0 0 0 0 1 0

.

The operator T (1,1,1) affords a convenient notion of conjunction (And). This And is

characterized as a function whose arguments are pairs of vectors in C2, and whose

values are vectors of the product space ⊗3C2. If |ψ〉 , |ϕ〉 ∈ C2, we define

And(|ψ〉 , |ϕ〉) = T (1,1,1)(|ψ〉 ⊗ |ϕ〉 ⊗ |0〉).

In the above definition, |0〉 represents an ancilla which increases the dimension of the

space, but renders the operator reversible. For the arguments |0〉 and |1〉 we obtain the

following typically reversible (one-to-one) table:

|00〉 → |000〉

|01〉 → |010〉

|10〉 → |100〉

|11〉 → |111〉 .

1.1.4 Semiclassical and genuinely quantum gates

A gate A is semiclassical if its outputs cannot be superposition states whenever its

inputs are not superposition states. The label “semiclassical” is being used since such

gates behave just like their respective Boolean counterparts whenever they are applied

to non-superposition inputs (elements of the computational basis); nevertheless, unlike

classical gates, they can also be applied to superposition states.

A typical example, beside the quantum And, is the quantum Not. For any n ≥ 1, the

negation on ⊗nC2 is the unitary operator Not(n) such that, for every element |x1, ..., xn〉


of the computational basis B(n),

Not(n)(|x1, ..., xn〉) = |x1, ..., xn−1〉 ⊗ |1− xn〉 .

We have that:

Not(n) =

{σx, if n = 1

I(n−1) ⊗ σx, otherwise;(1.1)

where σx is the Pauli matrix

(0 1

1 0

).

Let us see what happens if we apply Not(1) to a state |ψ〉 = a0 |0〉+ a1 |1〉 in C2:

Not(1)(|ψ〉) = σx(a0 |0〉+ a1 |1〉)

=

(0 1

1 0

)(a0

a1

)

=

(a1

a0

).

A gate A is genuinely quantum if it is not semiclassical; in other words, when there

exists a state |ψ〉 in the computational basis, such that A |ψ〉 can be a superposition

state. A remarkable case in point is√Not. For any n ≥ 1, the squareroot of the negation

on ⊗nC2 is the unitary operator√Not

(n)such that, for every element |x1, ..., xn〉 of the

computational basis B(n),

√Not

(n)(|x1, ..., xn〉) = |x1, ..., xn−1〉 ⊗

1

2((1 + i) |xn〉+ (1− i) |1− xn〉) .

If we apply, say,√Not

(1)to a state |ψ〉 = a0 |0〉+ a1 |1〉 in C2, the result is

√Not

(1)(|ψ〉) =

1

2

(1 + i 1− i1− i 1 + i

)(a0 |0〉+ a1 |1〉)

=

(1 + i 1− i1− i 1 + i

)(a0

a1

)

=

((a0 + a1) + i(a0 − a1)(a0 + a1)− i(a0 − a1)

).


The basic property of√Not

(n)is the following: for any |ψ〉 ∈ ⊗nC2,

√Not

(n)(√

Not(n)

(|ψ〉))

= Not(n) (|ψ〉) .

From a logical point of view, therefore, the squareroot of the negation can be regarded as

a kind of “tentative partial negation” that transforms precise pieces of information into

maximally uncertain ones. By calculating the probability of squareroot of not applied

to the two basis vectors of C2, we have

p(√Not

(1)(|0〉)) =

1

2= p(√Not

(1)(|1〉)).

True to form,√Not has no boolean counterpart. In the Mach-Zender interferometer

each beam splitter plays the role of a natural physical model of this gate.

Figure 1.2: The Mach-Zehnder interferometer

Let us proceed with another useful genuinely quantum gate. For any n ≥ 1, the square-

root of the identity on ⊗nC2 is the unitary operator√I(n)

such that for every element

|ψ〉 = |x1, ..., xn〉 of the computational basis B(n):

√I(n)

(|x1, ..., xn〉) = |x1, ..., xn−1〉 ⊗1√2

((−1)xn |xn〉+ |1− xn〉) .

We have that


√I(n)

=

{H, if n = 1;

I(n−1) ⊗H, otherwise;(1.2)

where H is the Hadamard matrix:

H =1√2

(1 1

1 −1

).

The basic property of√I(n)

is the following: for any |ψ〉 ∈ ⊗nC2,√I(n)(√

I(n)

(|ψ〉))

= |ψ〉. Logically speaking, thus,√I(n)

can be seen as a “tentative

partial assertion”.

1.1.5 Genuine entanglement gates

Within the set of genuinely quantum gates, we can isolate a notable subset: the computa-

tionally locally entangled gates. Let us consider a special case first. A unitary operator

C on ⊗nC2 is computationally entangled if there exists a vector |bi〉 of the computa-

tional basis B(n) such that C |bi〉 = |ψ〉, where |ψ〉 is an entangled state.4 Now, upon

inductively defining, for any two unitary operators U, V ∈ ⊗mC2,

U ⊗0 V = V

U ⊗n+1 V = U ⊗ (U ⊗n V )

we say that U is computationally locally entangled iff there exist m ≥ 0 and a com-

putationally entangled gate W such that U = I ⊗m W (clearly, any computationally

entangled gate is locally entangled, for it suffices to fix n = 0).

A relevant example is the following. For any n ≥ 1, the squareroot of swap on ⊗nC2 is the

unitary operator√Swp

(n) such that, for every element |x1, ..., xn〉 of the computational

basis B(n),

√Swp

(n)(|x1, ..., xn〉) =

12 ((1 + i) |xn−1xn〉+ (1− i) |xnxn−1〉) if n = 2;

|x1, ..., xn−2〉 ⊗ 12 ((1 + i) |xn−1xn〉+ (1− i) |xnxn−1〉) if n > 2.

4Remark that there exist unitary gates which may admit of entangled states as outputs, yet fail tobe computationally fully entangled. A case in point is the Xor gate, which yields an entanglement onlywhen applied to superposition states (cp. [31]).


The name “squareroot of swap” stems from the basic property of√Swp

(n): by applying

it twice to a given quregister, the target bits are “swapped”. The matrix representation

of√Swp

(2) is the following:

√Swp

(2)=

1 0 0 0

0 1+i2

1−i2 0

0 1−i2

1+i2 0

0 0 0 1

.

As we can see, if we apply the√Swp

(2) to the basis elements |10〉 and |01〉 we get

entangled states as outputs; it can be noticed that this gate is computationally en-

tangled. On the other hand,√Swp

(3) is computationally locally entangled in that√Swp

(3) = I ⊗1√Swp

(2). If we apply√Swp

(3) to an arbitrary element |x1x2x3〉 of

⊗3C2, our output is |x1〉 ⊗ 12 ((1 + i) |x2x3〉+ (1− i) |x3x2〉). It is essential to remark

that, although 12 ((1 + i) |x2x3〉+ (1− i) |x3x2〉) is an entangled state, the whole output

is a factorised state with respective factors in C2 and in ⊗2C2.

1.2 From pure to mixed states

The aim of this section is to draw a path from the standard approach to quantum

computing - which tends to consider pure states as information quantities, and quan-

tum gates as transformations of those units - to another approach where the informa-

tion units are not necessarily pure states and the transformations are not necessarily

reversible. This new model will be based on quantum operations acting on density

operators (qumixes)[4].

We have seen that a quregister is a pure state, and, as in the classical case, its evolution

is obtained by the application of a (quantum) gate. In classical computation, a gate is a

function f : {0, 1}n → {0, 1}, which fails, in general, to be reversible. On the contrary,

in quantum computation a quantum gate U is a unitary operator that transforms qureg-

isters into quregisters. The complexity of the output bit of a classical gate f is generally

different from the complexity of the input bit(s) since f is not generally injective5. On

the other hand, the output of a quantum gate “lives in the same Hilbert space” of the

input, since there is a one-to-one correspondence between inputs and outputs. Because

5 Note that, however, any irreversible map f : {0, 1}m → {0, 1}n can be turned into a reversiblefunction f♥ : {0, 1}m+n → {0, 1}n+m such that: ∀x1, ..., xn, xn+1, xn+m

f♥(x1, ..., xn, xn+1, xn+m) =(x1, ..., xn, f (x1,...,xm) +(xm+1...xm+n)

).


of the unitarity constraint, quantum gates are always reversible. By looking at the

output quregister, we can always trace it back to the corresponding input.

Looking at the classical And truth table, it is immediate to see that it represents an

irreversible transformation f :

(0, 0) → 0

(0, 1) → 0

(1, 0) → 0

(1, 1) → 1.

In virtue of note 5, from f we can obtain a reversible function f♥ : {0, 1}3 → {0, 1}3 .

This function f♥ is the classical Toffoli gate. The quantum generalization of the classical

Toffoli gate was already mentioned in Section 1.1.3.

So far, we have dealt only with transformations from pure states to pure states. Nonethe-

less, in describing a physical system, different situations may arise. In practice, a physical

state may not correspond to a pure state.

Let us consider a simple case. Let A be an observable, i.e. a hermitian operator,

whose spectral decomposition is∑k

amP|ψm〉, where P|ψm〉 = |ψm〉〈ψm| is the projection

operator onto the span of |ψm〉. The real numbers am represent the possible values that

A may assume. If the physical system is in a pure state |ψk〉 the probability pm of

getting the result am is: |〈ψm|ψk〉|2.

In general, however, the preparing instrument fluctuates in such a way that successive

preparations of the system may correspond to different states. Let sk be the relative

frequency |ψk〉 is prepared with (sk > 0,∑k

sk = 1). Then the probability value pm can

be replaced by the following expression:

∑k

sk |〈ψm | ψk〉|2 =∑k

sk 〈ψm | ψk〉 · 〈ψk | ψm〉† =

〈ψm|∑k

sk |ψk〉〈ψk|ψm〉 = 〈ψm | ρ | ψm〉,

where

ρ =∑k

sk |ψk〉〈ψk| .


It turns out that ρ is a density operator, i.e. a positive, self adjoint, trace class operator

ρ such that tr(ρ) = 1, where tr is the trace functional. From a physical point of view,

the operator ρ represents a mixed state of all possible pure states |ψk〉, each of them with

weight equal to sk. Any density operator ρ describes the mixture of its eigenvectors,

with the probabilities corresponding to respective eigenvalues.

A density operator in ⊗nC2, with n ∈ N, will be called a qumix, and represents a mixed

state. We will denote with D(⊗nC2) the set of all density operator in ⊗nC2.

Apparently, it is possible to formulate quantum mechanics in terms of density operators,

because pure states are just particular cases of density operators. If A is an observable,

in the “quregister world”, for a state |ψ〉, the expectation value of A is

〈A〉 = 〈ψ | A | ψ〉. (1.3)

On the other hand, in the “qumix world”, for ρ = |ψ〉〈ψ|, the expectation value of A is:

〈A〉 = tr(Aρ). (1.4)

Apparently, the correspondence between Equations 1.3 and 1.4 follows from the obser-

vation that tr(|ψ〉〈ψ|A) = 〈ψ|A |ψ〉.

From a computational point of view, we can draw a comparison between the truth-

probability of a qubit and of a qumix in C2. As mentioned above, the truth-probability

(false-probability) of a qubit |ψm〉 = a0 |0〉 + a1 |1〉 is |a1|2 (|a0|2). On the other hand,

the truth-probability (false probability) of a density operator ρ ∈ D(C2) is given by

p|1〉(ρ) = tr(P|1〉ρ) (p|0〉(ρ) = tr(P|0〉ρ)). Further, in case ρ = P|ψm〉, we have that

p|1〉(ρ) = |a1|2 (p|0〉(ρ) = |a0|2). From now on, for notational clarity, we will denote p|x〉,

P|x〉, by p and P(1)x , respectively, where x ∈ {0, 1}. More generally, let ρ be a density

operator on ⊗nC2. The truth-probability of ρ is defined as follows:

p(ρ) = tr(P(n)1 ρ), (1.5)

where P(n)1 stands for the extension of P

(1)1 to dimension n. More precisely, P

(n)1 =

In−1 ⊗ P (1)1 .

It should be noticed that for any density operator ρ ∈ D(H), there exists an orthonormal

sequence of quregisters {|ψi〉}i∈I in a Hilbert space H, and a set of positive real numbers


{λi}i∈I , with∑

i λi = 1, such that ρ =∑

i λi |ψi〉〈ψi|. In other words, every density

operator ρ in D(H) can be represented as a convex combination of pure states |ψi〉〈ψi|in H. Nonetheless, such a representation is, in general, not unique. Actually, if a density

operator ρ has degenerate eigenvalues, i.e. if the same eigenvalue corresponds to distinct

eigenvectors |ψ1〉, |ψ2〉 of ρ, then there are infinitely many different convex combinations

of orthonormal states {|ψi〉}i∈I , such that ρ =∑

i λi |ψi〉〈ψi|. For instance, let ρ ∈ D(H),

and let λ be a degenerate eigenvalue of ρ whose corresponding eigenvectors are |ψ1〉, |ψ2〉.Then, if we set

|ψ′1〉 = cosα|ψ1〉 − sinα|ψ2〉;

|ψ′2〉 = sinα|ψ1〉+ cosα|ψ2〉;

where α is a real number, it can be seen that

ρ = λ(|ψ1〉〈ψ1|+ |ψ2〉〈ψ2|) = λ(|ψ′1〉〈ψ′1|+ |ψ

′2〉〈ψ

′2|).

Nonetheless, even though the representation of a density operator is not unique, two

different representations are statistically indistinguishable. In fact, for any possible

representation ρi of a density operator ρ ∈ D(⊗nC2), we have that for any abservable A

is tr(Aρ) = tr(Aρi).

Since a density operator ρ on ⊗nC2 might not have a unique decomposition, it is not

possible to provide an epistemic interpretation of a mixed state. For, if

ρ = λ |ψ1〉〈ψ1|+ (1− λ) |ψ2〉〈ψ2| ,

we cannot conclude that “the state is either in |ψ1〉 or in |ψ2〉”. In fact, if we have a

different representation of ρ, say

λ′ |ψ′1〉〈ψ

′1|+ (1− λ′)|ψ2〉〈ψ

′2|,

we would get into a contradiction. Truly, the state ρ is a state apart. This observation

somehow resembles, in the context of density operators, what happens in the case of the

superposition principle for quregisters. Let us express the state of a polarised photon

by a density operator

ρ = λP upθ + (1− λ)P downθ ,


where θ is an arbitrary direction. It can be seen that if λ ∈ {0, 1}, then the density

operator ρ assumes one of the pure states P upθ or P downθ , and so, in these cases, the

photon is completely polarised. If λ ∈ (0, 1), then ρ is not completely polarised, and

it represents a mixed state. In particular, if λ = 12 , then ρ represents the completely

unpolarised state.

As in the case of qubits, it is possible to give a geometrical insight for density operators

in D(C2) also. It is known that the Pauli matrices σ1,σ2 and σ3 (see Appendix C) and

the identity matrix I are a basis for every density operator on C2. Whence, to every

density operator ρ on C2 one can associate a triple of real numbers (r1, r2, r3) such that

ρ =1

2(r1σ1 + r2σ2 + r3σ3 + I) ,

where r21 + r22 + r23 ≤ 1. Therefore, to every density operator on C2 one can uniquely

assign a point in Bloch sphere, and, vice-versa, to every point in the Bloch sphere one

can uniquely associate a density operator on C2. It should be noticed that the point

associated to a density operator ρ is a surface point if and only if ρ represents a pure state.

Figure 1.3: quregisters Vs. qumixes

Since, if we represent a density operator ρ on C2 by a convex combination

ρ = λP|ψ〉 + (1− λ)P|φ〉,


then, λ = 1 or λ = 0 if and only if r21 + r22 + r23 = 1. Whence, in this case, ρ stands

for a pure state, and it is consequently identified with a point on the surface of the

Bloch sphere. It can be verified that the observations above can be generalised to every

Hilbert space, with respect to its corresponding. Thus, in virtue of the arguments above,

it stems that density operators are a generalisation of quregisters, as well as pure states

are limit cases of mixed ones.

Figure 1.4 represents some different aspects of physical reality. On the upper part of the

diagram we have states describing macroscopical systems depicted, whilst on the lower

part are those describing microscopical ones. Further, on the left branch of the diagram

we find pure states, and on the right part are the mixed ones. So, we can distinguish

the following four sets of states:

Figure 1.4: The square

• a classical pure state is described by a probability function f : Ps → [0, 1], where

Ps is the phase space of the system;

• if we have n classical states, each of them represented by a probability function fi,

a classical mixture is represented by the following linear combination:

∑i

λifi with∑i

λi = 1;

• a quantum pure state is represented by a complete linear combination of the

elements of the computational basis;


• finally, a quantum mixed state is a density operator, representable as a convex

combination of quantum pure states.

1.3 Reversible and irreversible transformations

As we have seen, we exploited the notion of quantum gate to mathematically model the

evolution of a pure state S, represented by a quantum register |ψ〉, see Subsection 1.1.3.

Now, in case the information quantity at issue is no longer a maximal one, its evolution

is mathematically modelled by a generalisation of the notion of quantum gate: the so

called quantum operations[27].

Definition 1.2. A quantum operation is a trace preserving, completely positive, linear

map from density operators to density operators.

It is worth noting that a quantum gate can be naturally extended to a quantum opera-

tion. Let us consider a quantum gate U : ⊗nC2 → ⊗nC2. We can extend the gate U to

a quantum operation DU on the space of density operators D(⊗nC2) as follows, for any

ρ ∈ D(⊗nC2):

DUρ = UρU †.

Nonetheless, even if from any quantum gate U a quantum operation DU is derivable, not

all quantum operations are of this form, and some specifications are required. First of all,

in general, quantum operations can model irreversible transformations. This observation

boils down to the fact that if a quantum gate is an isometry, a quantum operation, in

general, is not such: pure states are not closed under the action of quantum operations.

It may happen that a non-unitary quantum operation transforms a pure state into a

genuine mixed state. Intuitively, a quantum operation may turn a maximal information

quantity into a non-maximal one.

In [4], Aharonov, Kitaev, and Nisan proposed the quantum circuit with mixed states

model, and proved that such a model is polynomially equivalent in computational power

to the standard unitary model. Nonetheless, at the same time, the quantum circuit with

mixed states model presents several innovative features, since it allows to treat situations

impossible to be dealt with the standard model based on reversible transformations on

maximal information units. Among these situations we can list, for example, measure-

ments in the middle of a computation, noise, and decoherence. For instance, in general,

it may happen that, if a measurement occurs in the middle of a computation process, the


state of the system might well be mixed,6 e.g. because of interactions of the system with

the environment, decoherence and so on, and therefore a maximal information input

state could be turned into a non maximal state.

Figure 1.5: quregisters Vs. qumixes II

Let us now list the extensions of the quantum gates encountered in Subsection 1.1.3, to

the quantum operation scenario.

For any qumixes σ ∈ D(⊗mC2

)and τ ∈ D

(⊗nC2

),

D√Not(m)σ =

√Not

(m)σ√Not

(m)†;

D√I(m)σ =

√I(m)σ√I(m)†

;

DNot = Not(m)σNot(m)†;

And(m,n,1) (σ, τ) = DT(m,n,1)

(σ, τ, P

(1)0

):= T (m,n,1)

(σ ⊗ τ ⊗ P (1)

0

)T (m,n,1)†.

As an example of irreversible transformation, we can mention the partial trace:

Definition 1.3. Let H1 and H2 be m-dimensional and n-dimensional vector spaces,

respectively. For any space H, let L(H) denote the space of all linear operators on H.

The partial trace trH1 is the unique linear operator trH1 : L(H1 ⊗H2) → L(H2) such

6The state of a system after the measurement is Mn|ψ〉√〈ψ|M+

n Mn|ψ〉, where Mn is the measurement oper-

ator, satisfying the completeness equation:∑m

M+nMn = I[31]


that, for any R ∈ L(H1), and for any S ∈ L(H2):

trH1(R⊗ S) = tr(R)S.

Let now us consider a density operator ρ on H1 ⊗ H2, where H1 and H2 are Hilbert

spaces. The partial trace of ρ with respect to the space H1, denoted by ρH2 , is called

the reduced state ρ on the space H2 is given by: ρH2 = trH1(ρ).

Chapter 2

Irreversible quantum

tranformations

In this chapter we focus our attention on two particular irreversible transformations: the

irreversible conjunction and the Lukasiewicz truncated sum. We show some properties

of these transformations and we show - in the last section - that both tranformations can

be exactly and approximately simulated, respectively, via certain quantum operations,

called polynomial quantum operations. Furthermore, in Section 2.2, we consider the class

of irreversible fuzzy-like gates and we show that, as long as probabilities are concerned,

there is no hope to simulate the behaviour of these gates in terms of unitary operations.

2.1 Quantum gates from statistical operators

As mentioned in the previous chapter, interestingly enough, qumixes and quregisters are

connected with the real closed unit interval [0, 1]. In fact, given a real number λ ∈ [0, 1]

and an n ∈ N+, an n-quregister |ψ〉λ and a qumix ρ(n)λ are uniquely determined as

follows:

• |ψ〉λ =

{√

1− λ |0〉+√λ |1〉 , if n = 1,

√(1− λ) kn

∑2n−1−1j=0 || j〉〉 |0〉+

√λkn

∑2n−1−1j=0 || j〉〉 |1〉, otherwise;

• ρ(n)λ = (1− λ) knP(n)0 + λknP

(n)1 .

where |ψ〉λ ∈ R(⊗nC2

)represents the maximal information that might correspond to

the truth with probability λ, while ρ(n)λ ∈ D

(⊗nC2

)represents a “mixture” of infor-

mation pieces that might correspond to the truth with probability λ. From a physical

21

Irreversible Quantum Transformations 22

point of view ρ(n)λ corresponds to a particular preparation of the system such that the

quantum system is in the state knP(n)0 with probability 1 − λ and in the state knP

(n)1

with probability λ.

As we mentioned in the previous section, we resort the following notation:

For any qumixes σ ∈ D(⊗mC2

)and τ ∈ D

(⊗mC2

),

D√Not(m)σ =

√Not

(m)σ√Not

(m)†;

D√I(m)σ =

√I(m)σ√I(m)†

;

DNot = Not(m)σNot(m)†;

And(m,n,1) (σ, τ) = DT(m,n,1)

(σ, τ, P

(1)0

):= T (m,n,1)

(σ ⊗ τ ⊗ P (1)

0

)T (m,n,1)†.

Let us list some relevant properties:

Lemma 2.1. [16]

1. ∀n ∈ N+∀λ ∈ [0, 1] p(ρ(n)λ

)= λ,

2. p(D√Notρ(n)λ

)= 1

2 ,

3. p(D√Iρ(n)λ

)= 1

2 .

An irreversible conjunction can now be defined on the set of all qumixes of D(C2).

Definition 2.2. (The irreversible conjunction)

Let σ, τ ∈ D(C2).

IAND (σ, τ) = ρ(1)p(σ)p(τ).

Some particular features of the irreversible conjunction IAND are the following:

Lemma 2.3. 1. IAND is associative and commutative,

2. IAND (ρ, P0) = P0,

3. IAND (ρ, P1) = ρp(ρ),

4. p (IAND (ρ, σ)) = p (ρ) p (σ),

5. p(D√Not (IAND (ρ, σ))

)= 1

2 ,

6. p(D√I (IAND (ρ, σ))

)= 1

2 .


2.1.1 The Lukasiewicz truncated sum and its properties

Another example of an irreversible quantum transformation is represented by a Lukasiewicz-

like disjunction, a sort of “quantum analogue” of the Lukasiewicz disjunction in fuzzy

logic [9].

Definition 2.4. (The Lukasiewicz disjunction)

Let σ ∈ D(⊗nC2

)and τ ∈ D

(⊗mC2

):

σ ⊕ τ = ρ(1)p(σ)⊕p(τ),

where ⊕ is the Lukasiewicz “truncated sum”: p (σ)⊕ p (τ) = min(p (σ) + p (τ) , 1).

As one can easily see, the Lukasiewicz disjunction can be inverted if and only if σ = P(n)0

and τ = P(m)0 .

Lemma 2.5. [9]

1. σ ⊕ τ =

ρ(1)p(σ)+p(τ), if p (σ) + p (τ) ≤ 1,

P(n)1 , otherwise;

2. p (σ ⊕ τ) = p (σ)⊕ p (τ);

3. p(D√Not (σ ⊕ τ)

)= 1

2 .

2.1.2 Fuzzy irreducibility of genuine quantum gates

In this section, we present two conflicting - at least to some extent - results concerning

our irreversible fuzzy-like gates. We shall see that, as far as probabilities are concerned,

there is no hope to simulate the behaviour of these gates by means of unitary operators.

As usual, by Boolean function we mean a function f : {0, 1}n → {0, 1}; of course, if

n = 2, the function is said to be binary. A binary fuzzy function, on the other hand, is

a function g : [0, 1]2 → [0, 1]. The key concept to be used in what follows is the notion

of fuzzy extension of a binary Boolean function.

Definition 2.6. Let f : {0, 1}2 → {0, 1} be a binary Boolean function. A fuzzy extension

of a f is any function g : [0, 1]2 → [0, 1] such that ∀x, y ∈ {0, 1} : g(x, y) = f(x, y).


In general the fuzzy extension of a binary Boolean function is not unique.

The notion of fuzzy extension of a Boolean function is natural enough: by means of it we

can partition binary fuzzy functions into “families” modulo their identity of behaviour

on the endpoints of the closed real unit interval. For example, the family of the fuzzy

“conjunctions” can be identified with the fuzzy extensions of Boolean conjunction; this

class contains as members, e.g., product, Lukasiewicz conjunction, and the min function.

Likewise, the family of the fuzzy “disjunctions” will contain Lukasiewicz disjunction, the

max function and the MYCIN sum g(x, y) = x+ y − xy.

Example 2.1. Given a Boolean function f : {0, 1}2 → {0, 1}, let us consider the func-

tion gf : [0, 1]2 → [0, 1] such that

∀x, y ∈ [0, 1] : gf (x, y) = x0y0f (0, 0) + x0y1f (0, 1) + x1y0f (1, 0) + x1y1f (1, 1)

where

x0 = 1− x,

x1 = x,

y0 = 1− y,

y0 = y.

It can be readily seen that the function gf is a fuzzy extension f , in fact gf turns out

to be the extension to the closed real interval [0, 1] of the normal disjunctive form of the

Boolean funtion f .

An interesting question now arises: which fuzzy extensions of binary Boolean functions

admit of a quantum computational counterpart? To address this problem properly, we

first need to exactly specify what it means for a fuzzy function to admit of a quantum

analogue. The next definition provides what is needed.

Definition 2.7. A binary fuzzy function g is said to admit a quantum computational

simulation iff there exists an n ≥ 1, a unitary operator Ug on ⊗n+2C2 and a quregister

|χ〉 in ⊗nC2 such that, for any pair |ϕ〉 , |ψ〉 of qubits in C2, the following condition is

satisfied:

p (Ug (|ϕ〉 |ψ〉 |χ〉)) = g(p (|ϕ〉) , p (|ψ〉)).

In plain words, and with a good deal of oversimplification, we might say that a bi-

nary fuzzy function g admits a quantum computational simulation whenever there is

an associated unitary operator U such that, for any qubits |ϕ〉 , |ψ〉, the probability of

U(|ϕ〉 , |ψ〉) is just the result of the application of g to the probabilities of |ϕ〉 and |ψ〉.


Definition 2.8. Let |ψ〉 =2n−1∑i=0

ai |i〉 be an n-quregister. The probability-value of |ψ〉 is

defined as follows:

p (|ψ〉) =∑i∈C1

|ai|2 ,

where C1 is generally defined as C(n)1 = {i| || i〉〉 = |x1, ..., xn〉 and n = 1}.

Lemma 2.9. Let |ψ〉 =∑

i∈C0ai |i〉 +

∑i∈C1

(bi + ci) |i〉 be an n-quregister such that∑i∈C1

bi |i〉 is orthogonal to∑

i∈C1ci |i〉. Then

p (|ψ〉) =∑i∈C1

|bi|2 +∑i∈C1

|ci|2 .

We now have the following result and corollary:

Theorem 2.10. [15] Let f be a binary Boolean function. The fuzzy function gf is the

unique quantum computationally simulable fuzzy extension of f .

Proof. As we have seen in Example 2.1 gf is a fuzzy extension of a given boolean function

f . We now prove that gf is quantum computationally simulable.

Let n = 3, |χ〉 = |0〉 ∈ C2, and for x, y, z ∈ {0, 1} define:

Ugf (|x〉 |y〉 |z〉) = |x〉 |y〉∣∣f(x, y)+z

⟩.

Consider the linear extension of Ugf , still denoted as Ugf . It can be easily seen that Ugf

is unitary. We now prove that, for any pair |ψ〉 , |φ〉 of qubits in C2:

p(Ugf (|ϕ〉 |ψ〉 |χ〉)

)= gf (p (|ϕ〉) , p (|ψ〉)).

Let |ψ〉 = a0 |0〉+ a1 |1〉 and |φ〉 = b0 |0〉+ b1 |1〉 be a pair of qubits. It turns out that

p(Ugf (|ϕ〉 |ψ〉 |χ〉)

)= p

(a0b0 |00〉 |f (0, 0)〉+ a0b1 |0, 1〉 |f (0, 1)〉+a1b0 |1, 0〉 |f (1, 0)〉+ a1b1 |11〉 |f (1, 1)〉

)= |a0b0|2 |00〉 |f (0, 0)〉+ |a0b1|2 |0, 1〉 |f (0, 1)〉+

|a1b0|2 |1, 0〉 |f (1, 0)〉+ |a1b1|2 |11〉 |f (1, 1)〉

= |1− a0|2 |1− b0|2 f (0, 0) + |1− a0|2 |b1|2 f (0, 1) +

|a1|2 |1− b0|2 f (1, 0) + |a1|2 |b1|2 f (1, 1)

= gf

(|ai|2 , |bj |2

)= gf (p (|ψ〉) , p (|φ〉)) .

We now prove the uniqueness of gf .


Let h : [0, 1]2 → [0, 1] be a quantum computationally simulable fuzzy extension of f . By

definition we have that for any x, y in {0, 1} h(x, y) = f(x, y) = gf (x, y).

Moreover ∃m > 1, a unitary operator Uh on ⊗n+2C2 and a m-quregister |χ〉 in ⊗mC2

such that, for any pair |ϕ〉 , |ψ〉 of qubits in C2, the following condition is satisfied:

p (Ug (|ϕ〉 |ψ〉 |χ〉)) = g(p (|ϕ〉) , p (|ψ〉)).

We now show that for any x, y ∈ [0, 1]: h(x, y) = gf (x, y).

Let x, y ∈ [0, 1]. Then there exist two qubits |ψ〉 , |φ〉 such that

|ψ〉 =√x0 |0〉+

√x1 |1〉 and |φ〉 =

√y0 |0〉+

√y1 |1〉

where x0 = 1−x, x1 = x, y0 = 1−y, y0 = y. Clearly, p (|ψ〉) = x and p (|φ〉) = y. Thus:

h(x, y) = h (p (|ψ〉) , p (|φ〉))

= p (Uh(|ψ〉 , |φ〉 , |δ〉))

= p(Uh

(√a0b0 |00〉 |δ〉+

√a0b1 |0, 1〉 |δ〉+

√a1b0 |1, 0〉 |δ〉+

√a1b1 |11〉 |δ〉

))= p

(√a0b0Uh (|00〉 |δ〉) +

√a0b1Uh (|0, 1〉 |δ〉) +

√a1b0Uh (|1, 0〉 |δ〉) +

√a1b1Uh (|11〉 |δ〉)

).

We now show that

p (Uh(|ψ〉 , |φ〉 , |δ〉)) = gf (x, y)

and, consequently, h(x, y) = gf (x, y).

Now, p (Uh(|0〉 , |0〉 , |δ〉)) = h (0, 0) = f (0, 0) = gf (0, 0). Therefore

Uh(|0〉 , |0〉 , |δ〉) =∑i∈C0

a00i (1− f(0, 0) |i〉) +∑i∈C1

b00i (f(0, 0) |i〉)

where∑i∈C0

∣∣a00i ∣∣2 (1− f(0, 0)) = 1− f(0, 0), and∑i∈C1

∣∣b00i ∣∣2 (f(0, 0)) = f(0, 0).

Similarly

Uh(|0〉 , |1〉 , |δ〉) =∑i∈C0

a01i (1− f(0, 1) |i〉) +∑i∈C1

b01i (f(0, 1) |i〉)

with∑i∈C0

∣∣a01i ∣∣2 (1− f(0, 1)) = 1− f(0, 1), and∑i∈C1

∣∣b01i ∣∣2 (f(0, 1)) = f(0, 1);

Uh(|1〉 , |0〉 , |δ〉) =∑i∈C0

a10i (1− f(1, 0) |i〉) +∑i∈C1

b10i (f(1, 0) |i〉)


with∑i∈C0

∣∣a10i ∣∣2 (1− f(1, 0)) = 1− f(1, 0), and∑i∈C1

∣∣b10i ∣∣2 (f(1, 0)) = f(1, 0);

Uh(|1〉 , |1〉 , |δ〉) =∑i∈C0

a11i (1− f(1, 1) |i〉) +∑i∈C1

b11i (f(1, 1) |i〉)

with∑i∈C0

∣∣a11i ∣∣2 (1− f(1, 1)) = 1− f(1, 1), and∑i∈C1

∣∣b11i ∣∣2 (f(1, 1)) = f(1, 1).

A routinary calculation shows that p (Uh(|ψ〉 , |φ〉 , |δ〉)) is equal to

p

∑i∈C0

[ √x0y0a

00i (1− f(0, 0)) +

√x0y1a

01i (1− f(0, 1)) +

√x1y0a

10i (1− f(1, 0)) +

√x1y1a

11i (1− f(1, 1))

]+

∑i∈C1

[ √x0y0b

00i f(0, 0) +

√x0y1b

01i f(0, 1)+

√x1y0b

10i f(1, 0) +

√x1y1b

11i f(1, 1)

].

The following four cases are possible:

1. f is a constant function;

2.1∑

i,j=0f (i, j) = 1;

3.1∑

i,j=0f (i, j) = 3;

4.1∑

i,j=0f (i, j) = 2.

1. Trivial

2. By hypothesis there are exactly two elements r, s ∈ {0, 1} such that f(r, s) = 1. Thus,

h (x, y) = p (Uh(|ψ〉 , |φ〉 , |δ〉))

=∑i∈C1

|√xrysbrsi |2

= xrysf (r, s)

since∑i∈C1

|brsi |2 = 1.

Thus we have that h (x, y) = x0y0f (0, 0) + x0y1f (0, 1) + x1y0f (1, 0) + x1y1f (1, 1) =

gf (x, y).

3. Along the lines of 2.


4. By hypothesis there are exactly two pairs (r, s), (u,w), for r, s, u, w ∈ {0, 1}, such

that (r, s) 6= (u,w) and f (r, s) = f (u,w) = 1. Thus,

Uh(|r〉 , |s〉 , |δ〉) =∑i∈C1

√xrysb

rsi |i〉

and

Uh(|u〉 , |w〉 , |δ〉) =∑i∈C1

√xuywb

uwi |i〉 .

The quregisters (|r〉 , |s〉 , |δ〉) and (|u〉 , |w〉 , |δ〉) are orthogonal because (r, s) 6= (u,w).

Moreover, since Uh is unitary∑i∈C1

√xrysb

rsi |i〉 is orthogonal to

∑i∈C1

√xuywb

uwi |i〉. In

virtue of Lemma 2.9 we have: p (Uh(|ψ〉 , |φ〉 , |δ〉)) =∑i∈C1

∣∣√xrysbrsi ∣∣2+ ∑i∈C1

∣∣√xuywbuwi ∣∣2.Consequently

h (x, y) = p (Uh(|ψ〉 , |φ〉 , |δ〉))

=∑i∈C1

xrys |brsi |2 +

∑i∈C1

xuyw |buwi |2

= xrys + xuyw

since∑i∈C1

|brsi |2 =

∑i∈C1

|buwi |2 = 1.

Then we obtain h (x, y) = x0y0f (0, 0) + x0y1f (0, 1) + x1y0f (1, 0) + x1y1f (1, 1) =

gf (x, y), concluding our proof.

Corollary 2.11.

(i) There are exactly 16 quantum computationally simulable fuzzy extension of the 16

binary Boolean functions;

(ii) the function f : [0, 1]2 → [0, 1] such that f (x, y) = xy is the unique fuzzy extension

of the Boolean conjunction which admits a quantum analogue;

(iii) the MYCIN sum (or probabilistic sum: x+ y − xy) is the unique quantum compu-

tationally simulable fuzzy extension of Boolean inclusive disjunction;

(iv) the function f : [0, 1]2 → [0, 1] such that f (x, y) = x+ y− 2xy is the unique fuzzy

extension of the Boolean excluded disjunction which admits a quantum analogue.

An important consequence of Corollary 2.11(ii)-(iii) is the fact that the Lukasiewicz

disjunction does not admit a quantum analogue, a fact that seems to cast a shadow on

the usefulness of the gates introduced above.


2.2 Representing same irreversible transformation as quan-

tum operation

In this section we introduce a particular kind of quantum operations, named polynomial

quantum operations (particular ρλ operations, where λ is a polynomial), and we focus

on the irreversible operations: IAND and the Lukasiewicz disjunction. We will see that

IAND is a proper quantum operation, representable via Kraus Theorem [27], while the

behaviour of the Lukasiewicz disjunction can be simulated up to a small approximation

via the polynomial quantum operations.

Recently, increasing attention has been paid to logical and algebraic structures arising

from quantum computation [9, 13, 24] that are related, from a probabilistic point of

view, to systems appearing in fuzzy logic. Examples of these algebraic structures are

represented by quasi MV-algebras and√′quasi MV-algebras [21, 22, 29]. Some opera-

tions in these algebras are an abstraction of irreversible operations; consequently, they

cannot be accommodated inside the usual model of quantum computation, based on

qubits and unitary transformations. Nonetheless, we will show that they can be framed

into the more general model of quantum computation adopted in this chapter.

2.2.1 Polynomial quantum operations

In this section, we will represent in a probabilistic way some classes of polynomials as

quantum operations. First of all, we introduce some notations and preliminary defini-

tions. Due to the fact that Pauli matrices σi, i ∈ {0, x, y, z}, form a basis for the set

of operators over C2, an arbitrary density operator ρ for n-qbits may be represented in

terms of tensor products of them in the following way:

ρ =1

2n

∑µ1...µn

Pµ1...µn(σµ1 ⊗ . . .⊗ σµn),

where µi ∈ {0, x, y, z} for each i = 1 . . . n. As usual, we have chosen units such that ~ =

1, and the real expansion coefficients Pµ1...µn are given by Pµ1...µn = tr(σµ1⊗ . . .⊗σµnρ).

Since the eigenvalues of the Pauli matrices are ±1, the expansion coefficients satisfy

|Pµ1...µn | ≤ 1 [28].


Taking into account the Born rule, for ρ ∈ D(⊗nC2) we have defined the probabilty

value of ρ as p(ρ) = tr(P(n)1 ρ), i.e. the expectation value of ρ in the state P

(n)1 where

P(n)1 = (⊗n−1I)⊗ |1〉〈1|. Let ρ ∈ D(C2) be such that ρ = 1

2(I + rxσx + ryσy + rzσz), i.e.

ρ =1

2

(1 + rz rx − iryrx + iry 1− rz

)=

(1− α β

β∗ α

).

Let us recall the fact that any real number λ (0 ≤ λ ≤ 1) uniquely determines a density

operator

ρλ = (1− λ)P0 + λP1 =1

2(I + (1− 2λ)σz) =

(1− λ 0

0 λ

).

It is easy to see that, if ρ ∈ D(C2), then p(ρ) = 1−rz2 and p(ρλ) = λ. Thus each density

operator ρ in D(C2) can be written as

ρ =

(1− p(ρ) a

a∗ p(ρ)

).

Let us further recall that a quantum operation is a linear operator E : L(H1) → L(H2)

representable via Kraus Theorem as E(ρ) =∑

iAiρA†i where Ai are operators satisfy-

ing∑

iA†iAi = I. If Ai are unitary operators, the correspondent quantum operation

is named unitary quantum operation. It can be seen that a quantum operation maps

density operators into density operators. The new model “density operators-quantum

operations” also called “quantum computation with mixed states” ([4, 38]) is equivalent

in computational power to the standard one but it subsumes irreversible transformations.

The term multi-index denotes an ordered n-tuple α = (α1, . . . αn) of non negative inte-

gers αi. The order of α is given by |α| = α1 + . . . . . .+ αn.

If x = (x1, . . . , xn) is an n-tuple of variables and α = (α1, . . . αn) a multi-index, the

monomial xα is defined by xα = xα11 xα2

2 . . . xαnn . In this language a real polynomial of

order k is a function p(x) =∑|α|≤k aαxα such that aα ∈ R. Let x = (x1, . . . , xn) and k

be a natural number. Then, we consider the set Dk(x) defined as

Dk(x) = {(1− x1)α1xβ11 . . . (1− xn)αnxβnn : αi + βi = k, 1 ≤ i ≤ n}.

Lemma 2.12. Let X1, . . . ,Xn be a family of matrices such that

Xi =

(1− xi bi

b∗i xi

)


and let us consider a tensor product X = (⊗kX1) ⊗ (⊗kX2) ⊗ . . . ⊗ (⊗kXn). Then we

have that

Diag(X) = Dk(x1, . . . , xn).

Proof. By induction on k we can prove that Diag(⊗kXi) = {h1h2 . . . hk : hj ∈ {(1 −xi), xi}, 1 ≤ j ≤ k} = {(1− x1)αxβ1 : α+ β = k}.

Thus, Diag((⊗kX1) ⊗ (⊗kX2) ⊗ . . . ⊗ (⊗kXn)) = = {(1 − x1)α1xβ11 . . . (1 − xn)αnxβnn :

αi + βi = k, i ∈ {1, . . . , n}}. Whence our claim follows.

Lemma 2.13. Let x = (x1, . . . , xn) and k be a natural number. Given any monomial

xα such that | α |≤ k, we have that:

1. xα =∑

y∈Dk(x)δyy;

2. 1− xα =∑

y∈Dk(x)γyy;

where δy and γy lies in {0, 1}.

Proof. For each i ∈ {1, . . . , n}, consider the matrix Xi given by

Xi =

(1− xi 0

0 xi

).

Let us prove 1. Let xα = xα11 xα2

2 . . . xαnn such that | α |≤ k. Thus, there exist s1, . . . , sn

such that αi + si = k.

Let W = (⊗s1X1) ⊗ (⊗s2X2) ⊗ . . . ⊗ (⊗snXn) and let consider the matrix Wxα. In

view of Lemma 2.12, Diag(Wxα) ⊆ Dk(x1, . . . , xn) since every element in Diag(Wxα)

is a monomial of order nk. Further, since tr(Wxα) = (trW)xα = 1xα = xα, we have

that xα = tr(Wxα) is the required polynomial expansion.

Now we prove 2. Let X = (⊗kX1)⊗(⊗kX2)⊗ . . .⊗(⊗kXn). By Lemma 2.12 Diag(X) =

Dk(x1, . . . , xn) and tr(X) = 1. Taking into account that xα =∑

y∈Dk(x) δyy, we define

γy = 1 if δy = 0 and γy = 0 if δy = 1. Therefore, 1 = tr(X) =∑

y∈Dk(x) δyy +∑y∈Dk(x) γyy == xα +

∑y∈Dk(x) γyy and 1− xα =

∑y∈Dk(x) γyy.


Definition 2.14. A quantum operation P : L(⊗nkC2)→ L(⊗nkC2) is called polynomial

quantum operation iff there exists a polynomial P (x1, . . . , xn) such that for each n-tuple

(σ1, . . . , σn) in D(C2) we have that:

p(P((⊗kσ1)⊗ . . .⊗ (⊗kσn))) = P (p(σ1), . . . , p(σn)).

Theorem 2.15. Let x = (x1, . . . , xn) be an n-tuple of variables and consider the set

Dk(x). Let P (x) =∑

y∈Dk(x)ayy be a polynomial such that y ∈ Dk(x), 0 ≤ ay ≤ 1 and

the restriction P (x) �[0,1]n satisfies that 0 ≤ P (x) �[0,1]n≤ 1.

Then there exists a polynomial quantum operation

P : L(⊗nkC2)→ L(⊗nkC2) such that for each n-tuple σ = (σ1, . . . , σn) in D(C2)

p(P((⊗kσ1)⊗ . . .⊗ (⊗kσn))) = P (p(σ1), . . . , p(σn)).

Moreover, P((⊗kσ1)⊗ . . .⊗ (⊗kσn)) = ( 12nk−1 ⊗nk−1 I)⊗ ρP (p(σ1),...,p(σn)).

Proof. Let σ1, . . . , σn density operators in D(C2). Assume that for any σi

σi =

(1− xi bi

b∗i xi

).

Hence, p(σi) = xi. It is clear that σ = (⊗kσ1) ⊗ . . . ⊗ (⊗kσn) is a matrix of order

2nk × 2nk and, by Lemma 2.12,

Diag(σ) = Dk(x1, . . . , xn). Thus, each y ∈ Dk(x) can be seen as the (i, i)-th entry

of Diag(σ). Further, the polynomial P (x) =∑

y∈Dk(x) ayy =∑2nk

j=1 ajyj is such that

every yj is the (j, j)-th entry of Diag(σ). Let, now, yj0 ∈ Diag(σ).

a) We want to place aj0yj0 in the (2s, 2s)-th entries of a 2nk × 2nk matrix.

Let us consider the 2nk× 2nk matrix A2sj0

=√

aj02nk−1D

2sj0

such that D2sj0

have 1 just in the

(2s, j0)-th entry and 0 in any other entry. It is not difficult to check that A2sj0σ(A2s

j0)† is

the required matrix. Moreover, one can verify that:

∑2s

A2sj0σ(A2s

j0 )† =1

2nk−1

0 0 0 0 . . .

0 aj0yj0 0 0 . . .

0 0 0 0 . . .

0 0 0 aj0yj0 . . ....

......

.... . .

.


b) Taking into account that 1 =∑

y∈Dk(x) y =∑2nk

j=1 yj , we have that:

1−2nk∑j=1

ajyj =

2nk∑j=1

yj −2nk∑j=1

ajyj =

2nk∑j=1

(1− aj)yj .

We want to place (1−aj0)yj0 in the (2s−1, 2s−1)-th entries of a 2nk×2nk matrix. Let

us consider the 2nk × 2nk matrix A2s−1j0

=√

1−aj02nk−1D

2s−1j0

such that D2s−1j0

have 1 just in

the (2s− 1, j0)-th entry and 0 in any other entry.

It is not difficult to check that A2s−1j0

σ(A2s−1j0

)† is the required matrix. Moreover, one

can verify that:

∑2s−1

A2s−1j0

σ(A2s−1j0

)† =1

2nk−1

(1− aj0)yj0 0 0 0 . . .

0 0 0 0 . . .

0 0 (1− aj0)yj0 0 . . .

0 0 0 0 . . ....

......

.... . .

.

Thus, we have that P =∑

j0

∑2sA

2sj0σ(A2s

j0)† +

∑j0

∑2s−1A

2s−1j0

σ(A2s−1j0

)† =

= (1

2kn−1⊗nk−1 I)⊗

(1−

∑2nk

j=1 ajyj 0

0∑2nk

j=1 ajyj

).

Let us consider A =∑

j0

∑2s(A

2sj0

)†A2sj0

+∑

j0

∑2s+1(A

2s+1j0

)†A2s+1j0

. Our task is now to

verify that A = I.

c) First of all, notice that the matrix (A2sj0

)†A2sj0

has the valueaj0

2nk−1 just in the (j0, j0)-

th entry and 0 in any other entry. Therefore, the matrix∑

2s(A2sj0

)†A2sj0

has the value2nk−1aj02nk−1 = aj0 in the (j0, j0)-th entry and all the other entries are equal to 0. Hence:

∑j0

∑2s

(A2sj0 )†A2s

j0 =

a1 0 0 0 . . .

0 a2 0 0 . . .

0 0 a3 0 . . .

0 0 0 a4 . . ....

......

.... . .

.


d) On the other hand matrix (A2s−1j0

)†A2s−1j0

has the value1−aj02nk−1 just in the (j0, j0)-th

entry and 0 in any other entry. Therefore, the matrix∑

2s−1(A2s−1j0

)†A2s−1j0

has the value2nk−1(1−aj0 )

2nk−1 = 1 − aj0 in the (j0, j0)-th entry and all the other entries are equal to 0.

Hence:

∑j0

∑2s−1

(A2s−1j0

)†A2s−1j0

=

1− a1 0 0 0 . . .

0 1− a2 0 0 . . .

0 0 1− a3 0 . . .

0 0 0 1− a4 . . ....

......

.... . .

.

Thus∑

j0

∑2s(A

2sj0

)†A2s−1j0

+∑

j0

∑2s−1(A

2s−1j0

)†A2s−1j0

= I and P is a quantum opera-

tion.

2.2.2 Representing IAND and Lukasiewicz transformations as quan-

tum operations

At the begining of this chapter, we introduced the two following irreversible transfor-

mations:

IAND(σ ⊗ τ) = ρp(σ)·p(τ); (The irreversible conjunction)

σ ⊕ τ = ρ(1)p(σ)⊕p(τ)

= ρ(1)min(1,p(σ)+p(τ)); ( Lukasiewicz truncated sum)

where x⊕ y = min{x+ y, 1}.

In [13], a quantum computational system based on irreversible transformations was in-

troduced. The standard model of this system is based on a particular algebraic structure

whose universe is D(C2) equipped with the irreversible conjunction and the Lukasiewicz

truncated sum. Providing a quantum computational representation of these structures

is useful to provide a physical meaning to those irreversible transformations, and also

such a representation allows us to think of a possible computational implementation of


those irreversible transformations.

First let us show that IAND is a proper polynomial quantum operation. Let us consider

the following matrices:

G1 =

1√2

0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

G2 =

0 1√2

0 0

0 0 0 0

0 0 0 0

0 0 0 0

G3 =

0 0 1√2

0

0 0 0 0

0 0 0 0

0 0 0 0

G4 =

0 0 0 0

0 0 0 01√2

0 0 0

0 0 0 0

G5 =

0 0 0 0

0 0 0 0

0 1√2

0 0

0 0 0 0

G6 =

0 0 0 0

0 0 0 0

0 0 1√2

0

0 0 0 0

G7 =

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 1√2

G8 =

0 0 0 0

0 0 0 1√2

0 0 0 0

0 0 0 0

.

It is straightforward to check that∑8

i=1Gi(τ⊗σ)G†i = 12I⊗ρp(τ)p(σ), where σ, τ ∈ D(C2).

By the Kraus representation theorem, it is a quantum operation and it represents the

IAND modulo a tensor power.

Clearly, the Lukasiewicz operation ⊕ is not a polynomial, then the idea is to give a poly-

nomial P (x, y) in some generating system Dk(x, y), such that P (x, y) can approximate

the Lukasiewicz truncated sum. The graph of the Lukasiewicz function is depicted in

Fig.2.1.

Now, let us consider the following polynomial:


Figure 2.1: The Lukasiewicz function

P (x, y) =5

12x(1− x) +

5

12y(1− x) +

5

12x(1− y) +

5

12y(1− y) +

1

2x+

1

2y, (2.1)

with x, y ∈ [0, 1], whose graph is the following:

Figure 2.2: The approximating polynomial

As a comparison between P (x, y) in 2.1 and the Lukasiewicz we obtain the graph rep-

resented in Fig. 2.3


Figure 2.3: A comparison

We can conclude that we obtained a very good accuracy in the approximation: the

worst case occours when x = y = 0.5 and in that case the approximation is 0.08.

This result provides a strong quantum computational motivation for the investigation

of algebraic structures equipped with the Lukasiewicz sum and build a bridge between

“classical”fuzzy logic and quantum computational logics.

Chapter 3

An approximately universal

system of quantum computational

gates

The classical circuit-model of computation, both in its reversible and in its irreversible

version, can be formulated by using a very small set of gates, called universal set of gates.

This property (termed functional universality) amounts to say that every gate can be

mathematically simulated by means of a convenient composition of gates belonging to

the universal set. For instance, in the irreversible case, the single gate NAND of the system

consisting of the two gates AND and NOT turn out to be functionally universal. In the

reversible case, such a role is played by a single gate: the Toffoli gate T (also called

controlled-controlled not).

Unlike the classical circuit model, quantum computation “originates” in a naturally

reversible way, because quantum gates are interpreted as unitary operators acting on

pure states (qubits or quregisters) of the Hilbert space associated with the quantum

circuit at issue. Being unitary, quantum gates represent reversible time-evolution of the

circuit in question. Since there are uncountably many unitary operators, there is no

hope to find any finite functionally universal set of quantum gates. The best we can do

is to recur to the notion of finite approximate universality [35]: a finite set of gates is

said to be approximately universal if and only if any quantum gate can be approximated

up to an arbitrary accuracy by a quantum circuit that consists of elements of this set.

Finding simpler and simpler sets of universal gates represents a crucial step in order to

try and realize concrete quantum computers. Interestingly enough, this does not involve

any serious loss in computational power; in fact, as proved by Solovay and Kitaev [31],

38

A universal system of quantum computational gates 39

shifting from a universal set to another one only causes a polylogarithmic overhead.

The existence of a three-element (approximately) universal set of quantum gates has

been proved by Deutsch in [12]. Many other universal sets were discovered afterwards,

culminating in the result obtained by Shi [35] and further investigated by Aharonov

[3]. These authors found a two-element universal set consisting of the (three-qubit)

Toffoli gate T and of the one-qubit Hadamard gate√I (also called the squareroot of the

identity). Unlike the classical reversible case, the Toffoli gate alone is not sufficient to

reproduce the behavior of all quantum gates. A gate exhibiting a “genuine” quantum

behavior needs to be added: a “good” example is represented by the operator√I. From

a foundational point of view, we can say that√I is just all the Toffoli gate needs to reach

quantum (approximate) universality, starting from classical (functional) universality.

The results mentioned so far are formulated in the framework of the usual approach to

quantum computation, essentially based on quregisters and unitary operators of conve-

nient Hilbert spaces. As discussed in Chapter 1, such a representation is unduly restric-

tive, as it does not encompass open systems, where phenomena such as interactions with

an environment, or measurement-processes may occur. In this case, the time-evolution

of quantum objects is no longer reversible. One can formulate a more general model of

quantum computational processes, where quregisters and unitary operators are replaced

by density operators (qumixes) and by unitary quantum operations, respectively see [4]

and [24].

From a physical point of view, using qumixes instead of quregisters has plenty of advan-

tages. In fact, physical systems are never completely isolated and are always somehow

interacting with the rest of the universe. Hence, quantum states are better represented

by qumixes (mixed states) instead of quregisters (pure states). Moreover (as shown by

Aharonov, Kitaev and Nisan [4]), taking into account quantum circuits with qumixes al-

lows us to treat critical problems (such as measurements in the middle of a computation,

decoherence, noise, and so on), which cannot be adequately represented in the framework

of the usual approach. It should be noticed, however, that the Aharonov-Kitaev-Nisan

model and the standard model are polynomially equivalent in computational power [4].

In this chapter we will investigate some algebraic properties of the Shi-Aharonov univer-

sal set of gates (in their quantum operational form). To this aim we will equip the set

of all qumixes with two quantum operations representing an appropriate generalization

of the Toffoli gate T and of the Hadamard gate√I, as already shown in Chapter 1.

We will show that the main algebraic properties of this structure can be also captured by

restricting the action of the two quantum operations to qumixes “living” in the simplest

Hilbert space, C2. In this way, the dimension of the Hilbert space associated with a

reversible quantum circuit is dramatically reduced. The price to pay is the loss of the


reversible nature of the two quantum operations.

3.1 The Toffoli and the Hadamard gates

We will now investigate the basic algebraic properties of the Toffoli and of the Hadamard

gates (the two elements of the Shi-Aharonov approximately universal set of gates). Let

us recall same basic notion already mentionated in the Chapter 1.

The Toffoli gate represents the classical part of the Shi-Aharonov system: a classically

universal gate, that permits us to define the reversible versions of all Boolean functions.

Definition 3.1. (The Toffoli gate)

For any n,m, p ≥ 1, the Toffoli gate is the linear operator T (n,m,p) defined on ⊗n+m+pC2

such that, for every element |x1, . . . , xn〉⊗|y1, . . . , ym〉⊗|z1, . . . , zp〉 of the computational

basis B(n+m+p),

T (n,m,p)(|x1, . . . , xn〉 ⊗ |y1, . . . , ym〉 ⊗ |z1, . . . , zp〉)

= |x1, . . . , xn〉 ⊗ |y1, . . . , ym〉 ⊗ |z1, . . . , zp−1, xnym+zp〉,

where + represents the sum modulo 2.

One can easily show that T (n,m,p) is a unitary operator.

The Boolean functions AND, NAND, NOT can be now defined in terms of the Toffoli gate.

Definition 3.2.

• For any |ψ〉 ∈ ⊗nC2 and for any |ϕ〉 ∈ ⊗mC2,

AND(|ψ〉, |ϕ〉) := T (n,m,1)(|ψ〉 ⊗ |ϕ〉 ⊗ |0〉);

• For any |ψ〉 ∈ ⊗nC2 and for any |ϕ〉 ∈ ⊗mC2,

NAND(|ψ〉, |ϕ〉) := T (n,m,1)(|ψ〉 ⊗ |ϕ〉 ⊗ |1〉);

• For any |ψ〉 ∈ ⊗nC2,

NOT(|ψ〉) := T (1,1,n)(|1〉 ⊗ |1〉 ⊗ |ψ〉).


Defining the Boolean negation NOT in terms of the Toffoli gate has, however, a shortcom-

ing that is determined by the increasing of the dimension of the Hilbert space. Namely,

if |ψ〉 belongs to ⊗nC2, then its negation NOT(|ψ〉) belongs to ⊗2n+1C2.

For computational aims, the following independent definition of the negation-gate is

more economical:

Definition 3.3. (The negation)

For any n ≥ 1, the negation on ⊗nC2 is the linear operator Not(n) such that, for every

element |x1, . . . , xn〉 of the computational basis B(n),

Not(n)(|x1, . . . , xn〉) = |x1, . . . , xn−1〉 ⊗ |1− xn〉.

We have:

Not(n) =

σx, if n = 1;

I(n−1) ⊗ σx, otherwise,

where σx is the “first” Pauli matrix, i.e.,

σx =

(0 1

1 0

).

The following Lemma can be easily proved.

Lemma 3.4.

1. T (n,m,p)Not(n+m+p) = Not(n+m+p)T (n,m,p);

2. T (n,m,p) =(I(n+m) − P (n)

1 ⊗ P (m)1

)⊗ I(p) + P

(n)1 ⊗ P (m)

1 ⊗ Not(p).

The Toffoli gate represents a classical reversible gate: whenever the input is a classical

register, then also the output will be a classical register. In other words, the gate is

incapable to “create” superpositions. The “genuine” quantum component of the Shi-

Aharonov system is represented by the Hadamard gate (also called the squareroot of the

identity).

Definition 3.5. (The squareroot of the identity)

For any n ≥ 1, the squareroot of the identity on ⊗nC2 is the linear operator√I(n)

such

that for every element |x1, . . . , xn〉 of the computational basis B(n):

√I(n)

(|x1, . . . , xn〉) = |x1, . . . , xn−1〉 ⊗1√2

((−1)xn |xn〉+ |1− xn〉) .


The basic property of√I(n)

is the following:

for any |ψ〉 ∈ ⊗nC2,√I(n)(√

I(n)

(|ψ〉))

= |ψ〉.

Clearly:

√I(n)

=

H, if n = 1;

I(n−1) ⊗H, otherwise,

where H is the Hadamard matrix:

H =1√2

(1 1

1 −1

).

By definition, gates are unitary operators whose domains consist of vectors of convenient

Hilbert spaces. At the same time, gates can be naturally generalized also to qumixes,

as already investigated in section 1.2 . Let us remind that DT(m,n,p)

and D√I(n)

will

represent the Toffoli and the Hadamard qumix-gates, respectively.

The following theorems describe some basic properties of our qumix-gates.

Theorem 3.6. [24]

1. p(DNot

(n)(ρ))

= 1− p(ρ);

2. p(DAND(ρ, σ)

)= p(ρ)p(σ);

3. p(DNAND(ρ, σ)

)= 1− p(ρ)p(σ).

Theorem 3.7. [16]

1. For any ρ ∈ D(⊗nC2): D√I(n)(D√I(n)(ρ)

)= ρ ;

2. ∀n ∈ N+ : p(D√I

(knP

(n)1

))= p

(D√I

(knP

(n)0

))= 1

2 , where kn := 12n−1 .

Theorem 3.8. Let ρ ∈ D(⊗nC2), σ ∈ D(⊗mC2) and τ ∈ D(⊗pC2). Then,

p(DT

(n,m,p)(ρ⊗ σ ⊗ τ)

)= (1− p(τ))p(ρ)p(σ) + p(τ)(1− p(ρ)p(σ)).


Proof. We have:

p(DT

(n,m,p)(ρ⊗ σ ⊗ τ)

)= tr

(P

(n+m+p)1 T (n,m,p)(ρ⊗ σ ⊗ τ)T (n,m,p)

)= tr

(P

(n+m+p)1

((I(n+m) − I(n) ⊗ P (m)

1

)⊗ I(p) + P

(n)1 ⊗ P (m)

1 ⊗ Not(p))

(ρ⊗ σ ⊗ τ)((I(n+m) − P (n)

1 ⊗ P (m)1

)⊗ I(p) + P

(n)1 ⊗ P (m)

1 ⊗ Not(p)))

(Theorem 3.4)

= tr(((

I(n+m) − I(n) ⊗ P (m)1

)⊗ P (p)

1 + P(n)1 ⊗ P (m)

1 ⊗ P (p)1 Not(p)

)(ρ⊗ σ ⊗ τ)((

I(n+m) − P (n)1 ⊗ P (m)

1

)⊗ I(p) + P

(n)1 ⊗ P (m)

1 ⊗ Not(p)))

(by definitions of P(n)1 and P

(n)0 )

= tr(((

I(n+m) − P (n)1 ⊗ P (m)

1

)⊗ I(p) + P

(n)1 ⊗ P (m)

1 ⊗ Not(p))

((I(n+m) − I(n) ⊗ P (m)

1

)⊗ P (p)

1 + P(n)1 ⊗ P (m)

1 ⊗ P (p)1 Not(p)

)(ρ⊗ σ ⊗ τ)

)= tr

(((I(n+m) − P (n)

1 ⊗ P (m)1

)⊗ P (p)

1 + P(n)1 ⊗ P (m)

1 ⊗ Not(p)P(p)1 Not(p)

)(ρ⊗ σ ⊗ τ)

)= tr

((I(n+m) − P (n)

1 ⊗ P (m)1

) (ρ⊗ σ)⊗ P (p)

1 τ)

+ tr(P

(m)1 ρ⊗ P (m)

1 σ ⊗ P (p)0 τ

)= tr

((I(n+m) − P (n)

1 ⊗ P (m)1

)(ρ⊗ σ)

)tr(P

(p)1 τ

)+tr

(P

(m)1 ρ

)tr(P

(m)1 σ

)tr(P

(p)0 τ

)= tr

((I(n+m) − P (n)

1 ⊗ P (m)1

)(ρ⊗ σ)

)tr(P

(p)1 τ

)+tr

(P

(m)1 ρ

)tr(P

(m)1 σ

)tr(P

(p)0 τ

)= (1− p(ρ)p(σ))p(τ) + p(ρ)p(σ)(1− p(τ)).

As a consequence of Theorem 3.8 and of Theorem 3.6, the probability-value p(DT(n,m,p)

(ρ⊗σ⊗τ)) can be regarded as a kind of weighted sum of p(DAND(ρ, σ)) and of p(DNAND(ρ, σ)),

with weight p(DNot(p)(τ)) and p(τ), respectively.

Theorem 3.9. Let ρ ∈ D(⊗nC2). Then,

1.√I(n)P

(n)1

√I(n)

= 12I

(n) − 12Not

(n);

2. p(D√I(n)(ρ)

)= 1

2 −12tr

(Not(n)ρ

).

Proof. (1) One can easily show that√I(1)P

(1)1

√I(1)

= HP(1)1 H = 1

2I(1)− 1

2X = 12I

(1)−12Not

(1). Thus, by definitions of P(n)1 and P

(n)0 , we can conclude that

√I(n)P

(n)1

√I(n)

=12I

(n) − 12Not

(n).

(2) The proof follows from (1).


Theorem 3.10. Let ρ ∈ D(⊗nC), σ ∈ D(⊗mC) and τ ∈ D(⊗pC2). Then,

p(D√I(n+m+p)

(DT

(n,m,p)(ρ⊗ σ ⊗ τ)

))= p

(D√I(p)(τ)

).

Proof.

p(D√I(n+m+p)

(DT

(n,m,p)(ρ⊗ σ ⊗ τ)

))= tr

(√I(n+m+p)

P(n+m+p)1

√I(n+m+p)DT (ρ⊗ σ ⊗ τ)

)=

1

2− 1

2tr(Not(n+m+p)T (n,m,p)(ρ⊗ σ ⊗ τ)T (n,m,p)

)(Theorem 3.9(1))

=1

2− 1

2tr(Not(n+m+p)

(I(n+m+p) − P (n)

1 ⊗ P (m)1 ⊗ I(p) + P

(n)1 ⊗ P (m)

1 ⊗ Not(p))

(ρ⊗ σ ⊗ τ)(I(n+m+p) − P (n)

1 ⊗ P (m)1 ⊗ I(p) + P

(n)1 ⊗ P (m)

1 ⊗ Not(p)))

(Lemma 3.4(2))

=1

2− 1

2tr((

Not(n+m+p) − P (n)1 ⊗ P (m)

1 ⊗ Not(p) + P(n)1 ⊗ P (m)

1 ⊗ Not(p)Not(p))

(ρ⊗ σ ⊗ τ)(I(n+m+p) − P (n)

1 ⊗ P (m)1 ⊗ I(p) + P

(n)1 ⊗ P (m)

1 ⊗ Not(p)))

=1

2− 1

2tr((

I(n+m+p) − P (n)1 ⊗ P (m)

1 ⊗ I(p) + P(n)1 ⊗ P (m)

1 ⊗ Not(p))

(Not(n+m+p) − P (n)

1 ⊗ P (m)1 ⊗ Not(p) + P

(n)1 ⊗ P (m)

1 ⊗ I(p))

(ρ⊗ σ ⊗ τ))

(since Not(p)Not(p) = I(p))

=1

2− 1

2tr(Not(n+m+p)(ρ⊗ σ ⊗ τ)

)= p

(D√I(p)(τ)

). (Theorem 3.9(2))

3.2 Reversible and irreversible quantum computational struc-

tures

We will now introduce an algebraic structure whose domain is the set of all possible

qumixes and whose operations are defined in terms of the Toffoli and of the Hadamard

gates.


Definition 3.11. (The Shi-Aharonov quantum computational algebra)

The Shi-Aharonov quantum computational algebra is the following structure(D ,T ,

√I, P (1)

0 , P(1)1 ,

1

2I(1)),

where:

• D is the set of all qumixes;

• T is a ternary operation defined for any ρ ∈ D(⊗nC2), for any σ ∈ D(⊗mC2) and

for any τ ∈ D(⊗pC2) as follows:

T(ρ, σ, τ) := DT(n,m,p)

(ρ⊗ σ ⊗ τ);

•√I is a unary operation defined for any ρ ∈ D(⊗nC2) as follows:

√I(ρ) := D√I(n)(ρ);

• P (1)0 , P

(1)1 , 1

2I(1) are three special elements of D(C2) that represent the privileged

true, false and indeterminate qumix, respectively.

The set D of all qumixes can be preordered by the relation � that is defined as follows.

Definition 3.12. (The qumix-preorder)

For any ρ ∈ D(⊗nC2) and any σ ∈ D(⊗mC2),

ρ � σ iff p(ρ) ≤ p(σ) and p(D√I(n)(ρ)

)≤ p

(D√I(m)

(σ)).

One can easily show that � is reflexive and transitive. This permits us to define, in the

expected way, an equivalence relation ≡ on the set D.

Definition 3.13. ρ ≡ σ iff ρ � σ and σ � ρ.

Consider now the set

[D]≡ := {[ρ]≡ : ρ ∈ D} .

Unlike qumixes (which are only preordered by �), the equivalence-classes of [D]≡ can

be partially ordered in a natural way.

Definition 3.14.

[ρ]≡ � [σ]≡ iff ρ � σ.


The relation � (which is well defined) is a partial order.

We will now consider a quotient-structure based on the quotient set [D]≡.

Theorem 3.15. ≡ is a congruence relation with respect to T and√I.

Proof. That√I is preserved by ≡ is a consequence of the definition of ≡. Suppose that

ρ1 ≡ ρ2 , σ1 ≡ σ2 and τ1 ≡ τ2. We have to show that T (ρ1, σ1, τ1) ≡ T (ρ2, σ2, τ2).

That p (T (ρ1, σ1, τ1)) = p (T (ρ2, σ2, τ2)) follows from the hypothesis and from Theorem

3.8. That p(√

I (T (ρ1, σ1, τ1)))

= p(√

I (T (ρ2, σ2, τ2)))

follows from the hypothesis

and from Theorem 3.10.

On this basis we can define, in the expected way, the operations T and√I on [D]≡. We

obtain the following quotient-structure:([D]≡ ,T ,

√I ,[P

(1)0

]≡,[P

(1)1

]≡,

[1

2I(1)]≡

).

3.3 The complex quantum computational algebra

We will now prove that the quotient of the Shi-Aharonov quantum computational algebra

is isomorphic to a structure based on a particular set of complex numbers (the closed

disc with center(12 ,

12

)and radius 1

2). Let

C1 :={

(a, b) : a, b ∈ R and (1− 2a)2 + (1− 2b)2 ≤ 1}.

Note that for all pairs (a, b) ∈ C1, both elements a and b belong to the real interval

[0, 1].

Before introducing an algebraic structure on the set C1, let us first recall some properties

of the set D(C2)

of all density operators of C2. As is well known, D(C2)

is in one-to-one

correspondence with the set of all points of the Poincare sphere. Consider a qumix τ of

D(C2)

and let (t1, t2, t3) be the point of the Poincare sphere uniquely associated to τ .

We have:

τ =1

2

(1 + t3 t1 − it2t1 + it2 1− t3

).

One can easily see that:

p (τ) =1− t3

2

and

p(D√I(τ)

)=

1− t12

.


Let (a, b) ∈ C1 and let ρ(a, b) be the density operator associated to the triplet (1 −2b, 0, 1− 2a). Thus,

ρ(a, b) =

(1− a 1

2 − b12 − b a

).

Clearly, p(ρ(a, b)) = a and p(√

I(ρ(a, b)))

= b.

On this basis, recalling Theorem 3.8, the following operations (TC1 and√IC1

) can be

naturally defined on the set C1:

Definition 3.16. (The pair Toffoli and the pair squareroot of the identity)

1. TC1 ((a1, a2) , (b1, b2) (c1, c2)) = ((1− c1) a1b1 + c1 (1− a1b1) , c2);

2.√IC1

(a1, a2) = (a2, a1).

Lemma 3.17. C1 is closed under TC1 and√IC1

.

Proof. That C1 is closed under√I follows from the definitions of C1 and

√I. We are left

with the task of proving that C1 is closed under IT. Let (a1, a2) , (b1, b2) (c1, c2) ∈ C1.

We have to show that

(1− 2 ((1− c1) a1b1 + c1 (1− a1b1)))2 + (1− 2c2)2 ≤ 1.

By hypothesis we have that (1− 2c2)2 ≤ 1− (1− 2c1)

2. Thus it holds that

(1− 2 ((1− c1) a1b1 + c1 (1− a1b1)))2 + (1− 2c2)2

≤ (1− 2 ((1− c1) a1b1 + c1 (1− a1b1)))2 + 1− (1− 2c1)2 .

In virtue of this observation it suffices to show that

(1− 2 ((1− c1) a1b1 + c1 (1− a1b1)))2 + 1− (1− 2c1)2 ≤ 1.

Now,

(1− 2 ((1− c1) a1b1 + c1 (1− a1b1)))2 + 1− (1− 2c1)2

= (1− 2a1b1 + 4a1b1c1 − 2c1)2 − 4c21 + 4c1

= [(1− 2c1) (1− 2a1b1)]2 − 4c21 + 4c1

= (1− 2c1)2 (1− 2a1b1)

2 − (1− 2c1)2 + 1.

We have now to prove that (1− 2c1)2 (1− 2a1b1)

2 − (1− 2c1)2 + 1 ≤ 1, i.e.

(1− 2c1)2[(1− 2a1b1)

2 − 1]≤ 0


thus, (1− 2a1b1)2 ≤ 1, which is always true in [0, 1].

Consider now the structure

C1 =(C1 ,TC1 ,

√IC1, 0 , 1 , 1/2

),

where 0 :=(0, 12), 1 :=

(1, 12)

and 1/2 :=(12 ,

12

).

We will prove that C1 is isomorphic to([D]≡ ,T ,

√I ,[P

(1)0

]≡,[P

(1)1

]≡,

[1

2I(1)]≡

).

Lemma 3.18.

1. ρ(TC1 ((a1, a2) , (b1, b2) , (c1, c2))

)= T ((ρ (a1, a2) , ρ (b1, b2) , ρ (c1, c2)));

2. ρ(√

IC1((a1, a2))

)=√I (ρ (a1, a2)).

Proof. Easy computation.

Theorem 3.19. The structure C1 =(C1 ,TC1 ,

√IC1

, 0 , 1 , 1/2)

is isomorphic to([D]≡ ,T ,

√I ,[P

(1)0

]≡,[P

(1)1

]≡,[12I

(1)]≡

).

Proof. (Along the lines of [9]).

Let h be the map of C1 into [D]≡ such that for any (a, b) ∈ C1:

h ((a, b)) := [ρ(a, b)]≡ .

The map is well defined (by definition of ≡) and it is an homomorphism by Lemma 3.18.

We now prove that h is injective. Suppose that h ((a1, a2)) = h ((b1, b2)). Then

[ρ(a1, a2)]≡ = [ρ(b1, b2)]≡. Thus,

p (ρ(a1, a2)) = p (ρ(b1, b2)) and p(√

I (ρ(a1, a2)))

= p(√

I (ρ(b1, b2))).

Therefore:

p (ρ(a1, a2)) = a1 = b1 = p (ρ(b1, b2))

and


p(√

I (ρ(a1, a2)))

= a2 = b2 = p(√

I (ρ(b1, b2))).

Hence, ρ(a1, a2) = ρ(b1, b2).

We now prove that h is surjective. Let ρ be a density operator in D(⊗nC2

)and let ρred

the reduced state of ρ to C2 (see [17]). We will show that

1. p (ρ) = p (ρred);

2. p(√

I(ρ))

= p(√

I (ρred))

.

(1) By definition of reduced state, we have that for any self-adjoint operator A of C2,

the following equality holds:

tr(I(n−1) ⊗Aρ

)= tr (Aρred) . (3.1)

Thus, p (ρ) = tr(P

(n)1 ρ

)= tr

(I(n−1) ⊗ P (1)

1 ρ)

= tr(P

(1)1 ρred

)= p (ρred). We now

prove (2).

p(√

I(ρ))

=1

2− 1

2tr(Not(n)ρ

)(Theorem 3.9(2))

=1

2− 1

2tr(I(n−1) ⊗ Not(1)ρ

)=

1

2− 1

2tr(Not(1)ρred

)(3.1)

=p(√

I (ρred)). (Theorem 3.9(2))

It follows that

[ρ]≡ = [ρred]≡ . (3.2)

Let (a, b, c) be the point of the Poincare sphere associated to ρred. Take(1−c2 , 1−a2

)∈ C1.

Since p(ρ(1−c2

1−a2

))= c and p

(√I(ρ(1−c2 , 1−a2

)))= a, we obtain:

[ρred]≡ = h

((1− c

2,1− a

2

)).

By 3.2, we can conclude that

[ρ]≡ = h

((1− c

2,1− a

2

)).

Consequently h is surjective.


On the basis of Theorem 3.19, one can say that the “logic” of the Shi-Aharonov system

of quantum computational gates is nothing but a complex-valued logic.

Appendix A

Preliminaries

We assume the reader has a basic knowledge of the fundamental notions of set theory,

abstract algebra and calculus. The approach to set theory is standard, and no particular

set of axioms is required.

We use classes as well as sets. Roughly speaking, a class is a collection so large that

subjecting it to the operations admissible for sets would lead to logical contradictions.

We often use the term family in reference to set whose members are sets.

In dealing with sets we use the following standard notations: membership (∈), set-builder

notation ({− : −−}), the empty set (∅), inclusion (⊆), proper inclusion (⊂), union (∪and

⋃), intersection (∩ and

⋂), complement (−), (ordered) n-tuples (〈x1, ..., xn〉), direct

(Cartesian) products of sets

(A×B,

∏i∈I

Ai

), direct powers of a set

(AI). We shall not

distinguish between (ordered) pairs and 2-tuples. We will denote the ordered pair of x

and y by 〈x, y〉, and sometimes by (x, y).

We now list a series of remarks introducing some notations and basic definitions.

1. The power set of a set A, the set of all subsets of A, will be denoted by P (A).

2. An is the set of all n-tuples each of whose terms belongs to A.

3. As regards relations:

(a) An n-ary relation on a set A is a subset of An.

(b) A 2-ary relation on a set A is called a binary relation.

(c) The relational product of two binary relations r and s on A is defined by:

〈a, b〉 ∈ r ◦ s iff for some c, 〈a, c〉 ∈ r and 〈c, b〉 ∈ s.

51

Preliminaries 52

(d) The transitive closure of a binary relation r on A is the binary relation r ∪(r ◦ r) ∪ (r ◦ r ◦ r) ∪ ....

4. As regards functions:

(a) A function f from a set A to a set B is a subset of B ×A such that for each

a ∈ A there is exactly one b ∈ B with 〈b, a〉 ∈ f . Synonyms for functions are

mapping, map and system. If f is a function from A to B we write f : A→ B,

and if 〈b, a〉 ∈ f we write f (a) = b.

(b) If f : A → B and g : B → C, then f and g are relations on A ∪ B ∪ C. We

write gf for their relational product g ◦ f .

(c) If f : A→ B, then ker(f), the kernel of f , is the binary relation{〈a0, a1〉 ∈ A2 : f (a0) = f (a1)

}. f is called injective, or one-to-one, iff 〈x, y〉 ∈

ker(f) implies x = y, for all x, y ∈ A.

(d) If f : A → B, X ⊆ A and Y ⊆ B, then f (X) = {f (x) : x ∈ X} (the f -

image of X) and f−1 (Y ) = {x ∈ A : f(x) ∈ Y } (the f -inverse image of Y ).

f : A→ B is said to be surjective, or said that f maps A onto B, iff f(A) = B.

(e) The function f : A→ B is called bijective iff it is both injective and surjective.

(f) If f : A → B, then we say that the domain of f is A, the co-domain of f is

B, and the range of f is the set f(A).

5. Z, Q, R, C denote respectively the set of all the integer numbers, the set of all

the rational numbers, the set of all the real number and the set of all complex

numbers.

6. The union of a family F of sets,⋃F , is defined by x ∈

⋃F iff x ∈ B, for some

B ∈ F . The intersection of a family F of sets,⋂F , is defined dually to the union.

7. A preorder over a set A is a binary relation � on A such that:

(a) � is reflexive over A; i.e. 〈x, x〉 ∈�, for any x ∈ A.

(b) � is transitive over A; i.e. 〈x, y〉 ∈� and 〈y, z〉 ∈� then 〈x, z〉 ∈�, for all

x, y, z ∈ A.

8. A partial order over a set A is a binary relation ≤ on A such that:

(a) ≤ is a preorder over A.

(b) ≤ is anti-symetric; i.e. 〈x, y〉 ∈≤ and 〈y, x〉 ∈≤ then x = y, for all x, y ∈ A.

For orders, and binary relations in general, we often prefer to write x ≤ y instead

of 〈x, y〉 ∈≤. Given an order over a nonempty set A, the pair 〈A,≤〉 is called a

(partially) ordered set.

Preliminaries 53

9. By a chain in an ordered set 〈A,≤〉 is meant a set B ⊆ A such that for all x, y ∈ Beither x ≤ y or y ≤ x. An upper bound of B is an element u ∈ A for which c ≤ u,

for all c ∈ B.

10. Zorn’s Lemma is the statement that if 〈A,≤〉 is an ordered set in which every

chain has an upper bound, then 〈A,≤〉 has a maximal element m; i.e. m ∈ A and

m ≤ x implies that m = x. We take this statement as an axiom.

11. A linearly ordered set, sometimes called a chain, is an ordered set 〈A,≤〉 such that

for all x, y ∈ A either x ≤ y or y ≤ x. A well-ordered set is a linearly ordered set

〈A,≤〉 such that every nonempty subset B ⊆ A has a least element l; i.e. l ∈ Band l ≤ x for all x ∈ A.

12. As regards ordinals:

(a) The ordinals are generated from the empty set ∅ using the operation of suc-

cessor (the successor of x is S(x) = x ∪ {x}) and union (the union of any set

of ordinals is an ordinal).

(b) 0 = ∅, 1 = S(0), 2 = S(1), ... The finite ordinals are 0, 1, 2, ... also called the

natural numbers or non-negative integers.

(c) Every set of ordinals is well-ordered by setting, for α and β ordinals, α ≤ β

iff α = β or α ∈ β.

(d) the least infinite ordinal is ω = {0, 1, 2...}, which is the set of all finite ordinals.

(e) It is useful to notice that n-tuples are functions having domain {0, 1, ..., n− 1}.

13. As regards cardinals:

(a) Two sets A and B have the same cardinality iff there is a bijection from A to

B.

(b) The cardinals are those ordinals k such that no ordinal β < k has the same

cardinality of k. The finite cardinals are just the finite ordinals, and ω is the

smallest infinite cardinal.

(c) The Well Ordering Theorem is the statement that every set has the same

cardinality as some ordinal. We take it as an axiom.

(d) The cardinality of a set A is the unique cardinal k such that A and k have

the same cardinality. The cardinality of A is denoted by |A|.

(e) The power set P(A) has the same cardinality as 2A.

(f) Operations of addition, multiplication and exponentiation of cardinals are

defined as follows for any sets A and B: |A|·|B| = |A×B|, |A|+|B| = |A∪B|,

Preliminaries 54

if A and B are disjoint (i.e. A ∩ B = ∅), and |A||B| = |AB|. Addition and

multiplication are trivial where infinite cardinals are involved. The cardinal

2ω is the cardinality of the real numbers.

(g) There is a unique one-to-one order-preserving function, denoted by the He-

brew letter aleph, from the class of all ordinal numbers onto the class of all

infinite cardinal numbers. The first few cardinal numbers, in ascending order,

are ℵ0(= ω),ℵ1, ...,ℵω, ... etc.

14. As regards equivalence relations:

(a) An equivalence relation over a set A is a binary relation ∼ on A that is

reflexive over A, transitive and symmetric; i.e. 〈x, y〉 ∈∼ iff 〈y, x〉 ∈∼.

(b) Given an equivalence relation over a set A and for x ∈ A, the equivalence

class of x modulo ∼ is the set x/∼ = {y ∈ A : x ∼ y}.1 The factor set of A

modulo ∼ is the set A/∼ = {x/∼ : x ∈ A}.

(c) Given an equivalence relation ∼ over A, A/∼ is a partition of A. That is,

A/∼ is a set of nonempty subsets of A, A =⋃A/∼, and each pair of distinct

sets U and V in A/ ∼ are disjoint.

(d) The set of all equivalence relations over A is denoted by Eq(A).

(e) 〈Eq(A),⊆〉 is an orderd set having greatest lower bounds and least upper

bounds for any subset of its elements. The greatest lower bound of S ⊆ Eq (A)

is⋂S. The least upper bound is the transitive closure of the

⋃S.

15. The equality symbol = is used to assert that two expressions name the same object.

The formal equality symbol ≈ is used to build equations, as for the associative law

x · (y · z) ≈ (x · y) · z which can just become true or false when we assign specific

values to the symbols and check if the two sides of the equation name the same

object.

All the remaining conventions are standard; possible brakes with widespread usage will

be explicitly emphatized in what follows.

1In what follows, sometimes, we will call the equivalence class of an element x modulo an equivalencerelation the block containing x.

Appendix B

Basic of linear algebra and

functional analysis

B.1 Hilbert spaces

We now turn to the peculiar backdrop of quantum theory: Hilbert spaces. But, before

introducing Hilbert spaces, it is appropriate to recall first some preliminary notions.1

Definition B.1. Let D be the field of the real or of the complex numbers. A pre-Hilbert

space (or inner space) over D is a vector space V over D, equipped with an inner product

〈. |.〉 : V × V → D

that satisfies the following conditions for any |ϕ〉 , |χ〉 , |ψ〉 ∈ V and any scalar a ∈ D:

1. 〈ϕ |ϕ〉 ≥ 0,

2. 〈ϕ |ϕ〉 = 0 iff |ϕ〉 = 0,

3. 〈ψ |aϕ〉 = a 〈ψ |ϕ〉,

4. 〈ϕ |ψ + χ〉 = 〈ϕ |ψ〉+ 〈ϕ |χ〉,

5. 〈ϕ |ψ〉 = 〈ψ |ϕ〉∗, where ∗ is the identity if D = R, and the complex conjugation if

D = C.

1Following the Dirac’s notation, in what follows the elements (vectors) of a vector space V will beindicated by |ϕ〉 , |χ〉 , |ψ〉 , ... (〈ϕ| , 〈χ| , 〈ψ| , ...), while a, b, c, ... will represent elements (scalars) of thefield D.

55

basic of linear algebra and functional analysis 56

The notion of inner product2 allows to generalize some geometrical notions of ordinary

3-dimensional spaces.

Definition B.2. The norm || |ϕ〉 || of a vector |ϕ〉 is the number 〈ϕ |ϕ〉12 .

Note that the norm of any vector is a real number greater or equal to 0.

A unit (or normalized) vector is a vector |ψ〉 such that || |ψ〉 || = 1.

Two vectors |ϕ〉 , |ψ〉 are called orthogonal iff 〈ϕ |ψ〉 = 0.

Definition B.3. A set {|ψ〉i}i∈I of vectors is called orthonormal iff its elements are

pairwise orthogonal unit vectors.

The norm ||.|| induces a metric d on the pre-Hilbert space V:

d(|ψ〉 , |ϕ〉) = || |ψ〉 − |ϕ〉 ||

We say that a sequence {|ψi〉}i∈N of vectors in V converges to a vector |ϕ〉 ∈ V iff

limi→∞

d(|ψ〉i , |ϕ〉) = 0

i.e. ∀ε > 0∃n ∈ N∀k > n : d(|ψ〉k , |ϕ〉) < ε.

A Cauchy sequence is a sequence {|ψi〉}i∈N of vectors in V such that

∀ε > 0∃n ∈ N∀h > n∀k > n : d(|ψ〉h , |ϕ〉k) < ε

It is obvious that whenever a sequence {|ψ〉i}i∈N of vectors in V converges to a vector

|ϕ〉, then {|ψ〉i}i∈N is a Cauchy sequence. The converse, on the contrary, is not always

true.

Spaces in which every Cauchy sequence converges to a vector are called metrically com-

plete.

Definition B.4. A pre-Hilbert space V with inner product 〈. |.〉 is said metrically com-

plete with respect to the metric d(|.〉 , |.〉) induced by 〈. |.〉 iff every Cauchy sequence of

vectors in V converges to a vector in V .

We can, finally, define the notion of Hilbert space:

Definition B.5. A Hilbert space is a metrically complete pre-Hilbert space.

2For notational clarity we will write the inner product of two vectors |ϕ〉, |ψ〉 by 〈ϕ |ψ〉 instead of〈〈ϕ| ||ψ〉〉.


Definition B.6. Let H be a Hilbert space over a field D,3 {|ψ〉i}i∈I be a set of vectors

in H and {ai}i∈I ⊆ F . A vector |ψ〉 is called a (Hilbert) linear combination (or super-

position) of {|ψ〉i}i∈I (with scalars {ai}i∈I ) iff ∀ε ∈ R+ there is a finite set J ⊆ I such

that for any finite subset K of I including J :

|| |ψ〉 −∑i∈K

ai |ψ〉i || < ε.

Clearly, if it exists, the linear combination of {|ϕ〉i}i∈I is unique.

Definition B.7. An orthonormal basis of H is a maximal orthonormal set {|ψ〉i}i∈I of

H.

It can be proven that any Hilbert space H has an orthonormal basis, and that all

the orthonormal bases have the same cardinality. The dimension of H is the cardinal

number of any orthonormal basis of H.

For a given n-dimensional Hilbert space H, its canonical orthonormal basis is the basis

obtained by taking the n basis vectors

{ei : 1 ≤ i ≤ n}

where ej is the vector with a 1 in the j-th coordinate and 0 elsewhere.

An Hilbert space H is said separable iff H has a countable orthonormal basis. In what

follows we will always refer to separable Hilbert spaces.

Definition B.8. A closed subspace of H is a subset X of vectors that satisfies the

following conditions:

1. X is a subspace of H,

2. X is closed under limits of Cauchy sequences.

B.1.1 Operators of a Hilbert space

Definition B.9. Let H be a Hilbert space. An operator of H is a map

A : Dom(A)→ H

where Dom(A), the domain of A, is a subset of H.

3Where D stands either for the real or for the complex field.


Definition B.10. A densely defined operator of H is an operator A that satisfies the

following condition:

∀ε ∈ R+∀ |ψ〉 ∈ H∃ |ϕ〉 ∈ Dom(A)[d(|ψ〉 , |ϕ〉) < ε]

where d represent the metric induced by 〈. |.〉.

Definition B.11. A linear operator on H is an operator A that satisfies the following

conditions

1. Dom(A) is a closed subspace of H,

2. ∀ |ϕ〉 , |ψ〉 ∈ Dom(A)∀a, b ∈ D : A(a |ϕ〉+ b |ψ〉) = aA |ϕ〉+ bA |ψ〉.

In other words, linear operators preserve linear combinations.

Definition B.12. A linear operator A is called a bounded operator iff there exists a

positive real number a such that ∀ |ϕ〉 ∈ H : ||A |ϕ〉 || ≤ a|| |ϕ〉 ||.

Definition B.13. A bounded operator is said to be positive iff ∀ |ψ〉 ∈ H : 〈ψ |Aψ〉 ≥ 0.

Definition B.14. Let A be a densely defined linear operator of H. The adjoint of A is

the unique operator A∗ such that

∀ψ ∈ Dom(A)∀ϕ ∈ Dom(A) : 〈Aψ |ϕ〉 = 〈ψ |A∗ϕ〉 .

Definition B.15. A self-adjoint (Hermitian) operator is a densely defined linear oper-

ator A such that A = A∗.

Note that if A is self-adjoint and everywhere defined, that is Dom(A) = H, then A is

bounded.

Definition B.16. A projection operator is an everywhere defined self-adjoint operator

P that satisfies the idempotence property: ∀ |ψ〉 ∈ H : PP |ψ〉 = P |ψ〉.

There are two special projections: O, I, the zero and the identity projections, respec-

tively, which are defined as follows:

O |ψ〉 = 0; I |ψ〉 = |ψ〉

for any |ψ〉 ∈ H. We will indicate by Π (H) the set of all projection operators of H.

It can be proven that the set of all closed subspaces of an Hilbert space H and the set

of all projections of H are in one-to-one correspondence.


Let X be a closed subspace of H. By the projection theorem (see [36]) every vector

|ψ〉 ∈ H can be expressed as a linear combination |ψ1〉+ |ψ2〉, where |ψ1〉 ∈ X and |ψ2〉is orthogonal to every vector in X. Accordingly, we can define an operator PX on Hsuch that

PX |ψ〉 = |ψ1〉

for any |ψ〉 ∈ H. It can be easily seen that PX is a projection operator of H.

We can also associate to any projection operator P its range:

XP = {|ψ〉 : ∃ |ϕ〉 (P |ϕ〉 = |ψ〉)}

which turns out to be a closed subspace of H. Moreover, for any closed subspace X and

for every projection P , it holds that:

X(PX) = X; P(XP ) = P.

Definition B.17. Let B (R) the set of all Borel sets4 of real numbers . A projection-

valued measure (the spectral measure) is a map

M : B (R)→ Π (H)

that satisfies the following conditions:

• M(∅) = O,

• M(I) = R,

• for any countable set {∆i}i∈I of pairwise disjoint Borel-sets:

M(⋃{∆i}i∈I) =

∑i

M(∆i)

where in the infinite case the series converges in the weak operator topology (see

[36]) of the set of all bounded operators of H.

It can be proved that there exists a one-to-one correspondence between the set of all

projection valued measures and the set of all self-adjoint operators of H. This is one of

the implications of the spectral theorem.

4A Borel set is any set in a topological space that can be formed from open sets (or, equivalently, fromclosed sets) through the operations of countable union, countable intersection, and relative complement.


Definition B.18. Let {|ψi〉}i∈I be any orthonormal basis for H and let A be a positive

operator. The trace of A, in symbols tr(A), is defined as follows:

tr(A) :=∑i

〈ψi |Aψi〉 .

For any positive operator A, there exists a unique positive operator B such that: B2 = A.

If A is a bounded operator then A∗A is positive.

Let |A| be the unique positive operator such that |A|2 = A∗A. A bounded operator A

is called trace class operator iff tr(|A|) <∞.

Definition B.19. A density operator is a positive, self adjoint, trace class operator ρ

such that tr(ρ) = 1.

Clearly, for any vector |ψ〉, the projection P|ψ〉 onto the unique 1-dimensional closed

subspace |ψ〉 containing |ψ〉 is a density operator.

Definition B.20. A unitary operator is a linear operator U such that:

• Dom(U) = H,

• UU∗ = U∗U = I.

One can prove that, in the finite dimensional case, unitary operators are exactly the

operators that preserve the inner product.

A relevant example of complex, self-adjoint unitary 2× 2 matrices is represented by the

so-called Pauli matrices:

I =

(1 0

0 1

), σx =

(0 1

1 0

), σy =

(0 −ii 0

), σz =

(1 0

0 −1

).

It is important to stress the fact that any unitary operator in C2 can be expressed as a

linear combination of the Pauli matrices.

Appendix C

The von Neumann’s

axiomatization of quantum theory

After summarizing the mathematical framework of quantum theory, we now present the

main axioms of the Von Neumann’s axiomatization of the theory.

Axiom 1. Physical systems the mathematical interpretation of any physical system

S is a separable Hilbert space over the complex field.

Axiom 2. Pure and mixed states any pure state of a system S is mathematically

represented by a unit vector of the space H. Non maximal information is matched

by mixed states (mixtures). They are mathematically represented by density op-

erators ρ of H. Clearly, pure states turn out to be limit-cases of mixed states; in

fact every unit vector |ψ〉 uniquely determines a density operator P|ψ〉, the pro-

jection associated to the 1- dimensional closed subspace spanned by |ψ〉. Density

operators that can not be represented in the form P|ψ〉 are called proper mixtures.

Axiom 3. Events any event that may hold for S is mathematically represented by a

projection operator P of H. Since the set of all projection operators and the set

of all closed subspace are in one-to-one correspondence, events can be equivalently

represented by closed subspaces of H.

Axiom 4. Observables any observable on S is mathematically represented by a projection-

valued measure M of H. For any projection valued measure M and any Borel set

∆, the intended physical interpretation of the projection value M(∆) will be the

event: “ the observable represented by M has a value that lies in ∆”. By the spec-

tral theorem, there is a 1-1 correspondence between projection valued measure and

hermitian operator. For, we will use interchangeably both terms.

61

The von Neumann’s axiomatization of quantum theory 62

Axiom 5. The Born probability the probability p(ρ, P ) that the system S in a state

ρ satisfies an event P is determined by the following rule, called Born-rule:

p(ρ, P ) = tr (ρP )

where tr represent the trace functional. It can be shown that for any state ρ and

for any projection P :

p(ρ, P ) ∈ [0, 1] .

Moreover, for any pure state |ψ〉 and any proiection PX :

p(P|ψ〉, PX

)= 1 iff |ψ〉 ∈ X iff PX |ψ〉 = |ψ〉 .

That is, a pure state |ψ〉 certainly verifies an event corresponding to the closed

subspace X iff |ψ〉 is an element of X.

A crucial relation between observables is represented by compatibility. Intuitively,

two observables are compatible when they are simultaneously measurables.

Definition C.1. Two observables M and N are compatible iff for any pair of Borel

sets ∆,Γ the two projections M(∆), N(Γ) commute:

M(∆)N(Γ) = N(Γ)M(∆).

It turns out that two observables M and N are compatible iff for any pair of Borel

sets ∆,Γ the operator M(∆)N(Γ) is a projection.

Axiom 6. The Schrœdinger equation the spontaneous time-evolution of the states

of a quantum system S is determined by the Schrœdinger equation. In agreement

with this equation, for any time interval [t0, t1], there exists a unitary operator

U[t0,t1] that maps pure states of S into the pure states of S. For any pure state

|ψ〉, U[t0,t1] |ψ〉 represent the state of the system at time t1, provided the system is

in state |ψ〉 at time t0. This map can also be extended to the case of mixed states.

Axiom 7. The reduction-postulate suppose the observer measures an observable

represented by the spectral measure M in the system S during the time inter-

val [t0, t1] by a non-destructive measurement process. Let ρ represent the state of

S at the initial time t0. Suppose the result of the measurement is the Borel set

∆. Then, soon after the measurement, at time t1, the observer will associate to

the system the following state:

ρ′ =M (∆) ρM (∆)

tr(ρM (∆)).


It can be seen that ρ′ assigns probability 1 to the event M (∆). In other words,

the performance of a measurement induces a state-transformation that takes into

account the information obtained by the measuring procedure.

Axiom 8. Compound systems suppose we have a compound quantum system S

consisting of two subsystems S1 and S2. Let H1 be the Hilbert space represent-

ing the mathematical interpretation of S1 while H2 represents the mathematical

interpretation of S2. The tensor product H1⊗H2 will represent the mathematical

interpretation of the compound system.

C.1 Quantum computational background

The aim of this chapter is to provide a survey of the basics of quantum computation up

to the introduction of quantum gates, focussing especially on the standard gates that

have been most extensively discussed in the literature.

C.1.1 Qubits and superposition states

As we have seen in the previous chapter, in quantum mechanics a physical system is

naturally associated to a Hilbert space. Let us recall again that a state, as given by

a vector in such a Hilbert space (see e.g. [10]), is pure if and only if it represents a

maximal information quantity, i.e. an information on the physical system that could

not be consistently augmented by any further observation.

Consider a two-dimensional Hilbert space H, and let {|0〉 , |1〉} be its canonical orthonor-

mal basis. The quantum computational counterpart of the bit - the basic information

quantity of classical information theory - is the quantum bit (qubit), i.e. any unit vector

|ψ〉 in C2. The general form of a qubit is:

|ψ〉 = a0 |0〉+ a1 |1〉

where a0, a1 are complex numbers such that |a0|2+|a1|2 = 1, as required by the unitarity

hypothesis. Qubits, therefore, correspond to pure states: in fact, as dictated by the Born

rule,

• |a0|2 yields the probability of the information described by the pure state |0〉,which, from a logical viewpoint, corresponds to falsity;


• |a1|2 yields the probability of the information described by the pure state |1〉,corresponding to truth.

Therefore, |0〉 and |1〉 represent maximal and certain pieces of information, while a

superposition |ψ〉 (in other words, a linear combination with nonzero coefficients of the

basis vectors |0〉 and |1〉) corresponds to a maximal but uncertain piece of information.

So far, so good. However, what physical meaning can we attach to superposition states?

A superposition |ψ〉 of the states |0〉 and |1〉 is a new state absolutely distinct from both

|0〉 and |1〉; this typically holistic phenomenon is known as the superposition principle

[5]. For example, consider an idealized atom with a single electron and two energy

levels: a ground state (identified with |0〉), which we suppose to be the current state of

the electron, and an excited state (identified with |1〉). By shining a light pulse of half

the duration as the one needed to perform a change of the energy level from |0〉 to |1〉,we can effect a “half-flip” between the two logical states. The ensuing state of the atom

is neither |0〉 nor |1〉, but rather a superposition of both states: |ψ〉 = a0 |0〉 + a1 |1〉.The electron is neither in the ground state, nor in the excited state, but ”halfway in

between”.

Suppose, now, that we measure the energy of such an electron. The measurement process

will not admit an uncertain result: the electron must be detected in either one of the

two levels. The respective probabilities that the electron will be detected in the ground

or in the excited level will be |a0|2 and |a1|2. That is, the electron has changed again

its energy level since the measurement procedure “has forced” |ψ〉 to collapse into only

one of the two possible states. In some sense (see e.g. [13]) the measurement procedure

did not produce any information about the way |ψ〉 was before the measurement, but

caused an irreversible change of the initial state |ψ〉.

The superposition principle is deeply connected with the notion of reality element of a

system, by which we mean a value of a given physical observable that it is possible to

forecast with certainty without performing any measurement. The following example

will show the strict tie between reality elements and the superposition principle. Suppose

we prepare a physical system in such a way that its spin value on the −→x axis, Sx, is

equal to +~2 .1 This system can be described by the wavefunction

|ψ〉1 =1√2

(1

1

).

1Where ~ = h2π

, and h = 6.62618× 10−34 joule second, is the Planck constant (see e.g. [10]).


Since

(1

1

)is the eigenvector of the Pauli matrix σx (see e.g. [10]) corresponding to the

eigenvalue +1 and representing the spin projection on the −→x axis, no interaction with

the system is needed to safely forecast that any spin measurement on the −→x axis will

assume the value +~2 . Therefore Sx = +~

2 is a reality element of the observed system.

It is immediate to verify that the same wavefunction ψ1 can be written, analogously, in

the following way:

|ψ〉2 =1√2

(1

0

)+

1√2

(0

1

).

Now,

(1

0

)and

(0

1

)are eigenvectors of the Pauli matrix σz, corresponding, respectively,

to the eigenvalues +1 and −1 and representing the spin projection on the −→z axis. Then

|ψ〉2 expresses a state for which it is not possible to anticipate the spin measure on the−→z axis. This fact means that the same wavefunction, if written as in |ψ〉1 , carries an

information about Sx, a reality element of the system, while if written as in |ψ〉2, it

cannot express information about any reality element of the system.

Appendix D

What is a quantum computer?

It was almost three quarters of a century after the discovery of quantum mechanics,

and half a century after the birth of information theory and the arrival of large-scale

digital computation, that people finally realized that quantum physics profoundly alters

the character of information processing and digital computation. For physicists this

development offers an exquisitely different way of using and thinking about the quan-

tum theory. For computer scientists it presents a surprising demonstration that the

abstract structure of computation cannot be divorced from the physics governing the

instrument that performs the computation. Quantum mechanics provides new compu-

tational paradigms that had not been imagined prior to the 1980s and whose power was

not fully appreciated until the mid 1990s.

First of all, a quantum computer or, more accurately, the abstract quantum computer

that one hopes someday to be able to embody in actual hardware is an extremely

simple example of a physical system. It is discrete, not continuous. It is made up out

of a finite number of units, each of which is the simplest possible kind of quantum-

mechanical system, a so-called two-state system, whose behavior, as we shall see, is

highly constrained and easily specified. Much of the analytical complexity of learning

quantum mechanics is connected with mastering the description of continuous (infinite-

state) systems. By restricting attention to collections of two-state systems (or even d

-state systems for finite d ) one can avoid much suffering. Of course one also looses much

wisdom, but hardly any of it at least at this stage of the art is relevant to the basic

theory of quantum computation. Second, and just as important, the most difficult part

of learning quantum mechanics is to get a good feeling for how the formalism can be

applied to actual phenomena. This almost invariably involves formulating oversimplified

abstract models of real physical systems, to which the quantum formalism can then be

applied. History tells us that it was necessary an extraordinary intuition to realize

66

What is a quantum computer? 67

what features of the phenomena are essential and must be represented in a model, and

what features are inessential and can be ignored. The theory of quantum computation,

however, is entirely concerned with an abstract model the easy part of the problem.

It is tempting to say that a quantum computer is one whose operation is governed by

the laws of quantum mechanics. But since the laws of quantum mechanics govern the

behavior of all physical phenomena, this temptation must be resisted. Every laptop

operates under the laws of quantum mechanics, but it is not a quantum computer. A

quantum computer is one whose operation exploits certain very special transformations

of its internal state. The laws of quantum mechanics allow these peculiar transforma-

tions to take place under very carefully controlled conditions. In a quantum computer

the physical systems that encode the individual logical bits must have no physical in-

teractions whatever that are not under the complete control of the program. All other

interactions, however irrelevant they might be in an ordinary computer which we shall

call classical introduce potentially catastrophic disruptions into the operation of a quan-

tum computer. Such damaging encounters can include interactions with the external

environment, such as air molecules bouncing off the physical systems that represent bits,

or the absorption of minute amounts of ambient radiant thermal energy. There can even

be distructive interactions between the computationally relevant features of the physical

systems that represent bits and other features of those same systems that are associated

with computationally irrelevant aspects of their internal structure. Such destructive

interactions, between what matters for the computation and what does not, result in

decoherence, which is fatal to a quantum computation. To avoid decoherence individual

bits cannot in general be encoded in physical systems of macroscopic size, because such

systems (except under very special circumstances) cannot be isolated from their own

irrelevant internal properties. Such isolation can be achieved if the bits are encoded in

a small number of states of a system of atomic size, where extra internal features do

not matter, either because they do not exist, or because they require unavailably high

energies to come into play. Such atomic-scale systems must also be decoupled from their

surroundings except for the completely controlled interactions that are associated with

the computational process itself.

Two things keep the situation from being hopeless. First, because the separation between

the discrete energy levels of a system on the atomic scale can be enormously larger

than the separation between the levels of a large system, the dynamical isolation of an

atomic system is easier to achieve. It can take a substantial kick to knock an atom

out of its state of lowest energy. The second reason for hope is the discovery that

errors induced by extraneous interactions can actually be corrected if they occur at a

sufficiently low rate. While error correction is routine for bits represented by classical

systems, quantum error correction is constrained by the formidable requirement that

What is a quantum computer? 68

it be done without knowing either the original or the corrupted state of the physical

systems that represent the bits. Remarkably, this turns out to be possible. Whether

or not it will ever become a practical technology, there is a beauty to the theory of

quantum computation that gives it a powerful appeal as a lovely branch of mathematics,

and as a strange generalization of the paradigm of classical computer science, which

had completely escaped the attention of computer scientists until the 1980s. The new

paradigm demonstrates that the theory of computation can depend profoundly on the

physics of the devices that carry it out. Quantum computation is also a valuable source

of examples that illustrate and illuminate, in novel ways, the mysterious phenomena

that quantum behavior can give rise to. For computer scientists the most striking thing

about quantum computation is that a quantum computer can be vastly more efficient

than anything ever imagined in the classical theory of computational complexity, for

certain computational tasks of considerable practical interest. The time it takes the

quantum computer to accomplish such tasks scales up much more slowly with the size

of the input than it does in any classical computer.

Appendix E

Bibliographical remarks

For a wide overview of the topics in Chapter 1 the reader is referred to ([23], [6], [26],

[26], [25]).

For a wide overview of the topics in Chapter 2 the reader is referred to ([2], [37]).

For a wide overview of the topics in Chapter 3 the reader is referred to ([14], [19], [14],

[19]).

For a wide overview of the topics in Appendix A the reader is referred to ([23], [8], [30]).

For a wide overview of the topics in Appendix B the reader is referred to ([34], [40],

[11]).

For a wide overview of the topics in Appendix C the reader is referred to ([25], [26], [8]).

For a wide overview of the topics in Appendix D the reader is referred to ([32], [30], [23],

[7], [1]).

69

Acknowledgements

First, I would warmly thank my supervisor Professor Roberto Giuntini and Professor

Maria Luisa Dalla Chiara for their precious teachings and suggestions.

I am very grateful to my teachers, collegues and friends Professor Tomasz Kowalski,

Professor Marco Giunti, Hector Freytes and especially Professor Francesco Paoli and

Antonio Ledda for their assistance and guidance. The core of the present thesis is joint

work with them and I was very privileged to do research with them.

I would like to personally thank Professor Francesco Ledda, Professor Michele Camerota,

Valentina Favrin and all my Ph.D. collegues for their support and valuable suggestions.

A particular acknowledgement goes to my mother: in some sense this thesis is dedicated

to her as well.

Last but not least I’d like to thank Francesca: she is in every page of this work.

70

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