A PTAS for the Classical Ising Spin Glass Problem on the Chimera Graph Structure

We present a polynomial time approximation scheme (PTAS) for the minimum value of the classical Ising Hamiltonian with linear terms on the Chimera graph structure as defined in the recent work of McGeoch and Wang. The result follows from a direct application of the techniques used by Bansal, Bravyi and Terhal who gave a PTAS for the same problem on planar and, in particular, grid graphs. We also show that on Chimera graphs, the trivial lower bound is within a constant factor of the optimum.Comment: 6 pages, corrected PTAS running tim

Glassy Chimeras could be blind to quantum speedup: Designing better benchmarks for quantum annealing machines

Recently, a programmable quantum annealing machine has been built that minimizes the cost function of hard optimization problems by adiabatically quenching quantum fluctuations. Tests performed by different research teams have shown that, indeed, the machine seems to exploit quantum effects. However experiments on a class of random-bond instances have not yet demonstrated an advantage over classical optimization algorithms on traditional computer hardware. Here we present evidence as to why this might be the case. These engineered quantum annealing machines effectively operate coupled to a decohering thermal bath. Therefore, we study the finite-temperature critical behavior of the standard benchmark problem used to assess the computational capabilities of these complex machines. We simulate both random-bond Ising models and spin glasses with bimodal and Gaussian disorder on the D-Wave Chimera topology. Our results show that while the worst-case complexity of finding a ground state of an Ising spin glass on the Chimera graph is not polynomial, the finite-temperature phase space is likely rather simple: Spin glasses on Chimera have only a zero-temperature transition. This means that benchmarking optimization methods using spin glasses on the Chimera graph might not be the best benchmark problems to test quantum speedup. We propose alternative benchmarks by embedding potentially harder problems on the Chimera topology. Finally, we also study the (reentrant) disorder-temperature phase diagram of the random-bond Ising model on the Chimera graph and show that a finite-temperature ferromagnetic phase is stable up to 19.85(15)% antiferromagnetic bonds. Beyond this threshold the system only displays a zero-temperature spin-glass phase. Our results therefore show that a careful design of the hardware architecture and benchmark problems is key when building quantum annealing machines.Comment: 8 pages, 5 figures, 1 tabl

A note on QUBO instances defined on Chimera graphs

McGeoch and Wang (2013) recently obtained optimal or near-optimal solutions to some quadratic unconstrained boolean optimization (QUBO) problem instances using a 439 qubit D-Wave Two quantum computing system in much less time than with the IBM ILOG CPLEX mixed-integer quadratic programming (MIQP) solver. The problems studied by McGeoch and Wang are defined on subgraphs -- with up to 439 nodes -- of Chimera graphs. We observe that after a standard reformulation of the QUBO problem as a mixed-integer linear program (MILP), the specific instances used by McGeoch and Wang can be solved to optimality with the CPLEX MILP solver in much less time than the time reported in McGeoch and Wang for the CPLEX MIQP solver. However, the solution time is still more than the time taken by the D-Wave computer in the McGeoch-Wang tests.Comment: Version 1 discussed computational results with random QUBO instances. McGeoch and Wang made an error in describing the instances they used; they did not use random QUBO instances but rather random Ising Model instances with fields (mapped to QUBO instances). The current version of the note reports on tests with the precise instances used by McGeoch and Wan

A Hybrid Quantum-Classical Paradigm to Mitigate Embedding Costs in Quantum Annealing

Despite rapid recent progress towards the development of quantum computers capable of providing computational advantages over classical computers, it seems likely that such computers will, initially at least, be required to run in a hybrid quantum-classical regime. This realisation has led to interest in hybrid quantum-classical algorithms allowing, for example, quantum computers to solve large problems despite having very limited numbers of qubits. Here we propose a hybrid paradigm for quantum annealers with the goal of mitigating a different limitation of such devices: the need to embed problem instances within the (often highly restricted) connectivity graph of the annealer. This embedding process can be costly to perform and may destroy any computational speedup. In order to solve many practical problems, it is moreover necessary to perform many, often related, such embeddings. We will show how, for such problems, a raw speedup that is negated by the embedding time can nonetheless be exploited to give a real speedup. As a proof-of-concept example we present an in-depth case study of a simple problem based on the maximum weight independent set problem. Although we do not observe a quantum speedup experimentally, the advantage of the hybrid approach is robustly verified, showing how a potential quantum speedup may be exploited and encouraging further efforts to apply the approach to problems of more practical interest.Comment: 30 pages, 6 figure

The Transverse-Field Ising Spin Glass Model on the Bethe Lattice with an Application to Adiabatic Quantum Computing

In this Ph.D. thesis we examine the Adiabatic Quantum Algorithm from the point of view of statistical and condensed matter physics. We do this by studying the transverse-field Ising spin glass model defined on the Bethe lattice, which is of independent interest to both the physics community and the quantum computation community. Using quantum Monte Carlo methods, we perform an extensive study of the the ground-state properties of the model, including the R\'enyi entanglement entropy, quantum Fisher information, Edwards--Anderson parameter, correlation functions. Through the finite-size scaling of these quantities we find multiple independent and coinciding estimates for the critical point of the glassy phase transition at zero temperature, which completes the phase diagram of the model as was previously known in the literature. We find volumetric bipartite and finite multipartite entanglement for all values of the transverse field considered, both in the paramagnetic and in the glassy phase, and at criticality. We discuss their implication with respect to quantum computing. By writing a perturbative expansion in the large transverse field regime we develop a mean-field quasiparticle theory that explains the numerical data. The emerging picture is that of degenerate bands of localized quasiparticle excitations on top of a vacuum. The perturbative energy corrections to these bands are given by pair creation/annihilation and hopping processes of the quasiparticles on the Bethe lattice. The transition to the glassy phase is explained as a crossing of the energy level of the vacuum with one of the bands, so that creation of quasiparticles becomes energetically favoured. We also study the localization properties of the model by employing the forward scattering approximation of the locator expansion, which we compute using a numerical transfer matrix technique. We obtain a lower bound for the mobility edge of the system. We find a localized region inside of the glassy phase and we discuss the consequences of its presence for the Adiabatic Quantum Algorithm