Quantum Optimization of Fully Connected Spin Glasses

Abstract

Many NP-hard problems can be seen as the task of finding a ground state of a disordered highly connected Ising spin glass. If solutions are sought by means of quantum annealing, it is often necessary to represent those graphs in the annealer's hardware by means of the graph-minor embedding technique, generating a final Hamiltonian consisting of coupled chains of ferromagnetically bound spins, whose binding energy is a free parameter. In order to investigate the effect of embedding on problems of interest, the fully connected Sherrington-Kirkpatrick model with random AE1 couplings is programmed on the D-Wave Two TM annealer using up to 270 qubits interacting on a Chimera-type graph. We present the best embedding prescriptions for encoding the Sherrington-Kirkpatrick problem in the Chimera graph. The results indicate that the optimal choice of embedding parameters could be associated with the emergence of the spin-glass phase of the embedded problem, whose presence was previously uncertain. This optimal parameter setting allows the performance of the quantum annealer to compete with (and potentially outperform, in the absence of analog control errors) optimized simulated annealing algorithms.

Open-System Quantum Annealing in Mean-Field Models with Exponential Degeneracy

“…. Quantum dynamics of the fully connected graphs is not a purely theoretical pursuit, it can be implemented experimentally on existing analog quantum annealing hardware [27], reducing the model to local interactions only [28,29]. Moreover, three-spin interactions may be implemented directly in superconducting circuits [30] motivated by the novel physics they could introduce [31].…”

Circuit design for multi-body interactions in superconducting quantum annealing systems with applications to a scalable architecture

“…[1][2][3][4][5][6][7][8][9] This interest is due in a large part to demonstration of the underlying principles of quantum annealing in condensed matter systems 10 and the more recent development of a programmable annealing device by D-Wave Systems Inc. 11,12 Although the niobium superconducting quantum interference device (SQUIDs) which are the basic building blocks of the D-Wave annealer display limited coherence, it has been used to demonstrate that quantum tunneling is an exploitable resource in a computational setting. 13 Furthermore the development of annealers using aluminum SQUIDs with orders-of-magnitude longer coherence lifetimes 14 may enable further improvements in the computational performance of future quantum annealers.…”

“…17 Because it is non-planar, minor embedding can be used to map a fully connected graph to the Chimera, although at a significant overhead. 7,18 The NP-completeness of finding the ground state of an arbitrary (two-body) Ising spin glass and therefore a Chimera graph guarantees that any NP-complete problem can be mapped to finding the ground state of a Hamiltonian which is a subgraph of a Chimera with polynomial overhead. For examples of how this can be done in practice, see ref.…”

“…[1][2][3][4][5][6][7][8][9] This interest is due in a large part to demonstration of the underlying principles of quantum annealing in condensed matter systems 10 and the more recent development of a programmable annealing device by D-Wave Systems Inc.…”

Comparative study of the performance of quantum annealing and simulated annealing

“…An early experimental study of a disordered magnet also revealed faster relaxations toward equilibrium through a quantum path than by a real thermal annealing path [9]. Recent studies concerning the D-Wave machine show mixed results [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24], and further careful investigations are necessary before firm conclusions are drawn.…”