Probing for quantum speedup in spin-glass problems with planted solutions

Abstract

The availability of quantum annealing devices with hundreds of qubits has made the experimental demonstration of a quantum speedup for optimization problems a coveted, albeit elusive goal. Going beyond earlier studies of random Ising problems, here we introduce a method to construct a set of frustrated Ising-model optimization problems with tunable hardness. We study the performance of a D-Wave Two device (DW2) with up to 503 qubits on these problems and compare it to a suite of classical algorithms, including a highly optimized algorithm designed to compete directly with the DW2. The problems are generated around predetermined ground-state configurations, called planted solutions, which makes them particularly suitable for benchmarking purposes. The problem set exhibits properties familiar from constraint satisfaction (SAT) problems, such as a peak in the typical hardness of the problems, determined by a tunable clause density parameter. We bound the hardness regime where the DW2 device either does not or might exhibit a quantum speedup for our problem set. While we do not find evidence for a speedup for the hardest and most frustrated problems in our problem set, we cannot rule out that a speedup might exist for some of the easier, less frustrated problems. Our empirical findings pertain to the specific D-Wave processor and problem set we studied and leave open the possibility that future processors might exhibit a quantum speedup on the same problem set.

Degeneracy, degree, and heavy tails in quantum annealing

“…13 shows performance scaling on U 3 1 , U 4 1 , U 5 1 , and U 6 1 instances. These results suggest that several threads of research that probed for computational advantage in U 6 1 problems [2, 9, 10] might be more successfully directed towards U 3 1 problems. Neither SA nor DW2X performance responds monotonically to increas- ing degree.…”

“…[12] ran SQA and SA on the same 200-qubit (C 5 ) U 6 1 instances, each containing 120 degree 6 (and therefore potentially floppy) qubits, and investigated the ground state probability p of each solver on a given instance. While they consider τ = 1/p, we consider the number of repetitions R needed to achieve 99% probability of finding a ground state [14]:…”

“…These recent works have proposed various important considerations such as error sensitivity, thermal hardness [10], ground state degeneracy, consistency of energy scale across a random ensemble [6,7], potential barrier shape [4,5], and the existence of classical phase transitions [8]. In parallel to this line of research is the consideration of suitable performance metrics for exact and approximate optimization [9,11,12].…”

“…Several previous efforts to quantify the performance of D-Wave hardware relative to physically motivated software solvers have focused on the median case performance for a given input class [2,6]. However, Ref.…”

“…The recent development of quantum annealing (QA) processors has spurred research into how to construct input instances-Ising spin instances-that might confirm or refute any purported computational advantage conferred by such hardware [1][2][3][4][5][6][7][8][9][10]. These recent works have proposed various important considerations such as error sensitivity, thermal hardness [10], ground state degeneracy, consistency of energy scale across a random ensemble [6,7], potential barrier shape [4,5], and the existence of classical phase transitions [8].…”