Quantum Annealing: Challenges & Prospects · Prototypical Hamming Weight Problem 5 10 15 5 10 15 Hamming weight Classical Energy The “spike” problem(1) (1) E. Farhi, J. Goldstone,

Quantum Annealing: Challenges & Prospects

Tameem Albash Information Sciences Institute @USC

29 November 2018

Information Sciences Institute

From Bits to Qubits

Goal is to solve an inherently classical problem with a quantum algorithm

�z |0i = + |0i �z |1i = � |1i

Classical variables will be binary (bits)

x = {0, 1}

" ⌘ 0 # ⌘ 1

A natural binary valued system is the

Ising spin

Quantum algorithm promotes bits to qubits

Computational basis: eigenstates of the Pauli-Z operator sz�z

Pick a basis for the qubits to associate

classical bit values to

| i = a|0i+ b|1i

Our Objective Today

Find the ground state of Ising Hamiltonians

Spins live on the vertices of the connectivity

graph

Weighted edges of the graph

correspond to Ising interaction

Total of 2N configurations for an N spin problem Computationally prohibitive to search for the ground state

2N N

Local fields hi, and Ising couplings Jij specify the problemhi Jij

HIsing =X

i

hi�zi +

X

hi,ji

Jij�zi �

zj

Or any Hamiltonian defined entirely in terms of sz (diagonal in the computational basis)

�z

Ising is Sufficient

The Ising problem on non-planar graphs

belongs to the complexity class

NP-hard(1)

Classic system for studying

magnetization, phase transitions,

glassiness

Physics Computer science

A very good algorithm for solving the Ising model is likely to be a good

algorithm for solving problems in NP

Algorithms to solve the Ising model is an important line of research

(1) F. Barahona, J. Phys. A: Math. Gen. 15 3241 (1982)

Why Solving NP Problems is ImportantNP problems are ubiquitous with with range of applications

These problems are typically very hard to solve, requiring a time that grows exponentially with the problem size

NP Problem ApplicationTraveling salesman Logistics, vehicle routing

Minimum Steiner tree Circuit layout, network design

Graph coloring Scheduling, register allocation

MAX-CLIQUE Social networks, bioinformatics

QUBO Machine learning, software V&V

0-1 Integer Linear Programming Natural language processing

Sub-graph isomorphism Chem-informatics, drug discovery

Job shop scheduling Manufacturing

MAX-2SAT Artificial intelligence

Optimum solution to optimization problem

Finding ground state to Ising problem

Can Quantum Computing Help?

Quantum annealing (QA)(1,2) is an adiabatic-paradigm quantum algorithm

to solve for the ground state of Ising problems

Can quantum computing help solve a class of Ising problems more efficiently?

No provable or demonstrated speedups for Ising-like Hamiltonians with

bounded locality

Provable speedups for oracular problems(3-5)

(3) J. Roland and N. Cerf, Phys. Rev. A 65, 042308 (2002) (4) I. Hen, Europhysics Letters 105 (5), 50005 (2014) (5) R. D. Somma, et al. , Phys. Rev. Lett. 109 (5), 050501 (2012)

Hamiltonians involve N-body operatorsN

No proof that a speedup is impossible

Good news Bad news

(1) T. Kadowaki, and H. Nishimori , Phys. Rev. E 58 (5), 5355 (1998) (2) E. Farhi,et al., Science 292 (5516), 472 (2001)

Outline

1. How should QA work? Closed-system QA

Open-system QA and

the D-Wave processors

2. Why has QA not worked?

3. How do we address these challenges? Beyond standard QA

What is Standard Quantum Annealing?

A continuous interpolation between two Hamiltonians

H0 Easily prepared

ground state

H1 Ground state

encodes solution

H0 H1

H0 = �X

i

�xi H1 = HIsing

Procedure for Quantum Annealing (QA):

Prepare system in the ground state

of H (t = 0) H(t = 0)

(1)Measure the state

in the sz basis�z

(3)

H(t) =

✓1� t

tf

◆H0 +

t

tfH1

t 2 [0, tf ]

Evolve according to H(t) for a

time tfH(t)

(2)

tf

[H0, H1] 6= 0

Success if measurement outcome is the ground state of H1H1

A Guarantee for QA to Find the SolutionAdiabatic theorem

if the interpolation is sufficiently slow then with high probability the final state of the system is the

ground state of H1H1

The efficiency of the algorithm is then determined by how tf must scale with system size Ntf N

0.495 0.500 0.505 0.510

0.495

0.500

0.505

0.510

�min

tf � 1

�3min

Adiabatic condition(1)

(1) S. Jansen,et al., J. Math. Phys. 48 (10), 102111 (2007)

t/tf

Spec

trum

of H

(t)

0.2 0.4 0.6 0.8 1.0

0.2

0.4

0.6

0.8

1.0 "0"1

H(t)

An Alternative Picture

Ground state of H0

Ground state of H1

H0 =Transverse field

H1= Ising

H0

H1

H0

H1

Quantum annealing tries to follow the ground state from the disordered phase to the ordered phase

�min

Transverse field strength g

Disordered phase

Ordered phase

Phase diagram of a transverse field Ising model

gcg

0

Transverse field

Ising model

An alternative pictureA Classical Analogue

Quench quantum fluctuations

Ground state of H0

Ground state of H1

Quench thermal fluctuations

High temperature state

Low temperature state

Simulated annealing (SA)(1)Quantum annealing (QA)

Quantum/classical

analogues

H0

H1

g

T

Disordered phase

Ordered phase

QA

SA

(1) S. Kirkpatrick et al., Science 220, 671–680 (1983) (2) T. Kadowaki, and H. Nishimori, Phys. Rev. E 58 (5), 5355 (1998)

(2)

High temperature state

Quench thermal fluctuations

Low temperature state

E

Configurations

e��E

�E

Simulated Annealing

Thermally hopping over a barrier at low temperatures is exponentially suppressed in the barrier height DE�E

T = ��1

Why Might QA Have an Advantage over SA

Efficiency to traverse the barrier is related to how the barrier width and height scale with problem size

Small energy gaps are associated with double-well energy barriers through which the system must tunnel

t

V

Semiclassical potential evolves with time State tries to stay in its local minimum

(1) S. Muthukrishnan, TA, and D. A. Lidar, Phys. Rev. X 6, 031010, (2016).

The semiclassical potential may have a double-well, even if the classical potential does not(1)

Illustrative Examples

Hamming Weight ProblemsA classical Hamiltonian H1 where the energy of a classical state

only depends on the Hamming weight of that state (equivalent to counting the number of down-pointing spins)

H1

Classical state Hamming weight

...

0

1}|"""i

|#""i|"#"i|""#i

|###i 3

E0

Energy

E1

E3

......

QA Hamiltonian with a transverse field is invariant under qubit permutations

Prototypical Hamming Weight Problem

5 10 15

5

10

15

Hamming weight

Cla

ssic

al E

nerg

y

The “spike” problem(1)

(1) E. Farhi, J. Goldstone, and S. Gutmann, arXiv:quant- ph/0201031 (2002).

(shown here for N = 16)N = 16

2

N

N

4

Barrier width constant with

problem size NN

Barrier height grows with the

problem size NN

(single-spin) SA requires exponential time in the system size to find the ground state

The “Spike” Problem

5 10 15

5

10

15

0.6 0.8 1.0 1.2 1.4 1.62.80

2.85

2.90

2.95

3.00

3.05

3.10

Exponential speedup over SA(1)

(1) E. Farhi, J. Goldstone, and S. Gutmann, arXiv:quant- ph/0201031 (2002).

Classical landscape (landscape at s = 1)s = 1

Quantum semi-classical landscape at s = 0.359s = 0.359

Double-well barrier height grows sub-linearly with N and width shrinks with increasing N

NN

Quantum minimum gap scales polynomially with NN

original global

minimum

new global

minimum

Recap: Closed System Quantum Annealing

Quantum annealing

Ground state of H1 encodes the

solution

Prepare system in the ground state

of H0

Adiabatic evolutionguarantees success

The changing energy landscape can be very different from the classical energy landscape

Can lead to quantum advantage over classical algorithms like simulated (thermal) annealing

Efficiency controlled by the scaling of the minimum gap

H1

H0

So What’s the Problem

Why not?

• No symmetries to help facilitate the analysis • Simulations are not possible at large sizes • Very good classical algorithms

Only way to test for speedup may be to build a device and see

(?)

The intuition gained has not yet led to the demonstration of a speedup

for physically realizable problems

Hamming weight Hamiltonians examples require N-body operators.N

The Realistic SettingA physical device naturally couples the system to additional degrees of freedom (the “bath”)

Conflict between running adiabatically (long time) and minimizing the interaction time with the bath

(non-unitary) population

transfer

decay of phase relationship between

specific states

Decoherence

Closed Open system quantum annealing

Unitary and dissipative dynamics

The most innocuous model of decoherence for QA (1) weak coupling and Markovian

Weakly Coupled Open Quantum Systems

Dissipative dynamics pushes system

towards the thermal state(2)

(1) TA and D. A. Lidar, Phys. Rev. A 91, 062320 (2015) (2) TA, S. Boixo, D. A. Lidar, P. Zanardi, New J. Phys. 14, 123016 (2012)

Key feature

limtf!1

⇢(t) ! ⇢G(t)

⇢G(t) =exp (��H(t))

Tr [exp (��H(t))]

(3)

(3) L. C. Venuti, TA, D. A. Lidar, and P. Zanardi Phys. Rev. A 93, 032118 (2016)

Minimum energy gap decreases with increasing problem size, so adiabatic theorem requires us to use

longer annealing times

Affect on QA

More time spent interacting with the environment means system

gets closer and closer to the thermal state

For a thermalizing quantum annealer

Thermal Quantum Annealer

Success depends on how much weight the thermal state has on the ground state

N = 1152 � = 1.47

Distribution becomes

more gaussian-like, with a mean that moves

away from the origin

hHIsingi � EGS

0 1000 2000 3000 40000

5

10

15

20

25

�(HIsing)hHIsingi � EGS

(1) TA, V. Martin-Mayor, I. Hen, Phys. Rev. Lett. 119, 110502 (2017).

At a fixed temperature, it becomes exponentially unlikely to sample the ground state with increasing problem size(1)

Analog Devices leads to Analog Errors

If the perturbation is sufficiently large, the ground state H_isin is different from the

ground state of H_isin

HIsing =X

i

hi�zi +

X

hi,ji

Jij�zi �

zj

Analog control of the Hamiltonian leads to misspecification of the final Hamiltonian

Even a perfect adiabatic evolution will result in the wrong answer!

Desired

Implemented

H0Ising =

X

i

(hi + �hi)�zi +

X

hi,ji

(Jij + �Jij)�zi �

zj

H0Ising

HIsing

Implementation Errors

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Define pS to be the probability that the ground state of HIsing matches one of the

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H0Ising

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Use instances at different sizes N with known ground states, and generate many noisy realizations of the same instances

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How does a fixed precision affect the success of QA with increasing

problem size?

Pick a noise model

Scaling with Implementation Errors

(1) TA, V. Martin-Mayor, I. Hen, arXiv:1806.03744.

0 5 10 15 20 25 3010

-3

10-2

10-1

100

Instances defined with a Chimera connectivity with range 3 and only two ground states(1)

For a fixed noise strength si, the probability of

the ground state not changing

decreases exponentially with problem

size N

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Different problem classes exhibit different dependence on si�<latexit sha1_base64="KSJKnHgEAaQvaBBEAVpBVLN/Gh8=">AAAB7XicbZDLSgMxFIZPvNZ6q7p0EyyCqzIjgi4LblxWsBdoh5JJM21sLkOSEcrQd3DjQhG3vo8738a0nYW2/hD4+M855Jw/TgW3Lgi+0dr6xubWdmmnvLu3f3BYOTpuWZ0ZyppUC206MbFMcMWajjvBOqlhRMaCtePx7azefmLGcq0e3CRlkSRDxRNOifNWq2f5UJJ+pRrUgrnwKoQFVKFQo1/56g00zSRTjgpibTcMUhflxDhOBZuWe5llKaFjMmRdj4pIZqN8vu0Un3tngBNt/FMOz93fEzmR1k5k7DslcSO7XJuZ/9W6mUtuopyrNHNM0cVHSSaw03h2Oh5ww6gTEw+EGu53xXREDKHOB1T2IYTLJ69C67IWer6/qtbrRRwlOIUzuIAQrqEOd9CAJlB4hGd4hTek0Qt6Rx+L1jVUzJzAH6HPH5vHjyE=</latexit><latexit sha1_base64="KSJKnHgEAaQvaBBEAVpBVLN/Gh8=">AAAB7XicbZDLSgMxFIZPvNZ6q7p0EyyCqzIjgi4LblxWsBdoh5JJM21sLkOSEcrQd3DjQhG3vo8738a0nYW2/hD4+M855Jw/TgW3Lgi+0dr6xubWdmmnvLu3f3BYOTpuWZ0ZyppUC206MbFMcMWajjvBOqlhRMaCtePx7azefmLGcq0e3CRlkSRDxRNOifNWq2f5UJJ+pRrUgrnwKoQFVKFQo1/56g00zSRTjgpibTcMUhflxDhOBZuWe5llKaFjMmRdj4pIZqN8vu0Un3tngBNt/FMOz93fEzmR1k5k7DslcSO7XJuZ/9W6mUtuopyrNHNM0cVHSSaw03h2Oh5ww6gTEw+EGu53xXREDKHOB1T2IYTLJ69C67IWer6/qtbrRRwlOIUzuIAQrqEOd9CAJlB4hGd4hTek0Qt6Rx+L1jVUzJzAH6HPH5vHjyE=</latexit><latexit sha1_base64="KSJKnHgEAaQvaBBEAVpBVLN/Gh8=">AAAB7XicbZDLSgMxFIZPvNZ6q7p0EyyCqzIjgi4LblxWsBdoh5JJM21sLkOSEcrQd3DjQhG3vo8738a0nYW2/hD4+M855Jw/TgW3Lgi+0dr6xubWdmmnvLu3f3BYOTpuWZ0ZyppUC206MbFMcMWajjvBOqlhRMaCtePx7azefmLGcq0e3CRlkSRDxRNOifNWq2f5UJJ+pRrUgrnwKoQFVKFQo1/56g00zSRTjgpibTcMUhflxDhOBZuWe5llKaFjMmRdj4pIZqN8vu0Un3tngBNt/FMOz93fEzmR1k5k7DslcSO7XJuZ/9W6mUtuopyrNHNM0cVHSSaw03h2Oh5ww6gTEw+EGu53xXREDKHOB1T2IYTLJ69C67IWer6/qtbrRRwlOIUzuIAQrqEOd9CAJlB4hGd4hTek0Qt6Rx+L1jVUzJzAH6HPH5vHjyE=</latexit><latexit sha1_base64="KSJKnHgEAaQvaBBEAVpBVLN/Gh8=">AAAB7XicbZDLSgMxFIZPvNZ6q7p0EyyCqzIjgi4LblxWsBdoh5JJM21sLkOSEcrQd3DjQhG3vo8738a0nYW2/hD4+M855Jw/TgW3Lgi+0dr6xubWdmmnvLu3f3BYOTpuWZ0ZyppUC206MbFMcMWajjvBOqlhRMaCtePx7azefmLGcq0e3CRlkSRDxRNOifNWq2f5UJJ+pRrUgrnwKoQFVKFQo1/56g00zSRTjgpibTcMUhflxDhOBZuWe5llKaFjMmRdj4pIZqN8vu0Un3tngBNt/FMOz93fEzmR1k5k7DslcSO7XJuZ/9W6mUtuopyrNHNM0cVHSSaw03h2Oh5ww6gTEw+EGu53xXREDKHOB1T2IYTLJ69C67IWer6/qtbrRRwlOIUzuIAQrqEOd9CAJlB4hGd4hTek0Qt6Rx+L1jVUzJzAH6HPH5vHjyE=</latexit>

Recap: Open System Quantum Annealing

Even in the optimistic setting of decoherenceTemperature must be scaled down with increasing problem size for thermalizing

quantum annealer

Scalable quantum annealing has no hope without fault tolerant quantum error correction,

which remains on open theoretical question

�(worst case)S ⇠ 1

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Implementation errors must be scaled down with increasing problem size,

even for an otherwise perfect device

T (worst case)S ⇠ 1

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Commercially Available Quantum AnnealersD-Wave Systems has been selling (purported)

quantum annealing processors since 2011

2011: D-Wave 1 “Rainier”2013: D-Wave 2 “Vesuvius”2015: D-Wave 2X “Washington”2017: D-Wave 2000Q

128 qubits512 qubits

1152 qubits2048 qubits

20mK17mK13mK

12-15mK

How the D-Wave Processor Works

HIsing =X

i

hi�zi +

X

hi,ji

Jij�zi �

zj

End user programs the Ising local fields {hi } and Ising couplers { Jij}

Ising connectivity limited to the physical qubit connectivity of the device

Square grid of 8 qubit unit cells

Unit cell

{hi}{Jij}

Benchmarking D-Wave Processors

DW1, L = 4 (128 qubits)

DW2, L = 8 (512 qubits)

DW2X, L = 12 (1152 qubits)

DW2000Q, L = 16 (2048 qubits)

Rønnow et al. (2014) Hen et al. (2015) King et al. (2015)

King et al. (2015) Denchev et al. (2016)

Boixo et al. (2014) King et al. (2017) TA et al. (2018)

Mandrà et al. (2018)

Several benchmarking studies over several generations of D-Wave devices defined on L ⨉ L Chimera graphs

ranging from 128 to 2048 qubits

Benchmarking standards set in Rønnow et al. (2014)Computational cost measured in terms of time-to-solution (TTS),

required time to run algorithm to find the ground state at least once with a 0.99 probability

At each problem size, algorithm parameters are optimized to minimize the TTS (averaged over instances), <TTS>hTTSi⇤

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Has it demonstrated a speedup?

No demonstration of a scaling advantage over classical algorithms(1)

(1) TA and D. A. Lidar, Phys. Rev. X 8, 031016 (2018).

Classical algorithms can be very efficient at solving this class of Ising problems on

the D-Wave connectivity graph(1)

⇠ exp(0.268pN)

⇠ exp(0.352pN)

DW2KQ: D-Wave 2000Q device

SA: Simulated Annealing

PIQMC: Path-Integral Monte

Carlo Annealing (aka Simulated

quantum annealing)

DW2KQ

SA

PIMC

PT-ICM

⇠ exp(0.110pN)

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PT-ICM: Parallel tempering with isoenergetic

cluster moves

The device temperature is too high

Temperature slow down?

Consider the scaling of path-integral Monte Carlo annealing at two different temperatures using the D-

Wave annealing schedule

⇠ exp(0.129pN)

⇠ exp(0.254pN)

Performance improves dramatically at lower temperatures

D-Wave operating

temperature corresponds to

beta = 0.5. D-Wave scaling is

exp(exp(0.268pN)

� = 0.51

Prefactor versus Scaling

I have so far focused on the scaling with problem size

Even if we don’t get a scaling advantage, we may get a wall-clock time advantage

(prefactor advantage)(1,2) Requires accounting of all time costs

(initial state preparation, measurement time, etc.)

(1) S. V. Isakov et al. PRL 117, 180402 (2016). (2) Z. Jiang et al. PRA 95, 012322– (2017). (3) S. Mandrà and H. Katzgraber, Quant. Sci. Technol. 3, 04LT01 (2018)

Other possible metrics: Power consumption per quality of solution(3)

Still no clear evidence of this advantage yet on current devices

‘Standard’ Quantum Annealing

To date, no theoretical or experimental evidence for a

quantum speedup for Ising-type Hamiltonians

Discussion so far has been about using transverse field Hamiltonian to drive the anneal

H(s) = (1� s)

�X

i

�xi

!+ sHIsing

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Nothing stops us from going beyond the standard setup

Beyond Standard QA

Ferromagnetic transverse interaction

Hc = ↵

X

hi,ji

�xi �

xj

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↵ = �1<latexit sha1_base64="AUvQBhHG1+R5sJ/7ScL6IUlqUMA=">AAAB83icbZBNS8NAEIYn9avWr6pHL4tF8GJJRNCLUPDisYL9gCaUyXbTLt0ky+5GKKF/w4sHRbz6Z7z5b9y2OWjrCwsP78wws28oBdfGdb+d0tr6xuZWebuys7u3f1A9PGrrNFOUtWgqUtUNUTPBE9Yy3AjWlYphHArWCcd3s3rniSnN0+TRTCQLYhwmPOIUjbV8H4UcIbklFx7pV2tu3Z2LrIJXQA0KNfvVL3+Q0ixmiaECte55rjRBjspwKti04meaSaRjHLKexQRjpoN8fvOUnFlnQKJU2ZcYMnd/T+QYaz2JQ9sZoxnp5drM/K/Wy0x0E+Q8kZlhCV0sijJBTEpmAZABV4waMbGAVHF7K6EjVEiNjaliQ/CWv7wK7cu6Z/nhqtZoFHGU4QRO4Rw8uIYG3EMTWkBBwjO8wpuTOS/Ou/OxaC05xcwx/JHz+QP18JBN</latexit><latexit sha1_base64="AUvQBhHG1+R5sJ/7ScL6IUlqUMA=">AAAB83icbZBNS8NAEIYn9avWr6pHL4tF8GJJRNCLUPDisYL9gCaUyXbTLt0ky+5GKKF/w4sHRbz6Z7z5b9y2OWjrCwsP78wws28oBdfGdb+d0tr6xuZWebuys7u3f1A9PGrrNFOUtWgqUtUNUTPBE9Yy3AjWlYphHArWCcd3s3rniSnN0+TRTCQLYhwmPOIUjbV8H4UcIbklFx7pV2tu3Z2LrIJXQA0KNfvVL3+Q0ixmiaECte55rjRBjspwKti04meaSaRjHLKexQRjpoN8fvOUnFlnQKJU2ZcYMnd/T+QYaz2JQ9sZoxnp5drM/K/Wy0x0E+Q8kZlhCV0sijJBTEpmAZABV4waMbGAVHF7K6EjVEiNjaliQ/CWv7wK7cu6Z/nhqtZoFHGU4QRO4Rw8uIYG3EMTWkBBwjO8wpuTOS/Ou/OxaC05xcwx/JHz+QP18JBN</latexit><latexit sha1_base64="AUvQBhHG1+R5sJ/7ScL6IUlqUMA=">AAAB83icbZBNS8NAEIYn9avWr6pHL4tF8GJJRNCLUPDisYL9gCaUyXbTLt0ky+5GKKF/w4sHRbz6Z7z5b9y2OWjrCwsP78wws28oBdfGdb+d0tr6xuZWebuys7u3f1A9PGrrNFOUtWgqUtUNUTPBE9Yy3AjWlYphHArWCcd3s3rniSnN0+TRTCQLYhwmPOIUjbV8H4UcIbklFx7pV2tu3Z2LrIJXQA0KNfvVL3+Q0ixmiaECte55rjRBjspwKti04meaSaRjHLKexQRjpoN8fvOUnFlnQKJU2ZcYMnd/T+QYaz2JQ9sZoxnp5drM/K/Wy0x0E+Q8kZlhCV0sijJBTEpmAZABV4waMbGAVHF7K6EjVEiNjaliQ/CWv7wK7cu6Z/nhqtZoFHGU4QRO4Rw8uIYG3EMTWkBBwjO8wpuTOS/Ou/OxaC05xcwx/JHz+QP18JBN</latexit><latexit sha1_base64="AUvQBhHG1+R5sJ/7ScL6IUlqUMA=">AAAB83icbZBNS8NAEIYn9avWr6pHL4tF8GJJRNCLUPDisYL9gCaUyXbTLt0ky+5GKKF/w4sHRbz6Z7z5b9y2OWjrCwsP78wws28oBdfGdb+d0tr6xuZWebuys7u3f1A9PGrrNFOUtWgqUtUNUTPBE9Yy3AjWlYphHArWCcd3s3rniSnN0+TRTCQLYhwmPOIUjbV8H4UcIbklFx7pV2tu3Z2LrIJXQA0KNfvVL3+Q0ixmiaECte55rjRBjspwKti04meaSaRjHLKexQRjpoN8fvOUnFlnQKJU2ZcYMnd/T+QYaz2JQ9sZoxnp5drM/K/Wy0x0E+Q8kZlhCV0sijJBTEpmAZABV4waMbGAVHF7K6EjVEiNjaliQ/CWv7wK7cu6Z/nhqtZoFHGU4QRO4Rw8uIYG3EMTWkBBwjO8wpuTOS/Ou/OxaC05xcwx/JHz+QP18JBN</latexit> ↵ = 1

<latexit sha1_base64="ch7DPOr5NGQBAWE+Ku+p92wy0tU=">AAAB8nicbZBNSwMxEIZn61etX1WPXoJF8FR2RdCLUPDisYKthXYps2m2Dc0mS5IVytKf4cWDIl79Nd78N6btHrT1hcDDOzNk5o1SwY31/W+vtLa+sblV3q7s7O7tH1QPj9pGZZqyFlVC6U6EhgkuWctyK1gn1QyTSLDHaHw7qz8+MW24kg92krIwwaHkMadondXtoUhHSG5IQPrVml/35yKrEBRQg0LNfvWrN1A0S5i0VKAx3cBPbZijtpwKNq30MsNSpGMcsq5DiQkzYT5feUrOnDMgsdLuSUvm7u+JHBNjJknkOhO0I7Ncm5n/1bqZja/DnMs0s0zSxUdxJohVZHY/GXDNqBUTB0g1d7sSOkKN1LqUKi6EYPnkVWhf1APH95e1RqOIowwncArnEMAVNOAOmtACCgqe4RXePOu9eO/ex6K15BUzx/BH3ucPiYuQFg==</latexit><latexit sha1_base64="ch7DPOr5NGQBAWE+Ku+p92wy0tU=">AAAB8nicbZBNSwMxEIZn61etX1WPXoJF8FR2RdCLUPDisYKthXYps2m2Dc0mS5IVytKf4cWDIl79Nd78N6btHrT1hcDDOzNk5o1SwY31/W+vtLa+sblV3q7s7O7tH1QPj9pGZZqyFlVC6U6EhgkuWctyK1gn1QyTSLDHaHw7qz8+MW24kg92krIwwaHkMadondXtoUhHSG5IQPrVml/35yKrEBRQg0LNfvWrN1A0S5i0VKAx3cBPbZijtpwKNq30MsNSpGMcsq5DiQkzYT5feUrOnDMgsdLuSUvm7u+JHBNjJknkOhO0I7Ncm5n/1bqZja/DnMs0s0zSxUdxJohVZHY/GXDNqBUTB0g1d7sSOkKN1LqUKi6EYPnkVWhf1APH95e1RqOIowwncArnEMAVNOAOmtACCgqe4RXePOu9eO/ex6K15BUzx/BH3ucPiYuQFg==</latexit><latexit sha1_base64="ch7DPOr5NGQBAWE+Ku+p92wy0tU=">AAAB8nicbZBNSwMxEIZn61etX1WPXoJF8FR2RdCLUPDisYKthXYps2m2Dc0mS5IVytKf4cWDIl79Nd78N6btHrT1hcDDOzNk5o1SwY31/W+vtLa+sblV3q7s7O7tH1QPj9pGZZqyFlVC6U6EhgkuWctyK1gn1QyTSLDHaHw7qz8+MW24kg92krIwwaHkMadondXtoUhHSG5IQPrVml/35yKrEBRQg0LNfvWrN1A0S5i0VKAx3cBPbZijtpwKNq30MsNSpGMcsq5DiQkzYT5feUrOnDMgsdLuSUvm7u+JHBNjJknkOhO0I7Ncm5n/1bqZja/DnMs0s0zSxUdxJohVZHY/GXDNqBUTB0g1d7sSOkKN1LqUKi6EYPnkVWhf1APH95e1RqOIowwncArnEMAVNOAOmtACCgqe4RXePOu9eO/ex6K15BUzx/BH3ucPiYuQFg==</latexit><latexit sha1_base64="ch7DPOr5NGQBAWE+Ku+p92wy0tU=">AAAB8nicbZBNSwMxEIZn61etX1WPXoJF8FR2RdCLUPDisYKthXYps2m2Dc0mS5IVytKf4cWDIl79Nd78N6btHrT1hcDDOzNk5o1SwY31/W+vtLa+sblV3q7s7O7tH1QPj9pGZZqyFlVC6U6EhgkuWctyK1gn1QyTSLDHaHw7qz8+MW24kg92krIwwaHkMadondXtoUhHSG5IQPrVml/35yKrEBRQg0LNfvWrN1A0S5i0VKAx3cBPbZijtpwKNq30MsNSpGMcsq5DiQkzYT5feUrOnDMgsdLuSUvm7u+JHBNjJknkOhO0I7Ncm5n/1bqZja/DnMs0s0zSxUdxJohVZHY/GXDNqBUTB0g1d7sSOkKN1LqUKi6EYPnkVWhf1APH95e1RqOIowwncArnEMAVNOAOmtACCgqe4RXePOu9eO/ex6K15BUzx/BH3ucPiYuQFg==</latexit>

Introduce intermediate ‘catalyst’ Hamiltonian Hc to help the anneal(1)Hc

<latexit sha1_base64="GIa1mWNdSoHfN2MqfUvXuJgA50c=">AAAB83icbVDLSgMxFL3js9ZX1aWbYBFclRkRdFlw02UF+4DOUDJppg1NMkMeQhn6G25cKOLWn3Hn35hpZ6GtBwKHc+7lnpw440wb3//2Nja3tnd2K3vV/YPDo+PayWlXp1YR2iEpT1U/xppyJmnHMMNpP1MUi5jTXjy9L/zeE1WapfLRzDIaCTyWLGEEGyeFrWEosJkokZP5sFb3G/4CaJ0EJalDifaw9hWOUmIFlYZwrPUg8DMT5VgZRjidV0OraYbJFI/pwFGJBdVRvsg8R5dOGaEkVe5Jgxbq740cC61nInaTRUK96hXif97AmuQuypnMrKGSLA8lliOToqIANGKKEsNnjmCimMuKyAQrTIyrqepKCFa/vE66143A8YeberNZ1lGBc7iAKwjgFprQgjZ0gEAGz/AKb571Xrx372M5uuGVO2fwB97nD03fkdc=</latexit><latexit sha1_base64="GIa1mWNdSoHfN2MqfUvXuJgA50c=">AAAB83icbVDLSgMxFL3js9ZX1aWbYBFclRkRdFlw02UF+4DOUDJppg1NMkMeQhn6G25cKOLWn3Hn35hpZ6GtBwKHc+7lnpw440wb3//2Nja3tnd2K3vV/YPDo+PayWlXp1YR2iEpT1U/xppyJmnHMMNpP1MUi5jTXjy9L/zeE1WapfLRzDIaCTyWLGEEGyeFrWEosJkokZP5sFb3G/4CaJ0EJalDifaw9hWOUmIFlYZwrPUg8DMT5VgZRjidV0OraYbJFI/pwFGJBdVRvsg8R5dOGaEkVe5Jgxbq740cC61nInaTRUK96hXif97AmuQuypnMrKGSLA8lliOToqIANGKKEsNnjmCimMuKyAQrTIyrqepKCFa/vE66143A8YeberNZ1lGBc7iAKwjgFprQgjZ0gEAGz/AKb571Xrx372M5uuGVO2fwB97nD03fkdc=</latexit><latexit sha1_base64="GIa1mWNdSoHfN2MqfUvXuJgA50c=">AAAB83icbVDLSgMxFL3js9ZX1aWbYBFclRkRdFlw02UF+4DOUDJppg1NMkMeQhn6G25cKOLWn3Hn35hpZ6GtBwKHc+7lnpw440wb3//2Nja3tnd2K3vV/YPDo+PayWlXp1YR2iEpT1U/xppyJmnHMMNpP1MUi5jTXjy9L/zeE1WapfLRzDIaCTyWLGEEGyeFrWEosJkokZP5sFb3G/4CaJ0EJalDifaw9hWOUmIFlYZwrPUg8DMT5VgZRjidV0OraYbJFI/pwFGJBdVRvsg8R5dOGaEkVe5Jgxbq740cC61nInaTRUK96hXif97AmuQuypnMrKGSLA8lliOToqIANGKKEsNnjmCimMuKyAQrTIyrqepKCFa/vE66143A8YeberNZ1lGBc7iAKwjgFprQgjZ0gEAGz/AKb571Xrx372M5uuGVO2fwB97nD03fkdc=</latexit><latexit sha1_base64="GIa1mWNdSoHfN2MqfUvXuJgA50c=">AAAB83icbVDLSgMxFL3js9ZX1aWbYBFclRkRdFlw02UF+4DOUDJppg1NMkMeQhn6G25cKOLWn3Hn35hpZ6GtBwKHc+7lnpw440wb3//2Nja3tnd2K3vV/YPDo+PayWlXp1YR2iEpT1U/xppyJmnHMMNpP1MUi5jTXjy9L/zeE1WapfLRzDIaCTyWLGEEGyeFrWEosJkokZP5sFb3G/4CaJ0EJalDifaw9hWOUmIFlYZwrPUg8DMT5VgZRjidV0OraYbJFI/pwFGJBdVRvsg8R5dOGaEkVe5Jgxbq740cC61nInaTRUK96hXif97AmuQuypnMrKGSLA8lliOToqIANGKKEsNnjmCimMuKyAQrTIyrqepKCFa/vE66143A8YeberNZ1lGBc7iAKwjgFprQgjZ0gEAGz/AKb571Xrx372M5uuGVO2fwB97nD03fkdc=</latexit>

Hamiltonian H(s) remains stoquastic(2)

Anti-ferromagnetic transverse interactionHamiltonian H(s) may be non-stoquastic(2)

(1) E. Crosson, et al., arXiv preprint arXiv:1401.7320 . (2) S. Bravyi, et al., Quant. Inf. Comp. 8, 0361 (2008). (3) L. Hormozi, et al., Phys. Rev. B 95, 184416 (2017).

In most cases, alpha=- is the better choice, and often alph1 is worse than alpha(1,3)

↵ = �1<latexit sha1_base64="AUvQBhHG1+R5sJ/7ScL6IUlqUMA=">AAAB83icbZBNS8NAEIYn9avWr6pHL4tF8GJJRNCLUPDisYL9gCaUyXbTLt0ky+5GKKF/w4sHRbz6Z7z5b9y2OWjrCwsP78wws28oBdfGdb+d0tr6xuZWebuys7u3f1A9PGrrNFOUtWgqUtUNUTPBE9Yy3AjWlYphHArWCcd3s3rniSnN0+TRTCQLYhwmPOIUjbV8H4UcIbklFx7pV2tu3Z2LrIJXQA0KNfvVL3+Q0ixmiaECte55rjRBjspwKti04meaSaRjHLKexQRjpoN8fvOUnFlnQKJU2ZcYMnd/T+QYaz2JQ9sZoxnp5drM/K/Wy0x0E+Q8kZlhCV0sijJBTEpmAZABV4waMbGAVHF7K6EjVEiNjaliQ/CWv7wK7cu6Z/nhqtZoFHGU4QRO4Rw8uIYG3EMTWkBBwjO8wpuTOS/Ou/OxaC05xcwx/JHz+QP18JBN</latexit><latexit sha1_base64="AUvQBhHG1+R5sJ/7ScL6IUlqUMA=">AAAB83icbZBNS8NAEIYn9avWr6pHL4tF8GJJRNCLUPDisYL9gCaUyXbTLt0ky+5GKKF/w4sHRbz6Z7z5b9y2OWjrCwsP78wws28oBdfGdb+d0tr6xuZWebuys7u3f1A9PGrrNFOUtWgqUtUNUTPBE9Yy3AjWlYphHArWCcd3s3rniSnN0+TRTCQLYhwmPOIUjbV8H4UcIbklFx7pV2tu3Z2LrIJXQA0KNfvVL3+Q0ixmiaECte55rjRBjspwKti04meaSaRjHLKexQRjpoN8fvOUnFlnQKJU2ZcYMnd/T+QYaz2JQ9sZoxnp5drM/K/Wy0x0E+Q8kZlhCV0sijJBTEpmAZABV4waMbGAVHF7K6EjVEiNjaliQ/CWv7wK7cu6Z/nhqtZoFHGU4QRO4Rw8uIYG3EMTWkBBwjO8wpuTOS/Ou/OxaC05xcwx/JHz+QP18JBN</latexit><latexit sha1_base64="AUvQBhHG1+R5sJ/7ScL6IUlqUMA=">AAAB83icbZBNS8NAEIYn9avWr6pHL4tF8GJJRNCLUPDisYL9gCaUyXbTLt0ky+5GKKF/w4sHRbz6Z7z5b9y2OWjrCwsP78wws28oBdfGdb+d0tr6xuZWebuys7u3f1A9PGrrNFOUtWgqUtUNUTPBE9Yy3AjWlYphHArWCcd3s3rniSnN0+TRTCQLYhwmPOIUjbV8H4UcIbklFx7pV2tu3Z2LrIJXQA0KNfvVL3+Q0ixmiaECte55rjRBjspwKti04meaSaRjHLKexQRjpoN8fvOUnFlnQKJU2ZcYMnd/T+QYaz2JQ9sZoxnp5drM/K/Wy0x0E+Q8kZlhCV0sijJBTEpmAZABV4waMbGAVHF7K6EjVEiNjaliQ/CWv7wK7cu6Z/nhqtZoFHGU4QRO4Rw8uIYG3EMTWkBBwjO8wpuTOS/Ou/OxaC05xcwx/JHz+QP18JBN</latexit><latexit sha1_base64="AUvQBhHG1+R5sJ/7ScL6IUlqUMA=">AAAB83icbZBNS8NAEIYn9avWr6pHL4tF8GJJRNCLUPDisYL9gCaUyXbTLt0ky+5GKKF/w4sHRbz6Z7z5b9y2OWjrCwsP78wws28oBdfGdb+d0tr6xuZWebuys7u3f1A9PGrrNFOUtWgqUtUNUTPBE9Yy3AjWlYphHArWCcd3s3rniSnN0+TRTCQLYhwmPOIUjbV8H4UcIbklFx7pV2tu3Z2LrIJXQA0KNfvVL3+Q0ixmiaECte55rjRBjspwKti04meaSaRjHLKexQRjpoN8fvOUnFlnQKJU2ZcYMnd/T+QYaz2JQ9sZoxnp5drM/K/Wy0x0E+Q8kZlhCV0sijJBTEpmAZABV4waMbGAVHF7K6EjVEiNjaliQ/CWv7wK7cu6Z/nhqtZoFHGU4QRO4Rw8uIYG3EMTWkBBwjO8wpuTOS/Ou/OxaC05xcwx/JHz+QP18JBN</latexit>

↵ = 0<latexit sha1_base64="W+XLPOvXPwtga+NFbt8AJ3p/QE8=">AAAB8XicbZBNS8NAEIYn9avWr6hHL4tF8FQSEfQiFLx4rGA/sA1lst20SzebsLsRSui/8OJBEa/+G2/+G7dtDtr6wsLDOzPszBumgmvjed9OaW19Y3OrvF3Z2d3bP3APj1o6yRRlTZqIRHVC1ExwyZqGG8E6qWIYh4K1w/HtrN5+YkrzRD6YScqCGIeSR5yisdZjD0U6QnJDvL5b9WreXGQV/AKqUKjRd796g4RmMZOGCtS663upCXJUhlPBppVeplmKdIxD1rUoMWY6yOcbT8mZdQYkSpR90pC5+3six1jrSRzazhjNSC/XZuZ/tW5mousg5zLNDJN08VGUCWISMjufDLhi1IiJBaSK210JHaFCamxIFRuCv3zyKrQuar7l+8tqvV7EUYYTOIVz8OEK6nAHDWgCBQnP8ApvjnZenHfnY9FacoqZY/gj5/MHL4GP6w==</latexit><latexit sha1_base64="W+XLPOvXPwtga+NFbt8AJ3p/QE8=">AAAB8XicbZBNS8NAEIYn9avWr6hHL4tF8FQSEfQiFLx4rGA/sA1lst20SzebsLsRSui/8OJBEa/+G2/+G7dtDtr6wsLDOzPszBumgmvjed9OaW19Y3OrvF3Z2d3bP3APj1o6yRRlTZqIRHVC1ExwyZqGG8E6qWIYh4K1w/HtrN5+YkrzRD6YScqCGIeSR5yisdZjD0U6QnJDvL5b9WreXGQV/AKqUKjRd796g4RmMZOGCtS663upCXJUhlPBppVeplmKdIxD1rUoMWY6yOcbT8mZdQYkSpR90pC5+3six1jrSRzazhjNSC/XZuZ/tW5mousg5zLNDJN08VGUCWISMjufDLhi1IiJBaSK210JHaFCamxIFRuCv3zyKrQuar7l+8tqvV7EUYYTOIVz8OEK6nAHDWgCBQnP8ApvjnZenHfnY9FacoqZY/gj5/MHL4GP6w==</latexit><latexit sha1_base64="W+XLPOvXPwtga+NFbt8AJ3p/QE8=">AAAB8XicbZBNS8NAEIYn9avWr6hHL4tF8FQSEfQiFLx4rGA/sA1lst20SzebsLsRSui/8OJBEa/+G2/+G7dtDtr6wsLDOzPszBumgmvjed9OaW19Y3OrvF3Z2d3bP3APj1o6yRRlTZqIRHVC1ExwyZqGG8E6qWIYh4K1w/HtrN5+YkrzRD6YScqCGIeSR5yisdZjD0U6QnJDvL5b9WreXGQV/AKqUKjRd796g4RmMZOGCtS663upCXJUhlPBppVeplmKdIxD1rUoMWY6yOcbT8mZdQYkSpR90pC5+3six1jrSRzazhjNSC/XZuZ/tW5mousg5zLNDJN08VGUCWISMjufDLhi1IiJBaSK210JHaFCamxIFRuCv3zyKrQuar7l+8tqvV7EUYYTOIVz8OEK6nAHDWgCBQnP8ApvjnZenHfnY9FacoqZY/gj5/MHL4GP6w==</latexit><latexit sha1_base64="W+XLPOvXPwtga+NFbt8AJ3p/QE8=">AAAB8XicbZBNS8NAEIYn9avWr6hHL4tF8FQSEfQiFLx4rGA/sA1lst20SzebsLsRSui/8OJBEa/+G2/+G7dtDtr6wsLDOzPszBumgmvjed9OaW19Y3OrvF3Z2d3bP3APj1o6yRRlTZqIRHVC1ExwyZqGG8E6qWIYh4K1w/HtrN5+YkrzRD6YScqCGIeSR5yisdZjD0U6QnJDvL5b9WreXGQV/AKqUKjRd796g4RmMZOGCtS663upCXJUhlPBppVeplmKdIxD1rUoMWY6yOcbT8mZdQYkSpR90pC5+3six1jrSRzazhjNSC/XZuZ/tW5mousg5zLNDJN08VGUCWISMjufDLhi1IiJBaSK210JHaFCamxIFRuCv3zyKrQuar7l+8tqvV7EUYYTOIVz8OEK6nAHDWgCBQnP8ApvjnZenHfnY9FacoqZY/gj5/MHL4GP6w==</latexit>

↵ = 1<latexit sha1_base64="ch7DPOr5NGQBAWE+Ku+p92wy0tU=">AAAB8nicbZBNSwMxEIZn61etX1WPXoJF8FR2RdCLUPDisYKthXYps2m2Dc0mS5IVytKf4cWDIl79Nd78N6btHrT1hcDDOzNk5o1SwY31/W+vtLa+sblV3q7s7O7tH1QPj9pGZZqyFlVC6U6EhgkuWctyK1gn1QyTSLDHaHw7qz8+MW24kg92krIwwaHkMadondXtoUhHSG5IQPrVml/35yKrEBRQg0LNfvWrN1A0S5i0VKAx3cBPbZijtpwKNq30MsNSpGMcsq5DiQkzYT5feUrOnDMgsdLuSUvm7u+JHBNjJknkOhO0I7Ncm5n/1bqZja/DnMs0s0zSxUdxJohVZHY/GXDNqBUTB0g1d7sSOkKN1LqUKi6EYPnkVWhf1APH95e1RqOIowwncArnEMAVNOAOmtACCgqe4RXePOu9eO/ex6K15BUzx/BH3ucPiYuQFg==</latexit><latexit sha1_base64="ch7DPOr5NGQBAWE+Ku+p92wy0tU=">AAAB8nicbZBNSwMxEIZn61etX1WPXoJF8FR2RdCLUPDisYKthXYps2m2Dc0mS5IVytKf4cWDIl79Nd78N6btHrT1hcDDOzNk5o1SwY31/W+vtLa+sblV3q7s7O7tH1QPj9pGZZqyFlVC6U6EhgkuWctyK1gn1QyTSLDHaHw7qz8+MW24kg92krIwwaHkMadondXtoUhHSG5IQPrVml/35yKrEBRQg0LNfvWrN1A0S5i0VKAx3cBPbZijtpwKNq30MsNSpGMcsq5DiQkzYT5feUrOnDMgsdLuSUvm7u+JHBNjJknkOhO0I7Ncm5n/1bqZja/DnMs0s0zSxUdxJohVZHY/GXDNqBUTB0g1d7sSOkKN1LqUKi6EYPnkVWhf1APH95e1RqOIowwncArnEMAVNOAOmtACCgqe4RXePOu9eO/ex6K15BUzx/BH3ucPiYuQFg==</latexit><latexit sha1_base64="ch7DPOr5NGQBAWE+Ku+p92wy0tU=">AAAB8nicbZBNSwMxEIZn61etX1WPXoJF8FR2RdCLUPDisYKthXYps2m2Dc0mS5IVytKf4cWDIl79Nd78N6btHrT1hcDDOzNk5o1SwY31/W+vtLa+sblV3q7s7O7tH1QPj9pGZZqyFlVC6U6EhgkuWctyK1gn1QyTSLDHaHw7qz8+MW24kg92krIwwaHkMadondXtoUhHSG5IQPrVml/35yKrEBRQg0LNfvWrN1A0S5i0VKAx3cBPbZijtpwKNq30MsNSpGMcsq5DiQkzYT5feUrOnDMgsdLuSUvm7u+JHBNjJknkOhO0I7Ncm5n/1bqZja/DnMs0s0zSxUdxJohVZHY/GXDNqBUTB0g1d7sSOkKN1LqUKi6EYPnkVWhf1APH95e1RqOIowwncArnEMAVNOAOmtACCgqe4RXePOu9eO/ex6K15BUzx/BH3ucPiYuQFg==</latexit><latexit sha1_base64="ch7DPOr5NGQBAWE+Ku+p92wy0tU=">AAAB8nicbZBNSwMxEIZn61etX1WPXoJF8FR2RdCLUPDisYKthXYps2m2Dc0mS5IVytKf4cWDIl79Nd78N6btHrT1hcDDOzNk5o1SwY31/W+vtLa+sblV3q7s7O7tH1QPj9pGZZqyFlVC6U6EhgkuWctyK1gn1QyTSLDHaHw7qz8+MW24kg92krIwwaHkMadondXtoUhHSG5IQPrVml/35yKrEBRQg0LNfvWrN1A0S5i0VKAx3cBPbZijtpwKNq30MsNSpGMcsq5DiQkzYT5feUrOnDMgsdLuSUvm7u+JHBNjJknkOhO0I7Ncm5n/1bqZja/DnMs0s0zSxUdxJohVZHY/GXDNqBUTB0g1d7sSOkKN1LqUKi6EYPnkVWhf1APH95e1RqOIowwncArnEMAVNOAOmtACCgqe4RXePOu9eO/ex6K15BUzx/BH3ucPiYuQFg==</latexit>

H(s)<latexit sha1_base64="lL2q5dAeoEiEreg6RShXJj/CW4Q=">AAAB63icbZDLSgMxFIZPvNZ6q7p0EyxC3ZQZEXRZcNNlBXuBdiiZNNOGJpkhyQhl6Cu4caGIW1/InW9jpp2Ftv4Q+PjPOeScP0wEN9bzvtHG5tb2zm5pr7x/cHh0XDk57Zg41ZS1aSxi3QuJYYIr1rbcCtZLNCMyFKwbTu/zeveJacNj9WhnCQskGSsecUpsbjVr5mpYqXp1byG8Dn4BVSjUGla+BqOYppIpSwUxpu97iQ0yoi2ngs3Lg9SwhNApGbO+Q0UkM0G22HWOL50zwlGs3VMWL9zfExmRxsxk6DolsROzWsvN/2r91EZ3QcZVklqm6PKjKBXYxjg/HI+4ZtSKmQNCNXe7YjohmlDr4im7EPzVk9ehc133HT/cVBuNIo4SnMMF1MCHW2hAE1rQBgoTeIZXeEMSvaB39LFs3UDFzBn8Efr8ATZcjaw=</latexit><latexit sha1_base64="lL2q5dAeoEiEreg6RShXJj/CW4Q=">AAAB63icbZDLSgMxFIZPvNZ6q7p0EyxC3ZQZEXRZcNNlBXuBdiiZNNOGJpkhyQhl6Cu4caGIW1/InW9jpp2Ftv4Q+PjPOeScP0wEN9bzvtHG5tb2zm5pr7x/cHh0XDk57Zg41ZS1aSxi3QuJYYIr1rbcCtZLNCMyFKwbTu/zeveJacNj9WhnCQskGSsecUpsbjVr5mpYqXp1byG8Dn4BVSjUGla+BqOYppIpSwUxpu97iQ0yoi2ngs3Lg9SwhNApGbO+Q0UkM0G22HWOL50zwlGs3VMWL9zfExmRxsxk6DolsROzWsvN/2r91EZ3QcZVklqm6PKjKBXYxjg/HI+4ZtSKmQNCNXe7YjohmlDr4im7EPzVk9ehc133HT/cVBuNIo4SnMMF1MCHW2hAE1rQBgoTeIZXeEMSvaB39LFs3UDFzBn8Efr8ATZcjaw=</latexit><latexit sha1_base64="lL2q5dAeoEiEreg6RShXJj/CW4Q=">AAAB63icbZDLSgMxFIZPvNZ6q7p0EyxC3ZQZEXRZcNNlBXuBdiiZNNOGJpkhyQhl6Cu4caGIW1/InW9jpp2Ftv4Q+PjPOeScP0wEN9bzvtHG5tb2zm5pr7x/cHh0XDk57Zg41ZS1aSxi3QuJYYIr1rbcCtZLNCMyFKwbTu/zeveJacNj9WhnCQskGSsecUpsbjVr5mpYqXp1byG8Dn4BVSjUGla+BqOYppIpSwUxpu97iQ0yoi2ngs3Lg9SwhNApGbO+Q0UkM0G22HWOL50zwlGs3VMWL9zfExmRxsxk6DolsROzWsvN/2r91EZ3QcZVklqm6PKjKBXYxjg/HI+4ZtSKmQNCNXe7YjohmlDr4im7EPzVk9ehc133HT/cVBuNIo4SnMMF1MCHW2hAE1rQBgoTeIZXeEMSvaB39LFs3UDFzBn8Efr8ATZcjaw=</latexit><latexit sha1_base64="lL2q5dAeoEiEreg6RShXJj/CW4Q=">AAAB63icbZDLSgMxFIZPvNZ6q7p0EyxC3ZQZEXRZcNNlBXuBdiiZNNOGJpkhyQhl6Cu4caGIW1/InW9jpp2Ftv4Q+PjPOeScP0wEN9bzvtHG5tb2zm5pr7x/cHh0XDk57Zg41ZS1aSxi3QuJYYIr1rbcCtZLNCMyFKwbTu/zeveJacNj9WhnCQskGSsecUpsbjVr5mpYqXp1byG8Dn4BVSjUGla+BqOYppIpSwUxpu97iQ0yoi2ngs3Lg9SwhNApGbO+Q0UkM0G22HWOL50zwlGs3VMWL9zfExmRxsxk6DolsROzWsvN/2r91EZ3QcZVklqm6PKjKBXYxjg/HI+4ZtSKmQNCNXe7YjohmlDr4im7EPzVk9ehc133HT/cVBuNIo4SnMMF1MCHW2hAE1rQBgoTeIZXeEMSvaB39LFs3UDFzBn8Efr8ATZcjaw=</latexit>

H(s)<latexit sha1_base64="lL2q5dAeoEiEreg6RShXJj/CW4Q=">AAAB63icbZDLSgMxFIZPvNZ6q7p0EyxC3ZQZEXRZcNNlBXuBdiiZNNOGJpkhyQhl6Cu4caGIW1/InW9jpp2Ftv4Q+PjPOeScP0wEN9bzvtHG5tb2zm5pr7x/cHh0XDk57Zg41ZS1aSxi3QuJYYIr1rbcCtZLNCMyFKwbTu/zeveJacNj9WhnCQskGSsecUpsbjVr5mpYqXp1byG8Dn4BVSjUGla+BqOYppIpSwUxpu97iQ0yoi2ngs3Lg9SwhNApGbO+Q0UkM0G22HWOL50zwlGs3VMWL9zfExmRxsxk6DolsROzWsvN/2r91EZ3QcZVklqm6PKjKBXYxjg/HI+4ZtSKmQNCNXe7YjohmlDr4im7EPzVk9ehc133HT/cVBuNIo4SnMMF1MCHW2hAE1rQBgoTeIZXeEMSvaB39LFs3UDFzBn8Efr8ATZcjaw=</latexit><latexit sha1_base64="lL2q5dAeoEiEreg6RShXJj/CW4Q=">AAAB63icbZDLSgMxFIZPvNZ6q7p0EyxC3ZQZEXRZcNNlBXuBdiiZNNOGJpkhyQhl6Cu4caGIW1/InW9jpp2Ftv4Q+PjPOeScP0wEN9bzvtHG5tb2zm5pr7x/cHh0XDk57Zg41ZS1aSxi3QuJYYIr1rbcCtZLNCMyFKwbTu/zeveJacNj9WhnCQskGSsecUpsbjVr5mpYqXp1byG8Dn4BVSjUGla+BqOYppIpSwUxpu97iQ0yoi2ngs3Lg9SwhNApGbO+Q0UkM0G22HWOL50zwlGs3VMWL9zfExmRxsxk6DolsROzWsvN/2r91EZ3QcZVklqm6PKjKBXYxjg/HI+4ZtSKmQNCNXe7YjohmlDr4im7EPzVk9ehc133HT/cVBuNIo4SnMMF1MCHW2hAE1rQBgoTeIZXeEMSvaB39LFs3UDFzBn8Efr8ATZcjaw=</latexit><latexit sha1_base64="lL2q5dAeoEiEreg6RShXJj/CW4Q=">AAAB63icbZDLSgMxFIZPvNZ6q7p0EyxC3ZQZEXRZcNNlBXuBdiiZNNOGJpkhyQhl6Cu4caGIW1/InW9jpp2Ftv4Q+PjPOeScP0wEN9bzvtHG5tb2zm5pr7x/cHh0XDk57Zg41ZS1aSxi3QuJYYIr1rbcCtZLNCMyFKwbTu/zeveJacNj9WhnCQskGSsecUpsbjVr5mpYqXp1byG8Dn4BVSjUGla+BqOYppIpSwUxpu97iQ0yoi2ngs3Lg9SwhNApGbO+Q0UkM0G22HWOL50zwlGs3VMWL9zfExmRxsxk6DolsROzWsvN/2r91EZ3QcZVklqm6PKjKBXYxjg/HI+4ZtSKmQNCNXe7YjohmlDr4im7EPzVk9ehc133HT/cVBuNIo4SnMMF1MCHW2hAE1rQBgoTeIZXeEMSvaB39LFs3UDFzBn8Efr8ATZcjaw=</latexit><latexit sha1_base64="lL2q5dAeoEiEreg6RShXJj/CW4Q=">AAAB63icbZDLSgMxFIZPvNZ6q7p0EyxC3ZQZEXRZcNNlBXuBdiiZNNOGJpkhyQhl6Cu4caGIW1/InW9jpp2Ftv4Q+PjPOeScP0wEN9bzvtHG5tb2zm5pr7x/cHh0XDk57Zg41ZS1aSxi3QuJYYIr1rbcCtZLNCMyFKwbTu/zeveJacNj9WhnCQskGSsecUpsbjVr5mpYqXp1byG8Dn4BVSjUGla+BqOYppIpSwUxpu97iQ0yoi2ngs3Lg9SwhNApGbO+Q0UkM0G22HWOL50zwlGs3VMWL9zfExmRxsxk6DolsROzWsvN/2r91EZ3QcZVklqm6PKjKBXYxjg/HI+4ZtSKmQNCNXe7YjohmlDr4im7EPzVk9ehc133HT/cVBuNIo4SnMMF1MCHW2hAE1rQBgoTeIZXeEMSvaB39LFs3UDFzBn8Efr8ATZcjaw=</latexit>

H(s) = (1� s)H0 + s(1� s)Hc + sHIsing<latexit sha1_base64="OFfUGp+2X9pkHABISt8i1y6FwU4=">AAACJnicbZBdS8MwFIbT+TXnV9VLb4JDmIijFUFvBgNv5t0E9wFbKWmWbmFpWpJUGKW/xhv/ijdeTES886eYbh3o5oGQN885h5zzehGjUlnWl1FYW9/Y3Cpul3Z29/YPzMOjtgxjgUkLhywUXQ9JwignLUUVI91IEBR4jHS88V2W7zwRIWnIH9UkIk6Ahpz6FCOlkWvWGhV5DmuwYl/qu+Fa8ALKxaMfIDUSQYLTjGqQLMi9pHyYpq5ZtqrWLOCqsHNRBnk0XXPaH4Q4DghXmCEpe7YVKSdBQlHMSFrqx5JECI/RkPS05Cgg0klma6bwTJMB9EOhD1dwRn93JCiQchJ4ujIbUy7nMvhfrhcr/9ZJKI9iRTief+THDKoQZp7BARUEKzbRAmFB9awQj5BAWGlnS9oEe3nlVdG+qtpaP1yX6/XcjiI4AaegAmxwA+qgAZqgBTB4Bq9gCt6NF+PN+DA+56UFI+85Bn/C+P4BRx+iKA==</latexit><latexit sha1_base64="OFfUGp+2X9pkHABISt8i1y6FwU4=">AAACJnicbZBdS8MwFIbT+TXnV9VLb4JDmIijFUFvBgNv5t0E9wFbKWmWbmFpWpJUGKW/xhv/ijdeTES886eYbh3o5oGQN885h5zzehGjUlnWl1FYW9/Y3Cpul3Z29/YPzMOjtgxjgUkLhywUXQ9JwignLUUVI91IEBR4jHS88V2W7zwRIWnIH9UkIk6Ahpz6FCOlkWvWGhV5DmuwYl/qu+Fa8ALKxaMfIDUSQYLTjGqQLMi9pHyYpq5ZtqrWLOCqsHNRBnk0XXPaH4Q4DghXmCEpe7YVKSdBQlHMSFrqx5JECI/RkPS05Cgg0klma6bwTJMB9EOhD1dwRn93JCiQchJ4ujIbUy7nMvhfrhcr/9ZJKI9iRTief+THDKoQZp7BARUEKzbRAmFB9awQj5BAWGlnS9oEe3nlVdG+qtpaP1yX6/XcjiI4AaegAmxwA+qgAZqgBTB4Bq9gCt6NF+PN+DA+56UFI+85Bn/C+P4BRx+iKA==</latexit><latexit sha1_base64="OFfUGp+2X9pkHABISt8i1y6FwU4=">AAACJnicbZBdS8MwFIbT+TXnV9VLb4JDmIijFUFvBgNv5t0E9wFbKWmWbmFpWpJUGKW/xhv/ijdeTES886eYbh3o5oGQN885h5zzehGjUlnWl1FYW9/Y3Cpul3Z29/YPzMOjtgxjgUkLhywUXQ9JwignLUUVI91IEBR4jHS88V2W7zwRIWnIH9UkIk6Ahpz6FCOlkWvWGhV5DmuwYl/qu+Fa8ALKxaMfIDUSQYLTjGqQLMi9pHyYpq5ZtqrWLOCqsHNRBnk0XXPaH4Q4DghXmCEpe7YVKSdBQlHMSFrqx5JECI/RkPS05Cgg0klma6bwTJMB9EOhD1dwRn93JCiQchJ4ujIbUy7nMvhfrhcr/9ZJKI9iRTief+THDKoQZp7BARUEKzbRAmFB9awQj5BAWGlnS9oEe3nlVdG+qtpaP1yX6/XcjiI4AaegAmxwA+qgAZqgBTB4Bq9gCt6NF+PN+DA+56UFI+85Bn/C+P4BRx+iKA==</latexit><latexit sha1_base64="OFfUGp+2X9pkHABISt8i1y6FwU4=">AAACJnicbZBdS8MwFIbT+TXnV9VLb4JDmIijFUFvBgNv5t0E9wFbKWmWbmFpWpJUGKW/xhv/ijdeTES886eYbh3o5oGQN885h5zzehGjUlnWl1FYW9/Y3Cpul3Z29/YPzMOjtgxjgUkLhywUXQ9JwignLUUVI91IEBR4jHS88V2W7zwRIWnIH9UkIk6Ahpz6FCOlkWvWGhV5DmuwYl/qu+Fa8ALKxaMfIDUSQYLTjGqQLMi9pHyYpq5ZtqrWLOCqsHNRBnk0XXPaH4Q4DghXmCEpe7YVKSdBQlHMSFrqx5JECI/RkPS05Cgg0klma6bwTJMB9EOhD1dwRn93JCiQchJ4ujIbUy7nMvhfrhcr/9ZJKI9iRTief+THDKoQZp7BARUEKzbRAmFB9awQj5BAWGlnS9oEe3nlVdG+qtpaP1yX6/XcjiI4AaegAmxwA+qgAZqgBTB4Bq9gCt6NF+PN+DA+56UFI+85Bn/C+P4BRx+iKA==</latexit>

Non-generic CounterexampleInfinite-range ferromagnetic p-spin model(1)

0 50 100 15010

-4

10-3

10-2

10-1

5 10 15 20 25

10-11

10-8

10-5

10-2

Exponentially closing gap Polynomially closing gap(1) Y Seki and H Nishimori, Phys. Rev. E 85, 051112 (2012). (2) TA, arXiv:1811.09980.

p = 6(2)p = 6<latexit sha1_base64="RPu0dDOedyuWp8aAVAwXju/DaEw=">AAAB6nicbZBNS8NAEIYn9avWr6pHL4tF8FQSkepFKHjxWNF+QBvKZjtpl242YXcjlNCf4MWDIl79Rd78N27bHLT1hYWHd2bYmTdIBNfGdb+dwtr6xuZWcbu0s7u3f1A+PGrpOFUMmywWseoEVKPgEpuGG4GdRCGNAoHtYHw7q7efUGkey0czSdCP6FDykDNqrPWQ3NT65Ypbdeciq+DlUIFcjX75qzeIWRqhNExQrbuemxg/o8pwJnBa6qUaE8rGdIhdi5JGqP1svuqUnFlnQMJY2ScNmbu/JzIaaT2JAtsZUTPSy7WZ+V+tm5rw2s+4TFKDki0+ClNBTExmd5MBV8iMmFigTHG7K2EjqigzNp2SDcFbPnkVWhdVz/L9ZaVez+Mowgmcwjl4cAV1uIMGNIHBEJ7hFd4c4bw4787HorXg5DPH8EfO5w/Sw415</latexit><latexit sha1_base64="RPu0dDOedyuWp8aAVAwXju/DaEw=">AAAB6nicbZBNS8NAEIYn9avWr6pHL4tF8FQSkepFKHjxWNF+QBvKZjtpl242YXcjlNCf4MWDIl79Rd78N27bHLT1hYWHd2bYmTdIBNfGdb+dwtr6xuZWcbu0s7u3f1A+PGrpOFUMmywWseoEVKPgEpuGG4GdRCGNAoHtYHw7q7efUGkey0czSdCP6FDykDNqrPWQ3NT65Ypbdeciq+DlUIFcjX75qzeIWRqhNExQrbuemxg/o8pwJnBa6qUaE8rGdIhdi5JGqP1svuqUnFlnQMJY2ScNmbu/JzIaaT2JAtsZUTPSy7WZ+V+tm5rw2s+4TFKDki0+ClNBTExmd5MBV8iMmFigTHG7K2EjqigzNp2SDcFbPnkVWhdVz/L9ZaVez+Mowgmcwjl4cAV1uIMGNIHBEJ7hFd4c4bw4787HorXg5DPH8EfO5w/Sw415</latexit><latexit sha1_base64="RPu0dDOedyuWp8aAVAwXju/DaEw=">AAAB6nicbZBNS8NAEIYn9avWr6pHL4tF8FQSkepFKHjxWNF+QBvKZjtpl242YXcjlNCf4MWDIl79Rd78N27bHLT1hYWHd2bYmTdIBNfGdb+dwtr6xuZWcbu0s7u3f1A+PGrpOFUMmywWseoEVKPgEpuGG4GdRCGNAoHtYHw7q7efUGkey0czSdCP6FDykDNqrPWQ3NT65Ypbdeciq+DlUIFcjX75qzeIWRqhNExQrbuemxg/o8pwJnBa6qUaE8rGdIhdi5JGqP1svuqUnFlnQMJY2ScNmbu/JzIaaT2JAtsZUTPSy7WZ+V+tm5rw2s+4TFKDki0+ClNBTExmd5MBV8iMmFigTHG7K2EjqigzNp2SDcFbPnkVWhdVz/L9ZaVez+Mowgmcwjl4cAV1uIMGNIHBEJ7hFd4c4bw4787HorXg5DPH8EfO5w/Sw415</latexit><latexit sha1_base64="RPu0dDOedyuWp8aAVAwXju/DaEw=">AAAB6nicbZBNS8NAEIYn9avWr6pHL4tF8FQSkepFKHjxWNF+QBvKZjtpl242YXcjlNCf4MWDIl79Rd78N27bHLT1hYWHd2bYmTdIBNfGdb+dwtr6xuZWcbu0s7u3f1A+PGrpOFUMmywWseoEVKPgEpuGG4GdRCGNAoHtYHw7q7efUGkey0czSdCP6FDykDNqrPWQ3NT65Ypbdeciq+DlUIFcjX75qzeIWRqhNExQrbuemxg/o8pwJnBa6qUaE8rGdIhdi5JGqP1svuqUnFlnQMJY2ScNmbu/JzIaaT2JAtsZUTPSy7WZ+V+tm5rw2s+4TFKDki0+ClNBTExmd5MBV8iMmFigTHG7K2EjqigzNp2SDcFbPnkVWhdVz/L9ZaVez+Mowgmcwjl4cAV1uIMGNIHBEJ7hFd4c4bw4787HorXg5DPH8EfO5w/Sw415</latexit>

↵ = �1<latexit sha1_base64="AUvQBhHG1+R5sJ/7ScL6IUlqUMA=">AAAB83icbZBNS8NAEIYn9avWr6pHL4tF8GJJRNCLUPDisYL9gCaUyXbTLt0ky+5GKKF/w4sHRbz6Z7z5b9y2OWjrCwsP78wws28oBdfGdb+d0tr6xuZWebuys7u3f1A9PGrrNFOUtWgqUtUNUTPBE9Yy3AjWlYphHArWCcd3s3rniSnN0+TRTCQLYhwmPOIUjbV8H4UcIbklFx7pV2tu3Z2LrIJXQA0KNfvVL3+Q0ixmiaECte55rjRBjspwKti04meaSaRjHLKexQRjpoN8fvOUnFlnQKJU2ZcYMnd/T+QYaz2JQ9sZoxnp5drM/K/Wy0x0E+Q8kZlhCV0sijJBTEpmAZABV4waMbGAVHF7K6EjVEiNjaliQ/CWv7wK7cu6Z/nhqtZoFHGU4QRO4Rw8uIYG3EMTWkBBwjO8wpuTOS/Ou/OxaC05xcwx/JHz+QP18JBN</latexit><latexit sha1_base64="AUvQBhHG1+R5sJ/7ScL6IUlqUMA=">AAAB83icbZBNS8NAEIYn9avWr6pHL4tF8GJJRNCLUPDisYL9gCaUyXbTLt0ky+5GKKF/w4sHRbz6Z7z5b9y2OWjrCwsP78wws28oBdfGdb+d0tr6xuZWebuys7u3f1A9PGrrNFOUtWgqUtUNUTPBE9Yy3AjWlYphHArWCcd3s3rniSnN0+TRTCQLYhwmPOIUjbV8H4UcIbklFx7pV2tu3Z2LrIJXQA0KNfvVL3+Q0ixmiaECte55rjRBjspwKti04meaSaRjHLKexQRjpoN8fvOUnFlnQKJU2ZcYMnd/T+QYaz2JQ9sZoxnp5drM/K/Wy0x0E+Q8kZlhCV0sijJBTEpmAZABV4waMbGAVHF7K6EjVEiNjaliQ/CWv7wK7cu6Z/nhqtZoFHGU4QRO4Rw8uIYG3EMTWkBBwjO8wpuTOS/Ou/OxaC05xcwx/JHz+QP18JBN</latexit><latexit sha1_base64="AUvQBhHG1+R5sJ/7ScL6IUlqUMA=">AAAB83icbZBNS8NAEIYn9avWr6pHL4tF8GJJRNCLUPDisYL9gCaUyXbTLt0ky+5GKKF/w4sHRbz6Z7z5b9y2OWjrCwsP78wws28oBdfGdb+d0tr6xuZWebuys7u3f1A9PGrrNFOUtWgqUtUNUTPBE9Yy3AjWlYphHArWCcd3s3rniSnN0+TRTCQLYhwmPOIUjbV8H4UcIbklFx7pV2tu3Z2LrIJXQA0KNfvVL3+Q0ixmiaECte55rjRBjspwKti04meaSaRjHLKexQRjpoN8fvOUnFlnQKJU2ZcYMnd/T+QYaz2JQ9sZoxnp5drM/K/Wy0x0E+Q8kZlhCV0sijJBTEpmAZABV4waMbGAVHF7K6EjVEiNjaliQ/CWv7wK7cu6Z/nhqtZoFHGU4QRO4Rw8uIYG3EMTWkBBwjO8wpuTOS/Ou/OxaC05xcwx/JHz+QP18JBN</latexit><latexit sha1_base64="AUvQBhHG1+R5sJ/7ScL6IUlqUMA=">AAAB83icbZBNS8NAEIYn9avWr6pHL4tF8GJJRNCLUPDisYL9gCaUyXbTLt0ky+5GKKF/w4sHRbz6Z7z5b9y2OWjrCwsP78wws28oBdfGdb+d0tr6xuZWebuys7u3f1A9PGrrNFOUtWgqUtUNUTPBE9Yy3AjWlYphHArWCcd3s3rniSnN0+TRTCQLYhwmPOIUjbV8H4UcIbklFx7pV2tu3Z2LrIJXQA0KNfvVL3+Q0ixmiaECte55rjRBjspwKti04meaSaRjHLKexQRjpoN8fvOUnFlnQKJU2ZcYMnd/T+QYaz2JQ9sZoxnp5drM/K/Wy0x0E+Q8kZlhCV0sijJBTEpmAZABV4waMbGAVHF7K6EjVEiNjaliQ/CWv7wK7cu6Z/nhqtZoFHGU4QRO4Rw8uIYG3EMTWkBBwjO8wpuTOS/Ou/OxaC05xcwx/JHz+QP18JBN</latexit> ↵ = 1


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Incremental steps vs One step

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(1) TA, arXiv:1811.09980.


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Ground state changes incrementally along the anneal for a = 1, as opposed

to suddenly for a = - 1↵ = 1



Antiferromagnetic transverse field gives the system more “room to spread”

A Geometrically Local Example

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(1) TA, arXiv:1811.09980.

Identical ferromagnetic rings, coupled at their ends

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(1) TA, arXiv:1811.09980.

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100

Exponential scaling of the gap for al = 0, -1, but looks like polynomial scaling for al =1

↵ = 0,�1<latexit sha1_base64="cwDZmW7n8QQz/Vq7ac8qrX6wuIY=">AAAB9XicbZBNS8NAEIYnftb6VfXoZbEIHrQkIuhFKHjxWMF+QBvLZLtpl242YXejlND/4cWDIl79L978N27bHLT1hYWHd2aY2TdIBNfGdb+dpeWV1bX1wkZxc2t7Z7e0t9/Qcaooq9NYxKoVoGaCS1Y33AjWShTDKBCsGQxvJvXmI1Oax/LejBLmR9iXPOQUjbUeOiiSAZJr4p6SM69bKrsVdyqyCF4OZchV65a+Or2YphGThgrUuu25ifEzVIZTwcbFTqpZgnSIfda2KDFi2s+mV4/JsXV6JIyVfdKQqft7IsNI61EU2M4IzUDP1ybmf7V2asIrP+MySQ2TdLYoTAUxMZlEQHpcMWrEyAJSxe2thA5QITU2qKINwZv/8iI0ziue5buLcrWax1GAQziCE/DgEqpwCzWoAwUFz/AKb86T8+K8Ox+z1iUnnzmAP3I+fwDSSZC9</latexit><latexit sha1_base64="cwDZmW7n8QQz/Vq7ac8qrX6wuIY=">AAAB9XicbZBNS8NAEIYnftb6VfXoZbEIHrQkIuhFKHjxWMF+QBvLZLtpl242YXejlND/4cWDIl79L978N27bHLT1hYWHd2aY2TdIBNfGdb+dpeWV1bX1wkZxc2t7Z7e0t9/Qcaooq9NYxKoVoGaCS1Y33AjWShTDKBCsGQxvJvXmI1Oax/LejBLmR9iXPOQUjbUeOiiSAZJr4p6SM69bKrsVdyqyCF4OZchV65a+Or2YphGThgrUuu25ifEzVIZTwcbFTqpZgnSIfda2KDFi2s+mV4/JsXV6JIyVfdKQqft7IsNI61EU2M4IzUDP1ybmf7V2asIrP+MySQ2TdLYoTAUxMZlEQHpcMWrEyAJSxe2thA5QITU2qKINwZv/8iI0ziue5buLcrWax1GAQziCE/DgEqpwCzWoAwUFz/AKb86T8+K8Ox+z1iUnnzmAP3I+fwDSSZC9</latexit><latexit sha1_base64="cwDZmW7n8QQz/Vq7ac8qrX6wuIY=">AAAB9XicbZBNS8NAEIYnftb6VfXoZbEIHrQkIuhFKHjxWMF+QBvLZLtpl242YXejlND/4cWDIl79L978N27bHLT1hYWHd2aY2TdIBNfGdb+dpeWV1bX1wkZxc2t7Z7e0t9/Qcaooq9NYxKoVoGaCS1Y33AjWShTDKBCsGQxvJvXmI1Oax/LejBLmR9iXPOQUjbUeOiiSAZJr4p6SM69bKrsVdyqyCF4OZchV65a+Or2YphGThgrUuu25ifEzVIZTwcbFTqpZgnSIfda2KDFi2s+mV4/JsXV6JIyVfdKQqft7IsNI61EU2M4IzUDP1ybmf7V2asIrP+MySQ2TdLYoTAUxMZlEQHpcMWrEyAJSxe2thA5QITU2qKINwZv/8iI0ziue5buLcrWax1GAQziCE/DgEqpwCzWoAwUFz/AKb86T8+K8Ox+z1iUnnzmAP3I+fwDSSZC9</latexit><latexit sha1_base64="cwDZmW7n8QQz/Vq7ac8qrX6wuIY=">AAAB9XicbZBNS8NAEIYnftb6VfXoZbEIHrQkIuhFKHjxWMF+QBvLZLtpl242YXejlND/4cWDIl79L978N27bHLT1hYWHd2aY2TdIBNfGdb+dpeWV1bX1wkZxc2t7Z7e0t9/Qcaooq9NYxKoVoGaCS1Y33AjWShTDKBCsGQxvJvXmI1Oax/LejBLmR9iXPOQUjbUeOiiSAZJr4p6SM69bKrsVdyqyCF4OZchV65a+Or2YphGThgrUuu25ifEzVIZTwcbFTqpZgnSIfda2KDFi2s+mV4/JsXV6JIyVfdKQqft7IsNI61EU2M4IzUDP1ybmf7V2asIrP+MySQ2TdLYoTAUxMZlEQHpcMWrEyAJSxe2thA5QITU2qKINwZv/8iI0ziue5buLcrWax1GAQziCE/DgEqpwCzWoAwUFz/AKb86T8+K8Ox+z1iUnnzmAP3I+fwDSSZC9</latexit>

↵ = 1<latexit sha1_base64="jS/5J8qmKK8n12RLjLWJZVS4jG8=">AAAB8XicbZBNS8NAEIYn9avWr6hHL4tF8FQSEfQiFLx4rGA/sA1lst20SzebsLsRSui/8OJBEa/+G2/+G7dtDtr6wsLDOzPszBumgmvjed9OaW19Y3OrvF3Z2d3bP3APj1o6yRRlTZqIRHVC1ExwyZqGG8E6qWIYh4K1w/HtrN5+YkrzRD6YScqCGIeSR5yisdZjD0U6QnJD/L5b9WreXGQV/AKqUKjRd796g4RmMZOGCtS663upCXJUhlPBppVeplmKdIxD1rUoMWY6yOcbT8mZdQYkSpR90pC5+3six1jrSRzazhjNSC/XZuZ/tW5mousg5zLNDJN08VGUCWISMjufDLhi1IiJBaSK210JHaFCamxIFRuCv3zyKrQuar7l+8tqvV7EUYYTOIVz8OEK6nAHDWgCBQnP8ApvjnZenHfnY9FacoqZY/gj5/MHMQWP7A==</latexit><latexit sha1_base64="jS/5J8qmKK8n12RLjLWJZVS4jG8=">AAAB8XicbZBNS8NAEIYn9avWr6hHL4tF8FQSEfQiFLx4rGA/sA1lst20SzebsLsRSui/8OJBEa/+G2/+G7dtDtr6wsLDOzPszBumgmvjed9OaW19Y3OrvF3Z2d3bP3APj1o6yRRlTZqIRHVC1ExwyZqGG8E6qWIYh4K1w/HtrN5+YkrzRD6YScqCGIeSR5yisdZjD0U6QnJD/L5b9WreXGQV/AKqUKjRd796g4RmMZOGCtS663upCXJUhlPBppVeplmKdIxD1rUoMWY6yOcbT8mZdQYkSpR90pC5+3six1jrSRzazhjNSC/XZuZ/tW5mousg5zLNDJN08VGUCWISMjufDLhi1IiJBaSK210JHaFCamxIFRuCv3zyKrQuar7l+8tqvV7EUYYTOIVz8OEK6nAHDWgCBQnP8ApvjnZenHfnY9FacoqZY/gj5/MHMQWP7A==</latexit><latexit sha1_base64="jS/5J8qmKK8n12RLjLWJZVS4jG8=">AAAB8XicbZBNS8NAEIYn9avWr6hHL4tF8FQSEfQiFLx4rGA/sA1lst20SzebsLsRSui/8OJBEa/+G2/+G7dtDtr6wsLDOzPszBumgmvjed9OaW19Y3OrvF3Z2d3bP3APj1o6yRRlTZqIRHVC1ExwyZqGG8E6qWIYh4K1w/HtrN5+YkrzRD6YScqCGIeSR5yisdZjD0U6QnJD/L5b9WreXGQV/AKqUKjRd796g4RmMZOGCtS663upCXJUhlPBppVeplmKdIxD1rUoMWY6yOcbT8mZdQYkSpR90pC5+3six1jrSRzazhjNSC/XZuZ/tW5mousg5zLNDJN08VGUCWISMjufDLhi1IiJBaSK210JHaFCamxIFRuCv3zyKrQuar7l+8tqvV7EUYYTOIVz8OEK6nAHDWgCBQnP8ApvjnZenHfnY9FacoqZY/gj5/MHMQWP7A==</latexit><latexit sha1_base64="jS/5J8qmKK8n12RLjLWJZVS4jG8=">AAAB8XicbZBNS8NAEIYn9avWr6hHL4tF8FQSEfQiFLx4rGA/sA1lst20SzebsLsRSui/8OJBEa/+G2/+G7dtDtr6wsLDOzPszBumgmvjed9OaW19Y3OrvF3Z2d3bP3APj1o6yRRlTZqIRHVC1ExwyZqGG8E6qWIYh4K1w/HtrN5+YkrzRD6YScqCGIeSR5yisdZjD0U6QnJD/L5b9WreXGQV/AKqUKjRd796g4RmMZOGCtS663upCXJUhlPBppVeplmKdIxD1rUoMWY6yOcbT8mZdQYkSpR90pC5+3six1jrSRzazhjNSC/XZuZ/tW5mousg5zLNDJN08VGUCWISMjufDLhi1IiJBaSK210JHaFCamxIFRuCv3zyKrQuar7l+8tqvV7EUYYTOIVz8OEK6nAHDWgCBQnP8ApvjnZenHfnY9FacoqZY/gj5/MHMQWP7A==</latexit>

Beyond Standard QA Recap

More exotic interactions open up unexplored parameter spaces for QA

�zi �

zj�

zk �z

i �zj�

zk�

zl

Beyond 2-body Ising Hamiltonions

(1) A. Perdomo-Ortiz et al., QIP 10, 33 (2011). (2) M. Ohkuwa et al., Phys. Rev. A 98, 022314 (2018) (3) D. Venturelli and A. Kondratyev, arXiv:1810.08584.

Catalyst Hamiltonians

Hc = ↵

X

hi,ji

�xi �

xj

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New annealing protocols like ‘reverse annealing’ (1-3)

Conclusions

Most of our analysis has been restricted to ‘trivial’ computational problems, where analytic/numerical

progress can be made.

We are living in exciting times; quantum information processing devices are coming online,

albeit noisy ones, and we have the opportunity to ask: what can do we do with such quantum computing devices?

We may be in a situation where the only way to find out is to build such devices and try.

Will this translate to real world advantages? Is QA doomed? Is ending on a classical Hamiltonian the problem?

Will we need to adopt universal AQC to see an advantage?

Documents

Quantum Annealing: Challenges & Prospects · Prototypical Hamming Weight Problem 5 10 15 5 10 15 Hamming weight Classical Energy The “spike” problem(1) (1) E. Farhi, J. Goldstone,