The brief history of quantum computation

G.J. MilburnCentre for Laser Science

Department of Physics, The University of Queensland

Outline of talk.

The brief history of quantum computation.

Deutsch and quantum parallelism. The Shor breakthrough. Physical realisation and future

technology. Measurement and computation. Quantum computers and the

foundations of physics.

Paths to a quantum computer.

Two tracks converge to quantum computation:

R.P. Feynman, 1982Simulating physics with computers, Int. J. Theor. Phys. 21, 467 (1982).

R. Landauer, 1961Irreversibility and heat generation in the

computing process.IBM J. Res. Dev. 5 , 183 (1961)

Landauer’s principle

To erase one bit of information we must dissipate energy

ΔE = kBT ln 2

Landauer’s principle: explanation

Is the molecule on L or R ?– one bit of information

To erase, compress to half volume

L R

ΔE = kBT lnVi

Vf= kBT ln2

Logical irreversibility fi physical irreversibility.

The NOT gate is reversible The AND gate is irreversible

the AND gate erases information. the AND gate is physically irreversible.

Reversible computation.

Charles Bennett, IBM, 1973. Logical reversibility of computation,

– IBM J. Res. Dev. 17, 525 (1973).

Reversible gates for universal computation.

Fredkin, Toffoli 1979. minimum of three inputs and three

outputs– eg. Fredkin gate

INPUT OUTPUTA B C A B C0 0 0 0 0 00 1 0 0 1 01 0 0 1 0 01 1 0 1 1 00 0 1 0 0 10 1 1 1 0 11 0 1 0 1 11 1 1 1 1 1

Feynman’s question.

The second track to quantum computation. R.P. Feynman, 1982

Simulating physics with computers, Int. J. Theor. Phys. 21, 467 (1982).

Can a quantum system be simulated exactly by a universal computer ?

NO !

Classical simulation: transport problem.

R particles on a 1-dim lattice of N sites.– note, for fields R=O (N)

How does the calculation scale with N,R ?

size of input ≈ N2 R

Simulate Boltzmann equation.

Classical probabilistic simulation.

Use random numbers to simulate coarse grained dynamics.

The statistics of random numbers is classical.

Cannot simulate a large quantum process.

The Feynman processor.

A physical computer operating by quantum rules.

could it compute more efficiently than a classical computer ?

Universal computation.

Turing machines.– See R. Penrose, The Emperor’s New Mind, page 71.

Church-Turing thesis:A computable function is one that is

computable by a universal Turing machine.

Computational efficiency.

N; a number to specify the input to a Turing machine.

Code as log N bits. Efficient algorithm :

# steps for input N ≤p(logN),∀N

Deutsch and quantum parallelism.

D. Deutsch, 1985Quantum theory, the Church-Turing principle

and the universal quantum computer.Proc. Roy. Soc. A400, 97, (1985).

Feynman-Deutsch principle:(Church-Turing principle)

‘Every finitely realisable physical system can be perfectly simulated by a universal model computing machine operating by finite means”

Deutsch processor.

Computational basis: Direct product Hilbert space of N two-level

systems: Quantum Turing machines:

remain in computational basis state at end of each step.

Quantum computer arbitrary superpositions of computational

basis...explore all 2N dimensions !

| SN ⟩⊗ |SN−1 ⟩⊗K |S1⟩; Si ∈{1,0}

Quantum parallelism.

Code binary string for input as an integer.

Quantum TM.

Quantum parallelism

k =S120 +S22

1 +K +SN2N−1 : (k=0,1...2N−1 )

f : | k⟩ input⊗|0⟩output→ |k⟩ input⊗ | f(k)⟩output

f : | k⟩input⊗ |0⟩outputk=0

2 N-1

∑ → |k⟩input⊗| f(k)⟩outputk=0

2N-1

∑

The qubit.

A single two-state system can store a single bit in computational basis.

Superpositions are allowed the qubit. 1

2(| 1⟩±|0⟩)

1

2(| 1⟩⟨1|+ |0⟩⟨0 |)

The elementary single qubit operation.

The Hadamard transform.

Quantum circuit:

| 0⟩→12(|0⟩+ |1⟩)

|1⟩→12(|0⟩−|1⟩)

H

time

A quantum optical example.

A two-state system with a single photon. use a ‘direction qubit’

Quantum parallel input.

prepare an even superposition of all 2N-1 binary strings.

⊗ | 0⟩→ | k⟩k= o

2N−1

∑

Universal quantum gates.

One-qubit gate: Hadamard gate

Two-qubit gate: quantum controlled NOT gate

Controlled NOT from spin-spin coupling.

Step 1: Hadamard transform of target,

Step 2: Spin-spin coupling to control,

Step 3: Hadamard transform of target,

| 0⟩T⊗|1⟩C → (|0⟩T+ |1⟩T )⊗|1⟩C

Uπ =exp−iπ |1⟩T ⟨1 |⊗ |1⟩C⟨1|( )

Uπ (|0⟩T+ |1⟩T )⊗|1⟩C → (|0⟩T−|1⟩T)⊗ |1⟩C

(| 0⟩T−|1⟩T )⊗|1⟩C → |1⟩T⊗|1⟩C

Quantum circuit for Controlled NOT.

H HU

control

target

Deutsch’s algorithm.

Is f EVEN, f(0) = f(1) or ODD, f(0) π f(1) ?

Only evaluate f once.

f : 0,1{ } → 0,1{ }

f-controlled NOT

f must be implemented reversibly.

x⟩⊗ y⟩→ x⟩⊗ y⊕ f(x)⟩

quantum circuit

H H

Uf|0> -|1> |0> -|1>

readout

Output states at readout.

(−1) f(0) f (0)⊕ f(1)⟩

output is 0 ⇒ f (0) = f(1) 1 output is⇒ (0)f ≠ (1)f

Implementation of Deutsch algorithm.

quant- ph/ 9801027 14 Jan 1998“Implementation of a Quantum

Algorithm to Solve Deutsch's Problem on a Nuclear Magnetic Resonance Quantum Computer” J. A. Jones & M. Mosca, Oxford

Shor algorithm.

Peter Shor, AT&T, 1994– a quantum algorithm to find prime factors of

large composites N– public key cryptography no longer safe !

Key step:– find the ‘period’ of the function:

(x is random, but GCD(x,N)=1)

f (a) =xa modN

Example.

Factor 15.

N =15,x=7

a : 1,2,3, 4,5,6, 7, 8,9,10,11,12,13,14,15,16,17,18,19,20f (a) : 7, 4,13,1,7,4,13,1,7,4, 13, 1, 7, 4, 13, 1, 7, 4, 13, 1,

Order=4 Calculate: Factors; GCD(48,15)=3,

GCD(50,15)=5

xorder ±1=48,50

Quantum factoring

Step 1: run algorithm

a⟩ 0⟩ → 0⟩1⟩ +1⟩ 7⟩ +

a∑ 2⟩ 4⟩ +3⟩13⟩ + 4⟩1⟩ +5⟩ 7⟩ + 6⟩ 4⟩ +K + 19⟩13⟩

Step 2: readout and discard output

ψ (4) ⟩=2⟩+6⟩+10⟩+14⟩+18⟩

Quantum factoring.

Step 3: Discrete Fourier transform.strong interference of ‘paths’

a⟩ → exp2πiab20

⎡ ⎣

⎤ ⎦b=0

19

∑ b⟩

Quantum factoring.

Step 4. Readout register.most likely to obtain a number c such that

c =p×20order

if GCD(p,order) =1 then infer order

Physical realisations.

Ion traps– Cirac & Zoller 1994, Phys. Rev. Lett, 74,4094.

Cavity QED– Turchette et al. 1995, Phys. Rev. Lett,75, 4710

NMR– Gershenfeld & Chuang 1997, Science, 275, 350

SQUID– Rouse et al.,1995 Phys. Rev. Lett, 75, 1614.

Quantum dots– Loss &di Vincenzo, cond-mat/9701055

Ion traps

Couple lowest centre-of-mass mode to internal electronic states of N ions.

Quantum computation at UQ

New measurement by QC

von Neumannmeasurement

quantum computation

Quantum computation at UQ

measure vibrational energy of trapped ions. d’Helon&GJM Phys. Rev. A. 54, 5141-5146

(1996). tomographic state reconstruction of

vibrational state d’Helon & GJM quant-ph/9705014

measurement as a quantum search algorithm Schneider,Wiseman,Munro & GJM,

quant-ph/9709042

Feynman-Deutsch principle and measurement.

The virtual graduate student: part one.

Feynman-Deutsch principle and measurement.

The virtual graduate student: part two.