NASA Technical Reports Server (NTRS) 20160013398: Spin Glass Patch Planting

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by  NASA Technical Reports Server (NTRS)

Spin glass patch planting
Wenlong Wang,1,∗ Salvatore Mandr`a,2,3,† and Helmut G. Katzgraber1,4,5
1Department of Physics and Astronomy, Texas A&M University, College Station, Texas 77843-4242, USA
2NASA Ames Research Center Quantum Artificial Intelligence Laboratory (QuAIL), Mail Stop 269-1, 94035 Moffett Field CA
3Stinger Ghaffarian Technologies Inc., 7701 Greenbelt Rd., Suite 400, Greenbelt, MD 20770
4Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, New Mexico 87501, USA
5Applied Mathematics Research Centre, Coventry University, Coventry, CV1 5FB, England
In this paper, we propose a patch planting method for creating arbitrarily large spin glass in-
stances with known ground states. The scaling of the computational complexity of these instances
with various block numbers and sizes is investigated and compared with random instances using
population annealing Monte Carlo and the quantum annealing DW2X machine. The method can
be useful for benchmarking tests for future generation quantum annealing machines, classical and
quantum mechanical optimization algorithms.
PACSnumbers: 75.50.Lk,75.40.Mg,05.50.+q,64.60.-i
I. INTRODUCTION Given the importance to compare heuristics that can
beso different fromeachother, it ismandatory todefine
a benchmark class with the properties to be i) represen-
Many optimization problems belong to the important
tative of the hardness of a typical NP-Hard problem, ii)
class of NP-Hard problems, for which it is believed that
scalable for large systems and for which iii) the ground
no algorithms exist to solve them in polynomial time.
state is known a priori. Indeed, if it easy to find bench-
Spin glass problems like the Edward-Anderson (EA)
markclassthatfulfilli)andii),itbecomeschallengingto
model [1] represent a sub-class of the NP-Hard class.
find problems with thousands of variables but for which
Hence, not only they share the same computational
the ground state is still known.
hardness of all the other members of the class, but also
In order to fulfill all the properties which are expected
any NP-Hard problem (or, in general, any problems
for a good benchmark class, we introduce a new method
belong the NP class) can be mapped onto a spin glass
called “patch & planting” (PP). The method consists in
Hamiltonian with only a polynomial overhead. A num-
patchingtogethersmallerproblems,forwhichtheground
ber of heuristics, as well as exhaustive search methods,
state can be numerically found, to form larger instance
have been designed and developed to solve these prob-
for which the ground state is still known but computa-
lems as efficiently as possible. These methods including
tionallyhardertofind. Thepaperisstructuresasfollows:
simulated annealing [2], parallel tempering [3–5], pop-
In Sec. II we introduce the PP model, showing how to
ulation annealing [6–9], genetic algorithms [10, 11], as
constructinstanceswithmillionsofqubits. InSec.III,we
wellasDPLLandbranch-and-cut[12]algorithms. Many
numericallyshowthatproblemscreatedwithourmethod
of these optimization algorithms only use local move to
are indeed hard to solve. Results are obtained either
eventually reach the ground state. However, in many
numerically using simulated annealing or population an-
cases,theuseofclusteralgorithmswithnonlocalupdates
nealing, or experimentally using the D-Wave 2X chip.
can greatly enhance the searching process when the
Concluding remarks are finally stated in Sec. IV.
energy landscape presents many metastable states with
small overlap [13, 14]. In the last two decades, quantum
heuristics have been also proposed as an alternative
II. MODELS AND METHODS
for classical heuristics, thanks to their potentiality to
exploit quantum superposition and quantum tunneling.
A. Edwards-Anderson model
Among the quantum technologies, adiabatic quantum
optimization (AQO) is one of the most used [15–29]
and the more viable from the manufacturing point of The Edward-Anderson (EA) model [1] is one of the
view [30]. The state-of-art AQO hardware is manufac- most known and well studied spin glass problems, for
tured by D-Wave System Inc., whose latest chip allows which many properties are known either analytically or
the quantum optimization of 1000 qubits. However, numerically. For example, it is known that EA has a fi-
whether AQO can be more efficient than classical al- nite spin glass temperature when three-dimensional lat-
gorithmsincertainproblemsisstillcontroversial[31–33]. ticeareconsidered. Onthecontrary,fortwo-dimensional
lattices,itisexpectedthatEAhasazerospinglasscriti-
caltemperature. TheEAisdefinedbythefollowingspin
Hamiltonian
(cid:88) (cid:88)
∗Electronicaddress: [email protected] H =− JijSiSj − hiSi, (1)
†Electronicaddress: [email protected] (cid:104)ij(cid:105) i
2
whereS ∈{±1}areIsingspinsandthefirstsumisover
i
nearest neighbors on a d−dimensional lattice of linear
size L. In this work, we only considered 3-dimensional
lattices. For simplicity, all the local magnetic fields are
set to zero, namely hi = 0. Nevertheless, our method GS1 GS2
works with external fields as well. The couplings J de-
ij
fine the “disorder” and they are randomly chosen from
a given distribution. Here, we considered independent
and identically distributed random J , extracted from a
ij
Gaussian distribution with zero mean and unitary vari-
ance. Thesetofthecouplingsalsounequivocaldefinean
“instance”.
Given the hardware limitation of the D-Wave quan-
tumchips,instancesfortheD-Wave2Xhasbeencreated
by planting and patching together unit cells following
the two-dimensional architecture of the Chimera graph.
GS3 GS4
Moreover, we used couplings randomly extracted from
the Sidon set {±5,±6,±7}. In both cases, we use free
boundaryconditions(FBC)fortheblockstoplantlarger
instances. FortheEAmodel,wealsocompareFBCwith
random instances with regular periodic boundary condi-
tions (PBC).
FIG. 1: A schematic diagram of our “patch & planting”
B. Patch planting method for a 2-dimensional lattice. Firstly, ground states
for each block are found assuming free boundary conditions.
As briefly described in the introduction, the “patch Then,agroundstateofeachblockischosen: thesurfacespins
& planting” (PP) method works by following the simple of±1areshownasredandbluecircles,respectively. Finally,
acouplingisaddedbetweenneighborspinsofadjacentblocks:
scheme:
aferromagneticcoupling(blackinfigure)ifthetwospinsare
1. Solvesomesmallproblems,calledblocks,usingfree thealigned,orananti-ferromagneticifthetwo-spinsareanti-
boundary conditions and find their ground states. aligned (pink in figure). The direct product of the ground
states of the blocks is then the ground state of the larger
2. For each block, choose a ground state. system.
3. Addcouplingsbetweenblockssothatalltheextra-
block couplings are “satisfied” with respect to the
chosen ground states. patchedthesystemusingeithertwo,threeorfourblocks
respectively. For the experiments, we used an annealing
Observethattheaforementionedschemealwayssucceeds
time of 20 µs, 100 gauges and 1000 readouts for each
sincealltheextra-couplingscanbeeitherpositiveorneg-
gauge. For the numerical simulations, we used a num-
ative to accordingly match aligned or anti-aligned spins.
ber of replicas (R), number of temperature steps (N ),
Moreover,sincealltheextra-blockcouplingsaresatisfied, T
number of sweeps (N ) and temperature (T ) respec-
thegroundstateforthelargerinstanceisnecessarilythe S 0
tively equal to R = 2×105, N = 301, N = 10 and
composition of all the ground states of the blocks. A T S
T = 0.1. To compute ρ , we used the following param-
schematic diagram of patch planting is shown in Fig. 1. 0 0
etes: R=106, N =301, N =10 and T =0.1.
In the next section, we study the scaling of the com- T S 0
putational complexity of random instances generated by
our PP method by varying both the size of blocks and
III. RESULTS
the number of blocks used for composing the larger in-
stances. For the EA model, we also compare the hard-
ness of planted instances with random systems. More A. Correlation between the optimization hardness
precisely, we use the entropic family size ρs [9] to char- and the entropic family size ρs
acterize the hardness of the instances.
Classical simulations have been performed by using The first crucial step in investigating the hardness of
population annealing while experiments have been per- instances is to find a good measure which is reliable and
formed using the latest D-Wave 2X chip with N = yet easy to measure. The success probability of simu-
1097 working qubits. The simulation details for the lated annealing might look a good choice. However, it
3−dimensional EA model are shown in Table I. For the becomesquicklyimpossibletocomputeevenformedium
Chimera graph, we used all the available qubits and size systems. Another possibility consists in looking at
3
(see Fig. 2 for the analysis of the distribution of Ξ by
TABLEI: Simulationparametersforthe3−dimensionalEA
varying the system size). Therefore, let us define the
model using population annealing Monte Carlo. Here, L is
0 logarithm of ρ
the block size, L is the linear system size, R is the standard s
number of replicas, T = 1/β the lowest temperature simu-
0 0
lated, NT is the number of temperature steps (evenly spaced Ξ=log10(ρs). (2)
in β) in the annealing schedule, BC is the type boundary
condition[eitherperiodicboundaryconditions(PBC)orfree
Figure3showsthecorrelationbetweenthesuccessprob-
boundaryconditions(FBC)],andnisthenumberofdisorder
realizations studied. For each replica, N = 10 sweeps have ability for SA (with β =1/T =5) and Ξ (data are from
S
been performed at each temperature. Data for PBC with Refs. [8, 36]). As expected, the probability of success
L=8 are taken from Ref. [8]. decreases by increasing Ξ. Indeed, SA struggles more
to find the ground state when the energy landscape is
L L R T N BC n
0 0 T more rugged. Therefore, Ξ results the best candidate to
4 4 4×103 0.2 101 FBC 345600
measure the hardness of optimization problems.
4 8 104 0.2 101 FBC 5000
4 12 5×104 0.2 201 FBC 5120
800 L=4
4 16 2×105 0.2 301 FBC 1877 600
400
4 20 106 0.2 401 FBC 194 200
5 5 104 0.2 101 FBC 345600 0 L=8
400
5 10 105 0.2 201 FBC 5000
200
6 6 2×104 0.2 101 FBC 41472
0
6 12 105 0.2 201 FBC 1752 L=12
) 400
8 8 5×104 0.2 201 FBC 23358 Ξ
N( 200
8 16 106 0.2 301 FBC 624
0
10 10 106 0.2 301 FBC 8000 400 L=16
10 20 2×106 0.2 401 FBC 260 200
8 8 105 0.2 101 PBC 5099 0
L=20
12 12 106 0.2 201 PBC 3812 40
20
0
1 2 3 4 5 6
very specialized classical algorithms as HFS [34, 35] is Ξ
for the D-Wave 2X. Nevertheless, the hardness so de-
fined would be based dynamics on the chosen algorithm FIG.2: (Coloronline)DistributionofΞwithfixedblocksize
and may not reflect the structure of the energy land- L0 =4 for various system sizes L=4,8,12,16 and 20.
scape. Therefore, in this work we relate the hardness
of instances through the entropic family size ρ of pop-
s
ulation annealing. Population annealing (PA) is closely
relatedtosimulatedannealing(SA)[9]exceptthatituses
0 L=4 L=6
apopulationofreplicasandthispopulationisresampled
-1
at each temperature step. Unlike SA, PA is designed to
simulatetheequilibriumGibbsdistributionateachtem- -0.5
)
perature that is traversed. In particular, the resampling SA -2
p
step ensures that the population stays close to the equi- g(10 -1
librium ensemble. lo -3
At low temperature, most of the original popula-
-1.5
tion is eliminated in the resampling steps and the fi-
-4
nal population is descended from a small subset of the
0.8 1 1.2 1.4 1.61.6 2 2.4 2.8 3.2
initial population. Let ni be the fraction of popula- Ξ Ξ
tion from replica or family i in the initial population,
(cid:80)
ρ = lim Rexp[ n logn ]. Hence, ρ represents the FIG. 3: (Color online) Correlation of the success probability
s i i s
R→∞ i ofsimulationannealingpSAandthelogofentropicfamilysize
characteristic size of survival families: the larger ρs is, ofpopulationannealingΞforL=4andL=6atβ =5[8,36].
themoreruggedtheenergylandscaperesults. Moreover, When Ξ is larger, it is also more difficult to find the ground
ρ correlatesstronglywiththeintegratedautocorrelation state. Note that p drops very rapidly as L increases, while
s SA
time of parallel tempering. it is much easier to measure Ξ.
Itiswellestablishedthatρ convergesquicklyinpopu-
s
lationsize,anditisapproximatelylog-normaldistributed
4
B. Patch & Planting (PP) for the 3D EA model 0.8
L =4
0
Ξ ∼Mα
In this section, we focus on the scaling properties of Ξ
0.6
for PP instances, by varying either the block sizes L or
0
the system size L. Moreover, we will demonstrate that
harder blocks can be used to patch harder instances. Ξ) 0.4
LetusdenotewithM =(L/L0)3 thenumberofblocks g(10
ofsizeL0. Itiswellknownthat,forrandominstances,ρs lo
grows exponentially with L [9]. Since one would expect 0.2
that having a problem of size L by patching M blocks
of size L cannot be harder than a problem composed of
0
0
a single block of size L, the largest size for the entropic
family after patching M blocks is then bounded
0 0.5 1 1.5 2
Ξ(M,L0)≤MΞ(L0), (3) log10(M)
where Ξ(M,L ) is the logarithm of the entropy family FIG.4: (Coloronline)Scalingofthelogarithmoftheentropic
0
size after patching M blocks and Ξ(1,L0) ≡ Ξ(L0). In family size Ξ=log10ρs by varying the number of blocks M.
Fig. 4, we show the scaling of Ξ by varying the number Thedashedlineisapower-lawfitoftheformΞ(L0)Mα,where
Ξ(L ) is the entropic family size for a single block. From the
of blocks M and a power-law fit of the form 0
fit, we obtain α=0.31(3).
Ξ(M,L )=Ξ(L )Mα. (4)
0 0
As one can see, the price to pay to have instances with
an exactly known ground state consists in having Ξ
scaling sub-linearly with the number of blocks M, with
0.5
α = 0.31(3). Nevertheless, it also proves that the PP
instancesbecomeharder byincreasingthethenumberof
blocks. Therefore, it is guaranteed that, for a sufficient 0.4
large number of blocks, PP instances are arbitrarily
hard. Fig. 5 shows the scaling of the parameter α by
0.3
increasing the size of blocks L , when the number of
0
blocks is kept fixed to M = 8. As one can see from α
the figure, α remains a constant for a wide range of 0.2
L implying that α is a characteristic constant for PP
0
problems.
0.1
One may also expect to have some benefit by using ei-
ther larger or harder blocks. Indeed, in both cases, the 0
final effect is to have a larger Ξ(L ) from which build 4 6 8 10
0
up the hardness of the PP instances. In Fig. 6 we show L0
the effect of having larger blocks by analyzing the dis-
tribution of Ξ at fixed size of the system L = 16 us- FIG. 5: (Color online) Power-law exponent α for the scaling
ofΞ∼Mα,byvaryingthesizeofblocksL butkeepingfixed
ing two different block sizes, respectively L = 4 (top 0
0 the number of blocks to M =8.
panel) and L =8 bottom panel. As one can see, PP in-
0
stances are consistently harder by using larger blocks.
Similarly, in Fig. 7 we show the distribution of Ξ by
patching PP instances with M =8 blocks of size L =6
0
by either using easy or hard blocks. To create the two
setsofeasy/hardblocks,wehavechosenrespectivelythe
8000 easiest and the 8000 harder, among 41472 blocks, Finally, it is interesting to compare the complexity of
and patched together 1000 instances for each of the two the patch blocks with random instances. The distribu-
sets. Also in this case, PP instances patched with hard tion of Ξ for L = 8 and L = 12 with different block
blocksareconsistentlyharderthanPPinstancespatched sizes and random instances are shown in Fig. 8. One
witheasyblocks,withthemeanvaluesbeingrespectively canseewhilethepatchedinstancesaregenerallysimpler
3.214(5) and 2.957(4). Therefore, systematic and prac- than random instances, they are not all trivial. There
tical technique developed for engineering hard spin glass are some overlaps in the distributions. Furthermore, the
in Ref. [37] could be also used in generating hard blocks complexity grows for a fixed system size as the blocks
in patch planting. size gets larger.
5
200 600
L=16L =4 L=8Random
0 400
150 200
0
100 400 L=8L0=4
200
50
0
L=12Random
N(Ξ) 0 L=16L =8 N(Ξ) 240000
0
0
150
L=12L =4
400 0
100 200
0
L=12L =6
50 400 0
200
0 0
1 2 3 4 5 6 1 2 3 4 5 6
Ξ Ξ
FIG. 6: (Color online) Comparison of the distribution of Ξ FIG. 8: (Color online) Comparison of the complexity of the
for L = 16 but with different block sizes L = 4 and 8, re- patched instances with random instances. The patched in-
0
spectively. There is a noticeable shift in the distributions of stances are generally simpler than random instances as ex-
Ξ. Therefore to patch complex instances, one should use as pected, but they are not all trivial. There are some overlaps
large block sizes as possible. in the distributions of Ξ. Furthermore, the complexity grows
for a fixed system size as the used blocks size gets larger.
100
L=12L =6, easyblocks
0
80
Note that there is no spin glass phase on the chimera
60 graph, compared with the 3D case [38]. There are 1097
)
(Ξ qubits currently active in DW2X, and we study random
N 40
systemsandpatchedsystemswith2,3,and4blocksthat
20 cuts the chimera graph into approximately equal parts
horizontally,thedirectionthatcutsfewbonds. Westud-
0 ied 1000 instances of each class. For patch planting, the
L=12L =6, hardblocks
0 1000instanceswerechosenfrom10000instances,andwe
80
have selected the hardest ones.
Ξ) 60 The distributions of Ξ, psucc and the correlation of
( the two quantities are shown in Figs. 9, 10 and 11, re-
N 40
spectively. With some data mining of only a factor of
20 10, the instances with two blocks (2B) have approxi-
mately the same complexity as the random ones, which
0
2.4 2.8 3.2 3.6 4 are harder than instances with three blocks (3B) and
Ξ four blocks (4B). Some statistics of the success proba-
bilities are shown in Table II. It is also interesting that
FIG. 7: (Color online) Distributions of Ξ for M = 8 blocks the correlation of Ξ and p is about the same as the
succ
of size L0 = 6 using either easy blocks (top panel) or hard 3D case, even though a spin glass phase is absent on the
blocks (bottom panel). As expected, the distribution shifts
chimera graph, and the success probability is measured
to the left when harder blocks are used.
usingquantumannealinginsteadofsimulatedannealing,
showing the complexities are intrinsic to the instances.
C. The chimera graph
IV. SUMMARY
We have also studied patch planting on the chimera
graph, for the largest size N = 1152 available for the In this paper, we introduced an idea of patch planting
DW2X machine. The purpose is to see the performance for creating arbitrary large and nontrivial spin glass in-
of the patch instances for quantum annealing by studied stances with known ground states. We studied in detail
the success probabilities p . In addition, the correla- ofthescalingofthecomplexityofthepatchedinstances,
succ
tion of p and Ξ are also checked for this new model. and compared them with random instances using popu-
succ
6
30
Random
TABLEII: SomestatisticsoftheDW2Xsuccessprobability 20
psucc in Fig. 10 for random instances and patched instances 10
withtwoblocks(2B),threeblocks(3B)andfourblocks(4B). 0
Thereare1000instanceseach,thethepatchedinstanceswere 30 105psucc
chosen from the hardest ones out of 10000 instances in each 2B, Hardblocks
20
class. p ,p and p are the minimum, maximum and
min max ave 10
apvsuecrca=ge0v.aluesofpsucc andf isthefractionofinstanceswith (p)succ 3000 105psucc
N 3B, Hardblocks
Statistic Random 2B 3B 4B 200
100
p 0 0 0 0
min
0
pmax 2.55×10−3 1.84×10−3 3.49×10−2 1.76×10−1 300 104psucc
p 2.04×10−5 1.29×10−5 1.15×10−3 1.27×10−2 4B, Hardblocks
ave 200
f 0.831 0.775 0.185 0.011 100
0
0 50 100 150 200 250 300
100
Random 103psucc
75
50
FIG.10: (Coloronline)Thedistributionsofthesuccessprob-
25
ability p on the chimera graph for N = 1152 for random
succ
0 2B, Hardblocks instances,2B,3Band4B.Thereare1000instanceseach,and
75 the patched instances were chosen from the hardest ones out
50 of 10000 instances in each class. Note that 2B and Random
25 have about the same complexity.
)
Ξ
( 0
N 3B, Hardblocks
75
50
-0.5
25 Random
0 -1 2B, Hardblocks
4B, Hardblocks 3B, Hardblocks
75 -1.5 4B, Hardblocks
50
-2
25
) -2.5
0 ucc
2 2.5 3 3.5 4 4.5 5 5.5 ps -3
(
Ξ 10
g -3.5
o
l
FIG. 9: (Color online) The distributions of Ξ on the chimera -4
graph for N = 1152 for random instances and patched in-
-4.5
stances with two blocks (2B), three blocks (3B) and four
-5
blocks (4B). There are 1000 instances each, and the patched
instances were chosen from the hardest ones out of 10000 in- -5.5
stances in each class. Note that 2B and Random have about 2 2.5 3 3.5 4 4.5 5 5.5
the same complexity. Ξ
FIG. 11: (Color online) Correlation of Ξ and p on the
succ
chimeragraphforN =1152forrandominstances,2B,3Band
4B.NotethatthefeatherisessentiallythesameasFig.3,even
lation annealing and the DW2X machine. Furthermore,
thoughthereisnospinglassphaseonthechimeragraphand
we have shown that to patch hard instances, one should
thesuccessprobabilityismeasuredusingquantumannealing
use as large blocks as possible and also as hard blocks
instead of simulated annealing, showing the complexities are
as possible. Our method is easy to use and could be
intrinsic to the instances.
used as benchmark instances for future generation quan-
tumannealingmachines,andclassicalaswellasquantum
mechanical optimization algorithms.
supported in part by the Office of the Director of Na-
tional Intelligence (ODNI), Intelligence Advanced Re-
Acknowledgments searchProjectsActivity(IARPA),viaMITLincolnLab-
oratory Air Force Contract No. FA8721-05-C-0002. The
W.W. and H.G.K. acknowledge support from the Na- views and conclusions contained herein are those of the
tional Science Foundation (Grant No. DMR-1151387). authorsandshouldnotbeinterpretedasnecessarilyrep-
We thank Jon Machta and Ethan Brown for helpful dis- resenting the official policies or endorsements, either ex-
cussions. The research of H.G.K. is based upon work pressed or implied, of ODNI, IARPA, or the U.S. Gov-
7
ernment. The U.S. Government is authorized to repro- thankTexasA&MUniversityforaccesstotheirAdaand
duce and distribute reprints for Governmental purpose Curie clusters.
notwithstanding any copyright annotation thereon. We
[1] S. F. Edwards and P. W. Anderson, Theory of spin quantum annealing, Nat. Commun. 4, 2067 (2013).
glasses, J. Phys. F: Met. Phys. 5, 965 (1975). [21] S. Boixo, T. F. Rønnow, S. V. Isakov, Z. Wang,
[2] S. Kirkpatrick, C. D. Gelatt, Jr., and M. P. Vecchi, D. Wecker, D. A. Lidar, J. M. Martinis, and M. Troyer,
Optimization by simulated annealing, Science 220, 671 Evidenceforquantumannealingwithmorethanonehun-
(1983). dred qubits, Nat. Phys, 10, 218 (2014).
[3] R. H. Swendsen and J.-S. Wang, Replica Monte Carlo [22] T. F. Rønnow, Z. Wang, J. Job, S. Boixo, S. V. Isakov,
simulation of spin-glasses, Phys. Rev. Lett. 57, 2607 D. Wecker, J. M. Martinis, D. A. Lidar, and M. Troyer,
(1986). Defining and detecting quantum speedup, Science 345,
[4] C.Geyer,in23rd Symposium on the Interface,editedby 420 (2014).
E.M.Keramidas(InterfaceFoundation,FairfaxStation, [23] S. Boixo, V. N. Smelyanskiy, A. Shabani, S. V. Isakov,
VA, 1991), p. 156. M.Dykman,V.S.Denchev,M.H.Amin,A.Y.Smirnov,
[5] K. Hukushima and K. Nemoto, Exchange Monte Carlo M. Mohseni, and H. Neven, Computational multiqubit
methodandapplicationtospinglasssimulations,J.Phys. tunnelling in programmable quantum annealers, Nat.
Soc. Jpn. 65, 1604 (1996). Comm. 7, 10327 (2016).
[6] K. Hukushima and Y. Iba, in The Monte Carlo method [24] D.A.LidarandT.A.Brun,eds.,QuantumErrorCorrec-
in the physical sciences: celebrating the 50th anniversary tion (CambridgeUniversityPress,Cambride,UK,2013).
of the Metropolis algorithm, edited by J. E. Gubernatis [25] K. L. Pudenz, T. Albash, and D. A. Lidar, Quantum
(AIP, 2003), vol. 690, p. 200. annealing correction for random Ising problems (2014),
[7] E.ZhouandX.Chen,inProceedings of the 2010 Winter arXiv:quant-phys/1408.4382.
Simulation Conference (WSC) (2010), p. 1211. [26] A. Perdomo-Ortiz, B. O’Gorman, J. Fluegemann,
[8] W.Wang,J.Machta,andH.G.Katzgraber,Comparing R. Biswas, and V. N. Smelyanskiy, Determination and
Monte Carlo methods for finding ground states of Ising correction of persistent biases in quantum annealers
spin glasses: Population annealing, simulated annealing, (2015), (arXiv:quant-phys/1503.05679).
and parallel tempering, Phys. Rev. E 92, 013303 (2015). [27] W. Vinci, T. Albash, G. Paz-Silva, I. Hen, and D. A.
[9] W.Wang,J.Machta,andH.G.Katzgraber,Population Lidar, Quantum annealing correction with minor embed-
annealing: Theory and application in spin glasses, Phys. ding, Phys. Rev. A 92, 042310 (2015).
Rev. E 92, 063307 (2015). [28] S.Mandra`,G.G.Guerreschi,andA.Aspuru-Guzik,Adi-
[10] A.K.HartmannandH.Rieger,OptimizationAlgorithms abatic quantum optimization in the presence of discrete
in Physics (Wiley-VCH, Berlin, 2001). noise: Reducing the problem dimensionality, Phys. Rev.
[11] A.K.HartmannandH.Rieger,New Optimization Algo- A 92, 062320 (2015).
rithms in Physics (Wiley-VCH, Berlin, 2004). [29] D. Venturelli, S. Mandra`, S. Knysh, B. O’Gorman,
[12] C. De Simone, M. Diehl, M. Ju¨nger, P. Mutzel, R. Biswas, and V. Smelyanskiy, Quantum Optimization
G. Reinelt, and G. Rinaldi, Exact ground states in spin ofFullyConnectedSpinGlasses,Phys.Rev.X5,031040
glasses: New experimental results with a branch-and-cut (2015).
algorithm, J. Stat. Phys. 80, 487 (1995). [30] M. W. Johnson, M. H. S. Amin, S. Gildert, T. Lanting,
[13] J. Houdayer, A cluster Monte Carlo algorithm for 2- F. Hamze, N. Dickson, R. Harris, A. J. Berkley, J. Jo-
dimensionalspinglasses,Eur.Phys.J.B.22,479(2001). hansson,P.Bunyk,etal.,Quantumannealingwithman-
[14] Z. Zhu, A. J. Ochoa, and H. G. Katzgraber, Efficient ufactured spins, Nature 473, 194 (2011).
ClusterAlgorithmforSpinGlassesinAnySpaceDimen- [31] B. Heim, T. F. Rønnow, S. V. Isakov, and M. Troyer,
sion, Phys. Rev. Lett. 115, 077201 (2015). Quantum versus classical annealing of ising spin glasses,
[15] T. Kadowaki and H. Nishimori, Quantum annealing in Science 348, 215 (2015).
thetransverseIsingmodel,Phys.Rev.E58,5355(1998). [32] S. Mandra`, Z. Zhu, W. Wang, A. Perdomo-Ortiz,
[16] H.Nishimori,Statistical Physics of Spin Glasses and In- and H. G. Katzgraber, Strengths and Weaknesses of
formation Processing: An Introduction (Oxford Univer- Weak-Strong Cluster Problems: A Detailed Overview
sity Press, New York, 2001). of State-of-the-art Classical Heuristics vs Quantum
[17] E.Farhi,J.Goldstone,S.Gutmann,J.Lapan,A.Lund- Approaches, Physical Review A 94, 022337 (2016),
gren, and D. Preda, A quantum adiabatic evolution al- (arXiv:1604.01746).
gorithm applied to random instances of an NP-complete [33] S.Mandra`,Z.Zhu,andH.G.Katzgraber,Exponentially-
problem, Science 292, 472 (2001). biased ground-state sampling of quantum annealing ma-
[18] G.Santoro,E.Martonˇa´k,R.Tosatti,andR.Car,Theory chines with transverse-field driving hamiltonians, arXiv
ofquantumannealingofanIsingspinglass,Science295, preprint arXiv:1606.07146 (2016).
2427 (2002). [34] F. Hamze and N. de Freitas, in Proceedings of the
[19] J. Roland and N. J. Cerf, Quantum search by local adia- 20th Conference on Uncertainty in Artificial Intelligence
batic evolution, Phys. Rev. A 65, 042308 (2002). (AUAI Press, Arlington, Virginia, United States, 2004),
[20] S. Boixo, T. Albash, F. M. Spedalieri, N. Chancellor, UAI ’04, p. 243, ISBN 0-9749039-0-6.
andD.A.Lidar,Experimentalsignatureofprogrammable [35] A. Selby, Efficient subgraph-based sampling of Ising-
8
type models with frustration (2014), (arXiv:cond- gineering of hard spin-glass instances, Phys. Rev. A 94,
mat/1409.3934). 012320 (2016).
[36] W. Wang, J. Machta, and H. G. Katzgraber, Evi- [38] H. G. Katzgraber, F. Hamze, and R. S. Andrist, Glassy
dence against a mean-field description of short-range Chimeras Could Be Blind to Quantum Speedup: De-
spin glasses revealed through thermal boundary condi- signing Better Benchmarks for Quantum Annealing Ma-
tions, Phys. Rev. B 90, 184412 (2014). chines, Phys. Rev. X 4, 021008 (2014).
[37] J. Marshall, V. Martin-Mayor, and I. Hen, Practical en-