624. WE-Heraeus-Seminar, Bad Honnef, September 19-22, 2016 Simulating Quantum Processes and Devices

Quantum annealing in imaginary time

Anders W Sandvik, Boston University

C. De Grandi, A. Polkovnikov, A. W. Sandvik, PRB 84, 224303 (2011) C.-W. Liu, A. Polkovnikov, A. W. Sandvik, PRB 87, 174302 (2013)C.-W. Liu, A. Polkovnikov, A. W. Sandvik, PRB 89, 054307 (2014)C.-W. Liu, A. Polkovnikov, A. W. Sandvik, PRL 114, 147203 (2015)

Cheng-Wei Liu (BU → HP)Anatoli Polkovnikov (BU)

Na Xu (BU)

OutlineIntroduction- Quantum annealing for quantum computing/optimization- Transverse-field Ising model, quantum phase transition

- Kibble-Zurek velocity scaling- Demonstration for classical Ising model

Dynamic scaling at phase transitions

- Exact numerical solutions for small systems- Dynamical quantum Monte Carlo in imaginary time

Quantum dynamics in real and imaginary time

- Scaling of success probability in real and imaginary time- Problem hardness in real and imaginary time

[Going all the way to the classical spin-glass ground state]

- Imaginary-time QMC and KZ scaling- Implications for quantum computing

Quantum spin glass transition

h

Tglass phase

Thermal and Quantum Annealing

Adiabatic Theorem:For small v, the system stays in the ground state of H[s(t)]

Useful paradigm for quantum computing?

Can quantum annealing be more efficient than thermal annealing?Kadowaki, Nishimory (PRE 1998), Farhi et a (Science 2002),….

Simulated (Thermal) AnnealingReduce T as a function of time in a Monte Carlo simulation- efficient way to equilibrate a simulation- powerful as optimization algorithm

Reduce quantum fluctuations as a function of time- start with simple quantum system H0 (s=0):- end with a complicated classical potential H1 (s=1)

Quantum Annealing

s = s(t) = vt, v = 1/tmax

H(s) = (1� s)H0 + sH1 [H0, H1] 6= 0

Quantum Annealing & Quantum ComputingThe D-wave “quantum annealer”; ~1000 flux qubits- Claimed to solve some hard optimization problems- Is it really doing quantum annealing?

- Is quantum annealing really better than simulated annealing (on a classical computer)?

Hamiltonian implemented in D-wave quantum annealer....

Hamiltonian of the D-Wave Device

→ Studies of dynamics of transverse-field Ising models

||Matthias Troyer

Performance of simulated annealing, simulated quantum annealing and D-Wave on hard spin glass instances

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FIG. 6. The 476-qubit DW2 device architecture andqubit connectivity.

Appendix C: The D-Wave Two Chip

1. The Chimera

The Chimera graph of the D-Wave Two (DW2) chipused in this study is shown in Fig. 6. The chip is an8⇥8 array of unit cells where each unit cell is a balancedK4,4 bipartite graph of superconducting flux qubits. Inthe ideal Chimera graph the degree of each (internal)vertex is 6. On our chip, only a subset of 476 qubits, isfunctional. The temperature of the device ⇠ 15mK.

The chip is designed to solve a very specific typeof problems, namely, Ising-type optimization problemswhere the cost function is that of the Ising Hamiltonian[see Eq. (1) of the main text]. The Ising spins, s

i

= ±1are the variables to be optimized over and the sets {J

ij

}and {h

i

} are the programmable parameters of the costfunction. In addition, hiji denotes a sum over all activeedges of the Chimera graph.

2. The D-Wave Two annealing schedule

The DW2 performs the annealing by implementing thetime-dependent Hamiltonian

H(t) = �A(t)X

i

�x

i

+B(t)HIsing , (C1)

with t 2 [0, ta] where the allowed range of annealing timesta, due to engineering restrictions, is between 20µs and20ms. The annealing schedules A(t) and B(t) used inthe device are shown in Fig. 7. There are four anneal-ing lines, and their synchronization becomes harder forfaster annealers. The filtering of the input control linesintroduces some additional distortion in the annealingcontrol.

FIG. 7. Annealing schedule of the D-Wave chip. Thefunctions A(t) and B(t) are the amplitudes of the (transverse-field) driver and classical Ising Hamiltonians, respectively.Also shown is the temperature in units of energy (kB = 1).

3. Gauge averaging on the D-Wave device

Calibration inaccuracies stemming mainly from thedigital to analog conversions of the problem parameters,cause the couplings J

ij

and hi

realized on the DW2 chipto be slightly di↵erent from the intended programmedvalues (with a typical ⇠ 5% variation). Therefore, in-stances encoded on the device will be generally di↵erentfrom the intended instances. Additionally, other, moresystematic biases exist which cause spins to prefer oneorientation over another regardless of the encoded pa-rameters. To neutralize these e↵ects, it is advantageousto perform multiple annealing rounds (or ‘programmingcycles’) on the device, where each such cycle correspondsto a di↵erent encoding or ‘gauge’ of the same probleminstance onto the couplers of the device [9]. To real-ize these di↵erent encodings, we use a gauge freedomin realizing the Ising spin glass: for each qubit we canfreely define which of the two qubits states correspondsto s

i

= +1 and si

= �1. More formally this correspondsto a gauge transformation that changes spins s

i

! ⌘i

si

,with ⌘

i

= ±1 and the couplings as Jij

! ⌘i

⌘j

Jij

andhi

! ⌘i

hi

. The simulations are invariant under such agauge transformation, but due to calibration errors whichbreak the gauge symmetry, the results returned by theDW2 are not.

4. Performance of the DW2 chip as a function ofannealing time

Since the DW2 chip is a putative quantum annealer, itis only natural to ask how its performance, namely, thetypical time-to-solution ts it yields, depends on annealingtime ta. Ideally, the longer ta is, the better the perfor-mance we expect [52]. However, in practice, decoheringinteractions with the environment are present which be-

D-wave Chimera-lattice setup, picture fromMartin-Mayor & Hen, arXiv:1502.02494

Interactions Jij are programmable- restricted to “Chimera lattice”

Solves optimization problems mappedonto frustrated Ising model

H1 =NX

i=1

NX

j=1

Jij�zi �

zj , �z

i 2 {�1,+1}

Tune the strength of the field

adiabatically from s=0 to s=1 s = s(t) = vt, v = 1/t

max

H(s) = (1� s)H0 + sH1

H0 = �NX

i=1

�x

i

= �NX

i=1

(�+i

+ ��i

)

[H0, H1] 6= 0

Quantum Phase Transition

Ground state changes qualitatively as s changes- trivial (easy to prepare) for s=0- complex (solution of hard optimization problem) at s=1

→ expect a quantum phase transition at some s=sc

- trivial x-oriented ferromagnet at s=0 (→→→)- z-oriented (↑↑↑or ↓↓↓, symmetry broken) at s=1- sc=1/2 in 1D, appr. 0.25 in 2D

as in the clean transverse-field Ising ferromagnet

(N ! 1)

How long does it take (versus problem size N)?

Have to pass through sc and beyond adiabatically

H(s) = �sX

hiji

�z

i

�z

i+1 � (1� s)NX

i=1

�x

i

One can expect a quantum phase transition in the system

H(s) = (1� s)H0 + sH1 [H0, H1] 6= 0

Quantum Dynamics

What can imaginary time tell us about real-time dynamics?

Time evolution | (t)i = U(t, t0)| (t0)i

Difficult to study numerically for a many-body system- exact diagonalization of small systems- DMRG/MPS/TEBD for 1D systems (moderate sizes and times)

| (⌧)i = U(⌧, ⌧0)| (⌧0)i U(⌧, ⌧0) = T⌧exp

�Z ⌧

⌧0

d⌧ 0H[s(⌧ 0)]

�

Can be implemented in Quantum Monte CarloDe Grandi, Polkovnikov, Sandvik, PRB2011

Schrödinger dynamics in imaginary time t=iτAlternative approach:

Time evolution operator with time-dependent H

U(t, t0) = Ttexp

i

Z t

t0

dt0H[s(t0)]

�

Real and imaginary time quantum dynamics

How different? Which one is more adiabatic?

Example: linear ramp of transverse-field Ising ferromagnet

Start from eigenstate of H(s=0) at t=0

- Instantaneous ground state: | 0(t)i = | 0(s[t])i

- Actual state during evolution: | (t)i

Distance between these states given by log-fidelity

� ln[F (t)] = �1

2ln�|h 0(t)| (t)i|2

�

Integrate Schrödinger equation numerically for small L- compare real and imaginary time

2D square-lattice system; N=L2

H(s) = �sX

hiji

�z

i

�z

j

� (1� s)NX

i=1

�x

i

s 2 [0, 1], s = vt

Example: 4×4 lattice

Main peakreflects quantumphase transitionat Sc≈0.25

0 0.2 0.4 0.6 0.8 1t/tmax

0.00

0.02

0.04

0.06

0.08

0.10

-ln(F)

real time imaginary time

v = 1/40

Example: 4×4 lattice

Main peakreflects quantumphase transitionat Sc≈0.25

Imaginary timemore efficient inreaching ground state for s→1

0 0.2 0.4 0.6 0.8 1t/tmax

0.000

0.005

0.010

0.015

0.020

0.025

-ln(F)

real time imaginary time

v = 1/80

Example: 4×4 lattice

Main peakreflects quantumphase transitionat Sc≈0.25

Imaginary timemore efficient inreaching ground state for s→1

0 0.2 0.4 0.6 0.8 1t/tmax

0.000

0.001

0.002

0.003

0.004

0.005

-ln(F)

real time imaginary time

v = 1/160

Example: 4×4 lattice

Dynamic exponent z is same in real and imaginary time De Grandi, Polkovnikov, Sandvik, PRB 2011

Differences between real and imaginary time are of order v2 or v3

(depends on observable)

Same dynamic susceptibilitiesaccessed in realand imaginarytime

0 0.2 0.4 0.6 0.8 1t/tmax

0.0000

0.0002

0.0004

0.0006

0.0008

0.0010

0.0012

-ln(F)

real time imaginary time

v = 1/320

Use imaginary time for large systems

Quantum Monte Carlo Algorithm

Simpler scheme: evolve with just a H-product(Liu, Polkovnikov, Sandvik, PRB 2013)

Implemented in quantum Monte Carlo as:

| (⌧)i =1X

n=0

Z ⌧

⌧0

d⌧n

Z ⌧n

⌧0

d⌧n�1 · · ·Z ⌧2

⌧0

d⌧1[�H(⌧n)] · · · [�H(⌧1)]| (0)i

| (⌧)i = U(⌧, ⌧0)| (⌧0)i

Schrödinger dynamic in imaginary time t=iτ

U(⌧, ⌧0) = T⌧exp

�Z ⌧

⌧0

d⌧ 0H[s(⌧ 0)]

�

How is this method implemented?

| (sM )i = H(sM ) · · ·H(s2)H(s1)| (0)i, si = i�s, �s =sMM

Time unit is ∝1/N, velocity is v / N�s

Difference in v-dependence between product evolution and imaginary-time Schrödinger dynamics is O(v2)- same critical scaling behavior, dynamic susceptibilities

h (0)|H(s1) · · ·H(s7)|H(s7) · · ·H(s1)| (0)i

QMC Algorithm Illustration

H1(i) = �(1� s)(�+i + ��

i )

H2(i, j) = �s(�zi �

zj + 1)

Transverse-field Ising model: 2 types of operators:

12345677654321 12345677654321 1234567765432112345677654321

Represented as “vertices”

Simple extension of ground-state projector QMC (fixed H)

Analyze results versus velocity v and system size

12345677654321

MC sampling of networks of vertices

N = 4

M = 7

Dynamic QMC Illustration

Computational advantage:All s=values in one simulation!

“Asymmetric” expectation values

hAik = h (0)|1Y

i=M

H(si)MY

i=k

H(si)AkY

i=1

H(si)| (0)ihAik = h (0)|1Y

i=M

H(si)MY

i=k

H(si)AkY

i=1

H(si)| (0)i

Collect data, do scaling analysis...

Same leading-order (in v)behavior as conventionalexpectation values

Test on clean 2D Ising modelin transverse field

Using H-product dynamics

Animation of single configuration

Dynamic Critical Exponent and GapDynamic exponent z at a phase transition- relates time and length scales

Continuous quantum phase transition- excitation gap at the transition depends on the system size and z as

� ⇠ 1

Lz=

1

Nz/d, (N = Ld)

At a continuous transition (classical or quantum):- large (divergent) correlation length

⇠r ⇠ |�|�⌫ , ⇠t ⇠ ⇠zr ⇠ |�|�⌫z δ = distance from critical point (in T or other param)

Classical (thermal) phase transition- Fluctuations regulated by temperature T>0Quantum (ground state, T=0) phase transition- Fluctuations regulated by parameter g in Hamiltonian

Classical and quantum phase transitions

There are many similarities between classical and quantum transitions- and also important differences

ord

er p

aram

eter

T [g]Tc [gc]

(a)

ord

er p

aram

eter

T [g]Tc [gc]

(b)

FIGURE 3. Temperature (T ) or coupling (g) dependence of the order parameter (e.g., the magnetizationof a ferromagnet) at a continuous (a) and a first-order (b) phase transition. A classical, thermal transitionoccurs at some temperature T = Tc, whereas a quantum phase transition occurs at some g= gc at T = 0.

where the spin correlations decay exponentially with distance [60].While quasi-1D antiferromagnets were actively studied experimentally already in the

1960s and 70s, these efforts were further stimulated by theoretical developments in the1980s. Haldane conjectured [55], based on a field-theory approach, that the Heisenbergchain has completely different physical properties for integer spin (S = 1,2, . . .) and“half-odd integer” spin (S = 1/2,3/2, . . .). It was known from Bethe’s solution that theS = 1/2 chain has a gapless excitation spectrum (related to the power-law decayingspin correlations). Haldane suggested the possibility of the S= 1 chain instead having aground state with exponentially decaying correlations and a gap to all excitations; a kindof spin liquid state [26]. This was counter to the expectation (based on, e.g., spin wavetheory) that increasing S should increase the tendency to ordering. Haldane’s conjecturestimulated intense research activities, theoretical as well as experimental, on the S = 1Heisenberg chain and 1D systems more broadly. There is now completely conclusiveevidence from numerical studies that Haldane was right [61, 62, 63]. Experimentally,there are also a number of quasi-one-dimensional S = 1/2 [64] and S = 1 [65] (andalso larger S [66]) compounds which show the predicted differences in the excitationspectrum. A rather complete and compelling theory of spin-S Heisenberg chains hasemerged (and includes also the VBS transitions for half-odd integer S), but even to thisdate various aspects of their unusual properties are still being worked out [67]. There arealso many other variants of spin chains, which are also attracting a lot of theoretical andexperimental attention (e.g., systems including various anisotropies, external fields [68],higher-order interactions [69], couplings to phonons [70, 71], long-range interactions[72, 73], etc.). In Sec. 4 we will use exact diagonalization methods to study the S= 1/2Heisenberg chain, as well as the extended variant with frustrated interactions (and alsoincluding long-range interactions). In Sec. 5 we will investigate longer chains using theSSE QMC method. We will also study ladder-systems consisting of several coupledchains [9], which, for an even number of chains, have properties similar to the Haldanestate (i.e., exponentially decaying spin correlations and gapped excitations).

2.4. Models with quantum phase transitions in two dimensions

The existence of different types of ground states implies that phase transitions canoccur in a system at T = 0 as some parameter in the hamiltonian is varied (which

146

Downloaded 27 Feb 2012 to 128.197.40.223. Redistribution subject to AIP license or copyright; see http://proceedings.aip.org/about/rights_permissions

In both cases phase transitions can be- first-order (discontinuous): finite correlation length ξ as g→gc or g→gc- continuous: correlation length diverges, ξ~|g-gc|-ν or ξ~|T-Tc|-ν

3

Classical (thermal) phase transition- Fluctuations regulated by temperature T>0Quantum (ground state, T=0) phase transition- Fluctuations regulated by parameter g in Hamiltonian

Classical and quantum phase transitions

There are many similarities between classical and quantum transitions- and also important differences

ord

er p

aram

eter

T [g]Tc [gc]

(a)

ord

er p

aram

eter

T [g]Tc [gc]

(b)

FIGURE 3. Temperature (T ) or coupling (g) dependence of the order parameter (e.g., the magnetizationof a ferromagnet) at a continuous (a) and a first-order (b) phase transition. A classical, thermal transitionoccurs at some temperature T = Tc, whereas a quantum phase transition occurs at some g= gc at T = 0.

where the spin correlations decay exponentially with distance [60].While quasi-1D antiferromagnets were actively studied experimentally already in the

1960s and 70s, these efforts were further stimulated by theoretical developments in the1980s. Haldane conjectured [55], based on a field-theory approach, that the Heisenbergchain has completely different physical properties for integer spin (S = 1,2, . . .) and“half-odd integer” spin (S = 1/2,3/2, . . .). It was known from Bethe’s solution that theS = 1/2 chain has a gapless excitation spectrum (related to the power-law decayingspin correlations). Haldane suggested the possibility of the S= 1 chain instead having aground state with exponentially decaying correlations and a gap to all excitations; a kindof spin liquid state [26]. This was counter to the expectation (based on, e.g., spin wavetheory) that increasing S should increase the tendency to ordering. Haldane’s conjecturestimulated intense research activities, theoretical as well as experimental, on the S = 1Heisenberg chain and 1D systems more broadly. There is now completely conclusiveevidence from numerical studies that Haldane was right [61, 62, 63]. Experimentally,there are also a number of quasi-one-dimensional S = 1/2 [64] and S = 1 [65] (andalso larger S [66]) compounds which show the predicted differences in the excitationspectrum. A rather complete and compelling theory of spin-S Heisenberg chains hasemerged (and includes also the VBS transitions for half-odd integer S), but even to thisdate various aspects of their unusual properties are still being worked out [67]. There arealso many other variants of spin chains, which are also attracting a lot of theoretical andexperimental attention (e.g., systems including various anisotropies, external fields [68],higher-order interactions [69], couplings to phonons [70, 71], long-range interactions[72, 73], etc.). In Sec. 4 we will use exact diagonalization methods to study the S= 1/2Heisenberg chain, as well as the extended variant with frustrated interactions (and alsoincluding long-range interactions). In Sec. 5 we will investigate longer chains using theSSE QMC method. We will also study ladder-systems consisting of several coupledchains [9], which, for an even number of chains, have properties similar to the Haldanestate (i.e., exponentially decaying spin correlations and gapped excitations).

2.4. Models with quantum phase transitions in two dimensions

The existence of different types of ground states implies that phase transitions canoccur in a system at T = 0 as some parameter in the hamiltonian is varied (which

146

Downloaded 27 Feb 2012 to 128.197.40.223. Redistribution subject to AIP license or copyright; see http://proceedings.aip.org/about/rights_permissions

In both cases phase transitions can be- first-order (discontinuous): finite correlation length ξ as g→gc or g→gc- continuous: correlation length diverges, ξ~|g-gc|-ν or ξ~|T-Tc|-ν

3

Exponentially small gap at a first-order(discontinuous) transition

� ⇠ e�aL

Exactly how does z enter in the adiabatic criterion?

Important issue for quantum annealing!P. Young et al. (PRL 2008)

Kibble-Zurek Velocity and ScalingThe adiabatic criterion for passing through a continuous phase transition involves exponents z and ν:

Kibble 1978- defects in early universeZurek 1981- classical phase transitionsPolkovnikov 2005 + others- quantum phase transitions

Same criterion for classicaland quantum phase transitions- adiabatic (quantum)- quasi-static (classical)

A(�, v,N) = N�/⌫0g(�N1/⌫0

, vNz0+1/⌫0), ⌫0 = d⌫, z = z/d

Will use for spin glasses of interest in quantum computing

Apply to well-understood classical system first...

Must have v < vKZ, withvKZ ⇠ L�(z+1/⌫)

Generalized finite-size scaling hypothesis

A(�, v, L) = L�/⌫g(�L1/⌫ , vLz+1/⌫)δ = distance from critical point (in T or other param)

Classical simulated annealing

Repeat many times, collect averages, analyze,....

2D classical IsingMetropolis MC

simulations

100 102 104 106 108

v Lz+1/ν

10-4

10-3

10-2

10-1

100

<m

2 > L

2β/ν

L = 12L = 24L = 48L = 64L = 96L = 128L = 192L = 256L = 500L = 1024polynomial fitpower-law fit

104 105 10610-3

10-2L = 128

Velocity Scaling, 2D Ising ModelRepeat process many times, average data for T=Tc

Used known 2D Ising exponentsβ=1/8, ν=1

Result: z ≈2.1767(5)consistent withvalues obtainedin other ways

Adjusted z foroptimal scalingcollapse

Liu, Polkovnikov,Sandvik, PRB 2014

Can also be done for quantum systems in imaginary time

hm2(� = 0, v, L)i = L�2�/⌫f(vLz+1/⌫)

0.15 0.20 0.25 0.30S

0.0

0.2

0.4

0.6

0.8

1.0

U L = 12L = 16L = 32L = 48L = 56L = 60

0.24 0.25

0.8

0.9

2D Transverse-Ising, Scaling Example

A(�, v, L) = L�/⌫g(�L1/⌫ , vLz+1/⌫)

Example: Binder cumulant

Step function should form,jump from U=0 to 1 at sc

- crossing points for finite system size

Do similar studies for quantum spin glasses

If z, ν known, sc not: use

vLz+1/⌫= constant

for 1-parameter scaling

z = 1, ⌫ ⇡ 0.70

N=8

3-regular graphs with anti-ferro couplingsN spins, randomly connected, coordination-number 3

- sc ≈ 0.37 from quantum cavity approximation- QMC consistent with this sc, power-law gaps at sc

The quantum model was studied by Farhi, Gosset, Hen, Sandvik, Shor, Young, Zamponi, PRA 2012

More detailed studies with quantum annealing

Analyze <q2> using QMC and velocity scaling

Edwards-Anderson spin-glass order parameter

q =1

N

NX

i=1

�zi (1)�

zi (2)

(1) and (2) are independent simulations (replicas)

Classical model has mean-field glass transition- Tc known exactly (Krazakala et al.)

Extracting Quantum-glass transition

Using Binder cumulantU(s, v,N) = U [(s� sc)N

1/⌫0, vNz0+1/⌫0

]

But now we don’t know the exponents. Use

v / N�↵, ↵ > z0 + 1/⌫0

- do several α- check for consistency

Consistent with previouswork, but smaller errors

Next, critical exponents...

sc = 0.3565 +/- 0.0012

Best result for α=17/12

4

0.000 0.001 0.002 0.003 0.004

1/N

0.335

0.340

0.345

0.350

0.355

s c(N)

0.33 0.34 0.35 0.36

s

0.0

0.1

0.2

0.3

0.4

U

192

256

320

384

448

512

576

N

FIG. 2: (Color online) Crossing points between Binder cumu-lants for 3-regular graphs with N and N+64 spins, extractedusing the curves shown in the inset. The results were obtainedin quenches with v ⇠ N�↵ for ↵ = 17/12. The curve in themain panel is a power-law fit for extrapolating sc.

were in good agreement with this estimate. The errorbars on these calculations is several percent.

We find sc

using r = 1 QAQMC with v / N�↵, where↵ exceeds the KZ exponent z0 + 1/⌫0 (which is unknownbut later computable for a posteriori verification). Thenhq2i ⇠ N�2�/⌫0

at sc

because f(x) in Eq. (8) approachesa constant when x ! 0. As illustrated in Fig. 2, quench-ing past the estimated s

c

, we use a curve crossing analysisof the Binder cumulant, U = (3� hq4i/hq2i2)/2, and ob-tain s

c

= 0.3565(12). This value agrees well with theprevious results but has smaller uncertainty.

Performing additional quenches to sc

using protocolswith both r = 1 and r = 2/3 in Eq. (5) we extract allthe critical exponents. An example of scaling collapsefor r = 1 is shown in Fig. 3. Here all exponents aretreated as adjustable parameters for obtaining optimaldata collapse. The vertical and horizontal scalings givethe ratio �/⌫0 and the KZ exponent z0+1/⌫0, respectively,and the slope in the linear regime is the exponent (9).Combining results for r = 1 and r = 2/3 we obtain theexponents � = 0.54(1), ⌫0 = 1.26(1), and z0 = 0.52(2).

Interestingly, the exponents are far from those ob-tained using Landau theory [41] and other methods [42]for large-d and fully connected (d = 1 [43]) Ising modelsin a transverse field; � = 1, ⌫0 = 2 and z0 = 1/4 (d

u

= 8)[41]. One might have expected the same mean-field ex-ponents for these systems, as in the classical case. AQMC calculation for the fully-connected model [44] wasnot in complete agreement with the Landau values. Itwas argued that z = 4, which, with d

u

= 8, agrees withour z0 ⇡ 1/2 for the 3-regular graphs. However, � wasclose to 1 and ⌫ = 1/4 (⌫0 = 2) was argued. It wouldbe interesting to study n-regular graphs and follow theexponents from n = 3 to large n.

Implications for quantum computing.—The critical ex-

10-1

100

101

102

103

104

v N z’+1/ν’

0.3

0.5

0.8

1.0

<q

2>

N 2

β/ν

’

128

256

384

512

768

1024

1536

N

FIG. 3: (Color online) Optimized scaling collapse of the or-der parameter in critical quenches of 3-regular graphs, givingthe exponents listed in the text. The line has slope given inEq. (9) and the points above it deviate due to high-velocitycross-overs [31] not captured by Eq. (8). For N ! 1 thelinear behavior should extend to infinity.

ponents contain information relevant to QA quantumcomputing. In the classical case the KZ exponent isz0 + 1/⌫0 = 1, while in the quantum system z0 + 1/⌫0 ⇡1.31. Thus, by Eq. (7) the adiabatic annealing time growsfaster with N in QA. Furthermore, since the critical or-der parameter scales asN�2�/⌫0

, the critical cluster is lessdense with QA, i.e., it is further from the state soughtwhen s ! 1 (the solution of the optimization problem).Thus, in both these respects QA performs worse than SAin passing through the critical point (while QA on thefully-connected model, with the exponents of Ref. [41],would reach s

c

faster than SA, though the critical clus-ter is still less dense). QA can be made faster than SAby following a protocol (5) with su�ciently large r, butthis may not be practical when s

c

is not known and thegoal is anyway to proceed beyond this point. While ourresults do not contain any quantitative information onthe process continuing from s

c

to s = 1, it is certainlydiscouraging that the initial stage of QA is ine�cient.

It would be interesting to run the D-Wave machine[11, 12] as well on a problem with a critical point andstudy velocity scaling. This would give valuable insightsinto the nature of the annealing process.

Acknowledgments.—We thank David Huse and A Pe-ter Young for stimulating discussions. This work wassupported by the NSF under grant No. PHY-1211284.

[1] S. Kirkpatrick, C. D. Gelatt Jr, M. P. Vecchi, Science220 671 (1983).

[2] V. Cerny, J. Optim. Theor. and Appl. 45: 41 (1985).[3] V. Granville, M. Krivanek, J.-P. Rasson, J.-P., IEEE

Trans. on Pattern Analysis and Machine Intelligence 16

↵ = 17/12

Study evolution to sc

- several system sizes N- several velocities

4

0.000 0.001 0.002 0.003 0.004

1/N

0.335

0.340

0.345

0.350

0.355

s c(N)

0.33 0.34 0.35 0.36

s

0.0

0.1

0.2

0.3

0.4

U

192

256

320

384

448

512

576

N

FIG. 2: (Color online) Crossing points between Binder cumu-lants for 3-regular graphs with N and N+64 spins, extractedusing the curves shown in the inset. The results were obtainedin quenches with v ⇠ N�↵ for ↵ = 17/12. The curve in themain panel is a power-law fit for extrapolating sc.

were in good agreement with this estimate. The errorbars on these calculations is several percent.

We find sc

using r = 1 QAQMC with v / N�↵, where↵ exceeds the KZ exponent z0 + 1/⌫0 (which is unknownbut later computable for a posteriori verification). Thenhq2i ⇠ N�2�/⌫0

at sc

because f(x) in Eq. (8) approachesa constant when x ! 0. As illustrated in Fig. 2, quench-ing past the estimated s

c

, we use a curve crossing analysisof the Binder cumulant, U = (3� hq4i/hq2i2)/2, and ob-tain s

c

= 0.3565(12). This value agrees well with theprevious results but has smaller uncertainty.

Performing additional quenches to sc

using protocolswith both r = 1 and r = 2/3 in Eq. (5) we extract allthe critical exponents. An example of scaling collapsefor r = 1 is shown in Fig. 3. Here all exponents aretreated as adjustable parameters for obtaining optimaldata collapse. The vertical and horizontal scalings givethe ratio �/⌫0 and the KZ exponent z0+1/⌫0, respectively,and the slope in the linear regime is the exponent (9).Combining results for r = 1 and r = 2/3 we obtain theexponents � = 0.54(1), ⌫0 = 1.26(1), and z0 = 0.52(2).

Interestingly, the exponents are far from those ob-tained using Landau theory [41] and other methods [42]for large-d and fully connected (d = 1 [43]) Ising modelsin a transverse field; � = 1, ⌫0 = 2 and z0 = 1/4 (d

u

= 8)[41]. One might have expected the same mean-field ex-ponents for these systems, as in the classical case. AQMC calculation for the fully-connected model [44] wasnot in complete agreement with the Landau values. Itwas argued that z = 4, which, with d

u

= 8, agrees withour z0 ⇡ 1/2 for the 3-regular graphs. However, � wasclose to 1 and ⌫ = 1/4 (⌫0 = 2) was argued. It wouldbe interesting to study n-regular graphs and follow theexponents from n = 3 to large n.

Implications for quantum computing.—The critical ex-

10-1

100

101

102

103

104

v N z’+1/ν’

0.3

0.5

0.8

1.0

<q

2>

N 2

β/ν

’

128

256

384

512

768

1024

1536

N

FIG. 3: (Color online) Optimized scaling collapse of the or-der parameter in critical quenches of 3-regular graphs, givingthe exponents listed in the text. The line has slope given inEq. (9) and the points above it deviate due to high-velocitycross-overs [31] not captured by Eq. (8). For N ! 1 thelinear behavior should extend to infinity.

ponents contain information relevant to QA quantumcomputing. In the classical case the KZ exponent isz0 + 1/⌫0 = 1, while in the quantum system z0 + 1/⌫0 ⇡1.31. Thus, by Eq. (7) the adiabatic annealing time growsfaster with N in QA. Furthermore, since the critical or-der parameter scales asN�2�/⌫0

, the critical cluster is lessdense with QA, i.e., it is further from the state soughtwhen s ! 1 (the solution of the optimization problem).Thus, in both these respects QA performs worse than SAin passing through the critical point (while QA on thefully-connected model, with the exponents of Ref. [41],would reach s

c

faster than SA, though the critical clus-ter is still less dense). QA can be made faster than SAby following a protocol (5) with su�ciently large r, butthis may not be practical when s

c

is not known and thegoal is anyway to proceed beyond this point. While ourresults do not contain any quantitative information onthe process continuing from s

c

to s = 1, it is certainlydiscouraging that the initial stage of QA is ine�cient.

It would be interesting to run the D-Wave machine[11, 12] as well on a problem with a critical point andstudy velocity scaling. This would give valuable insightsinto the nature of the annealing process.

Acknowledgments.—We thank David Huse and A Pe-ter Young for stimulating discussions. This work wassupported by the NSF under grant No. PHY-1211284.

[1] S. Kirkpatrick, C. D. Gelatt Jr, M. P. Vecchi, Science220 671 (1983).

[2] V. Cerny, J. Optim. Theor. and Appl. 45: 41 (1985).[3] V. Granville, M. Krivanek, J.-P. Rasson, J.-P., IEEE

Trans. on Pattern Analysis and Machine Intelligence 16

hq2(sc)i / N�2�/⌫0f(vNz0+1/⌫0

)

Velocity Scaling at the Glass Transition

β/ν‘ ≈ 0.43(2)z’+1/ν’ ≈ 1.3(2)

Significance of the exponents?

β/ν‘ ≈ 0.47(3)z’+1/ν’ ≈ 0.8(1)

Fully connected model (SK)- velocity scaling gives

Differ from valuesexpected for d=∞:(Read, Sachdev, Ye, 1995)

β/ν‘ = 1/2z’+1/ν’ = 3/4

Why is 3-regular model different?

Relevance to Quantum Computing

Reaching sc, the degree of ordering scales asp< hq2i > ⇠ N��/⌫0

�/⌫0 ⇡ 0.43

Classicalβ/ν‘ = 1/3z’+1/ν’ = 1

Let’s compare with the know classical exponents(finite-temperature transition of 3-regular random graphs)

Quantumβ/ν‘ ≈ 0.43z’+1/ν’ ≈ 1.3

h

Tglass phase

• It takes longer for quantum annealing to reach its critical point

• And the state is further from ordered (further from the optimal solution)

Proposal: Do velocity scaling with the D-wave machine!

The time needed to stay adiabatic up to sc scales as

t ⇠ Nz0+1/⌫ z0 + 1/⌫0 ⇡ 1.3