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

43 Pages·2014·2.74 MB·English

Performance of simulated annealing, simulated quantum
annealing and D-Wave on hard spin glass instances
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Collaborators
▪ Troels Rønnow (ETH)
▪ Sergei Isakov (ETH → Google)
▪ Sergio Boixo (USC → Google)
▪ Joshua Job (USC)
▪ Zhihui Wang (USC)
▪ Bettina Heim (ETH, BSc student)
▪ Damian Steiger (ETH)
▪ Ilia Zintchenko (ETH)
▪ Dave Wecker (Microsoft Research)
▪ John Martinis (UCSB)
▪ Daniel Lidar (USC)
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Spin glasses on the D-Wave chimera graph
!
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Worst-case complexity and spin glass physics
H = ∑ J s s + ∑ h s + const. with s = ±1
ij i j i i i
ij i
▪ NP-hard for non-planar lattices (Barahona 1982)
▪ No spin glass phase in finite-D lattices in a magnetic field (Young et al, 2004)
no magnetic field with magnetic field
polynomial NP-hard
2D planar
spin glass phase at T=0 no spin glass phase
NP-hard NP-hard
2D non-planar
spin glass phase at T=0 no spin glass phase
3D or higher NP-hard NP-hard
dimensions spin glass phase with no spin glass phase
NP-hard NP-hard
Infinite dimensions
spin glass phase with spin glass phase with
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Average case complexity
▪ In the absence of a spin glass phase correlations are short-ranged.
Can we thus solve typical spin glass problems locally?
We observe average case polynomial scaling for a new algorithm

see tomorrow’s talk by Ilia Zintchenko
▪ Spin glass T =0 in 2D. 

c
Is a 2D lattice the wrong system for realizing hard problems?


see following talk by Helmut Katzgraber 

Katzgraber, Hamze, Andrist, Phys. Rev. X (2014)

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!
Annealing
Simulated annealing
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Adiabatic quantum optimization
Quantum annealing
Simulated quantum annealing
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Long history of annealing

Annealing
A neolithic technology
Slowly cool metal or glass to improve its 

properties and get closer to the ground state

Simulated annealing
Kirkpatrick, Gelatt and Vecchi, Science (1983)
A classical optimization algorithm Image credit ANFF NSW node, University of New South Wales
Slowly cool a model in a Monte Carlo simulation

to find the solution to an optimization problem
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Quantum annealing for a transverse field Ising model


Kadowaki and Nishimori (1998)
Farhi, Goldstone, Gutmann and Sipser (2000)
Add a transverse magnetic field 
 (cid:7)(cid:4)
to induce quantum fluctuations (cid:7)(cid:3) (cid:23)(cid:18)(cid:8)(cid:9)(cid:8) (cid:22)
(cid:10)
(cid:24)(cid:18)(cid:8)(cid:9)(cid:8) (cid:22)
(cid:10)
(cid:7)(cid:1)
(cid:22)
(cid:21)
(cid:20)
(cid:19) (cid:6)
(cid:18)
(cid:17)
(cid:16)
(cid:15)
(cid:14) (cid:5)
H (t) = B(t)∑ J σ zσ z − A(t)∑σx (cid:13)
(cid:12)
(cid:11)
ij i j i
(cid:4)
i<j i
(cid:3)
Initial time t=0: all spins aligned with the transverse(cid:1) field
(cid:1) (cid:1)(cid:2)(cid:3) (cid:1)(cid:2)(cid:4) (cid:1)(cid:2)(cid:5) (cid:1)(cid:2)(cid:6) (cid:7)
(cid:8)(cid:9)(cid:8)
(cid:10)
Final time t=t : ground state of the Ising spin glass
f
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Quantum annealing
▪ Quantum annealing not necessarily stays adiabatic

Kadowaki and Nishimori (1998)
▪ Adiabatic quantum optimization is the special case of
perfectly coherent adiabatic evolution in the ground state

Farhi, Goldstone, Gutmann and Sipser (2000)
▪ Experimental quantum annealing (QA) 

Quantum mechanical evolution 

in a material or device, 

potentially at finite temperatures


Brooke, Bitko, Rosenbaum, Aeppli, Science (1999)
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Simulated quantum annealing
▪ “Schrödinger” dynamics (unitary)
▪ Exponential complexity on classical hardware
▪ Simulates the time evolution of a quantum system
▪ Unitary evolution in the ground state: U-QA

Kadowaki and Nishimori (1998)
▪ Open systems dynamics using master equations: OS-QA
▪ Quantum Monte Carlo dynamics (stochastic)
▪ Classical algorithms with polynomial complexity
▪ QMC samples the equilibrium thermal state of a quantum system
▪ Typically based on path integral Monte Carlo simulations: QMC-QA

Apolloni et al (1988), Santoro at al (2002)
▪ Mean-field MC version using coherence but no entanglement: MC-QA

Shin, Smolin, Smith, Vazirani, arXiv:1401.7087
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