Download Talk - Quantum Device Lab

Ants Remm, Roland Matt
Quantum Adiabatic Computation
Overview
 The problems solved
 The methods
 Simulated annealing
 Quantum adiabatic computation
 Simulated quantum annealing
 Results
 Comparing SA,QA, SQA
 Is there a quantum speedup?
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first part by Ants
second part by Roland
Ants Remm, Roland Matt | 29.04.2016 |
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The spin glass system
𝐽𝑖𝑗 𝜎𝑖𝑧 𝜎𝑗𝑧 +
𝐻=
𝑖<𝑗
ℎ𝑖 𝜎𝑖𝑧
𝑖
 A set of coupled qubits
 Couplings 𝐽𝑖𝑗 between neighbours or
all qubits, could be random
 Couplings between more that two
qubits also possible: 𝐽𝑖𝑗𝑘 𝜎𝑖𝑧 𝜎𝑗𝑧 𝜎𝑘𝑧
 Local fields ℎ𝑖
 Not ergodic – many local minima,
frustration
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Problem equivalence
 Many different optimization problems can be reduced to
finding the ground state of the spin glass system:




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The knapsack problem
Finding Hamiltonian cycles
Graph colouring
Image restoration, etc.
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Problem equivalence:
Image restoration
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Problem equivalence:
Fair partitioning
 Given a set of numbers 𝑛𝑖 we want to divide the
numbers into two groups of equal sum
 For example 40, 25, 5, 11, 15, 11, 5, 26, 14 :
40 + 5 + 11 + 15 + 5 = 25 + 11 + 26 + 14 = 76
 Equivalent Hamiltonian:
2
𝑛𝑖 𝜎𝑖𝑧
𝐻=
𝑖
𝐽𝑖𝑗 = 2𝑛𝑖 𝑛𝑗
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Ants Remm, Roland Matt | 29.04.2016 |
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Problem equivalence:
The 𝒌-satisfiability problem
 Given a set of Boolean clauses 𝐶 on 𝑁 bits, find a
configuration of the bits that satisfies all (the most)
clauses
 Each clause acts on 𝑘 bits and specifies for each of them
whether they should be 0 or 1.
 A Hamiltonian describing a clause for bits 𝑖1 , 𝑖2 , … , 𝑖𝑘 can
be written as
1 ± 𝜎𝑖𝑧𝑘
1 ± 𝜎𝑖𝑧1 1 ± 𝜎𝑖𝑧2
𝐻𝐶 =
…
2
2
2
 Entire system Hamiltonian to be minimized: 𝐻 = − 𝐻𝐶
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Simulated annealing
 An ensemble of classical systems, stochastically
exploring the state space
 Probability for a system to cross a potential barrier of
Δ𝐸
− 𝑘𝑇
height Δ𝐸 is proportional to 𝑒
 Slowly lower the temperature so that Boltzmann
−
𝐸
𝑘𝑇
distribution 𝑝(𝐸) ∝ 𝑒
is maintained
 At the end of the simulation only ground state is occupied
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Simulated annealing:
Why it works
 Probability of crossing the barrier
left to right (right to left)
𝑝1(2) ~𝑒
−
Δ𝐸1(2)
𝑘𝑇
 For 𝑘𝑇 > Δ𝐸1 , Δ𝐸2 the barrier is
irrelevant: easily crossed both ways
 For Δ𝐸1 < 𝑘𝑇 < Δ𝐸2 crossings from
right to left are dominant
 For 𝑘𝑇 < Δ𝐸1 , Δ𝐸2 potential wells are
isolated
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Δ𝐸1
Δ𝐸2
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Simulated annealing:
How it works: The Metropolis algorithm
 Prepare a random ensemble of system configurations
 Sequentially select a configuration from the ensemble and
take a random step with it (a bit flip in our example)
 If the energy change Δ𝐸 is
 negative: accept the step
 positive: accept the step with probability exp −
Δ𝐸
𝑘𝑇
.
 Repeated application of this step gives us the Boltzmann
distribution for the configurations in the ensemble
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Simulated annealing:
The shortcomings
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Thermal jump
Energy
 The spin glass system has
typically many local minima
 High probability of having
relatively high and narrow
potential barriers
 For simulated annealing, only
the height of the barrier matters
 Idea: what if we could tunnel
through the barrier
Quantum
tunnelling
Configuration
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Quantum adiabatic computation
 A single quantum system at zero temperature (totally
coherent)
 Start with the system in the ground state of a simple
Hamiltonian 𝐻0 , e.g.
𝜎𝑖𝑥
𝐻0 = −
𝑖
 Slowly change the Hamiltonian to the problem
Hamiltonian 𝐻𝑃 :
𝑡
𝑡
𝐻 𝑡 = 1 − 𝐻0 + 𝐻𝑃
𝑇
𝑇
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Quantum adiabatic computation:
The adiabatic theorem
 If the Hamiltonian is changing slow enough, the system
stays near its instantaneous ground state.
 How slow is slow enough:
𝜀
𝑇≫
𝑔min 2
 with 𝜀 = max 𝐸1 (𝑡) 𝐻𝑃 − 𝐻0 𝐸0 (𝑡)
0≤𝑡≤𝑇
 𝐸0 (𝑡) and 𝐸1 𝑡 the two lowest energy instantaneous eigenvalues
(-states)
 and 𝑔min the minimal gap between 𝐸0 (𝑡) and 𝐸1 𝑡 .
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Quantum adiabatic computation:
Motivation for the initial Hamiltonian
1
1
𝐻0 = −
2 2𝑁





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𝜎𝑖𝑥
𝑖
Simple to implement
Describes a transverse interaction field
Does not commute with the problem Hamiltonian
Avoids energy level crossings (degeneracy)
Ground state is the equal superposition state, spanning
all of the solution space
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Quantum adiabatic computation:
Single qubit example
 Let’s consider the simplest example: finding the ground
state of a single qubit
1
𝐻𝑃 = 1 − 𝜎 𝑧
2
1 𝑠
𝐻 𝑡 =
2 −𝑠
𝐸0,1
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1
𝐻0 = 1 − 𝜎 𝑥
2
−𝑠
2−𝑠
𝑡
𝑠=
𝑇
1
= 1 ∓ 1 − 2𝑠 + 2𝑠 2
2
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Quantum adiabatic computation:
Single qubit example
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Quantum adiabatic computation:
Single qubit example
 Commuting
Hamiltonians
means that
energy levels
cross
1
𝐻𝑃 = 1 − 𝜎 𝑧
2
1
𝐻0 = 1 + 𝜎 𝑧
2
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Quantum adiabatic computation:
NP-hard example
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Quantum adiabatic computation:
NP-hard example
 Random instances of a specific type of 3-SAT problem,
the “Exact cover”
 Each clause requires that exactly one of the three bits
involved must be 1
 Remember:
 𝐻𝐶 𝑧1 𝑧2 … 𝑧𝑁 = 1 ∙ 𝑧1 𝑧2 … 𝑧𝑁 if clause 𝐶 is satisfied
 𝐻𝐶 𝑧1 𝑧2 … 𝑧𝑁 = 0 ∙ 𝑧1 𝑧2 … 𝑧𝑁 otherwise
 Problem Hamiltonian: 𝐻𝑃 = − 𝐻𝐶
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Quantum adiabatic computation:
NP-hard example
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Quantum adiabatic computation:
NP-hard example
 From exponential to quadratic complexity
 But:
 Median does not indicate worst case scaling
 They looked at a specific subset of “Exact Cover”
problems
 Don’t know the behaviour for large 𝑁
 Requires coherent quantum evolution!
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Simulated quantum annealing
 Is a quantum Monte Carlo algorithm
 Same annealing schedule as the quantum annealer
 Start with a strong initial transverse field which goes to zero during
simulation
 Start ramping up the problem Hamiltonian from zero
 Monte Carlo dynamics instead of unitary evolution
 Could use discrete or continuous time path integrals
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Comparing SA, QA and SQA
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One needs to note the difference
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How the problem is set up
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Similarities between D-Wave and SQA
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Correlation plots
Another way of comparing algorithms
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Pitfalls of detecting quantum speed-up
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Speedup of DW2 compared to SA
• Scaling of the highest quantile is the
most informative
• High quantiles are hard for DW2
• Range 7 is harder than Range 1
• Overall positive slope shows that DW
is better
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We have a quantum device, but is there an
universal quantum speedup?
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References
 http://people.fas.harvard.edu/~lucas/zurichCStalk.pdf
 Cohen, Eliahu, and Boaz Tamir. "D-Wave and predecessors: From
simulated to quantum annealing." International Journal of Quantum
Information 12.03 (2014): 1430002.
 Farhi, Edward, et al. "Quantum computation by adiabatic
evolution." arXiv preprint quant-ph/0001106 (2000).
 Farhi, Edward, et al. "A quantum adiabatic evolution algorithm applied
to random instances of an NP-complete problem." Science 292.5516
(2001): 472-475.
 Rønnow, Troels F., et al. "Defining and detecting quantum
speedup."Science 345.6195 (2014): 420-424.
 Boixo, Sergio, et al. "Evidence for quantum annealing with more than
one hundred qubits." Nature Physics 10.3 (2014): 218-224
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Thank you!
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