Michael A. Nielsen

Michael A. Nielsen

Quantum Computation – towards quantum circuits and algorithms

Goals: 1. To explain the quantum circuit model of

computation.2. To explain Deutsch’s algorithm.3. To explain an alternate model of quantum

computation based upon measurement.

What does it What does it mean to mean to

compute?compute?

Quantum mechanics seems to be very hard to simulateon a classical computer.

Deutsch (1985): Can we justify C-T thesis using laws of physics?

Candidate universal computer: quantum computer

Might it be that computers exploiting quantum mechanicsare not efficiently simulatable on a Turing machine?

Might it be that such a computer can solve some problemsfaster than a probabilistic Turing machine?

Violation of strong C-T thesis!

What does it mean to What does it mean to compute?compute?

Church-Turing thesis: An algorithmic process orcomputation is what we can do on a Turing machine.

The Church-Turing-Deutsch principle

Church-Turing-Deutsch principle: Any physical process can be efficiently simulated on a quantum computer.

Research problem: Derive (or refute) the Church-Turing-Deutsch principle, starting from the laws ofphysics.

Models of quantum computationThere are many models of quantum computation.

Historically, the first was the quantum Turing machine,based on classical Turing machines.

A more convenient model is the quantum circuit model.

The quantum circuit model is mathematically equivalentto the quantum Turing machine model, but, so far,human intuition has worked better in the quantumcircuit model.There are also many other interesting alternate modelsof quantum computation!

Quantum circuit Quantum circuit model and model and

equivalence equivalence operations on operations on

symbolic unitary symbolic unitary matricesmatrices

Quantum circuit Quantum circuit modelmodel

Classical

Quantum

Unit: bit Unit: qubit1. Prepare n-bit input 1. Prepare n-qubit input

in the computational basis.2. 1- and 2-bit logic

gates2. Unitary 1- and 2-qubit

quantum logic gates3. Readout value of bits 3. Readout partial

information about qubits

1 2, ,..., nx x x

External control by a classical computer.

Single-qubit quantum logic gates

Hadamard gate

10 1 0 1 1 10 ; 1 ; 1 122 2H H H

H

Phase gateP

0 0 ; 1 1P P i

0 1 0 1 0X ; Y ; Z1 0 0 0 1i

i

Pauli gates

P P Z=1 00P

i

2P Z

Controlled-not gatec

t ct

cControl

Target

0100100000100001

UControlled-phase gate

Z , ( 1) ,ctc t c t

1 0Z 0 1

Exercise: Show that HZH = X.

Z

Z=

Symmetry makes the controlled-phase gate more natural for implementation!

X=

ZH H

CNOT isthe casewhen U=X

Toffoli gate

1c

t 21 cct

1cControl qubit 1

Target qubit

Control qubit 22c 2c

Worked Exercise: Show that all permutation matrices are unitary. Use this to show that any classical reversible gate has a corresponding unitary quantum gate.

Cf. the classical case: it is not possible to build up aToffoli gate from reversible one- and two-bit gates.

Challenge exercise: Show that the Toffoli gate can bebuilt up from controlled-not and single-qubit gates.

Now we are using Dirac notation to be used to complex problems

Use the quantum analogue of classical reversiblecomputation.

The quantum NANDx

1

y

1 x y

x

y

How to compute classical functions on quantum computers

x

0 x

x

The quantum fanout

Classical circuit

x ( )f xf

x

0 mxg

( )f x

fU

Quantum circuit

0 m

( )f xfU

x

z

†fU

xg

( )z f x

x

0 m

Canonical f orm: x z x ( )z f x

Removing garbage on quantum computers

Example: x z x parity( )z x

Given an “easy to compute” classical function, there isa routine procedure we can go through to translatethe classical circuit into a quantum circuit computingthe canonical form. The issue is, “how efficient?”

Example of using Example of using symbolic Dirac symbolic Dirac

Notation: Notation: Deutsch’s Deutsch’s problemproblem

Example: Deutsch’s problemExample: Deutsch’s problem Given a computing a f unction black : 0 box ,1 0,1f

Our task is to determine wheth constaner is ort bala nced?f we need to evaClassically boluate (0) and .th (1)ff we need only use the black box f or Quantumly once!( ) f

Classical black boxx

z ( )z f x

x

f

x

z ( )z f x

x

fU

Quantum black box

( ) 0 :f x

0 1 0 1x x

( ) 1:f x

0 1 1 0 0 1x x x

( )0 1 1 0 1f xx x

Putting information in the phase

x

0 12

fU

( )1 f xx x

Putting information in the phase is a very important trick of quantum computing

Observe that this is counterintuitive with notions how signals are propagated in circuits

F(x) can be multi-bit function

Phase is Phase is propagated propagated to inputs and to inputs and hidden in hidden in themthem

Quantum algorithm for Deutsch’s problem

0

0 12 fU

H H

0 0 1

(0) (1)1 0 1 1ff

(0) (1)1 0 1 + 1 0 1ff

(0) (1) (0) (1)1 1 0 + 1 1 1ff ff

constant all amplitude in 0 .f

balanced all amplitude in 1 .f

Quantum parallelism

Research problem: What makes quantum computerspowerful?

Auxiliary Auxiliary slidesslides

Universality in the quantum circuit model

Suppose U is an arbitrary unitary transformation onn qubits.Then U can be composed from controlled-not gatesand single-qubit quantum gates.

Classically, any function f(x) can be computed using justnand and fanout; we say those operations are universal for classical computation.

Just as in the classical case, a counting argument canbe used to show that there are unitaries U that takeexponentially many gates to implement.Research problem: Explicitly construct a

class Un ofunitary operations taking exponentially many gatesto implement.

Summary of the quantum circuit model

An -bit string, , representing an instance of some pI np robu : m.t len x

is a number to be fExamp actoe: .l redx

0 , where is some computablI nit e f uial state nction f .: om m n

A circuit of single-qubit and controlled-not gates isapplied to the qubits. The sequence of gates applied is underthe control of an external classical computer, and may dependupon the

Circ

pr

uit:

oblem instance .x

Some fi xed subset of the qubits is measured in thecomputational basis at the end of the computation, and theoutput constitutes the solution to the p

Reado

rob

ut:

lem. For a decision problem, just the fi rst qubit would be

read out, to indicate "yes" oExamp

r "le:

no".

QP: The class of decision problems solvable by a quantum circuitof polynomial size, with polynomial classical overhead.

Quantum complexity Quantum complexity classesclasses

How does QP compare with P?BQP: The class of decision problems for which there is a polynomial quantum circuit which outputs the correct answer (“yes” or “no”) with probability at least ¾.BPP: The analogous classical complexity class.Research problem: Prove that BQP is strictly larger than BPP.

Research problem: What is the relationship of BQP to NP?

What is known: BPP BQP PSPACE

When will quantum computers be built?

Alternate models for quantum Alternate models for quantum computationcomputation

Standard model: prepare a computational basis state, then do a sequence of one- and two-qubit unitary gates,then measure in the computational basis.

Research problem: Find alternate models of quantumcomputation.Research problem: Study the relative power of thealternate models. Can we find one that is physicallyrealistic and more powerful than the standard model?

Research problem: Even if the alternate modelsare no more powerful than the standard model, canwe use them to stimulate new approaches toimplementations, to error-correction, to algorithms(“high-level programming languages”), or to quantumcomputational complexity?

Topological quantum computer: One creates pairs of“quasiparticles” in a lattice, moves those pairs around thelattice, and then brings the pair together to annihilate. This results in a unitary operation being implementedon the state of the lattice, an operation that depends only on the topology of the path traversed by the quasiparticles!

Overview: Alternate models for quantum computation

Quantum computation via entanglement and single-qubit measurements: One first creates a particular,fixed entangled state of a large lattice of qubits. Thecomputation is then performed by doing a sequence ofsingle-qubit measurements.

Quantum computation as equation-solving: It can beshown that quantum computation is actually equivalentto counting the number of solutions to certain setsof quadratic equations (modulo 8)!

Overview: Alternate models for quantum computation

Quantum computation via measurement alone:A quantum computation can be done simply by a sequence of two-qubit measurements. (No unitarydynamics required, except quantum memory!)

Further reading on the last model:

D. W. Leung, http://xxx.lanl.gov/abs/quant-ph/0111122

• 2007