an introduction to quantum computing gosn203 (ai); gosn204 (os) professor john fulcher christopher...

An Introduction to Quantum An Introduction to Quantum ComputingComputing

GOSN203 (AI); GOSN204 (OS)GOSN203 (AI); GOSN204 (OS)

Professor John FULCHERProfessor John FULCHER

Christopher Newport UniversityChristopher Newport UniversityApril 2004April 2004

ReferencesReferences:: ** E. Riefel & W. Polak (2000) “An Introduction to E. Riefel & W. Polak (2000) “An Introduction to

Quantum Computing for Non-Physicists” Quantum Computing for Non-Physicists” ACM ACM Computing SurveysComputing Surveys 2222(3) 300-335 (3) 300-335 * * HANDOUT#1HANDOUT#1

** J. Mullins (2002) “Making Unbreakable Code” J. Mullins (2002) “Making Unbreakable Code” IEEE IEEE SpectrumSpectrum May 40-45. May 40-45. * * HANDOUT#2HANDOUT#2

http://www.pcs.cnu.edu/~mzhang/PCS450_550/http://www.pcs.cnu.edu/~mzhang/PCS450_550/QuantumComp1(2).ppt QuantumComp1(2).ppt (Lecture Notes: MS-(Lecture Notes: MS-PowerPoint) PowerPoint)

C. Bennett, G. Brassad & A. Ekert (1992) “Quantum C. Bennett, G. Brassad & A. Ekert (1992) “Quantum Cryptography” Cryptography” Scientific AmericanScientific American 267267(4) 26-33(4) 26-33

C. Williams & S. Clearwater (1998) C. Williams & S. Clearwater (1998) Explorations in Explorations in Quantum ComputingQuantum Computing Springer (+ CDROM – Springer (+ CDROM – Mathematica)Mathematica)

……the story so far:the story so far:

Key Quantum PhenomenaKey Quantum Phenomena

Key Quantum Computing PhenomenaKey Quantum Computing Phenomena:: 1. 1. SuperpositionSuperposition of of allall possible states possible states

simultaneously. Hence an n-Qubit simultaneously. Hence an n-Qubit memory memory registerregister can exist in a can exist in a superposition of superposition of allall 2 2nn possible possible configurations: |f> = a|0> + b|1> configurations: |f> = a|0> + b|1>

i.e. a Quantum Computer = a i.e. a Quantum Computer = a massively massively parallelparallel computer ( computer (howeverhowever, it is impossible to , it is impossible to observe these parallel computations observe these parallel computations individually). individually).


Key Quantum Computing Key Quantum Computing PhenomenaPhenomena:: 2. 2. InterferenceInterference – since a QC can work on – since a QC can work on

severalseveral classical inputs at once, they classical inputs at once, they can interfere with/influence one another can interfere with/influence one another (either constructively or destructively):(either constructively or destructively):

|f> = |0 1> + |1 0> |f> = |0 1> + |1 0> a a netnet computational state that reveals a computational state that reveals a

joint/collective property of joint/collective property of allall the the computations i.e. computations i.e. quantum parallelismquantum parallelism..


Key Quantum Computing PhenomenaKey Quantum Computing Phenomena:: 3. 3. EntanglementEntanglement – 2 or more Qubits emerge – 2 or more Qubits emerge

from an interaction in a definite joint quantum from an interaction in a definite joint quantum state that state that cannotcannot be expected in terms of a be expected in terms of a product of definite product of definite individualindividual quantum states: | quantum states: |f> = |0 1> + |1 0> f> = |0 1> + |1 0>

Moreover, they retain a lingering, instantaneous Moreover, they retain a lingering, instantaneous influence on each other, influence on each other, irrespectiveirrespective of their distance of their distance of separation of separation quantum teleportationquantum teleportation (for which (for which there is no classical counterpart!)there is no classical counterpart!); ; quantum factoringquantum factoring relies on entanglement to create a repeating relies on entanglement to create a repeating sequence of numbers whose period reveals the sequence of numbers whose period reveals the factors of a large integer.factors of a large integer.


Key Quantum Computing Key Quantum Computing PhenomenaPhenomena:: 4. 4. Non-determinismNon-determinism = inability to predict = inability to predict

the quantum state into which a the quantum state into which a superposed state will collapse upon superposed state will collapse upon being measuredbeing measured

quantum key distributionquantum key distribution, which relies on , which relies on non-determinism to guarantee that any non-determinism to guarantee that any eavesdropping will be detected. eavesdropping will be detected.


Key Quantum Computing Key Quantum Computing PhenomenaPhenomena:: 5. 5. Non-clonabilityNon-clonability, since it is impossible , since it is impossible

to copy an unknown quantum state to copy an unknown quantum state exactlyexactly (Heisenberg Uncertainty (Heisenberg Uncertainty Principle).Principle).

It is impossible to measure pairs of It is impossible to measure pairs of quantities simultaneously (e.g. position & quantities simultaneously (e.g. position & momentum) momentum) quantum cryptographyquantum cryptography relies relies on non-clonability to guarantee security.on non-clonability to guarantee security.


Key Quantum Computing Key Quantum Computing PhenomenaPhenomena:: 6. 6. Non-localityNon-locality: : quantum teleportationquantum teleportation

relies on non-locality (as well as relies on non-locality (as well as entanglement) to disassemble and re-entanglement) to disassemble and re-assemble the quantum state to be assemble the quantum state to be teleported.teleported.


Key Quantum Computing PhenomenaKey Quantum Computing Phenomena:: 7. 7. reversiblereversible: and thus no power : and thus no power

dissipation:dissipation: HH |in> = |out>; |in> = |out>; H H -1-1 |out> = |in> |out> = |in> An operation is logically An operation is logically reversiblereversible if it if it

can be undone can be undone (run backwards)(run backwards) – i.e. if its – i.e. if its inputsinputs can always be deduced from the can always be deduced from the outputsoutputs..

cf. cf. classicalclassical computations, which are computations, which are irreversibleirreversible & thus & thus dissipativedissipative..

Quantum Spin StatesQuantum Spin States

2-state quantum system used to 2-state quantum system used to encode a encode a QubitQubit:: The The (solid)(solid) angle between the vector & angle between the vector &

the vertical axis the vertical axis (= phase)(= phase) is determined is determined by the relative contributions of the |by the relative contributions of the |00> > and |and |11> (eigen)> (eigen)statesstates

Q: Q: Where’s the power in Where’s the power in QCQCs?s?A:A: Superposition Superposition

SuperpositionSuperposition = = simultaneoussimultaneous existence in existence in manymany states, states, not justnot just |0> |0> andand |1> |1>

but:but:

QubitQubit:: phase = (solid) phase = (solid) magnitude = cmagnitude = c00|0> + c|0> + c11|1>|1>

where probabilities |cwhere probabilities |c00||22 + |c + |c11||22 = 1 = 1 (i.e. 100%)(i.e. 100%) & |c& |cii| = sqrt (x| = sqrt (xii

22 + y + yii22) ) (i.e. complex numbers)(i.e. complex numbers)

Quantum GatesQuantum GatesAny quantum computation can be reduced to a sequence of 1 and 2 qubit operations:

H |in> = H1 H2 H3 .... Hn |in>

Conventional operations: NOT , AND

Quantum operations: qNOT , CNOT

NOT CONTROLLED NOT IN OUT IN OUT|0|1 |00|00|1|0 |01|01 |10|11 |11|10

+Phase shifts

Quantum GatesQuantum Gates

e.g. a e.g. a rootroot(NOT) gate:(NOT) gate:

rootNOT rootNOT

|input> |output> = NOT |input>

superposition of bits(thus unlike any classical gate)

WWalsh-alsh-HHadamard Gateadamard Gate

e.g. if we apply the WH-gate to each e.g. if we apply the WH-gate to each of of nn qubitqubits individually, we obtain the s individually, we obtain the superpositionsuperposition of the 2 of the 2nn numbers that numbers that can be represented in can be represented in nn bits. bits. Thus we can effectively load Thus we can effectively load

exponentiallyexponentially many (2 many (2nn) numbers into a ) numbers into a quantum computerquantum computer using only using only polynomialpolynomial many ( many (nn) basic gate ) basic gate operations.operations.

Controlled-NOT GateControlled-NOT Gate

A particular 2-A particular 2-qubit gate is of gate is of paramount importance in quantum paramount importance in quantum computing, & this is the computing, & this is the controlled-NOT controlled-NOT gategate:: UUCNCN|00> = |00>|00> = |00> UUCNCN|01> = |01>|01> = |01> UUCNCN|10> = |11>|10> = |11> UUCNCN|11> = |10>|11> = |10>

ie. apply NOT to (flip) ie. apply NOT to (flip) secondsecond bit if bit if firstfirst qubit = 1. = 1.

|x> |x>

|y> |x xor y>

Controlled-NOT GateControlled-NOT Gate

NOTE that this operation involves no NOTE that this operation involves no measurements whatsoever – i.e. we measurements whatsoever – i.e. we do not need to do not need to measuremeasure qubits in s in order to bring about “controlled” order to bring about “controlled” operations.operations.

|x> |x>

|y> |x xor y>

xor:

control

1 0 1 not

0 1 1

Universal Quantum GatesUniversal Quantum Gates

Universal GatesUniversal Gates: the : the (infinite)(infinite) set of all 1- set of all 1-qubit rotations, together with the rotations, together with the controlled-NOT gate, is enough to achieve controlled-NOT gate, is enough to achieve anyany imaginable quantum computation. imaginable quantum computation.

i.e. we can perform i.e. we can perform anyany quantum quantum computationcomputation by connecting just 1- by connecting just 1-qubit rotation gates and controlled-NOT gates rotation gates and controlled-NOT gates (cf. any (cf. any classical computationclassical computation can be can be realized using realized using justjust AND and NOT gates) AND and NOT gates) Barenco (1995) and DiVincenzo (1995) Barenco (1995) and DiVincenzo (1995)

independently showed that a 2-independently showed that a 2-qubit gate is gate is universal for quantum computation. universal for quantum computation.

Reversible ComputationReversible Computation

QuantumQuantum realizations of sets of realizations of sets of reversiblereversible gates which are gates which are universaluniversal for for allall Boolean circuits. Boolean circuits. Recall that a Recall that a quantum circuitquantum circuit is is

composed of quantum wires & composed of quantum wires & elementary quantum gates; each wire elementary quantum gates; each wire represents a path of a single represents a path of a single qubitqubit & is & is described by a state in the 2D described by a state in the 2D Hilbert Hilbert SpaceSpace CC22..

Quantum CalculusQuantum Calculus

A A Hilbert SpaceHilbert Space is a mathematical is a mathematical model for representing state space model for representing state space vectors.vectors.

The The statestate of a of a quantum systemquantum system can be can be described by a described by a column vectorcolumn vector (| (|> > “ket”) in a “ket”) in a Hilbert SpaceHilbert Space of of wave wave functionsfunctions.. As the system As the system evolvesevolves, its , its state vectorstate vector

rotates with its base anchored to the origin rotates with its base anchored to the origin of the axes.of the axes.

Tensor ProductsTensor Products

Systems of Systems of more than onemore than one qubitqubit need need a a Hilbert SpaceHilbert Space which captures the which captures the interactioninteraction (entanglement) of the (entanglement) of the qubits.qubits.

A 2-qubit system can be represented A 2-qubit system can be represented by a unit vector in the by a unit vector in the tensor producttensor product of 2 copies of of 2 copies of CC22 (i.e. the space (i.e. the space CC22 CC22).).

Quantum CalculusQuantum Calculus

In general, a system containing exactly In general, a system containing exactly nn >= 2 >= 2 qubitqubits is represented by s is represented by nn copies of copies of CC22 tensoredtensored together. Thus the state space together. Thus the state space is 2is 2nn-dimensional.-dimensional.

Now in contrast to a Now in contrast to a classicalclassical system system, , which can be completely defined by which can be completely defined by describing the state of each individual describing the state of each individual componentcomponent, in a , in a quantumquantum system system, the , the state state cannotcannot always be described by always be described by considering only the component pieces.considering only the component pieces.

Entangled StatesEntangled States

e.g. the state 1/root(2)(|00> + |11>) e.g. the state 1/root(2)(|00> + |11>) cannotcannot be decomposed into separate be decomposed into separate states for each of the 2 states for each of the 2 qubitqubits.s. i.e. we cannot express this state as a i.e. we cannot express this state as a

tensor product of two single tensor product of two single qubitqubits.s. A state that A state that can’tcan’t be expressed as a be expressed as a

tensor product is called an tensor product is called an entangledentangled statestate. .

Not covered in lectures…Not covered in lectures…

Quantum Memory RegistersQuantum Memory Registers Quantum Error CorrectionQuantum Error Correction

Symmetry, entanglement, “ancilla” Symmetry, entanglement, “ancilla” qubits (Shor)qubits (Shor)

Fault tolerantFault tolerant ( (QQuantum) uantum) CComputersomputers [ref. [ref. Handout#1Handout#1, references], references]

This Lecture…This Lecture…

Quantum AlgorithmsQuantum Algorithms Quantum Key DistributionQuantum Key Distribution (Teleportation)?(Teleportation)? Quantum ComputerQuantum Computer Hardware Hardware QCQC Applications (OS; AI)? Applications (OS; AI)?

Quantum ParallelismQuantum Parallelism

The principal advantage of a The principal advantage of a quantum computerquantum computer over a over a classical classical computercomputer is that it can use a is that it can use a technique called technique called quantum parallelismquantum parallelism to compute certain joint properties of to compute certain joint properties of several several superposedsuperposed computations computations ((severalseveral answers to different classical answers to different classical

computation) in the time it takes a computation) in the time it takes a classical computer to find just classical computer to find just oneone of of these answers…these answers…

Quantum ParallelismQuantum Parallelism

……moreover, the moreover, the quantum computerquantum computer can do this can do this withoutwithout needing to reveal needing to reveal the answer to any one of those the answer to any one of those computations individually.computations individually.

This gives the This gives the quantum computerquantum computer the the potential to be potential to be vastlyvastly more more efficientefficient than a than a classical computerclassical computer at certain at certain computational tasks.computational tasks.

Quantum AlgorithmsQuantum Algorithms

1. The 1. The Deutsch-Jozsa problemDeutsch-Jozsa problem: Is a boolean : Is a boolean function function ff:{0,1} :{0,1} {0,1} {0,1} eveneven (i.e. always gives (i.e. always gives the same output)the same output) and/or and/or balancedbalanced (gives one (gives one output on half of the inputs, & another output output on half of the inputs, & another output on the other half)?on the other half)? Exploits Exploits superpositionsuperposition withoutwithout need for need for

measurementmeasurement 2. 2. SimonSimon (1994): a (1994): a quantum memory registerquantum memory register

could be used to evolve into a could be used to evolve into a superpositionsuperposition representing the representing the FFourier ourier TTransformransform.. Measurement Measurement sample sample period (of sines, cosines) period (of sines, cosines)

Quantum AlgorithmsQuantum Algorithms

3. 3. ShorShor (1994): (1994): factoringfactoring of of large large composite integerscomposite integers can be achieved by can be achieved by finding the finding the periodperiod (= (= QCQC’s “killer ’s “killer app”lication)app”lication) exploits a technique similar to Simon’s exploits a technique similar to Simon’s FFourier ourier

TTransform sampling.ransform sampling. 4. 4. GroverGrover (1996) showed that (1996) showed that unstructuredunstructured

searchsearch can be solved with bounded can be solved with bounded probability in O(rootprobability in O(rootNN) on a ) on a QQuantum uantum CComputer.omputer.

1. Deutsch-Joza Problem1. Deutsch-Joza Problem

The The Deutsch-Jozsa problemDeutsch-Jozsa problem: Consider a : Consider a boolean function boolean function f f :{0,1} -> {0,1}.:{0,1} -> {0,1}.

Is Is ff(0) = (0) = ff(1) or (1) or ff(0) not= (0) not= ff(1)?(1)? ClassicalClassical test: 2 computations & 1 test: 2 computations & 1

comparison.comparison. Can we do better on a Can we do better on a QCQC? Yes!? Yes! The key to a The key to a Quantum ComputerQuantum Computer solution solution is is

that we do not need to actually that we do not need to actually calculatecalculate ff(x), simply determine whether they are (x), simply determine whether they are the same. the same.


SupposeSuppose we possess a quantum we possess a quantum “black box” which computes ‘“black box” which computes ‘ff’ (a big ’ (a big if!)if!) Consider the transformation UConsider the transformation Uff which which

applies to 2 qubits |x> and |y> and applies to 2 qubits |x> and |y> and produces |x>|y produces |x>|y mod2mod2 ff(x)).(x)).

This transformation flips the second bit This transformation flips the second bit if if ff acting on the first bit is 1, & does acting on the first bit is 1, & does nothing if nothing if ff acting on the first qubit is 0. acting on the first qubit is 0.


Now since the black box is “quantum”, Now since the black box is “quantum”, we can choose the input state to be a we can choose the input state to be a superpositionsuperposition of |0> and |1>, say of |0> and |1>, say 1/root2(|0>+|1>) and 1/root2(|0>-|1>)1/root2(|0>+|1>) and 1/root2(|0>-|1>)…… Perform a measurement that projects the Perform a measurement that projects the

first qubit onto the basis 1/root2(|0>+|1>), first qubit onto the basis 1/root2(|0>+|1>), 1/root2(|0>-|1>)1/root2(|0>-|1>)

we will obtain 1/root2(|0>+|1>) if the we will obtain 1/root2(|0>+|1>) if the function is balanced, 1/root2(|0>-|1>) if not. function is balanced, 1/root2(|0>-|1>) if not.


We can achieve this because a We can achieve this because a quantum computerquantum computer can be in a can be in a blendblend of states: we can compute of states: we can compute ff(0) and (0) and ff(1), but more importantly, extract (1), but more importantly, extract information about information about ff which tells us which tells us whether whether ff(0) is equal to (0) is equal to ff(1) or not.(1) or not.


Solution of the Solution of the Deutsch-Jozsa Deutsch-Jozsa problemproblem on a on a quantum computerquantum computer:: Step#1Step#1. Initialize the 2-qubit . Initialize the 2-qubit register in the state |0>|1>.register in the state |0>|1>.

Step#2Step#2. Apply the . Apply the WWalsh-alsh-HHadamard adamard operation W to each qubit:operation W to each qubit: |0>||0>|1> 1> 1/ 1/rootroot2(|0> + |1>) 2(|0> + |1>) superimposed superimposed withwith 1/ 1/rootroot2(|0> - |1>)2(|0> - |1>)


Solution of the Solution of the Deutsch-Jozsa Deutsch-Jozsa problemproblem on a on a quantum computerquantum computer:: Step#3Step#3. Apply the operation . Apply the operation UU (which requires (which requires ff to be evaluated to be evaluated onceonce only):only): 1/1/rootroot2(|0> + |1>) 2(|0> + |1>) superimposed superimposed withwith 1/ 1/rootroot2(|0> - |1>) 2(|0> - |1>) 1/ 1/rootroot2((-2((-11))ff(0)(0)||0> + (-10> + (-1))ff(1)(1)|1>) |1>) superimposed withsuperimposed with 1/1/rootroot2(|0> - |1>)2(|0> - |1>)


Solution of the Solution of the Deutsch-Jozsa Deutsch-Jozsa problemproblem on a on a quantum computerquantum computer:: Step#4Step#4. Apply the operation . Apply the operation VV (which does (which does notnot require require ff to be to be evaluated):evaluated): 1/1/rootroot2((-12((-1))ff(0)(0)||0> + (-10> + (-1))ff(1)(1)||1>) 1>) superimposed withsuperimposed with 1/ 1/rootroot2(|0> - |2(|0> - |1>) 1>) 1/ 1/rootroot2((-12((-1))ff(0)(0)+(-1+(-1))ff(1)(1)||0> + (-0> + (-11))ff(0)(0)-(-1-(-1))ff(1)(1)||1>) 1>) superimposed withsuperimposed with 1/1/rootroot2(|0> - |1>)2(|0> - |1>)


Solution of the Solution of the Deutsch-Jozsa Deutsch-Jozsa problemproblem on a on a quantum computerquantum computer:: Step#5Step#5. Measure the bit value in . Measure the bit value in the first qubit:the first qubit:

If it is 0, If it is 0, ff(0) = (0) = ff(1);(1); If it is 1, If it is 1, ff(0) not= (0) not= ff(1).(1).


NOTE: in essence, this NOTE: in essence, this quantum quantum algorithmalgorithm exploits exploits superpositionsuperposition and and interferenceinterference to extract a joint to extract a joint property of both function values – property of both function values – ff(0) and (0) and ff(1) – (1) – withoutwithout having to having to calculate either function value calculate either function value explicitly.explicitly.

2. Fourier Transform on QC2. Fourier Transform on QC

AnyAny mathematical function can be mathematical function can be described as a weighted sum of described as a weighted sum of certain basis (elementary) functions certain basis (elementary) functions such as sines & cosines (or real & such as sines & cosines (or real & imaginary exponential functions): imaginary exponential functions): sin(sin(xx), sin(2), sin(2xx), …cos(), …cos(xx), cos(), cos(2x2x)…)…(the (the more terms, the better the more terms, the better the approximation)approximation) Fourier SeriesFourier Series recall erecall eii = cos = cos + + i i sin sin (circle) (circle)


Fourier seriesFourier series = (integral) = (integral) representation of continuous (linear) representation of continuous (linear) functions functions DDiscrete iscrete FFourier ourier TTransform for sampled functions ransform for sampled functions FFast ast FFourier ourier TTransformransform ( (= a more = a more efficient algorithm – 2efficient algorithm – 2nn terms terms)) In In DDigital igital SSignal ignal PProcessing, the rocessing, the FFTFFT

transforms a signal from the transforms a signal from the time-domaintime-domain to the to the frequency domainfrequency domain (& (& Inverse Inverse FFTFFT from from ff-domain to -domain to tt-domain)-domain)


ObservationObservation: if a time-varying signal : if a time-varying signal is very is very spikyspiky, this means it can be , this means it can be represented by just a represented by just a fewfew sines & sines & cosines, with precisely defined cosines, with precisely defined periods. periods.


D. Simon (1994): a quantum D. Simon (1994): a quantum computation could cause the state of computation could cause the state of a a quantum memory registerquantum memory register to to evolve into a evolve into a superpositionsuperposition representing the representing the Fourier TransformFourier Transform.. By By readingreading this memory register, we this memory register, we

would would most likelymost likely obtain a result obtain a result corresponding to where the corresponding to where the probabilityprobability amplitude was most highly concentrated amplitude was most highly concentrated – i.e. where the – i.e. where the Fourier TransformFourier Transform is is most strongly most strongly spikedspiked. .


Thus a quantum measurement Thus a quantum measurement returns a returns a samplesample from the from the Fourier Fourier TransformTransform, which provides us with , which provides us with some information about the periodic some information about the periodic sine & cosine functions which make sine & cosine functions which make up our original function. up our original function.

3. Shor’s Factoring Algorithm3. Shor’s Factoring Algorithm

P. P. ShorShor (AT&T) was wanting to (AT&T) was wanting to demonstrate that a demonstrate that a quantum computerquantum computer could be used to solve a could be used to solve a real problemreal problem, as , as opposed to the opposed to the contrivedcontrived problems problems demonstrated up to that time (mid 1990s).demonstrated up to that time (mid 1990s).

ShorShor: if you can relate the : if you can relate the (real)(real) problem problem of finding the of finding the factorsfactors of a of a large composite large composite integerinteger to that of finding the to that of finding the periodperiod, then , then you can exploit a technique similar to you can exploit a technique similar to Simon’s sampling of a Simon’s sampling of a FTFT. .


Shor (1994) showed that a Shor (1994) showed that a quantum quantum computercomputer could be used to factor a could be used to factor a large integer large integer super-efficientlysuper-efficiently.. This was big news, especially in security This was big news, especially in security

& banking circles (since all of a sudden & banking circles (since all of a sudden RSA public cryptography is rendered RSA public cryptography is rendered eminently breakable)! eminently breakable)!


MultiplyingMultiplying large prime numbers together large prime numbers together is computationally easy: e.g. 127 * 229 = ?is computationally easy: e.g. 127 * 229 = ?

By contrast, no conventional (classical) By contrast, no conventional (classical) polynomial algorithm exists for polynomial algorithm exists for factoringfactoring large prime numbers (exhaustive search large prime numbers (exhaustive search only) only) (in fact it is thought to be practically (in fact it is thought to be practically

impossible) impossible) hence used as the basis for hence used as the basis for (RSA) (RSA) public key cryptographypublic key cryptography. . e.g. ? * ? = e.g. ? * ? = 29,08329,083

& this is only for & this is only for fivefive digits – imagine 400, say! digits – imagine 400, say!


Classical Factoring AlgorithmsClassical Factoring Algorithms:: The time required to find the factors is The time required to find the factors is

strongly believed (but has never been strongly believed (but has never been proved) to be proved) to be superpolynomialsuperpolynomial in log( in log(nn); ); i.e. as i.e. as nn increases, the worst case time increases, the worst case time grows grows fasterfaster than any power of log( than any power of log(nn).).

Quadratic Sieve was the best known Quadratic Sieve was the best known technique in 1997 technique in 1997 (network of 1,000 (network of 1,000 workstations).workstations).


Shor’s exciting new result was that a Shor’s exciting new result was that a quantum computer could factor in quantum computer could factor in polynomialpolynomial time – O[(ln time – O[(ln n n))33] ] factoring of a 400-digit number in factoring of a 400-digit number in under 3 years (cf. 10under 3 years (cf. 101010 years on a years on a classical computer)!classical computer)!

3. Shor’s Factoring Algorithm3. Shor’s Factoring AlgorithmPublic Key Cryptography (RSA)

P1 P2 C [easy] : t = P(N)

C P1 P2 [hard] : t = exp(N)

Eg: Factorization of a 129-digit number

(RSA-129) ~2000 computers processing for 8 months

Shor’s Algorithm - Finds prime factors

Peter Shor (AT&T Bell Labs, 1994)

C P1 P2 [easy] : t = P(N)

With a QC could solve RSA-129 in seconds!


Shor’s quantum factoring algorithmShor’s quantum factoring algorithm relies on a result from relies on a result from number theorynumber theory that relates the that relates the periodperiod of a particular of a particular periodic function to the periodic function to the factorsfactors of an of an integer:integer: Given a number Given a number nn, choose a related , choose a related

function function ffnn(a)(a) = = xxa a mod mod nn, such that the , such that the GGreatest reatest CCommon ommon DDivisor of ivisor of xx and and nn = = 1.1.

Both mod & GCD can be computed Both mod & GCD can be computed efficiently (even on a efficiently (even on a classicalclassical computer). computer).


Step-1Step-1: pick a number : pick a number qq such that such that 2n2n22 =< =< qq =< 3n =< 3n22..

Step-2Step-2: pick a random integer : pick a random integer xx whose whose GGreatest reatest CCommon ommon DDivisor with ivisor with nn is 1. is 1.


Step-3Step-3: repeat the following steps (a) : repeat the following steps (a) through (g) through (g) aboutabout log( log(qq) times, using ) times, using the same random number x each the same random number x each time:time: (a) create a quantum memory register, (a) create a quantum memory register,

& partition the qubits into two sets & partition the qubits into two sets called register1 & register2, called register1 & register2,


Step-3Step-3: repeat the following steps (a) : repeat the following steps (a) through (g) through (g) aboutabout log( log(qq) times, using ) times, using the same random number x each the same random number x each time:time: (a) create a quantum memory register, & (a) create a quantum memory register, &

partition the qubits into two sets called partition the qubits into two sets called register1 & register2,register1 & register2,

(b) load register1 with all integers in the (b) load register1 with all integers in the range 0 to range 0 to qq-1, & load register2 with all -1, & load register2 with all zeroes, zeroes,


Step-3Step-3: repeat the following steps (a) : repeat the following steps (a) through (g) through (g) aboutabout log( log(qq) times, using the ) times, using the same random number x each time:same random number x each time: (a) create a quantum memory register, & (a) create a quantum memory register, &

partition the qubits into two sets called partition the qubits into two sets called register1 & register2,register1 & register2,

(b) load register1 with all integers in the range (b) load register1 with all integers in the range 0 to 0 to qq-1, & load register2 with all zeroes,-1, & load register2 with all zeroes,

(c) now compute, in (c) now compute, in quantum parallelquantum parallel, the , the function function xxaa mod mod nn of each number in register1, of each number in register1, & place result in register2. & place result in register2.


Step-3Step-3: repeat the following steps (a) : repeat the following steps (a) through (g) through (g) aboutabout log( log(qq) times, using ) times, using the same random number x each the same random number x each time:time: (d) measure the state of register2, (d) measure the state of register2,

obtaining some result obtaining some result kk. This has the . This has the effect of projecting out the state of effect of projecting out the state of register1 to be a register1 to be a superpositionsuperposition of just of just those values of a such that those values of a such that xxaa mod mod nn = = kk, ,


Step-3Step-3: repeat the following steps (a) : repeat the following steps (a) through (g) through (g) aboutabout log( log(qq) times, using ) times, using the same random number x each time:the same random number x each time: (d) measure the state of register2, (d) measure the state of register2,

obtaining some result obtaining some result kk. This has the effect . This has the effect of projecting out the state of register1 to of projecting out the state of register1 to be a be a superpositionsuperposition of just those values of a of just those values of a such that such that xxaa mod mod nn = = kk,,

(e) next compute the (e) next compute the FFourier ourier TTransform of ransform of the projected state in register1, the projected state in register1,


Step-3Step-3: repeat the following steps (a) : repeat the following steps (a) through (g) through (g) aboutabout log( log(qq) times, using ) times, using the same random number x each the same random number x each time:time: (f) measure the state of register1. This (f) measure the state of register1. This

effectively samples from the effectively samples from the FFourier ourier TTransform and returns some number ransform and returns some number c’c’ that is some multiple that is some multiple of of q/rq/r, where , where rr is is the desired period; i.e. the desired period; i.e. c’/qc’/q ~ l/ ~ l/rr for for some positive integer some positive integer . .


Step-3Step-3: repeat the following steps (a) : repeat the following steps (a) through (g) through (g) aboutabout log( log(qq) times, using the ) times, using the same random number x each time:same random number x each time: (f) measure the state of register1. This (f) measure the state of register1. This

effectively samples from the effectively samples from the FFourier ourier TTransform ransform and returns some number and returns some number c’c’ that is some that is some multiple multiple of of q/rq/r, where , where rr is the desired period; is the desired period; i.e. i.e. c’/qc’/q ~ l/ ~ l/rr for some positive integer for some positive integer ..

(g) to determine the (g) to determine the periodperiod rr, we need to , we need to estimate estimate . This is accomplished using a . This is accomplished using a continued fraction technique. continued fraction technique.


Step-4Step-4: by repeating steps (a) : by repeating steps (a) through (g) we create a set of through (g) we create a set of samples of the samples of the DDiscrete iscrete FFourier ourier TTransform in register1. This gives ransform in register1. This gives samples of multitudes of 1/samples of multitudes of 1/rr as as 11//rr, , 22//rr, , 33//rr…for various integers …for various integers ii.. After a few repetitions of the algorithm, After a few repetitions of the algorithm,

we have enough samples of the we have enough samples of the contents of register1 to compute what contents of register1 to compute what i i

must be and hence to guess must be and hence to guess r.r.


Step-5Step-5: when : when rr is known the factors is known the factors of of nn can be obtained from GCD( can be obtained from GCD(xxr/2r/2 – – 1,1,nn) and GCD() and GCD(xxr/2r/2 + 1, + 1,nn).). ((GGreatest reatest CCommon ommon DDemoninator)emoninator)

4. Grover’s Search Algorithm4. Grover’s Search Algorithm

Generally speaking, a solution Generally speaking, a solution search search spacespace has no special structure, which has no special structure, which prevents the development of efficient prevents the development of efficient algorithms.algorithms. e.g. e.g. (structured)(structured): you know someone’s : you know someone’s

namename - find their - find their telephone numbertelephone number in the in the city’s directory.city’s directory.

e.g. e.g. (unstructured)(unstructured): you know someone’s : you know someone’s telephone numbertelephone number – find their – find their namename!!


In order to search a simple In order to search a simple unstructuredunstructured filefile, a computer would have to run , a computer would have to run through, on average, through, on average, halfhalf of the data to of the data to locate an locate an xx satisfying satisfying P(x)P(x).. No shortcuts are possible, thus No shortcuts are possible, thus randomly randomly

testingtesting the predicate the predicate PP is the best strategy is the best strategy that can be adopted on a conventional that can be adopted on a conventional computer computer O( O(NN) for a search space = ) for a search space = NN (omitting time to test (omitting time to test PP))


Now while no shortcuts are possible Now while no shortcuts are possible on a on a conventional computerconventional computer, we can , we can do much better on a do much better on a quantum quantum computercomputer.. Grover (1996) showed that Grover (1996) showed that unstructuredunstructured

searchsearch can be solved with bounded can be solved with bounded probability in O(rootprobability in O(rootNN) on a ) on a QQuantum uantum CComputeromputer. .


Now whilst the resulting speedup of Now whilst the resulting speedup of our our undirected searchundirected search of a city’s of a city’s telephone directory (i.e. O(telephone directory (i.e. O(NN) ) O(rootO(rootNN)) is not particularly dramatic, )) is not particularly dramatic, it it isis in the case of in the case of data encryptiondata encryption.. Consider the Consider the DData ata EEncryption ncryption SStandard: tandard:

enciphering & deciphering are both enciphering & deciphering are both accomplished using a 56-bit accomplished using a 56-bit keykey, known , known onlyonly to the legitimate sender & receiver. to the legitimate sender & receiver.


The goal of an The goal of an eeavesdropperavesdropper, having , having intercepted matching pairs of intercepted matching pairs of plainplain and and ciphercipher text, is to find the key text, is to find the key that maps one onto the other.that maps one onto the other. This problem can be described as a This problem can be described as a

“virtual phone directory”, in which each “virtual phone directory”, in which each possible key is a “possible key is a “namename”, and the ”, and the enciphered text the corresponding enciphered text the corresponding ““phone numberphone number”. ”.


An exhaustive search would try 2An exhaustive search would try 25555 keys keys before hitting the correct one, which would before hitting the correct one, which would take over a year even if 1 billion keys are take over a year even if 1 billion keys are checked every second (on a conventional checked every second (on a conventional computer)!computer)!

By comparison, By comparison, Grover’s algorithmGrover’s algorithm can solve can solve the problem, after quantum-DES enciphering the problem, after quantum-DES enciphering the known clear text in just 185 million times.the known clear text in just 185 million times. Thus in principle, Grover’s algorithm can be used Thus in principle, Grover’s algorithm can be used

to break classical cryptographic systems such as to break classical cryptographic systems such as DES! DES!


Grover’s algorithm searches an Grover’s algorithm searches an unstructured listunstructured list of size of size NN to find one to find one item satisfying a given condition.item satisfying a given condition.

Let Let nn be such that 2 be such that 2nn >= >= NN.. Assume the Assume the predicatepredicate PP is is

implemented by a implemented by a quantum gatequantum gate UUPP||xx,0> -> |,0> -> |xx,,P(x)P(x)>, where “true” is >, where “true” is encoded as 1. encoded as 1.


Step-1Step-1: start with an equally : start with an equally weighted weighted superpositionsuperposition of all N = 2 of all N = 2nn possible indices. possible indices. Any one of which could be the target Any one of which could be the target

entry in the quantum “telephone entry in the quantum “telephone directory”. directory”.


Step-1Step-1: start with an equally weighted : start with an equally weighted superpositionsuperposition of all N = 2 of all N = 2nn possible indices. possible indices. Any one of which could be the target entry in Any one of which could be the target entry in

the quantum “telephone directory”.the quantum “telephone directory”. Step-2Step-2: Pick an : Pick an (almost)(almost) arbitrary unitary arbitrary unitary

operator. The operator has to have some operator. The operator has to have some non-zero overlap between the starting state non-zero overlap between the starting state and the target.and the target. The easiest way to ensure this is to pick an The easiest way to ensure this is to pick an

operator with no zero entries in its unitary operator with no zero entries in its unitary matrix. matrix.


Step-3Step-3: construct a special : construct a special amplitude-amplification operator Q amplitude-amplification operator Q from the quantum “telephone from the quantum “telephone directory” oracle and the arbitrary directory” oracle and the arbitrary unitary operator. unitary operator.


Step-3Step-3: construct a special amplitude-: construct a special amplitude-amplification operator Q from the amplification operator Q from the quantum “telephone directory” oracle quantum “telephone directory” oracle and the arbitrary unitary operator.and the arbitrary unitary operator.

Step-4Step-4: Iterate : Iterate QQ aboutabout ( (/4)rootN /4)rootN times starting with the state U|s> and times starting with the state U|s> and then measure.then measure. The measurement outcome is the target The measurement outcome is the target

index, with probability ~1 (i.e. near index, with probability ~1 (i.e. near certainty). certainty).


Grover’s algorithmGrover’s algorithm is is optimaloptimal up to a up to a constant factor; no constant factor; no quantum quantum algorithmalgorithm can perform an can perform an unstructured searchunstructured search faster. faster.


If there is only a unique If there is only a unique xxoo such that such that P(xP(xoo)) is true, then after ( is true, then after (/8/8)2)2n/2n/2 iterations of steps iterations of steps 22 through through 44 the the failure rate is ½.failure rate is ½. After iterating (After iterating (/4/4)2)2n/2n/2 times the failure times the failure

rate drops to 2rate drops to 2-n-n.. However However additionaladditional iterations will iterations will

increaseincrease the failure rate! the failure rate! e.g. after (e.g. after (/2/2)2)2n/2n/2 iterations, the failure iterations, the failure

rate is close to 1.rate is close to 1.


This is an important feature of many This is an important feature of many quantum algorithmsquantum algorithms, & has little , & has little counterpart in conventional counterpart in conventional computers.computers.

ie. ie. repeatingrepeating quantum procedures quantum procedures may improve results for a while, but may improve results for a while, but after some repetitions the results will after some repetitions the results will get worse again! get worse again!


Quantum procedures are Quantum procedures are unitary unitary transformationstransformations, which are rotations of , which are rotations of complex space; complex space; repeatedrepeated applications of a applications of a quantum transform may rotate the state quantum transform may rotate the state closer & closer to the desired state for a closer & closer to the desired state for a while, but eventually it will rotate while, but eventually it will rotate pastpast the the desired state & get further & further away desired state & get further & further away from it.from it. Thus to obtain useful results from a repeated Thus to obtain useful results from a repeated

application of a quantum transformation, it is application of a quantum transformation, it is paramount to know when to paramount to know when to stopstop! !

Quantum Key DistributionQuantum Key Distribution

Relies on Quantum Mechanical effects:Relies on Quantum Mechanical effects: Heisenberg Uncertainty PrincipleHeisenberg Uncertainty Principle precludes precludes

exact, simultaneous measurements.exact, simultaneous measurements. PolarizationPolarization: according to Quantum Theory, a : according to Quantum Theory, a

single photon passing through a polarizer will single photon passing through a polarizer will either emerge with its electric field oscillating either emerge with its electric field oscillating in the desired plane, or not at all.in the desired plane, or not at all.

NOTE: here quantum states = light (photon) NOTE: here quantum states = light (photon) polarizations, rather than spin states.polarizations, rather than spin states.


Encoding a (Encoding a (0 00 0 11 …) bit stream …) bit stream within a stream of polarized photons:within a stream of polarized photons:

Calcite (birefringent) Crystal

Vertically polarized photons

Horizontally polarized photons


But what happens when But what happens when diagonally diagonally polarizedpolarized light passes through light passes through vertically-oriented calcite, say?vertically-oriented calcite, say? The The Heisenberg Uncertainty PrincipleHeisenberg Uncertainty Principle

says that says that somesome photons will have their photons will have their polarizations shifted and polarizations shifted and somesome won’t, won’t, depending on the angle of their axis depending on the angle of their axis relative to the calcite crystal’s.relative to the calcite crystal’s.


In order to read the encoded bit In order to read the encoded bit stream, we need to stream, we need to measuremeasure the the polarization of each photon.polarization of each photon. However if we choose the wrong However if we choose the wrong

orientation (axis) with our calcite crystal orientation (axis) with our calcite crystal detector, then we only have a 50:50 detector, then we only have a 50:50 chance of getting the chance of getting the correctcorrect answer. answer.


But can we But can we measuremeasure bothboth the rectilinear the rectilinear (0(0oo/90/90oo) ) andand the diagonal (45 the diagonal (45oo/135/135oo) ) polarizationspolarizations (say) simultaneously? (say) simultaneously? NO!NO! because any attempt to because any attempt to measuremeasure one one

polarization polarization necessarilynecessarily perturbsperturbs (in fact (in fact randomizesrandomizes) the other polarization ) the other polarization (Heisenberg)(Heisenberg)

i.e. attempted i.e. attempted eavesdroppingeavesdropping will disturb will disturb the encoded bit pattern & become the encoded bit pattern & become immediately obvious to both sender & immediately obvious to both sender & receiver. receiver.


AAlice & lice & BBob want to establish a ob want to establish a secret secret keykey

AA chooses a chooses a random sequence of bitsrandom sequence of bits out out of which she & of which she & BB will construct a will construct a keykey.. Initially neither A nor B has a particular key Initially neither A nor B has a particular key

in mind; it will in mind; it will emergeemerge out of the out of the communication protocol they use.communication protocol they use.

Thus the Thus the exactexact bit sequencebit sequence is not important is not important – all that matters is that they – all that matters is that they & only they& only they come to learn the come to learn the common (private) bit common (private) bit subsetsubset = = keykey..


Quantum Key DistributionQuantum Key Distribution in the in the absenceabsence of of eavesdroppingeavesdropping:: AAlice & lice & BBob need to first agree on (a) ob need to first agree on (a)

the probability of detecting the probability of detecting eavesdropping & (b) the number of key eavesdropping & (b) the number of key bits bits #photons - e.g. 75% & 4-bits #photons - e.g. 75% & 4-bits AA::

1 1 1 1 1 0 0 1 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 1 1 1 1 1 1 0 0 1 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 1 0 1 10 1 1

x + x x x x x + x x + + + + x x + + x ++ x x + + x x x + x x x x x + x x + + + + x x + + x ++ x x + + x x + x x+ x x

\ - \ \ \ / / - / \ | | | | \ / | | / - | / / | | / \ | \ \ \ - \ \ \ / / - / \ | | | | \ / | | / - | / / | | / \ | \ \ row#2: + rectilinear x diagonal polarizations row#3: open communications channel


Quantum Key DistributionQuantum Key Distribution in the in the absenceabsence of of eavesdroppingeavesdropping:: Upon receipt of the photons, Upon receipt of the photons, BBob ob

chooses an orientation for his calcite chooses an orientation for his calcite crystal (row#2) with which he measures crystal (row#2) with which he measures their polarization their polarization BB::

\ - \ \ \ / / - / \ | | | | \ / | | / - | / / | | / \ | \ \\ - \ \ \ / / - / \ | | | | \ / | | / - | / / | | / \ | \ \ ++ + x + x ++ x x x x ++ xx ++ x + + x + + xx xx++ x x xx + x + + x + xx ++++ + + xx ++ x x

+ x + x ++ 00 1 1 1 1 0 0 1 1 1 1 0 0 00 11 1 0 0 0 1 0 0 0 11 00 0 0 0 0 1 0 0 0 0 1 11 11 11 0 0 11 11 1 1

0 1 0 1 00

row#3: reconstructed bit stream


Quantum Key DistributionQuantum Key Distribution in the in the absenceabsence of of eavesdroppingeavesdropping:: Now Now AAlice & lice & BBob enter into a public ob enter into a public

(insecure) communication in which (insecure) communication in which AA divulges to divulges to BB the the polarizer orientationpolarizer orientation of of a a subsetsubset of bits; likewise B divulges to A of bits; likewise B divulges to A the calcite orientations he used to the calcite orientations he used to decode the same set of bitsdecode the same set of bits


Quantum Key DistributionQuantum Key Distribution in the in the presencepresence of of eavesdroppingeavesdropping:: (i) (i) AAlice encodes her bits into a stream lice encodes her bits into a stream

of polarized photons of polarized photons (as previously)(as previously) (ii) (ii) EEve(sdropper) intercepts/measures ve(sdropper) intercepts/measures

these photons, just as these photons, just as BBob did ob did previously:previously:

row#1 = row#1 = (polarized)(polarized) photons; photons; row#2 = calcite orientations;row#2 = calcite orientations; row#3 = row#3 = (encoded) (encoded) bitsbits


Quantum Key DistributionQuantum Key Distribution in the in the presencepresence of of eavesdroppingeavesdropping:: (iii) (iii) EEve ve reretransmits photons to transmits photons to BBob, ob,

using using anyany polarizer orientations polarizer orientations (but most likely the (but most likely the samesame sequence she sequence she

used during decoding)used during decoding) (iv) (iv) BB, unaware of , unaware of EE’s presence, decodes ’s presence, decodes

the polarized photon stream in the usual the polarized photon stream in the usual manner.manner.


Quantum Key DistributionQuantum Key Distribution in the in the presencepresence of of eavesdroppingeavesdropping:: (iv) (iv) BB, unaware of , unaware of EE’s presence, decodes the ’s presence, decodes the

polarized photon stream in the usual manner.polarized photon stream in the usual manner. (v) (v) AA & & BB now compare orientations of their now compare orientations of their

polarizer & calcite crystals with measured polarizer & calcite crystals with measured (decoded) bit values, on a subset of the (decoded) bit values, on a subset of the photon/bit stream:photon/bit stream:

where they agree on polarizer orientation, they where they agree on polarizer orientation, they should should alsoalso agree on the measured/decoded bit; agree on the measured/decoded bit; where they where they don’tdon’t agree, then this reflects the agree, then this reflects the presence of (an) presence of (an) EE!!

TeleportationTeleportation

The fictional version of The fictional version of teleportationteleportation (= a 3-stage process)(= a 3-stage process)::

(i) dissociation(i) dissociation (ii) information transmission(ii) information transmission (iii) reconstitution(iii) reconstitution


In contrast with a In contrast with a faxfax transmission, where transmission, where the original object remains the original object remains intactintact at the at the transmitter location & only a transmitter location & only a replicareplica (facsimile) is constructed at the receiver (facsimile) is constructed at the receiver location,location,

with with teleportationteleportation, the original object is , the original object is destroyeddestroyed once the necessary information once the necessary information is extracted,is extracted,

& moreover an & moreover an exactexact replica is replica is reconstructed at the receiver destination! reconstructed at the receiver destination!


Quantum teleportationQuantum teleportation is the is the transmission of transmission of quantumquantum information information to a distant location.to a distant location.

The objective is to The objective is to transmittransmit the the quantum statequantum state of a particle using of a particle using classicalclassical bits, then bits, then reconstructreconstruct the the quantum state at the receiver.quantum state at the receiver. i.e. is it possible to send qubits without i.e. is it possible to send qubits without

sending qubits?!sending qubits?!


Let’s assume that Let’s assume that AAlice wishes to lice wishes to communicate (through communicate (through classicalclassical channels) with channels) with BBob a single ob a single qubitqubit of of unknown state unknown state = a|0> + b|1> = a|0> + b|1>

AA can can neitherneither measuremeasure this quantum this quantum state state nornor cloneclone it. it. It would appear the only way to send It would appear the only way to send BB

the qubit would be to the qubit would be to eithereither send him the send him the physical qubit, physical qubit, oror to swap the state into to swap the state into another quantum system, then send this another quantum system, then send this system to system to BB..


AAlice & lice & BBob use an ob use an entangled pairentangled pair:: = 1/root(2)(|00> + |11>);= 1/root(2)(|00> + |11>); AA controls the controls the firstfirst half of the pair & half of the pair &

BB the the secondsecond..


The The input stateinput state is is = = = (a|0> + b|1>) 1/root(2)(|00> + |= (a|0> + b|1>) 1/root(2)(|00> + |

11>)11>) = 1/root(2)(a|0> |00> + a|0> |11> = 1/root(2)(a|0> |00> + a|0> |11>

…… = 1/root(2)(a|000> + a|011> + b|100> = 1/root(2)(a|000> + a|011> + b|100>

+ b|111>) + b|111>)


AAlice now applies the lice now applies the transformationtransformation::(H I I)*(C(H I I)*(CNOTNOT I) I)

to this state (i.e. to this state (i.e. )) The third bit is left unchanged; only The third bit is left unchanged; only

the first two bits belong to the first two bits belong to AA – the – the rightmost bit belongs to rightmost bit belongs to BB. .


Applying (H I I) we have:Applying (H I I) we have: 1/root(2)H I I(a|000>+a|011>+b|1/root(2)H I I(a|000>+a|011>+b|

110>+b|101>)110>+b|101>) ……

½(a(|000> + |110> + |011> + |111>) ½(a(|000> + |110> + |011> + |111>) + b(|010> - |110> + |001> - |101>))+ b(|010> - |110> + |001> - |101>))

& by regrouping terms& by regrouping terms ½(|0>(a(|0> + b|1>) + |01>(a|1> + b|½(|0>(a(|0> + b|1>) + |01>(a|1> + b|

0>) + |10>(a|0> - b|1>) + |11(a|1> - b|0>) + |10>(a|0> - b|1>) + |11(a|1> - b|0>))) 0>)))


AAlice then measures her lice then measures her qubitsqubits, , obtaining four possible results: |00>, obtaining four possible results: |00>, |01>, |10> or |11>, with equal |01>, |10> or |11>, with equal probability (¼).probability (¼).

Depending on the result of the Depending on the result of the measurement, the quantum state of measurement, the quantum state of BBob’s ob’s qubitqubit is projected to a|0>+b| is projected to a|0>+b|1>, a|1>+b|0>, a|0>-b|1>, a|1>-b|1>, a|1>+b|0>, a|0>-b|1>, a|1>-b|0>, respectively. 0>, respectively.


BBob will know what has happened, & ob will know what has happened, & can apply the decoding can apply the decoding transformation T e {I,X,Y,Z} to fix his transformation T e {I,X,Y,Z} to fix his qubitqubit..

The final output state is The final output state is = a|0> + = a|0> + b|1>, which is the unknown b|1>, which is the unknown qubitqubit that that AAlice wanted to send. lice wanted to send.


e.g.e.g. received bitsreceived bits statestate transformationtransformation

resultresult 0000 a|0> + b|1>a|0> + b|1> II a|0> + b|1>a|0> + b|1> 0101 a|1> + b|0>a|1> + b|0> XX a|0> + b|1>a|0> + b|1> 1010 a|0> - b|1>a|0> - b|1> ZZ a|0> + b|1>a|0> + b|1> 1111 a|1> - b|0>a|1> - b|0> YY a|0> + b|1> a|0> + b|1>

Hardware Quantum Computers?Hardware Quantum Computers?

Dreaming up a quantum computer proposal is relatively easy; proposing a quantum computer that can be easily constructed is hard!

2 inherent difficulties: 1. is Quantum Mechanics correct? 2. what about decoherence & quantum

noise? (Quantum Error Correction?)

Requirements for Quantum Requirements for Quantum ComputationComputation

(DiVincenzo criteria):(DiVincenzo criteria): 1. robustly represent quantum 1. robustly represent quantum

informationinformation a a scalablescalable physical system with well- physical system with well-

characterized qubitscharacterized qubits 2. prepare an initial state2. prepare an initial state 3. decoherence times >> logic gate times3. decoherence times >> logic gate times 4. a universal set of logic operations4. a universal set of logic operations 5. high probability readout5. high probability readout

strong strong (projective)(projective) measurements measurements

1. Ion Traps1. Ion Traps

NISTOXFORD

LOS ALAMOS

Fluorescence from trapped Be Ions

Hyperfine states and vibrational modes of an atom form qubits

Manipulated by laser pulses

Main drawback: weakness of phonon mediated spin-spin Coupling, susceptible to decoherence.

2. 2. NNuclear uclear MMagnetic agnetic RResonance (NMR)esonance (NMR)

MITIBM

LOS ALAMOS

•Qubits are Spin of nuclei•rf pulse perform arbitary rotations•Coupling between spins is dipolar and hyperfine•Read-out: ensemble average induction.

Main drawback: Not scalable. Why?

3. Optical QC3. Optical QC

CQCT (UNSW et.al.)LOS ALAMOS

•Qubits formed from location betweentwo modes, or polarisation.•Single photons are manipulated by beam splitters, mirrors, phase shifters and non-linear Kerr media.•Read-out: Photomultipler.•Main drawback: Coupling is difficult!

4. Superconductors4. Superconductors

NECDELFT

ChalmersYale et.al.

•Charge Qubits•Flux (SQUID) Qubits.•Phase Qubits.

5. Solid State QCs5. Solid State QCs

20 nanometres

Metal Electrodes

Insulator

Silicon

Substrate

Benefits

• Clearly scalable

• Compatible with Si MOS - integrated control electronics

• Can borrow Si MOS technology- material quality- gate technology- interconnect architectures

Challenges

• Single spin readout is difficult

• Completely new nanofabrication technologies must be developed- single donor positioning never done before CQCT = UNSW

UQUMelb

MarylandLos Alamos

The State of the ArtThe State of the Art (2002) (2002)Implementation qubit 1 qubit

operation 2 qubit operation

Max Nq

Ion Trap Ion YES YES 10 – 100 ?

NMR Atom YES YES 10 –100 ?

Optical Photon YES ?? ??

Superconducting Flux, charge

YES 2002 ? 106 ?

Silicon Atom 2002 ? 2002 – 2004 ?

109 ?

A A quantum computerquantum computer can complete can complete calculations, such as factorizing large calculations, such as factorizing large numbers, much faster than even the most numbers, much faster than even the most powerful existing supercomputer, while powerful existing supercomputer, while other potential applications include other potential applications include determining the properties of proteins and determining the properties of proteins and molecules, and solving biochemical, molecules, and solving biochemical, biological, environmental, and climatology biological, environmental, and climatology problems.problems.

Quantum computerQuantum computers could also decode s could also decode practically any encrypted message, practically any encrypted message, though quantum cryptography itself though quantum cryptography itself promises to be unbreakable.promises to be unbreakable.

Quantum Computing ApplicationsQuantum Computing Applications

Quantum Computing ApplicationsQuantum Computing Applications

Shor’s prime factorization: encryption Grover’s Exhaustive Search has many

potential applications, including genomics. Quantum Simulation: Quantum Chemistry,

drug design, fundamental physics …etc.

but recall Amdahl’s Law (%(%SSerial + %erial + %PParallel)! arallel)! i.e. not i.e. not everyevery problem will benefit from (or problem will benefit from (or even be suited to) Quantum Parallelism!even be suited to) Quantum Parallelism!

Relevance to Operating Relevance to Operating Systems/Artificial IntelligenceSystems/Artificial Intelligence???? OSOS – security… – security… AIAI – quantum parallel search; QNNs… – quantum parallel search; QNNs…

Quantum ComputingQuantum Computing

2003/4 news items courtesy of 2003/4 news items courtesy of ACM ACM TechNewsTechNews weekly postings weekly postings

Quantum ComputersQuantum Computers

Shepelyansky et.al. (France) believe that a Shepelyansky et.al. (France) believe that a quantum computer could be capable of storing far quantum computer could be capable of storing far more information than all modern supercomputers more information than all modern supercomputers by employing 50 qubits. And it would only take 18 by employing 50 qubits. And it would only take 18 qubits to encode the voice of HAL 9000, the qubits to encode the voice of HAL 9000, the autonomous computer from the film "2001: A autonomous computer from the film "2001: A Space Odyssey," in a quantum computer's wave Space Odyssey," in a quantum computer's wave function, according to the researchers.function, according to the researchers.

Real-time audio communications require the Real-time audio communications require the reduction or compression of sound signals, and one reduction or compression of sound signals, and one form of audio compression - MP3 - can quickly form of audio compression - MP3 - can quickly access the audio-signal spectrum using a Fast access the audio-signal spectrum using a Fast Fourier Transform. Audio signals could be Fourier Transform. Audio signals could be transmitted even more rapidly through the transmitted even more rapidly through the Quantum Fourier TransformQuantum Fourier Transform, the quantum , the quantum equivalent of MP3.equivalent of MP3.

Quantum MP3 Quantum MP3 (10/09/03)(10/09/03)

Japanese scientists at NEC and the Institute of Physical Japanese scientists at NEC and the Institute of Physical and Chemical Research have successfully built a and Chemical Research have successfully built a fundamental element of a quantum computer--a fundamental element of a quantum computer--a ""quantum gatequantum gate" that could be a component of the " that could be a component of the quantum equivalent of a computer chip. NEC research quantum equivalent of a computer chip. NEC research fellow Tsai Jaw-Shen reports that the gate can only fellow Tsai Jaw-Shen reports that the gate can only function in extremely low temperatures because it relies function in extremely low temperatures because it relies on superconductivity, though he hopes that the on superconductivity, though he hopes that the operating temperature can be raised to a level more operating temperature can be raised to a level more comparable to that of conventional computers.comparable to that of conventional computers.

He wages that a considerable amount of time must pass He wages that a considerable amount of time must pass before before quantum computersquantum computers become a reality, and become a reality, and estimates that only 10 percent of the job has been estimates that only 10 percent of the job has been accomplished thus far. "The single Qubit [quantum bit] accomplished thus far. "The single Qubit [quantum bit] was completed in 1999 and the two Qubit operation has was completed in 1999 and the two Qubit operation has been completed this year," Nakamura notes. "But we been completed this year," Nakamura notes. "But we have to integrate these two components."have to integrate these two components."

Quantum Gate Quantum Gate (12/09/03)(12/09/03)

Experts believe even the best existing digital security system will Experts believe even the best existing digital security system will ultimately be defeated by hackers, and the only unbeatable solution is ultimately be defeated by hackers, and the only unbeatable solution is quantum cryptographyquantum cryptography, in which the keys used to encrypt and decrypt , in which the keys used to encrypt and decrypt data are encoded within light particles so sensitive that even the data are encoded within light particles so sensitive that even the slightest attempt to monitor their transmission will change their slightest attempt to monitor their transmission will change their encoded state and alert users to the intrusion. Researchers are encoded state and alert users to the intrusion. Researchers are hopeful that the encoding of binary bits on photons, electrons, and hopeful that the encoding of binary bits on photons, electrons, and other quantum particles will be a reality before 2020, thus enabling other quantum particles will be a reality before 2020, thus enabling computers to carry out multiple calculations concurrently.computers to carry out multiple calculations concurrently.

Commercial Commercial quantum cryptographyquantum cryptography products were recently introduced products were recently introduced by Geneva-based id Quantique and New York-based MagiQ by Geneva-based id Quantique and New York-based MagiQ Technologies, while NEC, Hewlett-Packard, Toshiba, and other large Technologies, while NEC, Hewlett-Packard, Toshiba, and other large companies are planning to roll out products of their own. Such companies are planning to roll out products of their own. Such products' commercial appeal will be restricted until certain challenges products' commercial appeal will be restricted until certain challenges are met: For one thing, quantum encrypted data sent over fiber-optic are met: For one thing, quantum encrypted data sent over fiber-optic cable has a limited range, and requires computers directly connected cable has a limited range, and requires computers directly connected to each other. Quantum repeaters are also required to expand to each other. Quantum repeaters are also required to expand transmission range and make quantum encryption workable in a transmission range and make quantum encryption workable in a networking environment, and both NEC and Hewlett-Packard are networking environment, and both NEC and Hewlett-Packard are pursuing this goal. Wireless quantum key transmission is also being pursuing this goal. Wireless quantum key transmission is also being developed in Europe and the US.developed in Europe and the US.

Quantum Cryptography Quantum Cryptography (1/28/03)(1/28/03)

Quantum communication has long been publicized as completely hack-proof, but Quantum communication has long been publicized as completely hack-proof, but quantum hackingquantum hacking is an area of research that engineers are exploring in parallel with is an area of research that engineers are exploring in parallel with the development of true quantum networks--and they are uncovering possible the development of true quantum networks--and they are uncovering possible exploits that exploits that quantum encryptionquantum encryption designers never anticipated. "The models that tell designers never anticipated. "The models that tell us quantum cryptography is hot stuff are drastically simplified," explains Harvard us quantum cryptography is hot stuff are drastically simplified," explains Harvard University's John Myers. Quantum communication encryption's basic incarnation is University's John Myers. Quantum communication encryption's basic incarnation is the BB84 scheme devised by IBM's Charles Bennett and the University of Montreal's the BB84 scheme devised by IBM's Charles Bennett and the University of Montreal's Gilles Brassard, in which a message sender (Alice) and receiver (Bob) use both a Gilles Brassard, in which a message sender (Alice) and receiver (Bob) use both a public link and a quantum communication link to set up a secret quantum key used public link and a quantum communication link to set up a secret quantum key used to encrypt messages that an eavesdropper (Eve) cannot guess without being to encrypt messages that an eavesdropper (Eve) cannot guess without being detected, since Eve's measurement of Alice's photons disturbs their quantum state. detected, since Eve's measurement of Alice's photons disturbs their quantum state. However, engineers have found several practical techniques that eavesdroppers However, engineers have found several practical techniques that eavesdroppers could use to correctly guess the key: In a photon number-splitting attack designed could use to correctly guess the key: In a photon number-splitting attack designed by Nicolas Gisin of the University of Geneva, Alice's laser accidentally releases two by Nicolas Gisin of the University of Geneva, Alice's laser accidentally releases two or three photons instead of just one, and Eve diverts and measures these extra or three photons instead of just one, and Eve diverts and measures these extra photons without Alice and Bob knowing. In another quantum hack, known as a photons without Alice and Bob knowing. In another quantum hack, known as a frying attack, Eve sends an intense pulse of laser light into Bob's 1 photon detector, frying attack, Eve sends an intense pulse of laser light into Bob's 1 photon detector, rendering it inoperative and making Bob capable of only receiving 0s; Alice and rendering it inoperative and making Bob capable of only receiving 0s; Alice and Bob's key will therefore be all 0s, which means that their data will be unencrypted Bob's key will therefore be all 0s, which means that their data will be unencrypted without their realizing it. "In general, I do not think that a real quantum without their realizing it. "In general, I do not think that a real quantum cryptography system will ever be 100 percent secure, because a real system will cryptography system will ever be 100 percent secure, because a real system will always implement an approximation of the theorist's system," states Gisin. Military always implement an approximation of the theorist's system," states Gisin. Military and intelligence agencies as well as financial firms are employing commercial and intelligence agencies as well as financial firms are employing commercial quantum communication products, but establishing secure quantum quantum communication products, but establishing secure quantum communication in a public Internet is a more complex proposition, especially since communication in a public Internet is a more complex proposition, especially since there is such a wide variety of quantum communication schemes.there is such a wide variety of quantum communication schemes.

Quantum Hacking Quantum Hacking (11/29/03)(11/29/03)

A A quantum computerquantum computer carries such promised capabilities as ultrafast carries such promised capabilities as ultrafast database searches and a "virtual lab" where the behavior of materials can database searches and a "virtual lab" where the behavior of materials can be predicted without actually fabricating them, but a practical quantum be predicted without actually fabricating them, but a practical quantum computer must be immune to decoherence, in which computations are computer must be immune to decoherence, in which computations are undone because even the slightest disturbance results in data leakage. undone because even the slightest disturbance results in data leakage. Microsoft Research's Alexei Kitaev and Michael Freedman, along with Microsoft Research's Alexei Kitaev and Michael Freedman, along with Zhenghan Wang and Michael Larson of Indiana University, may have Zhenghan Wang and Michael Larson of Indiana University, may have solved the problem with their outline of a solved the problem with their outline of a topological quantum computertopological quantum computer that could be constructed out of existing technology. The operating that could be constructed out of existing technology. The operating principle of the device is the manipulation of quantum particles--non-principle of the device is the manipulation of quantum particles--non-Abelian anyons--into braids that exist in both time and space. These Abelian anyons--into braids that exist in both time and space. These anyons' "world lines" can be weaved around each other into knots that anyons' "world lines" can be weaved around each other into knots that encode information; this braiding could be accomplished with an encode information; this braiding could be accomplished with an instrument similar to a scanning tunneling microscope. "The state of the instrument similar to a scanning tunneling microscope. "The state of the quantum computer is stored in the conserved charges that the anyons quantum computer is stored in the conserved charges that the anyons carry," notes Caltech's John Preskill. "Even if you hit an anyon with a carry," notes Caltech's John Preskill. "Even if you hit an anyon with a hammer, you can't change that charge, so the state stored in the hammer, you can't change that charge, so the state stored in the computer is quite robust." Bringing the anyons together in pairs allows computer is quite robust." Bringing the anyons together in pairs allows topological charges to be read off: Those with equal and opposite charges topological charges to be read off: Those with equal and opposite charges annihilate each other, creating a "0" output, and those with unbalanced annihilate each other, creating a "0" output, and those with unbalanced charges merge into a new anyon, resulting in a "1" output. charges merge into a new anyon, resulting in a "1" output. The topological The topological quantum computer is still speculative, since the existence of non-Abelian quantum computer is still speculative, since the existence of non-Abelian anyons has yet to be provedanyons has yet to be proved (sic!)(sic!)..

Topological QCs Topological QCs (1/24/04)(1/24/04)

Start-ups MagiQ Technologies and ID Quantique announced Start-ups MagiQ Technologies and ID Quantique announced quantum cryptography hardwarequantum cryptography hardware late last year, but most enterprise late last year, but most enterprise networks will not be able to take advantage of the technology. networks will not be able to take advantage of the technology. However, the continued development of quantum cryptography However, the continued development of quantum cryptography over the next few years is expected to make the advancement over the next few years is expected to make the advancement more beneficial to enterprise networks. Quantum cryptography more beneficial to enterprise networks. Quantum cryptography uses objects that are in different places at one time to create the uses objects that are in different places at one time to create the same random numbers in two locations, enabling the two identical same random numbers in two locations, enabling the two identical sets of random numbers to be used as symmetric encryption keys sets of random numbers to be used as symmetric encryption keys or one-time pads. The problem of creating and distributing or one-time pads. The problem of creating and distributing encryption would be solved because the keys would never be used encryption would be solved because the keys would never be used again. Nonetheless, dedicated fiber cable is needed for quantum again. Nonetheless, dedicated fiber cable is needed for quantum key distribution through a network, and fully optical switches for key distribution through a network, and fully optical switches for multiplexing entangled photons with ordinary data remain a few multiplexing entangled photons with ordinary data remain a few years away. Moreover, repeaters can not be used, prompting years away. Moreover, repeaters can not be used, prompting MagiQ to experiment with using Free Space Optics lasers to send MagiQ to experiment with using Free Space Optics lasers to send photons through a wireless link. Existing key distribution systems photons through a wireless link. Existing key distribution systems are unable to distribute a one-time pad, which makes them are unable to distribute a one-time pad, which makes them susceptible to outright mathematical attacks. A quantum computer susceptible to outright mathematical attacks. A quantum computer could break encryption that reuses keys, could break encryption that reuses keys, but a working computer but a working computer will not be here for decades!!!will not be here for decades!!!

QC Hardware? QC Hardware? (Feb’04)(Feb’04)

an introduction to quantum computing gosn203 (ai); gosn204 (os) professor john fulcher christopher...

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