Threshold theorem in isolated quantum dynamics with stochastic control errors.

We investigate the effect of stochastic control errors in the time-dependent Hamiltonian on isolated quantum dynamics. The control errors are formulated as time-dependent stochastic noise in the Schrödinger equation. For a class of stochastic control errors, we establish a threshold theorem that provides a sufficient condition to obtain the target state, which should be determined in noiseless isolated quantum dynamics, as a relation between the number of measurements and noise strength. The theorem guarantees that if the sum of the noise strengths is less than the inverse of computational time, the target state can be obtained through a constant-order number of measurements. If the opposite is true, the number of measurements to guarantee obtaining the target state increases exponentially with computational time. Our threshold theorem can be applied to any isolated quantum dynamics such as quantum annealing and adiabatic quantum computation. This article is part of the theme issue 'Quantum annealing and computation: challenges and perspectives'.

Travel time optimization on multi-AGV routing by reverse annealing.

Quantum annealing has been actively researched since D-Wave Systems produced the first commercial machine in 2011. Controlling a large fleet of automated guided vehicles is one of the real-world applications utilizing quantum annealing. In this study, we propose a formulation to control the traveling routes to minimize the travel time. We validate our formulation through simulation in a virtual plant and authenticate the effectiveness for faster distribution compared to a greedy algorithm that does not consider the overall detour distance. Furthermore, we utilize reverse annealing to maximize the advantage of the D-Wave's quantum annealer. Starting from relatively good solutions obtained by a fast greedy algorithm, reverse annealing searches for better solutions around them. Our reverse annealing method improves the performance compared to standard quantum annealing alone and performs up to 10 times faster than a commercial classical solver, Gurobi. This study extends a use of optimization with general problem solvers in the application of multi-AGV systems and reveals the potential of reverse annealing as an optimizer.

Benchmark of quantum-inspired heuristic solvers for quadratic unconstrained binary optimization.

Recently, inspired by quantum annealing, many solvers specialized for unconstrained binary quadratic programming problems have been developed. For further improvement and application of these solvers, it is important to clarify the differences in their performance for various types of problems. In this study, the performance of four quadratic unconstrained binary optimization problem solvers, namely D-Wave Hybrid Solver Service (HSS), Toshiba Simulated Bifurcation Machine (SBM), Fujitsu Digital Annealer (DA), and simulated annealing on a personal computer, was benchmarked. The problems used for benchmarking were instances of real problems in MQLib, instances of the SAT-UNSAT phase transition point of random not-all-equal 3-SAT (NAE 3-SAT), and the Ising spin glass Sherrington-Kirkpatrick (SK) model. Concerning MQLib instances, the HSS performance ranked first; for NAE 3-SAT, DA performance ranked first; and regarding the SK model, SBM performance ranked first. These results may help understand the strengths and weaknesses of these solvers.

Assessment of image generation by quantum annealer.

Quantum annealing was originally proposed as an approach for solving combinatorial optimization problems using quantum effects. D-Wave Systems has released a production model of quantum annealing hardware. However, the inherent noise and various environmental factors in the hardware hamper the determination of optimal solutions. In addition, the freezing effect in regions with weak quantum fluctuations generates outputs approximately following a Gibbs-Boltzmann distribution at an extremely low temperature. Thus, a quantum annealer may also serve as a fast sampler for the Ising spin-glass problem, and several studies have investigated Boltzmann machine learning using a quantum annealer. Previous developments have focused on comparing the performance in the standard distance of the resulting distributions between conventional methods in classical computers and sampling by a quantum annealer. In this study, we focused on the performance of a quantum annealer as a generative model from a different aspect. To evaluate its performance, we prepared a discriminator given by a neural network trained on an a priori dataset. The evaluation results show a higher performance of quantum annealer compared with the classical approach for Boltzmann machine learning in training of the generative model. However the generation of the data suffers from the remanent quantum fluctuation in the quantum annealer. The quality of the generated images from the quantum annealer gets worse than the ideal case of the quantum annealing and the classical Monte-Carlo sampling.

Simple relation between frustration and transition points in diluted spin glasses.

We investigate a possible relation between frustration and phase-transition points in two-dimensional spin glasses at zero temperature. The relation consists of a condition on the average number of frustrated plaquettes and was reported to provide very good predictions for the critical points at zero temperature, for several two-dimensional lattices. Although there has been no proof of the relation, the good correspondence in several lattices suggests the validity of the relation and an important role of frustration in the phase transitions. To examine the relation further, we present a natural extension of the relation to diluted lattices and verify its effectiveness for bond-diluted square lattices. We then confirm that the resulting points are in good agreement with the phase-transition points in a wide range of dilution rate. Our result supports the suggestion from R. Miyazaki [J. Phys. Soc. Jpn. 82, 094001 (2013)JUPSAU0031-901510.7566/JPSJ.82.094001] for nondiluted lattices on the importance of frustration to the phase transition of two-dimensional spin glasses at zero temperature.

Breaking limitation of quantum annealer in solving optimization problems under constraints.

Quantum annealing is a generic solver for optimization problems that uses fictitious quantum fluctuation. The most groundbreaking progress in the research field of quantum annealing is its hardware implementation, i.e., the so-called quantum annealer, using artificial spins. However, the connectivity between the artificial spins is sparse and limited on a special network known as the chimera graph. Several embedding techniques have been proposed, but the number of logical spins, which represents the optimization problems to be solved, is drastically reduced. In particular, an optimization problem including fully or even partly connected spins suffers from low embeddable size on the chimera graph. In the present study, we propose an alternative approach to solve a large-scale optimization problem on the chimera graph via a well-known method in statistical mechanics called the Hubbard-Stratonovich transformation or its variants. The proposed method can be used to deal with a fully connected Ising model without embedding on the chimera graph and leads to nontrivial results of the optimization problem. We tested the proposed method with a number of partition problems involving solving linear equations and the traffic flow optimization problem in Sendai and Kyoto cities in Japan.

Restored interlaced volumetric imaging increases image quality and scanning speed during intravital imaging in living mice.

Dynamic intravital imaging is essential for revealing ongoing biological phenomena within living organisms and is influenced primarily by several factors: motion artifacts, optical properties and spatial resolution. Conventional imaging quality within a volume, however, is degraded by involuntary movements and trades off between the imaged volume, imaging speed and quality. To balance such trade-offs incurred by two-photon excitation microscopy during intravital imaging, we developed a unique combination of interlaced scanning and a simple image restoration algorithm based on biological signal sparsity and a graph Laplacian matrix. This method increases the scanning speed by a factor of four for a field size of 212 µm × 106 µm × 130 µm, and significantly improves the quality of four-dimensional dynamic volumetric data by preventing irregular artifacts due to the movement observed with conventional methods. Our data suggest this method is robust enough to be applied to multiple types of soft tissue.

Efficient partition of integer optimization problems with one-hot encoding.

Quantum annealing is a heuristic algorithm for solving combinatorial optimization problems, and hardware for implementing this algorithm has been developed by D-Wave Systems Inc. The current version of the D-Wave quantum annealer can solve unconstrained binary optimization problems with a limited number of binary variables. However, the cost functions of several practical problems are defined by a large number of integer variables. To solve these problems using the quantum annealer, integer variables are generally binarized with one-hot encoding, and the binarized problem is partitioned into small subproblems. However, the entire search space of the binarized problem is considerably larger than that of the original integer problem and is dominated by infeasible solutions. Therefore, to efficiently solve large optimization problems with one-hot encoding, partitioning methods that extract subproblems with as many feasible solutions as possible are required. In this study, we propose two partitioning methods and demonstrate that they result in improved solutions.

Dynamics of order parameters of nonstoquastic Hamiltonians in the adaptive quantum Monte Carlo method.

We derive macroscopically deterministic flow equations with regard to the order parameters of the ferromagnetic p-spin model with infinite-range interactions. The p-spin model has a first-order phase transition for p>2. In the case of p≥5, the p-spin model with antiferromagnetic XX interaction has a second-order phase transition in a certain region. In this case, however, the model becomes a nonstoquastic Hamiltonian, resulting in a negative sign problem. To simulate the p-spin model with antiferromagnetic XX interaction, we utilize the adaptive quantum Monte Carlo method. By using this method, we can regard the effect of the antiferromagnetic XX interaction as fluctuations of the transverse magnetic field. A previous study [J. Inoue, J. Phys. Conf. Ser. 233, 012010 (2010)1742-659610.1088/1742-6596/233/1/012010] derived deterministic flow equations of the order parameters in the quantum Monte Carlo method. In this study, we derive macroscopically deterministic flow equations for the magnetization and transverse magnetization from the master equation in the adaptive quantum Monte Carlo method. Under the Suzuki-Trotter decomposition, we consider the Glauber-type stochastic process. We solve these differential equations by using the Runge-Kutta method, and we verify that these results are consistent with the saddle-point solution of mean-field theory. Finally, we analyze the stability of the equilibrium solutions obtained by the differential equations.

Improving solutions by embedding larger subproblems in a D-Wave quantum annealer.

Quantum annealing is a heuristic algorithm that solves combinatorial optimization problems, and D-Wave Systems Inc. has developed hardware implementation of this algorithm. However, in general, we cannot embed all the logical variables of a large-scale problem, since the number of available qubits is limited. In order to handle a large problem, qbsolv has been proposed as a method for partitioning the original large problem into subproblems that are embeddable in the D-Wave quantum annealer, and it then iteratively optimizes the subproblems using the quantum annealer. Multiple logical variables in the subproblem are simultaneously updated in this iterative solver, and using this approach we expect to obtain better solutions than can be obtained by conventional local search algorithms. Although embedding of large subproblems is essential for improving the accuracy of solutions in this scheme, the size of the subproblems are small in qbsolv since the subproblems are basically embedded by using an embedding of a complete graph even for sparse problem graphs. This means that the resource of the D-Wave quantum annealer is not exploited efficiently. In this paper, we propose a fast algorithm for embedding larger subproblems, and we show that better solutions are obtained efficiently by embedding larger subproblems.

Optimization of neural networks via finite-value quantum fluctuations.

We numerically test an optimization method for deep neural networks (DNNs) using quantum fluctuations inspired by quantum annealing. For efficient optimization, our method utilizes the quantum tunneling effect beyond the potential barriers. The path integral formulation of the DNN optimization generates an attracting force to simulate the quantum tunneling effect. In the standard quantum annealing method, the quantum fluctuations will vanish at the last stage of optimization. In this study, we propose a learning protocol that utilizes a finite value for quantum fluctuations strength to obtain higher generalization performance, which is a type of robustness. We demonstrate the performance of our method using two well-known open datasets: the MNIST dataset and the Olivetti face dataset. Although computational costs prevent us from testing our method on large datasets with high-dimensional data, results show that our method can enhance generalization performance by induction of the finite value for quantum fluctuations.

Quantum Speed Limit is Not Quantum.

The quantum speed limit (QSL), or the energy-time uncertainty relation, describes the fundamental maximum rate for quantum time evolution and has been regarded as being unique in quantum mechanics. In this study, we obtain a classical speed limit corresponding to the QSL using Hilbert space for the classical Liouville equation. Thus, classical mechanics has a fundamental speed limit, and the QSL is not a purely quantum phenomenon but a universal dynamical property of Hilbert space. Furthermore, we obtain similar speed limits for the imaginary-time Schrödinger equations such as the classical master equation.

Sparse modeling approach to analytical continuation of imaginary-time quantum Monte Carlo data.

A data-science approach to solving the ill-conditioned inverse problem for analytical continuation is proposed. The root of the problem lies in the fact that even tiny noise of imaginary-time input data has a serious impact on the inferred real-frequency spectra. By means of a modern regularization technique, we eliminate redundant degrees of freedom that essentially carry the noise, leaving only relevant information unaffected by the noise. The resultant spectrum is represented with minimal bases and thus a stable analytical continuation is achieved. This framework further provides a tool for analyzing to what extent the Monte Carlo data need to be accurate to resolve details of an expected spectral function.

Energy landscape analysis of neuroimaging data.

Computational neuroscience models have been used for understanding neural dynamics in the brain and how they may be altered when physiological or other conditions change. We review and develop a data-driven approach to neuroimaging data called the energy landscape analysis. The methods are rooted in statistical physics theory, in particular the Ising model, also known as the (pairwise) maximum entropy model and Boltzmann machine. The methods have been applied to fitting electrophysiological data in neuroscience for a decade, but their use in neuroimaging data is still in its infancy. We first review the methods and discuss some algorithms and technical aspects. Then, we apply the methods to functional magnetic resonance imaging data recorded from healthy individuals to inspect the relationship between the accuracy of fitting, the size of the brain system to be analysed and the data length.This article is part of the themed issue 'Mathematical methods in medicine: neuroscience, cardiology and pathology'.

Quantum Monte Carlo simulation of a particular class of non-stoquastic Hamiltonians in quantum annealing.

Quantum annealing is a generic solver of the optimization problem that uses fictitious quantum fluctuation. Its simulation in classical computing is often performed using the quantum Monte Carlo simulation via the Suzuki-Trotter decomposition. However, the negative sign problem sometimes emerges in the simulation of quantum annealing with an elaborate driver Hamiltonian, since it belongs to a class of non-stoquastic Hamiltonians. In the present study, we propose an alternative way to avoid the negative sign problem involved in a particular class of the non-stoquastic Hamiltonians. To check the validity of the method, we demonstrate our method by applying it to a simple problem that includes the anti-ferromagnetic XX interaction, which is a typical instance of the non-stoquastic Hamiltonians.

Conflict between fastest relaxation of a Markov process and detailed balance condition.

We consider the optimization of Markovian dynamics to pursue the fastest convergence to the stationary state. The brachistochrone method is applied to the continuous-time master equation for finite-size systems. The principle of least action leads to a brachistochrone equation for the transition-rate matrix. Three-state systems are explicitly analyzed, and we find that the solution violates the detailed balance condition. The properties of the solution are studied in detail to observe the optimality of the solution. We also discuss the counterdiabatic driving for the Markovian dynamics. The transition-rate matrix is then divided into two parts, and the state is given by an eigenstate of the first part. The second part violates the detailed balance condition and plays the role of a counterdiabatic term.

Langevin dynamics neglecting detailed balance condition.

An improved method for driving a system into a desired distribution, for example, the Gibbs-Boltzmann distribution, is proposed, which makes use of an artificial relaxation process. The standard techniques for achieving the Gibbs-Boltzmann distribution involve numerical simulations under the detailed balance condition. In contrast, in the present study we formulate the Langevin dynamics, for which the corresponding Fokker-Planck operator includes an asymmetric component violating the detailed balance condition. This leads to shifts in the eigenvalues and results in the acceleration of the relaxation toward the steady state. The numerical implementation demonstrates faster convergence and shorter correlation time, and the technique of biased event sampling, Nemoto-Sasa theory, further highlights the efficacy of our method.

Full-order fluctuation-dissipation relation for a class of nonequilibrium steady states.

Acceleration of relaxation toward a fixed stationary distribution via violation of detailed balance was reported in the context of a Markov chain Monte Carlo method recently. Inspired by this result, systematic methods to violate detailed balance in Langevin dynamics were formulated by using exponential and rotational nonconservative forces. In the present paper, we accentuate that such specific nonconservative forces relate to the large deviation of total heat in an equilibrium state. The response to these nonconservative forces can be described by the intrinsic fluctuation of the total heat in the equilibrium state. Consequently, the fluctuation-dissipation relation for nonequilibrium steady states is derived without recourse to a linear response approximation.

Violation of detailed balance accelerates relaxation.

Recent studies have experienced the acceleration of convergence in Markov chain Monte Carlo methods implemented by the systems without detailed balance condition (DBC). However, such advantage of the violation of DBC has not been confirmed in general. We investigate the effect of the absence of DBC on the convergence toward equilibrium. Surprisingly, it is shown that the DBC violation always makes the relaxation faster. Our result implies the existence of a kind of thermodynamic inequality that connects the nonequilibrium process relaxing toward steady state with the relaxation process which has the same probability distribution as its equilibrium state.

Duality with real-space renormalization and its application to bond percolation.

We obtain the exact solution of the bond-percolation thresholds with inhomogeneous probabilities on the square lattice. Our method is based on the duality analysis with real-space renormalization, which is a profound technique invented in the spin-glass theory. Our formulation is a more straightforward way than that of a very recent study on the same problem [R. M. Ziff et al., J. Phys. A: Math. Theor. 45, 494005 (2012)]. The resultant generic formulas from our derivation can give several estimations for the bond-percolation thresholds on other lattices rather than the square lattice.