Quantum and classical annealing in spin glasses and quantum computingC.-W. Liu, A. Polkovnikov, A. W. Sandvik, arXiv:1409.7192

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Quantum and classical annealing in spin glasses and quantum computing

C.-W. Liu, A. Polkovnikov, A. W. Sandvik, arXiv:1409.7192

Anders W Sandvik, Boston University

Cheng-Wei Liu (BU)

Anatoli Polkovnikov (BU)

NATIONAL TAIWAN UNIVERSITY, COLLOQUIUM, MARCH 10, 2015

1 Tuesday, March 10, 15

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Outline

• Classical and quantum spin glasses

• Quantum annealing for quantum computing Classical (thermal) fluctuations

versus

Quantum fluctuations (tunneling)

• Computational studies of

model systems (spin glasses)

• Relevance for adiabatic quantum computing

• Dynamical critical scaling

• Monte Carlo simulations and simulated annealing

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Example:

Particles with hard and soft cores (2 dim)

Monte Carlo Simulations

What happens when the temperature is lowered ?

P ( {r

ⁱ

}) / e

^E/T

, E = X

r1,r2

V (r

_i

r

_j

)

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Monte Carlo Simulations

Transition into liquid state has taken place Slow movement & growth of droplets

Is there a better way to reach equilibrium at low T?

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Simulated Annealing

Annealing: Removal of

crystal defects by heating followed by slow cooling Simulated Annealing:

MC simulation with slowly decreasing T - Can help to reach equilibrium faster

Optimization method:

express optimization of many parameters as

minimization of a

cost function, treat as

energy in MC simulation

Similar scheme in quantum mechanics?

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Quantum Annealing

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

H _classical = V (x) H _quantum = ~ ² 2m

d ² dx ² Adiabatic Theorem:

If the velocity v is small enough

the system stays in the ground state of H[s(t)] at all times

Can quantum annealing be more efficient than thermal annealing?

At t=t

max

we then know the minimum of V(x): (x) = (x x ₀ )

Useful paradigm for quantum computing?

H(s) = sH _classical + (1 s)H _quantum

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

Ray, Chakrabarty,Chakrabarty (PRB 1989), Kadowaki, Nishimory (PRE 1998),...

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Quantum Annealing & Quantum Computing

D-wave “quantum annealer”; 512 flux q-bits

- 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....

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Spin Glasses

J₂₃=-1 J₃₄=+1

J₄₁=-1

J₁₂=-1

J₂₃=-1 J₃₄=+1

J₄₁=-1

J₁₂=-1

Figure 1: A frustrated 4-spin Ising system with three ferromagnetic (solid lines) and one antiferromagnetic (dashed line) interaction. Solid and open circles correspond to up and down spins, respectively. The configurations shown, and the ones obtained from them by flipping all spins, are the ones with the lowest energy; E = 2. One of the interactions (bonds) is always “unsatisfied”.

Spin glasses

A spin glass is a spin system in which the interactions are random and frustrated. Frustration refers to the inability of the interacting spins to minimize simultaneously the energy of all bonds (interacting pairs). An example is shown in Fig. 1. Here three out of the four interactions are ferromagnetic, J₁₂ = J₂₃ = J₄₁ = 1, whereas one is antiferromagnetic, J₃₄ = +1. In such a 4-spin systems with all interactions equal, all J_ij = 1 or all +1, the lowest-energy configurations are the ones minimizing all the bond energies, _{i j} = 1, independently of the sign of the interaction.

There are two such configurations. In contrast, in the example shown in the figure, the energy is

E( ) = _{1 2} _{2 3} + _{3 4} _{4 1}, (2)

and in the states with minimum energy, E = 2, one of the bonds must be in the high-energy state (an ”unsatisfied” interaction). There are four lowest-energy configurations; the ones shown plus the two obtained by flipping all their spins.

In an Ising spin glass with a large number of spins the number of lowest-energy configurations (ground states) grows exponentially with increasing number of spins. It is in general very difficult to find those configurations. At finite temperature, a spin glass model may exhibit a glass transition, below which in practice all the configurations cannot be sampled in a Monte Carlo simulation utilizing flips of individual spins. The system “gets stuck” around a local energy minimum from which it cannot escape within reasonable simulation times. There are also spin glasses in nature, and they also exhibit glass transitions. The behavior is similar to that of amorphous materials such as normal glass; thus the name “spin glass”.

Spin glass models are often studied using simulated annealing methods. One interesting aspect is to study the glass transition. Normally the transition temperature T_g is not known, at least not to very high precision, and it is then useful to start the simulation at high temperature and slowly cool it. Above the glass transition such a simulation will be able to explore the full configuration space of the system—it is said to be ergodic. However, as the transition temperature is approached, the colling rate has to be decreased exponentially fast in order for the simulation to be ergodic. For a very large system one can in practice not achieve ergodic sampling below some temperature close to the glass transition. For ergodic sampling of a finite system at T < T_g, exponentially longer simulation times are required with increasing system size N . In practice, the glass transition can be seen in results obtained in several di↵erent annealing runs: For T > T_g all simulations will give similar results for measured quantities, whereas for T < T_g di↵erent results will be obtained in

2 J₂₃=-1

J₃₄=+1

J₄₁=-1

J₁₂=-1

J₂₃=-1 J₃₄=+1

J₄₁=-1

J₁₂=-1

Figure 1: A frustrated 4-spin Ising system with three ferromagnetic (solid lines) and one antiferromagnetic (dashed line) interaction. Solid and open circles correspond to up and down spins, respectively. The configurations shown, and the ones obtained from them by flipping all spins, are the ones with the lowest energy; E = 2. One of the interactions (bonds) is always “unsatisfied”.

Spin glasses

A spin glass is a spin system in which the interactions are random and frustrated. Frustration refers to the inability of the interacting spins to minimize simultaneously the energy of all bonds (interacting pairs). An example is shown in Fig. 1. Here three out of the four interactions are ferromagnetic, J₁₂ = J₂₃ = J₄₁ = 1, whereas one is antiferromagnetic, J₃₄ = +1. In such a 4-spin systems with all interactions equal, all J_ij = 1 or all +1, the lowest-energy configurations are the ones minimizing all the bond energies, _{i j} = 1, independently of the sign of the interaction.

There are two such configurations. In contrast, in the example shown in the figure, the energy is

E( ) = _{1 2} _{2 3} + _{3 4} _{4 1}, (2)

and in the states with minimum energy, E = 2, one of the bonds must be in the high-energy state (an ”unsatisfied” interaction). There are four lowest-energy configurations; the ones shown plus the two obtained by flipping all their spins.

In an Ising spin glass with a large number of spins the number of lowest-energy configurations (ground states) grows exponentially with increasing number of spins. It is in general very difficult to find those configurations. At finite temperature, a spin glass model may exhibit a glass transition, below which in practice all the configurations cannot be sampled in a Monte Carlo simulation utilizing flips of individual spins. The system “gets stuck” around a local energy minimum from which it cannot escape within reasonable simulation times. There are also spin glasses in nature, and they also exhibit glass transitions. The behavior is similar to that of amorphous materials such as normal glass; thus the name “spin glass”.

Spin glass models are often studied using simulated annealing methods. One interesting aspect is to study the glass transition. Normally the transition temperature T_g is not known, at least not to very high precision, and it is then useful to start the simulation at high temperature and slowly cool it. Above the glass transition such a simulation will be able to explore the full configuration space of the system—it is said to be ergodic. However, as the transition temperature is approached, the colling rate has to be decreased exponentially fast in order for the simulation to be ergodic. For a very large system one can in practice not achieve ergodic sampling below some temperature close to the glass transition. For ergodic sampling of a finite system at T < T_g, exponentially longer simulation times are required with increasing system size N . In practice, the glass transition can be seen in results obtained in several di↵erent annealing runs: For T > T_g all simulations will give similar results for measured quantities, whereas for T < T_g di↵erent results will be obtained in

2

Hard to find ground states if the interactions are highly frustrated (spin glass phase)

- many states with same or almost same energy Many (almost all) optimization problems can be mapped onto some general model

- hard problems correspond to spin glass physics Quantum fluctuations (quantum spin glasses)

- add transversal field Ising (H → H + H quantum )

H _quantum = h

X N i=1

i x = h

X N i=1

( _i ⁺ + _i )

Ising models with frustrated interactions H =

X N i=1

X N j=1

J _ij _i ^z _j ^z , _i ^z 2 { 1, +1}

The D-wave machine is based on this model on a special lattice

Nature of ground states of H depends on h and {J ij }

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Quantum Phase Transition

There must be a quantum phase transition in the system

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=s c

- trivial x-oriented ferromagnet at s=0 (→→→)

- z-oriented (↑↑↑or ↓↓↓, symmetry broken) at s=1 - s c =1/2 (exact solution, mapping to free fermions)

Simple example: 1D transverse-field Ising ferromagnet (N ! 1)

h = s

X N

i=1

i z z

i+1 (1 s)

X N

i=1

i x

Let’s look at a simpler problem first...

H(s) = sH _classical + (1 s)H _quantum

Have to pass through s c and beyond adiabatically - how long does it take? s = s(t) = vt, v = 1/t _max

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Landau-Zener Problem

Single spin in magnetic field, with mixing term

H = h ^z ✏ ^x = h ^z ✏( ⁺ + )

-1 -0.5 0 0.5 1

h

-1.0 -0.5 0.0 0.5 1.0

E l

"

#

Eigen energies are

E = ± p

h ² + ✏ ²

Time-evolution:

h(t) = h ₀ + vt

To stay adiabatic when crossing h=0, the velocity must be

v < ² (time > ² )

Suggests the smallest gap is important in general

- but states above the gap play role in many-body system Smallest gap: Δ=2ε

What can we expect at a quantum phase transition?

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Dynamic Critical Exponent and Gap

Dynamic 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

L ^z = 1

N ^z/d , (N = L ^d )

At a continuous transition (classical or quantum):

- large (divergent) correlation length

⇠ _r ⇠ | | ^⌫ , ⇠ _t ⇠ ⇠ _r ^z ⇠ | | ^⌫z δ = distance from critical point (in T or other param)

Classical (thermal) phase transition

- Fluctuations regulated by temperature T>0 Quantum (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

order parameter

T [g]

T_c [g_c] (a)

order parameter

T [g] T_c [g_c]

(b)

FIGURE 3. Temperature (T ) or coupling (g) dependence of the order parameter (e.g., the magnetization of a ferromagnet) at a continuous (a) and a first-order (b) phase transition. A classical, thermal transition occurs at some temperature T = T_c, whereas a quantum phase transition occurs at some g = g_c 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 the 1980s. Haldane conjectured [55], based on a field-theory approach, that the Heisenberg chain 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 the S = 1/2 chain has a gapless excitation spectrum (related to the power-law decaying spin correlations). Haldane suggested the possibility of the S = 1 chain instead having a ground state with exponentially decaying correlations and a gap to all excitations; a kind of spin liquid state [26]. This was counter to the expectation (based on, e.g., spin wave theory) that increasing S should increase the tendency to ordering. Haldane’s conjecture stimulated intense research activities, theoretical as well as experimental, on the S = 1 Heisenberg chain and 1D systems more broadly. There is now completely conclusive evidence 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] (and also larger S [66]) compounds which show the predicted differences in the excitation spectrum. A rather complete and compelling theory of spin-S Heisenberg chains has emerged (and includes also the VBS transitions for half-odd integer S), but even to this date various aspects of their unusual properties are still being worked out [67]. There are also many other variants of spin chains, which are also attracting a lot of theoretical and experimental 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 = 1/2 Heisenberg chain, as well as the extended variant with frustrated interactions (and also including long-range interactions). In Sec. 5 we will investigate longer chains using the SSE QMC method. We will also study ladder-systems consisting of several coupled chains [9], which, for an even number of chains, have properties similar to the Haldane state (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 can occur in a system at T = 0 as some parameter in the hamiltonian is varied (which

146

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In both cases phase transitions can be

- first-order (discontinuous): finite correlation length ξ as g→gc org→gc

- continuous: correlation length diverges, ξ~|g-gc|^-ν or ξ~|T-Tc|^-ν

3

Classical (thermal) phase transition

- Fluctuations regulated by temperature T>0 Quantum (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

order parameter

T [g]

T_c [g_c] (a)

order parameter

T [g]

T_c [g_c] (b)

FIGURE 3. Temperature (T ) or coupling (g) dependence of the order parameter (e.g., the magnetization of a ferromagnet) at a continuous (a) and a first-order (b) phase transition. A classical, thermal transition occurs at some temperature T = T_c, whereas a quantum phase transition occurs at some g = g_c 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 the 1980s. Haldane conjectured [55], based on a field-theory approach, that the Heisenberg chain 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 the S = 1/2 chain has a gapless excitation spectrum (related to the power-law decaying spin correlations). Haldane suggested the possibility of the S = 1 chain instead having a ground state with exponentially decaying correlations and a gap to all excitations; a kind of spin liquid state [26]. This was counter to the expectation (based on, e.g., spin wave theory) that increasing S should increase the tendency to ordering. Haldane’s conjecture stimulated intense research activities, theoretical as well as experimental, on the S = 1 Heisenberg chain and 1D systems more broadly. There is now completely conclusive evidence 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] (and also larger S [66]) compounds which show the predicted differences in the excitation spectrum. A rather complete and compelling theory of spin-S Heisenberg chains has emerged (and includes also the VBS transitions for half-odd integer S), but even to this date various aspects of their unusual properties are still being worked out [67]. There are also many other variants of spin chains, which are also attracting a lot of theoretical and experimental 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 = 1/2 Heisenberg chain, as well as the extended variant with frustrated interactions (and also including long-range interactions). In Sec. 5 we will investigate longer chains using the SSE QMC method. We will also study ladder-systems consisting of several coupled chains [9], which, for an even number of chains, have properties similar to the Haldane state (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 can occur 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 org→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)

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Kibble-Zurek Velocity and Scaling

The adiabatic criterion for passing through a continuous phase transition involves more than z

Kibble 1978

- defects in early universe Zurek 1981

- classical phase transitions Polkovnikov 2005

- quantum phase transitions

Same criterion for classical

and quantum phase transitions - adiabatic (quantum)

- quasi-static (classical)

Generalized finite-size scaling hypothesis

A( , v, L) = L ^/⌫ g( L ^1/⌫ , vL ^z+1/⌫ )

A( , v, N ) = N ^/⌫

⁰

g( N ^1/⌫

⁰

, vN ^z

⁰

^+1/⌫

⁰

), ⌫ ⁰ = d⌫, z = z/d

Will use for spin glasses of interest in quantum computing Apply to well-understood clean system first...

Must have v < v KZ , with

v _KZ ⇠ L ^(z+1/⌫)

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Fast and Slow Classical Ising Dynamics

L = 12 L = 24 L = 48 L = 64 L = 96 L = 128 L = 192 L = 256 L = 500 L = 1024

polynomial fit power-law fit

10⁴ 10⁵ 10⁶ 10^-3

10^-2

L = 128

Velocity Scaling, 2D Ising Model

Repeat process many times, average data for T=T c

Used known 2D Ising exponents β=1/8, ν=1

Result: z ≈2.17 consistent with values obtained in other ways

Adjusted z for optimal scaling collapse

Liu, Polkovnikov, Sandvik, PRB 2014

Can we do something like this for quantum models?

hm ² ( = 0, v, L) i = L ^{2 /⌫} f (vL ^z+1/⌫ )

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Quantum Evolution in Imaginary Time

| (⌧)i = U(⌧, ⌧ ⁰ ) | (⌧ ⁰ ) i

Schrödinger dynamic at imaginary time t=-iτ

Dynamical exponent z same as in real time!

(DeGrandi, Polkovnikov, Sandvik, PRB2011)

• Can be implemented in quantum Monte Carlo

Simpler scheme: evolve with just a H-product

(Liu, Polkovnikov, Sandvik, PRB2013)

| (⌧)i =

X 1 n=0

Z ⌧

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h (0)|H(s ¹ ) · · · H(s ⁷ ) |H(s ⁷ ) · · · H(s ¹ ) | (0)i

QMC Algorithm Illustration

H ₁ (i) = (1 s)( _i ⁺ + _i ) H ₂ (i, j) = s( _i ^z _j ^z + 1)

Transverse-field Ising model: 2 types of operators:

1 2 3

4 5 6

7 7 6

5 4 3

2 1 3 4 5 6 7 7 6 5 4 3 2 1 2

1 ¹ ² ³ ⁴ ⁵ ⁶ ⁷ ⁷ ⁶ ⁵ ⁴ ³ ² ¹

1 2 3 4 5 6 7 7 6 5 4 3 2 1

Represented as “vertices”

Similar to ground-state projector QMC

How to define (imaginary) time in this method?

1 2

3 4

5 6

7 7

6 5

4 3

2 1

MC sampling of networks of vertices

N = 4 M = 7

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Time and velocity Definitions

The parameter in H changes as

s _i = i _s , _s = s _M M

Def reproduces v-dependence in imag-time Schrödinger dynamics to order v (enough for scaling)

Time unit is ∝ 1/N, velocity is

v / N ^s

To this order we can use

“asymmetric” expectation values

All s in one simulation!

hAi ^k = h (0)|

Y 1

i=M

H(s _i )

Y M

i=k

H(s _i )A

Y k

i=1

H(s _i ) | (0)i

hAi

^k

_i

) | (0)i

Collect data, do scaling analysis...

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0.15 0.20 0.25 0.30

S

0.0 0.2 0.4 0.6 0.8 1.0

U

^{L = 12}

L = 16 L = 32 L = 48 L = 56 L = 60

0.24 0.25

0.8 0.9

2D Transverse-Ising, Scaling Example

A( , v, L) = L ^/⌫ g( L ^1/⌫ , vL ^z+1/⌫ )

If z, ν known, s c not: use vL ^z+1/⌫ = constant

for 1-parameter scaling

Example: Binder cumulant

Should have step from U=0 to U=1 at s c

- crossing points for finite system size

Do similar studies for quantum spin glasses

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Note on QMC Simulation Dynamics

Recent work claimed the D-wave machine shows behavior similar to

“simulated quantum annealing”

[S. Boixio, M. Troyer et al., Nat. Phys. 2014]

H(s) evolved in simulation time Is this the same as Hamiltonian quantum dynamics?

NO! Only accesses the dynamics of the QMC method

3 The 3-regular graphs have d = 1 and we will use N for

the size. To convert to unprimed exponents the upper critical dimension should then be used; d = du.

The existence of a characteristic velocity suggests a generalized finite-size scaling form for singular quantities at the critical point. For quantities calculated at the final time t_f when s = s_c, and when v / v^KZ or lower, the order parameter takes the form

hq²i ⇠ N ^{2 /⌫}⁰f (vN^z⁰^r+1/⌫⁰), (8) and we can extract , ⌫⁰, z⁰ by analyzing results for two di↵erent values of the quench parameter r [31].

Hamiltonian versus simulation dynamics.—Before pre- senting QAQMC results for the 3-regular graphs, let us comment on stochastic simulation-time dynamics and the method of changing H as a function of the simulation time. This approach is normally considered with thermal QMC simulations [15, 37] but can also be implemented in the QAQMC. To illustrate this we use the ferromagnetic d = 1 TFIM. We use a relatively large number of operators in the operator sequence in (4), m = 4N² (sufficient for ground-state convergence at all s in equilibrium), and keep s the same for all operators. The simulation starts at s = 0 and s is changed linearly at velocity v until sc = 1/2 is reached. At this stage the magnetization is calculated. The procedure is repeated many times to obtain hm²zi. The velocity is defined using a time unit of a sweep of either local updates (a Metropolis procedure where small segments of spins are flipped) or cluster updates (a generalization of the Swendsen-Wang, SW, cluster updates [38, 39]) throughout the system. Using the scaling ansatz (8) for hm²zi, we extract the dynamic exponent characterizing the approach to the critical point with the local and cluster updates, and compare with the exponent computed with QAQMC with s is evolv- ing within the operator string in Eq. (4). In the latter case there is no dependence on the type of MC updates (but cluster updates give results with smaller statistical errors for a given simulation time) and we should detect Hamiltonian dynamics with z = 1.

The scaling analysis for all the cases is presented in Fig. 1. The static exponents are known (those of the d = 2 classical Ising model), = 1/8 and ⌫ = 1, and we use these to produce scaling plots according to the form (8). We suspect that the simulation-time dynamics should be the same as in the classical d = 2 Ising model with local and SW updates, and therefore test scaling with z = 2.17 and z = 0.30, respectively (as recently computed using KZ scaling in Ref. 31). In all cases the data collapse is very good for sufficiently large systems and low velocities. The straight lines in the log-log plots have slopes given by the exponent

x = d 2 /⌫

zr + 1/⌫ = 1 2 /⌫⁰

z⁰r + 1/⌫⁰, (9)

10^-2 10⁰ 10² 10⁴

10^-2 10^-1 10⁰

<m2 > N 2β/ν

8 32 128 512 2048

10^-2 10⁰ 10² 10⁴ 10⁶ 10⁸ 10¹⁰

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N=8

3-regular graphs

Ising spin glass with coordination-number 3 - N spins, randomly connected to each other - all antiferromagnetic couplings

- frustration because of closed odd-length loops

• ^s ^c ≈ 0.37 from quantum cavity approximation

• QMC consistent with this s c , power-law gaps at s c

The quantum model was studied by

Farhi, Gosset, Hen, Sandvik, Shor, Young, Zamponi, PRA 2012

More detailed studies with quantum annealing...

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Spin-Glass order Parameter

Spin glasses are massively-degenerate - many “frozen” states

- replica symmetry breaking (going into one state)

Edwards-Anderson order parameter q = 1

N

X N i=1

i z (1) _i ^z (2)

(1) and (2) are from independent simulations (replicas) - with same random interactions

- |q| large if the two replicas are in similar states

<q ² > > 0 for N →∞ in spin-glass phase (disorder average) Cannot use a standard order parameter such as <m ² >

- nor any Fourier mode

- since no periodic ordering pattern

Analyze <q ² > using QMC and velocity scaling

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Extracting Quantum-glass transition

Using Binder cumulant

U (s, v, N ) = U [(s s _c )N ^1/⌫

⁰

, vN ^z

⁰

^+1/⌫

⁰

]

But now we don’t know the exponents. Use

v / N ^↵ , ↵ > z ⁰ + 1/⌫ ⁰ - do several α

- check for consistency

Consistent with previous work, but smaller errors

Next, critical exponents...

s c = 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, extracted using the curves shown in the inset. The results were obtained in quenches with v ⇠ N

^↵

for ↵ = 17/12. The curve in the main panel is a power-law fit for extrapolating s

c

.

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

We find s

_c

using r = 1 QAQMC with v / N

^↵

, where

↵ exceeds the KZ exponent z

⁰

+ 1/⌫

⁰

(which is unknown but later computable for a posteriori verification). Then hq

²

i ⇠ N

^{2 /⌫}⁰

at s

_c

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

_c

, we use a curve crossing analysis of the Binder cumulant, U = (3 hq

⁴

i/hq

2

> N

2β/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, giving the exponents listed in the text. The line has slope given in Eq. (9) and the points above it deviate due to high-velocity cross-overs [31] not captured by Eq. (8). For N ! 1 the linear behavior should extend to infinity.

_c

faster than SA, though the critical clus- ter is still less dense). QA can be made faster than SA by following a protocol (5) with sufficiently large r, but this may not be practical when s

_c

is not known and the goal is anyway to proceed beyond this point. While our results do not contain any quantitative information on the process continuing from s

_c

to s = 1, it is certainly discouraging that the initial stage of QA is inefficient.

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

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

[1] S. Kirkpatrick, C. D. Gelatt Jr, M. P. Vecchi, Science 220 671 (1983).

[2] V. Cern´ y, 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

(23)

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, extracted using the curves shown in the inset. The results were obtained in quenches with v ⇠ N

^↵

for ↵ = 17/12. The curve in the main panel is a power-law fit for extrapolating s

c

.

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

We find s _c using r = 1 QAQMC with v / N ^↵ , where

↵ exceeds the KZ exponent z ⁰ + 1/⌫ ⁰ (which is unknown but later computable for a posteriori verification). Then hq ² i ⇠ N ^{2 /⌫}

⁰

at s _c because f (x) in Eq. (8) approaches a constant when x ! 0. As illustrated in Fig. 2, quench- ing past the estimated s _c , we use a curve crossing analysis of the Binder cumulant, U = (3 hq ⁴ i/hq ² i ² )/2, and ob- tain s _c = 0.3565(12). This value agrees well with the previous results but has smaller uncertainty.

Performing additional quenches to s _c using protocols with both r = 1 and r = 2/3 in Eq. (5) we extract all the critical exponents. An example of scaling collapse for r = 1 is shown in Fig. 3. Here all exponents are treated as adjustable parameters for obtaining optimal data collapse. The vertical and horizontal scalings give the ratio /⌫ ⁰ and the KZ exponent z ⁰ +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 the exponents = 0.54(1), ⌫ ⁰ = 1.26(1), and z ⁰ = 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 models in a transverse field; = 1, ⌫ ⁰ = 2 and z ⁰ = 1/4 (d _u = 8) [41]. One might have expected the same mean-field ex- ponents for these systems, as in the classical case. A QMC calculation for the fully-connected model [44] was not in complete agreement with the Landau values. It was argued that z = 4, which, with d _u = 8, agrees with our z ⁰ ⇡ 1/2 for the 3-regular graphs. However, was close to 1 and ⌫ = 1/4 (⌫ ⁰ = 2) was argued. It would be interesting to study n-regular graphs and follow the exponents from n = 3 to large n.

Implications for quantum computing.—The critical ex-

2

> N

2β/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, giving the exponents listed in the text. The line has slope given in Eq. (9) and the points above it deviate due to high-velocity cross-overs [31] not captured by Eq. (8). For N ! 1 the linear behavior should extend to infinity.

ponents contain information relevant to QA quantum computing. In the classical case the KZ exponent is z ⁰ + 1/⌫ ⁰ = 1, while in the quantum system z ⁰ + 1/⌫ ⁰ ⇡ 1.31. Thus, by Eq. (7) the adiabatic annealing time grows faster with N in QA. Furthermore, since the critical or- der parameter scales as N ^{2 /⌫}

⁰

, the critical cluster is less dense with QA, i.e., it is further from the state sought when s ! 1 (the solution of the optimization problem).

Thus, in both these respects QA performs worse than SA in passing through the critical point (while QA on the fully-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 SA by following a protocol (5) with sufficiently large r, but this may not be practical when s _c is not known and the goal is anyway to proceed beyond this point. While our results do not contain any quantitative information on the process continuing from s _c to s = 1, it is certainly discouraging that the initial stage of QA is inefficient.

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

Quantum and classical annealing in spin glasses and quantum computingC.-W. Liu, A. Polkovnikov, A. W. Sandvik, arXiv:1409.7192