The Quantum Adiabatic Algorithm applied to random optimization problems: the quantum spin glass perspective

Among various algorithms designed to exploit the specific properties of quantum computers with respect to classical ones, the quantum adiabatic algorithm is a versatile proposition to find the minimal value of an arbitrary cost function (ground state energy). Random optimization problems provide a natural testbed to compare its efficiency with that of classical algorithms. These problems correspond to mean field spin glasses that have been extensively studied in the classical case. This paper reviews recent analytical works that extended these studies to incorporate the effect of quantum fluctuations, and presents also some original results in this direction.Comment: 151 pages, 21 figure

Taming a non-convex landscape with dynamical long-range order: memcomputing Ising benchmarks

Recent work on quantum annealing has emphasized the role of collective behavior in solving optimization problems. By enabling transitions of clusters of variables, such solvers are able to navigate their state space and locate solutions more efficiently despite having only local connections between elements. However, collective behavior is not exclusive to quantum annealers, and classical solvers that display collective dynamics should also possess an advantage in navigating a non-convex landscape. Here, we give evidence that a benchmark derived from quantum annealing studies is solvable in polynomial time using digital memcomputing machines, which utilize a collection of dynamical components with memory to represent the structure of the underlying optimization problem. To illustrate the role of memory and clarify the structure of these solvers we propose a simple model of these machines that demonstrates the emergence of long-range order. This model, when applied to finding the ground state of the Ising frustrated-loop benchmarks, undergoes a transient phase of avalanches which can span the entire lattice and demonstrates a connection between long-range behavior and their probability of success. These results establish the advantages of computational approaches based on collective dynamics of continuous dynamical systems

Strongly Refuting Random CSPs Below the Spectral Threshold

Random constraint satisfaction problems (CSPs) are known to exhibit threshold phenomena: given a uniformly random instance of a CSP with $n$ variables and $m$ clauses, there is a value of $m = \Omega(n)$ beyond which the CSP will be unsatisfiable with high probability. Strong refutation is the problem of certifying that no variable assignment satisfies more than a constant fraction of clauses; this is the natural algorithmic problem in the unsatisfiable regime (when $m/n = \omega(1)$). Intuitively, strong refutation should become easier as the clause density $m/n$ grows, because the contradictions introduced by the random clauses become more locally apparent. For CSPs such as $k$-SAT and $k$-XOR, there is a long-standing gap between the clause density at which efficient strong refutation algorithms are known, $m/n \ge \widetilde O(n^{k/2-1})$, and the clause density at which instances become unsatisfiable with high probability, $m/n = \omega (1)$. In this paper, we give spectral and sum-of-squares algorithms for strongly refuting random $k$-XOR instances with clause density $m/n \ge \widetilde O(n^{(k/2-1)(1-\delta)})$ in time $\exp(\widetilde O(n^{\delta}))$ or in $\widetilde O(n^{\delta})$ rounds of the sum-of-squares hierarchy, for any $\delta \in [0,1)$ and any integer $k \ge 3$. Our algorithms provide a smooth transition between the clause density at which polynomial-time algorithms are known at $\delta = 0$, and brute-force refutation at the satisfiability threshold when $\delta = 1$. We also leverage our $k$-XOR results to obtain strong refutation algorithms for SAT (or any other Boolean CSP) at similar clause densities. Our algorithms match the known sum-of-squares lower bounds due to Grigoriev and Schonebeck, up to logarithmic factors. Additionally, we extend our techniques to give new results for certifying upper bounds on the injective tensor norm of random tensors