Computational Bottlenecks of Quantum Annealing

A promising approach to solving hard binary optimisation problems is quantum adiabatic annealing (QA) in a transverse magnetic field. An instantaneous ground state --- initially a symmetric superposition of all possible assignments of $N$ qubits --- is closely tracked as it becomes more and more localised near the global minimum of the classical energy. Regions where the energy gap to excited states is small (e.g. at the phase transition) are the algorithm's bottlenecks. Here I show how for large problems the complexity becomes dominated by $O(\log N)$ bottlenecks inside the spin glass phase, where the gap scales as a stretched exponential. For smaller $N$, only the gap at the critical point is relevant, where it scales polynomially, as long as the phase transition is second order. This phenomenon is demonstrated rigorously for the two-pattern Gaussian Hopfield Model. Qualitative comparison with the Sherrington-Kirkpatrick Model leads to similar conclusions.Comment: 9 pages of main text + 3 pages of supplementary info, 5 figures; added discussion, updated reference

Blackbox: A procedure for parallel optimization of expensive black-box functions

This note provides a description of a procedure that is designed to efficiently optimize expensive black-box functions. It uses the response surface methodology by incorporating radial basis functions as the response model. A simple method based on a Latin hypercube is used for initial sampling. A modified version of CORS algorithm with space rescaling is used for the subsequent sampling. The procedure is able to scale on multicore processors by performing multiple function evaluations in parallel. The source code of the procedure is written in Python.Comment: 8 pages, 3 figure

On the relevance of avoided crossings away from quantum critical point to the complexity of quantum adiabatic algorithm

Two recent preprints [B. Altshuler, H. Krovi, and J. Roland, "Quantum adiabatic optimization fails for random instances of NP-complete problems", arXiv:0908.2782 and "Anderson localization casts clouds over adiabatic quantum optimization", arXiv:0912.0746] argue that random 4th order perturbative corrections to the energies of local minima of random instances of NP-complete problem lead to avoided crossings that cause the failure of quantum adiabatic algorithm (due to exponentially small gap) close to the end, for very small transverse field that scales as an inverse power of instance size N. The theoretical portion of this work does not to take into account the exponential degeneracy of the ground and excited states at zero field. A corrected analysis shows that unlike those in the middle of the spectrum, avoided crossings at the edge would require high [O(1)] transverse fields, at which point the perturbation theory may become divergent due to quantum phase transition. This effect manifests itself only in large instances [exp(0.02 N) >> 1], which might be the reason it had not been observed in the authors' numerical work. While we dispute the proposed mechanism of failure of quantum adiabatic algorithm, we cannot draw any conclusions on its ultimate complexity.Comment: 8 pages, 5 figure

Adiabatic Quantum Computing in systems with constant inter-qubit couplings

We propose an approach suitable for solving NP-complete problems via adiabatic quantum computation with an architecture based on a lattice of interacting spins (qubits) driven by locally adjustable effective magnetic fields. Interactions between qubits are assumed constant and instance-independent, programming is done only by changing local magnetic fields. Implementations using qubits coupled by magnetic-, electric-dipole and exchange interactions are discussed.Comment: 10 pages, 10 figures, reference adde

Quantum Adiabatic Evolution Algorithm and Quantum Phase Transition in 3-Satisfiability Problem

In this paper we show that the performance of the quantum adiabatic algorithm is determined by phase transitions in underlying problem in the presence of transverse magnetic field $\Gamma$. We show that the quantum version of random Satisfiability problem with 3 bits in a clause (3-SAT) has a first-order quantum phase transition. We analyze the phase diagram $\gamma=\gamma(\Gamma)$ where $\gamma$ is an average number of clauses per binary variable in 3-SAT. The results are obtained in a closed form assuming replica symmetry and neglecting time correlations at small values of the transverse field $\Gamma$. In the limit of $\Gamma=0$ the value of $\gamma(0)\approx$ 5.18 corresponds to that given by the replica symmetric treatment of a classical random 3-SAT problem. We demonstrate the qualitative similarity between classical and quantum versions of this problem.Comment: 30 pages, 7 figure

Comparative Study of the Performance of Quantum Annealing and Simulated Annealing

Relations of simulated annealing and quantum annealing are studied by a mapping from the transition matrix of classical Markovian dynamics of the Ising model to a quantum Hamiltonian and vice versa. It is shown that these two operators, the transition matrix and the Hamiltonian, share the eigenvalue spectrum. Thus, if simulated annealing with slow temperature change does not encounter a difficulty caused by an exponentially long relaxation time at a first-order phase transition, the same is true for the corresponding process of quantum annealing in the adiabatic limit. One of the important differences between the classical-to-quantum mapping and the converse quantum-to-classical mapping is that the Markovian dynamics of a short-range Ising model is mapped to a short-range quantum system, but the converse mapping from a short-range quantum system to a classical one results in long-range interactions. This leads to a difference in efficiencies that simulated annealing can be efficiently simulated by quantum annealing but the converse is not necessarily true. We conclude that quantum annealing is easier to implement and is more flexible than simulated annealing. We also point out that the present mapping can be extended to accommodate explicit time dependence of temperature, which is used to justify the quantum-mechanical analysis of simulated annealing by Somma, Batista, and Ortiz. Additionally, an alternative method to solve the non-equilibrium dynamics of the one-dimensional Ising model is provided through the classical-to-quantum mapping.Comment: 19 page

Estimation of Phase and Diffusion: Combining Quantum Statistics and Classical Noise

Coherent ensembles of $N$ qubits present an advantage in quantum phase estimation over separable mixtures, but coherence decay due to classical phase diffusion reduces overall precision. In some contexts, the strength of diffusion may be the parameter of interest. We examine estimation of both phase and diffusion in large spin systems using a novel mathematical formulation. For the first time, we show a closed form expression for the quantum Fisher information for estimation of a unitary parameter in a noisy environment. The optimal probe state has a non-Gaussian profile and differs also from the canonical phase state; it saturates a new tight precision bound. For noise below a critical threshold, entanglement always leads to enhanced precision, but the shot-noise limit is beaten only by a constant factor, independent of $N$. We provide upper and lower bounds to this factor, valid in low and high noise regimes. Unlike other noise types, it is shown for $N \gg 1$ that phase and diffusion can be measured simultaneously and optimally.Comment: 7 pages, 3 figure

Size dependence of the minimum excitation gap in the Quantum Adiabatic Algorithm

We study the typical (median) value of the minimum gap in the quantum version of the Exact Cover problem using Quantum Monte Carlo simulations, in order to understand the complexity of the quantum adiabatic algorithm (QAA) for much larger sizes than before. For a range of sizes, N <= 128, where the classical Davis-Putnam algorithm shows exponential median complexity, the QAA shows polynomial median complexity. The bottleneck of the algorithm is an isolated avoided crossing point of a Landau-Zener type (collision between the two lowest energy levels only).Comment: 4 pages, 5 figure

True Limits to Precision via Unique Quantum Probe

Quantum instruments derived from composite systems allow greater measurement precision than their classical counterparts due to coherences maintained between N components; spins, atoms or photons. Decoherence that plagues real-world devices can be particle loss, or thermal excitation and relaxation, or dephasing due to external noise sources -- and also due to prior parameter uncertainty. All these adversely affect precision estimation of time, phase or frequency. We develop a novel technique uncovering the uniquely optimal probe states of the N `qubits' alongside new tight bounds on precision under local and collective mechanisms of these noise types above. For large quantum ensembles where numerical techniques fail, the problem reduces by analogy to finding the ground state of a 1-D particle in a potential well; the shape of the well is dictated by the type and strength of decoherence. The formalism is applied to prototypical Mach-Zehnder and Ramsey interferometers to discover the ultimate performance of real-world instruments.Comment: 11 pages including Methods, 1 Appendix, 4 figures, 2 table

Approximating satisfiability transition by suppressing fluctuations

Using methods and ideas from statistical mechanics, we propose a simple method for obtaining rigorous upper bounds for satisfiability transition in random boolean expressions composed of N variables and M clauses with K variables per clause. Determining the location of satisfiability threshold $\alpha_c=M/N$ for a number of difficult combinatorial problems is a major open problem in the theory of random graphs. The method is based on identification of the core -- a subexpression (subgraph) that has the same satisfiability properties as the original expression. We formulate self-consistency equations that determine macroscopic parameters of the core and compute an improved annealing bound. We illustrate the method for three sample problems: K-XOR-SAT, K-SAT and positive 1-in-K-SAT.Comment: 31 pages, 6 figure