quantum annealing: challenges & prospects · prototypical hamming weight problem 5 10 15 5 10...
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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
ground states of HIsingHIsing<latexit sha1_base64="sE3OVEpO/zIL42w5S93VxmGsVUM=">AAAB+XicbVBNSwMxFHxbv2r9WvXoJVgET2VXBD0WvNRbBVsL7bJk02wbmmSXJFsoS/+JFw+KePWfePPfmG33oK0DgWHmPd5kopQzbTzv26lsbG5t71R3a3v7B4dH7vFJVyeZIrRDEp6oXoQ15UzSjmGG016qKBYRp0/R5K7wn6ZUaZbIRzNLaSDwSLKYEWysFLpuKxwIbMZK5PeaydE8dOtew1sArRO/JHUo0Q7dr8EwIZmg0hCOte77XmqCHCvDCKfz2iDTNMVkgke0b6nEguogXySfowurDFGcKPukQQv190aOhdYzEdnJIqVe9QrxP6+fmfg2yJlMM0MlWR6KM45Mgooa0JApSgyfWYKJYjYrImOsMDG2rJotwV/98jrpXjV8yx+u681mWUcVzuAcLsGHG2hCC9rQAQJTeIZXeHNy58V5dz6WoxWn3DmFP3A+fwDYIJPH</latexit><latexit sha1_base64="sE3OVEpO/zIL42w5S93VxmGsVUM=">AAAB+XicbVBNSwMxFHxbv2r9WvXoJVgET2VXBD0WvNRbBVsL7bJk02wbmmSXJFsoS/+JFw+KePWfePPfmG33oK0DgWHmPd5kopQzbTzv26lsbG5t71R3a3v7B4dH7vFJVyeZIrRDEp6oXoQ15UzSjmGG016qKBYRp0/R5K7wn6ZUaZbIRzNLaSDwSLKYEWysFLpuKxwIbMZK5PeaydE8dOtew1sArRO/JHUo0Q7dr8EwIZmg0hCOte77XmqCHCvDCKfz2iDTNMVkgke0b6nEguogXySfowurDFGcKPukQQv190aOhdYzEdnJIqVe9QrxP6+fmfg2yJlMM0MlWR6KM45Mgooa0JApSgyfWYKJYjYrImOsMDG2rJotwV/98jrpXjV8yx+u681mWUcVzuAcLsGHG2hCC9rQAQJTeIZXeHNy58V5dz6WoxWn3DmFP3A+fwDYIJPH</latexit><latexit sha1_base64="sE3OVEpO/zIL42w5S93VxmGsVUM=">AAAB+XicbVBNSwMxFHxbv2r9WvXoJVgET2VXBD0WvNRbBVsL7bJk02wbmmSXJFsoS/+JFw+KePWfePPfmG33oK0DgWHmPd5kopQzbTzv26lsbG5t71R3a3v7B4dH7vFJVyeZIrRDEp6oXoQ15UzSjmGG016qKBYRp0/R5K7wn6ZUaZbIRzNLaSDwSLKYEWysFLpuKxwIbMZK5PeaydE8dOtew1sArRO/JHUo0Q7dr8EwIZmg0hCOte77XmqCHCvDCKfz2iDTNMVkgke0b6nEguogXySfowurDFGcKPukQQv190aOhdYzEdnJIqVe9QrxP6+fmfg2yJlMM0MlWR6KM45Mgooa0JApSgyfWYKJYjYrImOsMDG2rJotwV/98jrpXjV8yx+u681mWUcVzuAcLsGHG2hCC9rQAQJTeIZXeHNy58V5dz6WoxWn3DmFP3A+fwDYIJPH</latexit><latexit sha1_base64="sE3OVEpO/zIL42w5S93VxmGsVUM=">AAAB+XicbVBNSwMxFHxbv2r9WvXoJVgET2VXBD0WvNRbBVsL7bJk02wbmmSXJFsoS/+JFw+KePWfePPfmG33oK0DgWHmPd5kopQzbTzv26lsbG5t71R3a3v7B4dH7vFJVyeZIrRDEp6oXoQ15UzSjmGG016qKBYRp0/R5K7wn6ZUaZbIRzNLaSDwSLKYEWysFLpuKxwIbMZK5PeaydE8dOtew1sArRO/JHUo0Q7dr8EwIZmg0hCOte77XmqCHCvDCKfz2iDTNMVkgke0b6nEguogXySfowurDFGcKPukQQv190aOhdYzEdnJIqVe9QrxP6+fmfg2yJlMM0MlWR6KM45Mgooa0JApSgyfWYKJYjYrImOsMDG2rJotwV/98jrpXjV8yx+u681mWUcVzuAcLsGHG2hCC9rQAQJTeIZXeHNy58V5dz6WoxWn3DmFP3A+fwDYIJPH</latexit>
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
<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>
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<latexit sha1_base64="Sxb3OToY+00fEbS/51pktI56xd0=">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</latexit><latexit sha1_base64="Sxb3OToY+00fEbS/51pktI56xd0=">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</latexit><latexit sha1_base64="Sxb3OToY+00fEbS/51pktI56xd0=">AAACfXicbZFda9swFIZl76vNPppul705LBSSLQ52GGQ3g8BuetnB0hbixMiKnIhKspGOS1OTf9Ff1rv9ld20SuqNrckBoZf3PQ+SjtJCCoth+Mvznz1/8fLV3n7j9Zu37w6ah+/PbF4axkcsl7m5SKnlUmg+QoGSXxSGU5VKfp5efl/n51fcWJHrn7gs+ETRuRaZYBSdlTRvT9q2G0sHzGgHvkHQjgLbgdiWKhFuE3NFEzG9hs9gwWV/W/W0CqIVxJJn2N5qj42YL7Az7UPguBpaM1FQrHYzN/AHKpJmK+yFm4JtEdWiReo6TZp38SxnpeIamaTWjqOwwElFDQom+aoRl5YXlF3SOR87qanidlJtpreCY+fMIMuNWxph4/5LVFRZu1Sp61QUF/ZptjZ3ZeMSs6+TSuiiRK7Z40FZKQFzWH8FzIThDOXSCcqMcHcFtqCGMnQf1nBDiJ4+eVuc9XuR0z++tIbDehx75Ih8JG0SkQEZkhNySkaEkd8eeB3vk3fvH/tdv/fY6ns184H8V/7gASrqvaQ=</latexit><latexit sha1_base64="Sxb3OToY+00fEbS/51pktI56xd0=">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</latexit>
Incremental steps vs One step
-5 0 5-1
0
1
-5 0 5-1
0
1
-5 0 5-1
0
1
-5 0 5-1
0
1
-5 0 5-1
0
1
-5 0 5-1
0
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-5 0 5-1
0
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-5 0 5-1
0
1
-5 0 5-1
0
1
-5 0 5-1
0
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-5 0 5-1
0
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-5 0 5-1
0
1
(1) TA, arXiv:1811.09980.
↵ = �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>
Ground state changes incrementally along the anneal for a = 1, as opposed
to suddenly for a = - 1↵ = 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>
↵ = �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>
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
A Geometrically Local Example
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(1) TA, arXiv:1811.09980.
8 10 12 14 16 18 2010
-5
10-4
10-3
10-2
10-1
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 �
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zk �z
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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
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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?