Quantum vs Classical Annealing: Seeking a Fair …hen/DWaveUsersMeeting_Sep14_ItayHen.pdf · Quantum vs Classical Annealing: Seeking a Fair Comparison . ... recent compelling evidence

Itay Hen Sep 10, 2014 D-Wave Users Colloquium

Quantum vs Classical Annealing: Seeking a Fair Comparison

Itay Hen Information Sciences Institute, USC

D-Wave Users Colloquium Sunnyvale, CA

September 10, 2014

Joint work with Victor Martin-Mayor, UCM


introduction: quantum annealing and the D-Wave chip

temperature chaos, parallel tempering and classical hardness.

some results

conclusions and outlook

Outline


Introduction


simulated (thermal) annealing: an optimization technique based on gradually reducing thermal fluctuations to find the minimizing configuration of a given cost function. used mainly for problems where the search space is discrete, e.g., combinatorial optimization problems with many local minima. thermal fluctuations are used to escape local minima.

quantum annealing: an optimization technique based on gradually reducing the magnitude of quantum fluctuations to find global minima, presumably uses quantum tunneling to traverse energy barriers.

Introduction


The D-Wave Two Chip

the D-Wave Two chip is presumably a quantum annealer.

designed to solve optimization problems of a very specific type.

the Chimera architecture

architecture of the chip and physical constraints are very limiting.

but, even within these limitations there is enough room for solving non-trivial problems.


however, it is essential to ask the following questions:

How Quantum is it?

The D-Wave Two Chip

recent compelling evidence by Lanting et al. (PRX, April 2014) that entanglement exists and can be indirectly detected midway through the run, using “probe” qubits.

unpublished preliminary results suggesting that the SSSV model does not reproduce the quantum signature (but Master Equation simulations do).


however, it is essential to ask the following questions:

How Quantum is it?

Can this “quantumness” be used?

The D-Wave Two Chip

recent compelling evidence by Lanting et al. (PRX, April 2014) that entanglement exists and can be indirectly detected midway through the run, using “probe” qubits.


Benchmarking so far

benchmarking tests have been inconclusive so far.

random MAX 2-SAT instances on the Chimera: - S. Santra et al., “MAX 2-SAT with up to 108 qubits” (New J. Phys., 2014).

random Ising models (no fields ℎ𝑖𝑖 = 0, 𝐽𝐽𝑖𝑖𝑖𝑖 = ±1 or similar): - S. Boixo, et al., “Quantum Annealing With More Than One Hundred Qubits”, (Nature Phys., 2014). - T. F. Rønnow et al., “Defining and detecting quantum speedup”., (Science, 2014).

no evidence for speedup so far.

(results of newer benchmarks look interesting– see talk by Josh Job).


Benchmarking so far

the usefulness of the D-Wave chip depends on the answers to the following two questions:

need to find hard instances (and if possible, find a way to design such).

can we find a “good parameter” for measuring classical hardness?

turns out there is one, for “thermal hardness” (but not controllable).

is “classically-hard” different from “quantum-hard?”

can this difference be detected by the D-Wave chip?


Temperature chaos, parallel tempering and classical hardness


Generating hard problems

possible issues with benchmarking problems tested so far (i.e., random couplings 𝐽𝐽 = ±1, ±2, or spin-glass benchmarks):

problems tested so far may have been too easy (Katzgraber, Hamze, and Andrist, PRX, April, 2014):

a large number of ground states.

classical phase transition to a spin glass occurs at 𝑇𝑇 = 0.

does that mean that all classes of problems on the Chimera are typically easy?



can we generate 3D-like instances on the Chimera? I don’t know.

the Chimera graph is essentially 2D.

if so, we expect spin-glass phase transitions to occur at 𝑇𝑇 = 0.

does that mean that typical instances are classically easy to solve?



can we generate 3D-like instances on the Chimera? I don’t know.

the Chimera graph is essentially 2D.

if so, we expect spin-glass phase transitions to occur at 𝑇𝑇 = 0.

does that mean that typical instances are classically easy to solve?

“chaotic” change of the thermodynamically-dominant configurations when temperature is slightly changed. goes back to Bray and Moore (1984), Fisher and Huse (1986).

causes thermal annealing algorithms to fail, but let’s back up a little.

as far as thermal-annealing-type algorithms are concerned, not necessarily! we could still have “temperature chaos”


thermal annealing uses the Metropolis algorithm:

endless loop of:

pick a spin at random

flip it, calculate energy change ∆𝐸𝐸.

if ∆𝐸𝐸 < 0, accept change, go back.

if ∆𝐸𝐸 ≥ 0, accept change with 𝑝𝑝 = 𝑒𝑒−∆𝐸𝐸/𝑇𝑇, otherwise discard change, go back.

for bipartite graphs, half of the spins may be updates simultaneously.

Thermal annealing


thermal energy

Thermal annealing

simulated annealing (simplest protocol):

high 𝑇𝑇: easy exploration


thermal energy

simulated annealing (simplest protocol):

high 𝑇𝑇: easy exploration

𝑇𝑇-lowering schedule: configuration can get trapped at nearby local minimum.

an outdated algorithm.

Thermal annealing


thermal energy

Parallel tempering

parallel tempering:

𝑇𝑇 may be raised or lowered:

low 𝑇𝑇: local exploration.


thermal energy

parallel tempering:



high 𝑇𝑇: global exploration.

Parallel tempering


thermal energy

Parallel tempering

parallel tempering:



high 𝑇𝑇: global exploration.

lesser chances of getting trapped better solution.


Parallel tempering

𝑁𝑁𝑇𝑇 temperatures: simultaneous simulation of 𝑁𝑁𝑇𝑇 replicas (one at each temperature).

periodically, replicas attempt to exchange their temperature. the rule preserves detailed balance.

swap acceptance rates can be fixed at a reasonable value by suitable choices of 𝑁𝑁𝑇𝑇 and density of temperatures.


Random Walk in temperatures of a replica

Parallel tempering

sounds perfect! what can go wrong?

each replica performs a random walk in “temperature space”.

the simulation is long enough if all the replicas visit all the temperatures several times. 𝜏𝜏, the “mixing time”, can be very long!


Temperature chaos

one can detect equilibrium using the equilibration of the parallel tempering walks in “temperature space”.

𝜏𝜏 can be found by looking at equilibrium autocorrelation functions of the replica random walk.

𝜏𝜏, the mixing time, can be used as a measure for the hardness of instances. corresponds to the amount of time the parallel tempering needs to equilibrate. t

C(t)

under what circumstances will 𝜏𝜏 be large?


Temperature chaos

𝑇𝑇 𝑇𝑇 − ∆𝑇𝑇

“chaotic” change of the thermodynamically-dominant configurations when temperature is slightly changed. goes back to Bray and Moore (1984), Fisher and Huse (1986).

temperature chaos is a general feature of spin-glasses.

relevant minima, completely different at nearby temperatures.

temperature random walk refuses to go across.

mixing time 𝜏𝜏 is long.


Temperature chaos


temperature chaos is generic for large problem sizes 𝑁𝑁.

in practice, for small 𝑁𝑁:

the large majority of problem: instances are easy (small 𝜏𝜏).

for some instances however, 𝜏𝜏 will be inordinately large.

the larger 𝑁𝑁 is, the larger the fraction of large-𝜏𝜏 instances will be.

difficult to assess algorithmic scaling with 𝑁𝑁 .


Some results


temperature chaos is known to exist in 3D.

does it exist in 2D as well?

Middleton et al. (2011): temperature chaos may even exist on a square lattice near 𝑇𝑇 = 0, but this was found for large 𝑁𝑁 = 2.6 × 103.

can even exist for problems in complexity class P.

Results



is there chaos in the 2D-like 𝑁𝑁 = 503 chimera lattice?

we generated millions of instances with random 𝐽𝐽 = ±1 and calculated the mixing time for each.

for most instances, mixing time is short, but…

Results


is there chaos in the 2D-like 𝑁𝑁 = 503 chimera lattice?

we generated millions of instances with random 𝐽𝐽 = ±1 and calculated the mixing time for each.

for most instances, mixing time is short, but…

look at the tail of the distribution

by extrapolation: two in 104 instances will have 𝜏𝜏 ≫ 108.

Results


temperature chaos is rare for 𝑁𝑁 = 503.

however, it is expected to be in abundance for larger 𝑁𝑁.

it is therefore important to see how instances exhibiting temperature chaos are “handled” by the D-Wave chip.

Results


example: energy (per particle) for an instance with 𝜏𝜏 ≈ 106.

as temperature is lowered, the system seems to equilibrate,

but at 𝑇𝑇 ≈ 0.2 there is a sudden drop in energy (a better configuration has been found).

Temperature chaos


this is the interesting temperature range.

both for D-Wave.

and for simulations.

Temperature chaos

example: energy (per particle) for an instance with 𝜏𝜏 ≈ 106.

as temperature is lowered, the system seems to equilibrate,

but at 𝑇𝑇 ≈ 0.2 there is a sudden drop in energy (a better configuration has been found).


this is the interesting temperature range.

both for D-Wave.

and for simulations.

Temperature chaos

D-Wave’s temperature seems to be low enough to detect “temperature chaos” for this instance.

there could be however “worse” instances. ≈D-Wave’s 𝑇𝑇


Temperature chaos

five instances with 𝜏𝜏 ≈ 1000: distribution of spin-overlap (left) and distribution of link overlap (right) between ground-state and first excitations.

excited states are not very far from ground states problems are easy.

Q (link overlap) q (spin overlap)

num

ber o

f con

figs.


Temperature chaos

five instances with 𝜏𝜏 ≈ 1000000: distribution of spin-overlap (left) and distribution of link overlap (right) between ground-state and first excitations.

excited states are very far from ground states problems are hard.

num

ber o

f con

figs.

Q (link overlap) q (spin overlap)


presence of Temperature Chaos kills thermal annealing.

this doesn’t mean that better non-thermal algorithms cannot exist. they do for Chimera but not in general.

for the Chimera we have the Hamze-de Freitas-Selby (HFS) algorithm:

can be specifically tailored to solve problems defined on D-Wave’s Chimera architecture, this algorithm uses the tree-like features of the Chimera graph.

however, we must remember that for other problems and architectures thermal annealing may be the only way.

Non-thermal algorithms


Results

performance of D-Wave on “hard” instances with τ ≈ 106.

these take several hours to equilibrate using parallel tempering.

some of the instances were solved rather quickly!

other instances could not be solved (perhaps more runs would have done the trick).

∞ (complete failure)


Results

meaningful algorithmic classification at fixed 𝑁𝑁 : 𝜏𝜏-scaling.

parallel tempering: time to solution (tts)~𝜏𝜏1 (by definition),

Selby’s heuristic (2D-like): tts~𝜏𝜏≈0.3 (much better than PT).

D-wave: tts~𝜏𝜏1.75

however, not yet schedule-optimized

worse than PT!

why? Important to understand.


Results

this needs to be investigated, could be for several reasons:

temperature may still be too high.

programming errors in the 𝐽𝐽 values may have devastating effects.


Results

this needs to be investigated, could be for several reasons:

temperature may still be too high.

programming errors in the 𝐽𝐽 values may have devastating effects.

also interesting: Selby’s average time on hard instances is a ~0.1 second! orders of magnitude longer than tts of random instances.

(still PT is > 1000 times slower.)


Conclusions and outlook



there are rare instances that are classically (thermally) very hard to solve, very long mixing times, even for lattices as small and as sparse as the 503-qubit Chimera!

we tried to answer the following important question: is quantum hardness different from classical hardness?

as far as D-Wave is concerned, Yes! In fact, QA scales worse!

we need to understand why, because there will be many more chaotic instances as the size of the problems grows.

could be for several reasons.



D-Wave’s temperature may be too high. if temperature is above onset of temperature chaos, then D-Wave may not reach true GS configurations.

it could be that the error in the programming of the 𝐽𝐽’s is too large.

these matters can be investigated via simulations, specifically, by “Quantum parallel tempering” (an ideal quantum annealer). T (temp)

Γ (transverse field)

𝑇𝑇𝑚𝑚𝑖𝑖𝑚𝑚

𝑇𝑇𝑚𝑚𝑚𝑚𝑚𝑚

Γ𝑚𝑚𝑖𝑖𝑚𝑚 Γ𝑚𝑚𝑚𝑚𝑚𝑚

classical PT

quantum PT

classical-quantum PT



T (temp)

Γ (transverse field)

𝑇𝑇𝑚𝑚𝑖𝑖𝑚𝑚

𝑇𝑇𝑚𝑚𝑚𝑚𝑚𝑚

Γ𝑚𝑚𝑖𝑖𝑚𝑚 Γ𝑚𝑚𝑚𝑚𝑚𝑚

classical PT

quantum PT

classical-quantum PT

these matters can be investigated via simulations, specifically, by “Quantum parallel tempering” (an ideal quantum annealer).

compare mixing-times of classical PT with mixing-times of quantum PT, with and without noisy J’s, and at different temperatures.

this will help us understand the less-than-favorable 𝜏𝜏-scaling of D-Wave. we could then offer improvements.



another important question: how strong is the effect of noisy 𝐽𝐽’s on success rates?

this can be tested on the chip by looking at variance of success rates as a function of 𝜏𝜏. if the variance correlates with 𝜏𝜏:

we can then answer the question: how should the error in 𝐽𝐽 scale with 𝑁𝑁 to prevent “𝐽𝐽-chaos”?

need to study “𝐽𝐽-chaos”: effects of noisy 𝐽𝐽’s on the minimizing configuration. small changes in J may lead to a completely different ground state configuration.


Quantum vs Classical Annealing: Seeking a Fair Comparison

Itay Hen Information Sciences Institute, USC

D-Wave Users Colloquium Sunnyvale, CA

September 10, 2014

Joint work with Victor Martin-Mayor, UCM

Thank You!

Documents

Quantum vs Classical Annealing: Seeking a Fair …hen/DWaveUsersMeeting_Sep14_ItayHen.pdf · Quantum vs Classical Annealing: Seeking a Fair Comparison . ... recent compelling evidence