quantum computers algorithms and applications. simulating classical operations 2/41 dušan gajević

Quantum Computers

Algorithms and applications

Simulating classical operations

2/41Dušan Gajević


• Is it possible to simulateclassical computer operations on a quantum computer?– Pure states correspond to classical bits– To implement any classical computer operation

a universal set of gates is needed – (NOT, AND), (NOT, OR), (NAND), (NOR)…• Quantum computers are reversible,

yet, in any of these sets there is at least one irreversible operation

• Obviously, the real question is,can a reversible computer simulate an irreversible operation?

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Simulating classical operations– Toffoli gate can be used to implement classical irreversible operations

– Is there anything elsea reversible circuit might needto simulate a classical computer?

NOT AND NAND

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Simulating classical operations– In fact,

classical computer contains two more “gates” (elements)that are easily overlooked

– Using Toffoli gate

FANOUT ERASE

…but not the actual ERASE

Even a type of erase…

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Simulating classical operations– In reversible computing,

it is allowed to erase duplicates of information,but it is not allowed to erase the last copy

• It is possible to simulateany classical irreversible computation using only Toffoli gates,but it requires a use of some extra bits

• Put another way,Toffoli gate is a universal gatefor classical reversible computing

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• Computation based on reversible logiccan take no energy to compute!

Offtopic: Energy dissipation

Run forwardto get an answer…

…copy that answer… …undo the computation

This will recover all the energy spentexcept the small amount used for copying an answer

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Offtopic: Energy dissipation– Energy must be dissipated

to initialize the systemor to make a permanent record of an answer• These operations

set a new value in a memory registerregardless of what the state was,hence, they are irreversible

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• We saw it’s possible to simulateirreversible operations on a reversible computer

• Hence,it is possible to simulate classical operationson a quantum computer– But this is not what quantum computers are intended for –

it is not much of a benefit…

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Quantum parallelism

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Quantum parallelism

• A fundamental feature of many quantum algorithms– Suppose a binary function f(x)

– Let’s introduce a following unitary transformationsingle bit in – single bit out

XOR (addition modulo 2)


Quantum parallelism– And use it in a following circuit

…state after applying the introduced transformation

As if f(x) is evaluatedfor 2 values of x simultaneously


– Now a modified binary function f(x)

– and a corresponding unitary transformation

Quantum parallelism

n-bit input

single bit output


Quantum parallelism– in a modified circuit Hadamard transform –

applied on n qubits


Quantum parallelism

As if f(x) is evaluatedfor 2n values of xsimultaneously


Quantum parallelism

• To achieve this sort of parallelism in classical computing multiple circuits are needed,while here a single circuit is employed– but, after measurement,

only one evaluation of f(x) remains

• Quantum computingrequires more than quantum parallelism alone to work


“I do not like it, and I am sorry I had anything to do with it”– Erwin Schrödinger


Deutsch-Jozsa algorithm



• Problem formulation

f(x) is allowed to be either:- constant or- balanced

A black box…

2n possible input values

output has the same value for all inputs

output is:- 0 for half of the function inputs,- 1 for the other half

The goal is to determinewhether the function f(x)

is constant or balanced



• Classical solution:– We are checking the output for every possible input value

• If we come across a different output –we know the function is balanced

• Otherwise, when we check at least half plus onepossible inputs to be equal (2n/2 + 1) –we know the function is constant

• Time complexity of the classical solution is exponential



• Deutsch-Jozsa algorithm

classical bits

……

…

the measurement


Deutsch-Jozsa algorithm– What makes the resulting state so special?

A single amplitude of this statedepends on f(x) evaluatedfor all values of x!


Deutsch-Jozsa algorithm– Of the special interest

is the amplitude of the state



– there are three different possibilities

The function is constantf(x)=0

The function is constantf(x)=1

The function is balancedhalf f(x)=1 – half f(x)=0

function constantresults in 0…0

function balancedresults in ≠ 0…0



• Deutsch-Jozsa algorithm:– Compared to the classical solution displayed before,

it offers exponential speedup– Unlike most quantum algorithms,

it is deterministic– But it is of little practical use –

it has no known applications

• Nevertheless, it suggestsquantum computers are capable of solving some problemsmuch more efficiently than conventional computers


ApplicationsFactorization of large numbers


• The simplest solution is to start from 2and check for every consequent number whether it’s a factor– Exponential time complexity

• But even the best known classical algorithmshave superpolynomial time complexity

Factorization of large numbers

prime numbersn digits



• The fact there is no computer todaywhich could efficiently factor numbersis widely used in cryptography for secure data transmission (RSA public key routine)



• Shor’s algorithm –in 1994, Peter Shor published a quantum algorithmfor factoring large numbers– Like most quantum algorithms, it is non-deterministic– It has polynomial time complexity– Shor also published a polynomial-time quantum algorithm

for discrete logarithms

• These algorithms sparked a huge amount of interest in quantum computing


ApplicationsQuantum search


Quantum search

• Classically, and on average,how many operations are requiredfor searching an unsorted database with N elements?

• Grover’s algorithm –in 1996, Lov Grover formulated a quantum algorithmfor searching an unstructured database in O(N1/2) time– Another non-deterministic quantum algorithm


Quantum search

• The quadratic speedup might not look impressivecompared to exponential speedup offered by algorithmsbased on quantum Fourier transform– But for large search spaces…

• Quantum search algorithms, also,have a wider range of applicationthan algorithms based on quantum Fourier transformation


ApplicationsQuantum simulation


Quantum simulation

• Simulation of naturally occurring quantum systems –another possible application of quantum computers

• Why are classical computers so bad at simulating quantum systems?– For a single qubit, a hydrogen atom,

two complex coefficients are needed to specify the state of the system– For 2 hydrogen atoms, 4 complex coefficients are needed– For 3 atoms, 8 coefficients– …– For N atoms, 2N coefficients

Number grows exponentially!


Quantum simulation– For only 30 qubits,

more than a billion complex coefficients are needed!

• Amount of resources neededto simulate a quantum system on a quantum computergrows linearly with the growthof the number of simulated elements– But the extraction of desired information still poses a problem,

due to collapse caused by measurement

Obviously, quantum systems are really goodat spending classical resources


Technical challenges


Quantum decoherence

• There are a lot of technical challengesin building a large-scale quantum computer

• One of the greatest challengesis minimizing quantum decoherence– Quantum system can get entangled with the environment

and evolve as if the environment “measured” the quantum system– Decoherence is irreversible,

as it is non-unitary

• Quantum systems have to be isolatedas much as possible from its environmentas any unintentional interaction with the outside worldcan disturb the fragile state of the system


Advances


Advances

• In 2010,company named D-Wave Systemsannounced D-Wave One,the first commercially available quantum computer system– 128-qubit quantum processor acting as a co-processor, accelerator

• Intended for solving discrete optimization problems– Front-ended on a network as a standard server– Adiabatic model, unlike gate model explained earlier

• Quantum annealing– Operating in an extreme environment– Low power consumption -

power demand expected to remain constant with scaling up


Advances

• In 2013,D-Wave Systems launched 512-qubit D-Wave Two

• Price?• Controversy


References• University of California, Berkeley,

Qubits and Quantum Measurement and Entanglement, lecture notes,http://www-inst.eecs.berkeley.edu/~cs191/sp12/

• Michael A. Nielsen, Isaac L. Chuang,Quantum Computation and Quantum Information, Cambridge University Press, Cambridge, UK, 2010.

• Colin P. Williams, Explorations in Quantum Computing, Springer, London, 2011.• Samuel L. Braunstein, Quantum Computation Tutorial, electronic document

University of York, York, UK• Bernhard Ömer, A Procedural Formalism for Quantum Computing,

electronic document, Technical University of Vienna, Vienna, Austria, 1998.• Artur Ekert, Patrick Hayden, Hitoshi Inamori,

Basic Concepts in Quantum Computation, electronic document,Centre for Quantum Computation, University of Oxford, Oxford, UK, 2008.

• Wikipedia, the free encyclopedia, 2014.


http://www-inst.eecs.berkeley.edu/~cs191/sp12/

quantum computers algorithms and applications. simulating classical operations 2/41 dušan gajević

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