the brief history of quantum computation g.j. milburn centre for laser science department of...
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The brief history of quantum computation
G.J. MilburnCentre for Laser Science
Department of Physics, The University of Queensland
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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.
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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)
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Landauer’s principle
To erase one bit of information we must dissipate energy
ΔE = kBT ln 2
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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
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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.
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Reversible computation.
Charles Bennett, IBM, 1973. Logical reversibility of computation,
– IBM J. Res. Dev. 17, 525 (1973).
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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
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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 !
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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.
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Classical probabilistic simulation.
Use random numbers to simulate coarse grained dynamics.
The statistics of random numbers is classical.
Cannot simulate a large quantum process.
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The Feynman processor.
A physical computer operating by quantum rules.
could it compute more efficiently than a classical computer ?
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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.
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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
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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”
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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}
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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
∑
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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 |)
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The elementary single qubit operation.
The Hadamard transform.
Quantum circuit:
| 0⟩→12(|0⟩+ |1⟩)
|1⟩→12(|0⟩−|1⟩)
H
time
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A quantum optical example.
A two-state system with a single photon. use a ‘direction qubit’
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Quantum parallel input.
prepare an even superposition of all 2N-1 binary strings.
⊗ | 0⟩→ | k⟩k= o
2N−1
∑
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Universal quantum gates.
One-qubit gate: Hadamard gate
Two-qubit gate: quantum controlled NOT gate
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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
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Quantum circuit for Controlled NOT.
H HU
control
target
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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{ }
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f-controlled NOT
f must be implemented reversibly.
x⟩⊗ y⟩→ x⟩⊗ y⊕ f(x)⟩
quantum circuit
H H
Uf|0> -|1> |0> -|1>
readout
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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
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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
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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
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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
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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⟩
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Quantum factoring.
Step 3: Discrete Fourier transform.strong interference of ‘paths’
a⟩ → exp2πiab20
⎡ ⎣
⎤ ⎦b=0
19
∑ b⟩
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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
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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
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Ion traps
Couple lowest centre-of-mass mode to internal electronic states of N ions.
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Quantum computation at UQ
New measurement by QC
von Neumannmeasurement
quantum computation
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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
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Feynman-Deutsch principle and measurement.
The virtual graduate student: part one.
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Feynman-Deutsch principle and measurement.
The virtual graduate student: part two.