arXiv subjects

Hidetoshi Nishimori

Publications and source records attributed to Hidetoshi Nishimori.

At least 55 records · Page 3Linked to original sources

arXiv2017

Relation between quantum fluctuations and the performance enhancement of quantum annealing in a nonstoquastic Hamiltonian

We study the relation between quantum fluctuations and the significant enhancement of the performance of quantum annealing in a mean-field Hamiltonian. First-order quantum phase transitions were shown to be reduced to second order by antiferromagnetic transverse interactions in a mean-field-type many-body-interacting Ising spin system in a transverse field, which means an exponential speedup of quantum annealing by adiabatic quantum computation. We investigate if and how quantum effects manifest themselves around these first- and second-order phase transitions to understand if the antiferromagnetic transverse interactions appended to the conventional transverse-field Ising model induce notable quantum effects. By measuring the proximity of the semiclassical spin-coherent state to the true ground state as well as the magnitude of the concurrence representing entanglement, we conclude that significant quantum fluctuations exist around second-order transitions, whereas quantum effects are much less prominent at first-order transitions. Although the location of the transition point can be predicted by the classical picture, system properties near the transition need quantum-mechanical descriptions for a second-order transition but not necessarily for first order. It is also found that quantum fluctuations are large within the ferromagnetic phase after a second-order transition from the paramagnetic phase. These results suggest that the antiferromagnetic transverse interactions induce marked quantum effects, and this fact would be related to closely to the significant enhancement of the performance of quantum annealing.

quant-ph↗

arXiv2017

Exponential Enhancement of the Efficiency of Quantum Annealing by Non-Stochastic Hamiltonians

Non-stoquastic Hamiltonians have both positive and negative signs in off-diagonal elements in their matrix representation in the standard computational basis and thus cannot be simulated efficiently by the standard quantum Monte Carlo method due to the sign problem. We describe our analytical studies of this type of Hamiltonians with infinite-range non-random as well as random interactions from the perspective of possible enhancement of the efficiency of quantum annealing or adiabatic quantum computing. It is shown that multi-body transverse interactions like $XX$ and $XXXXX$ with positive coefficients appended to a stoquastic transverse-field Ising model render the Hamiltonian non-stoquastic and reduce a first-order quantum phase transition in the simple transverse-field case to a second-order transition. This implies that the efficiency of quantum annealing is exponentially enhanced, because a first-order transition has an exponentially small energy gap (and therefore exponentially long computation time) whereas a second-order transition has a polynomially decaying gap (polynomial computation time). The examples presented here represent rare instances where strong quantum effects, in the sense that they cannot be efficiently simulated in the standard quantum Monte Carlo, have analytically been shown to exponentially enhance the efficiency of quantum annealing for combinatorial optimization problems.

quant-ph↗

arXiv2016

Quantum annealing correction at finite temperature: ferromagnetic $p$-spin models

The performance of open-system quantum annealing is adversely affected by thermal excitations out of the ground state. While the presence of energy gaps between the ground and excited states suppresses such excitations, error correction techniques are required to ensure full scalability of quantum annealing. Quantum annealing correction (QAC) is a method that aims to improve the performance of quantum annealers when control over only the problem (final) Hamiltonian is possible, along with decoding. Building on our earlier work [S. Matsuura et al., Phys. Rev. Lett. 116, 220501 (2016)], we study QAC using analytical tools of statistical physics by considering the effects of temperature and a transverse field on the penalty qubits in the ferromagnetic $p$-body infinite-range transverse-field Ising model. We analyze the effect of QAC on second ($p=2$) and first ($p\geq 3$) order phase transitions, and construct the phase diagram as a function of temperature and penalty strength. Our analysis reveals that for sufficiently low temperatures and in the absence of a transverse field on the penalty qubit, QAC breaks up a single, large free energy barrier into multiple smaller ones. We find theoretical evidence for an optimal penalty strength in the case of a transverse field on the penalty qubit, a feature observed in QAC experiments. Our results provide further compelling evidence that QAC provides an advantage over unencoded quantum annealing.

quant-ph↗

arXiv2016

Critical properties of dissipative quantum spin systems in finite dimensions

We study the critical properties of finite-dimensional dissipative quantum spin systems with uniform ferromagnetic interactions. Starting from the transverse-field Ising model coupled to a bath of harmonic oscillators with Ohmic spectral density, we generalize its classical representation to classical spin systems with $O(n)$ symmetry and then take the large-$n$ limit to reduce the system to the spherical model. The exact solution to the resulting spherical model with long-range interactions along the imaginary-time axis shows a phase transition with static critical exponents coinciding with those of the conventional short-range spherical model in $d+2$ dimensions, where $d$ is the spatial dimensionality of the original quantum system. This implies the dynamical exponent to be $z=2$. These conclusions are consistent with the results of Monte Carlo simulations and renormalization group calculations for dissipative transverse-field Ising and $O(n)$ models in one and two dimensions. The present approach therefore serves as a useful tool to analytically investigate the properties of quantum phase transitions of the dissipative transverse-field Ising and related models. Our method may also offer a platform to study more complex phase transitions in dissipative finite-dimensional quantum spin systems, which recently receive renewed interest under the context of quantum annealing in a noisy environment.

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arXiv2016

Retrieving the ground state of spin glasses using thermal noise: Performance of quantum annealing at finite temperatures

We study the problem to infer the ground state of a spin-glass Hamiltonian using data from another Hamiltonian with interactions disturbed by noise from the original Hamiltonian, motivated by the ground-state inference in quantum annealing on a noisy device. It is shown that the average Hamming distance between the inferred spin configuration and the true ground state is minimized when the temperature of the noisy system is kept at a finite value, and not at zero temperature. We present a spin-glass generalization of a well-established result that the ground state of a purely ferromagnetic Hamiltonian is best inferred at a finite temperature in the sense of smallest Hamming distance when the original ferromagnetic interactions are disturbed by noise. We use the numerical transfer-matrix method to establish the existence of an optimal finite temperature in one- and two-dimensional systems. Our numerical results are supported by mean-field calculations, which give an explicit expression of the optimal temperature to infer the spin-glass ground state as a function of variances of the distributions of the original interactions and the noise. The mean-field prediction is in qualitative agreement with numerical data. Implications on postprocessing of quantum annealing on a noisy device are discussed.

cond-mat.dis-nn↗

arXiv2016

Mean Field Analysis of Quantum Annealing Correction

Quantum annealing correction (QAC) is a method that combines encoding with energy penalties and decoding to suppress and correct errors that degrade the performance of quantum annealers in solving optimization problems. While QAC has been experimentally demonstrated to successfully error-correct a range of optimization problems, a clear understanding of its operating mechanism has been lacking. Here we bridge this gap using tools from quantum statistical mechanics. We study analytically tractable models using a mean-field analysis, specifically the $p$-body ferromagnetic infinite-range transverse-field Ising model as well as the quantum Hopfield model. We demonstrate that for $p=2$, where the phase transition is of second order, QAC pushes the transition to increasingly larger transverse field strengths. For $p\ge3$, where the phase transition is of first order, QAC softens the closing of the gap for small energy penalty values and prevents its closure for sufficiently large energy penalty values. Thus QAC provides protection from excitations that occur near the quantum critical point. We find similar results for the Hopfield model, thus demonstrating that our conclusions hold in the presence of disorder.

quant-ph↗

arXiv2015

Anomalous behavior of the energy gap in the one-dimensional quantum XY model

We show for the one-dimensional quantum $XY$ model with $s=1/2$ that the energy gap between the ground and first excited states behaves anomalously as a function of the system size at the first-order quantum phase transition. Although it is generally the case that the energy gap closes exponentially at a quantum first-order transition, the gap in the present model behaves non-monotonically as a function of the system size, apparently very irregularly in some cases. This property of the gap is similar to that of the infinite-range quantum $XY$ model, in which the gap closes polynomially, exponentially, or even factorially fast depending on the choice of the series of system sizes toward the thermodynamic limit. This observation is surprising in consideration of the apparent maturity of our understanding of the one-dimensional quantum $XY$ model. Our result is also important from the viewpoint of quantum annealing, where the rate of gap closing determines the efficiency of computation.

quant-ph↗

arXiv2015

Quantum Annealing with Antiferromagnetic Transverse Interactions for the Hopfield Model

We investigate quantum annealing with antiferromagnetic transverse interactions for the generalized Hopfield model with $k$-body interactions. The goal is to study the effectiveness of antiferromagnetic interactions, which were shown to help us avoid problematic first-order quantum phase transitions in pure ferromagnetic systems, in random systems. We estimate the efficiency of quantum annealing by analyzing phase diagrams for two cases where the number of embedded patterns is finite or extensively large. The phase diagrams of the model with finite patterns show that there exist annealing paths that avoid first-order transitions at least for $5 \le k \le 21$. The same is true for the extensive case with $k=4$ and $5$. In contrast, it is impossible to avoid first-order transitions for the case of finite patterns with $k=3$ and the case of extensive number of patterns with $k=2$ and $3$. The spin-glass phase hampers the quantum annealing process in the case of $k=2$ and extensive patterns. These results indicate that quantum annealing with antiferromagnetic transverse interactions is efficient also for certain random spin systems.

cond-mat.stat-mech↗

arXiv2015

Anomalous behavior of the energy gap in the one-dimensional quantum XY model

We re-examine the well-studied one dimensional spin-1/2 $XY$ model to reveal its nontrivial energy spectrum, in particular the energy gap between the ground state and the first excited state. In the case of the isotropic $XY$ model -- the $XX$ model -- the gap behaves very irregularly as a function of the system size at a second order transition point. This is in stark contrast to the usual power-law decay of the gap and is reminiscent of the similar behavior at the first order phase transition in the infinite-range quantum $XY$ model. The gap also shows nontrivial oscillatory behavior for the phase transitions in the anisotropic model in the incommensurate phase. We observe a close relation between this anomalous behavior of the gap and the correlation functions. These results, those for the isotropic case in particular, are important from the viewpoint of quantum annealing where the efficiency of computation is strongly affected by the size dependence of the energy gap.

quant-ph↗

arXiv2015

Adiabatic approximation for the imaginary-time Schroedinger equation and its application to simulated annealing

We formulate an adiabatic approximation for the imaginary-time Schroedinger equation. The obtained adiabatic condition consists of two inequalities, one of which coincides with the conventional adiabatic condition for the real-time Schroedinger equation, but the other does not. We apply this adiabatic approximation to the analysis of Markovian dynamics of the classical Ising model, which can be formulated as the imaginary-time Schrödinger equation, to obtain an asymptotic formula for the probability that the system reaches the ground state in the limit of a long annealing time in simulated annealing. Using this form, we amend the theory of Somma, Batista, and Ortiz for a convergence condition for simulated annealing.

quant-ph↗

arXiv2014

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.

quant-ph↗

arXiv2014

Mean-field theory is exact for the random-field model with long-range interactions

We study the random-field Ising model with long-range interactions and show the exactness of the mean-field theory under certain mild conditions. This is a generalization of the result of Mori for the non-random and spin-glass cases. To treat random fields, we evoke the self-averaging property of a function of random fields, without recourse to the replica method. The result is that the mean-field theory gives the exact expression of the canonical free energy for systems with power-decaying interactions if the power is smaller than or equal to the spatial dimension.

cond-mat.stat-mech↗

arXiv2013

Energy gap at first-order quantum phase transitions: An anomalous case

We show that the rate of closing of the energy gap between the ground state and the first excited state, as a function of system size, behaves in many qualitatively different ways at first-order quantum phase transitions of the infinite-range quantum XY model. Examples include polynomial, exponential and even factorially fast closing of the energy gap, all of which coexist along a single axis of the phase diagram representing the transverse field. This variety emerges depending on whether or not the transverse field assumes a rational number as well as on how the series of system size is chosen toward the thermodynamic limit. We conclude that there is no generically applicable rule to relate the rate of gap closing and the order of quantum phase transitions as is often implied in many studies, in particular in relation to the computational complexity of quantum annealing in its implementation as quantum adiabatic computation.

quant-ph↗

arXiv2013

Real-space renormalization-group approach to the random transverse-field Ising model in finite dimensions

The transverse-field Ising models with random exchange interactions in finite dimensions are investigated by means of a real-space renormalization-group method. The scheme yields the exact values of the critical point and critical exponent nu in one dimension and some previous results in the case of random ferromagnetic interactions are reproduced in two and three dimensions. We apply the scheme to spin glasses in transverse fields in two and three dimensions, which have not been analyzed very extensively. The phase diagrams and the critical exponent nu are obtained and evidence for the existence of an infinite-randomness fixed point in these models is found.

cond-mat.dis-nn↗

arXiv2013

Nonadiabatic Quantum Annealing for One-Dimensional Trasverse-Field Ising Model

We propose a nonadiabatic approach to quantum annealing, in which we repeat quantum annealing in nonadiabatic time scales, and collect the final states of many realizations to find the ground state among them. In this way, we replace the diffculty of long annealing time in adiabatic quantum annealing by another problem of the number of nonsidabatic (short-time) trials. The one-dimensional transverse-field Ising model is used to test this idea, and it is shown that nonadiabatic quantum annealing has the same computational complexity to find the ground state as the conventional adiabatic annealing does. This result implies that the nonadiabatic method may be used to replace adiabatic annealing to avoid the effects of external disturbances, to which the adiabatic method is more prone than the nonadiabatic counterpart.

cond-mat.stat-mech↗

arXiv2012

Ensemble Inequivalence in the Spherical Spin Glass Model with Nonlinear Interactions

We investigate the ensemble inequivalence of the spherical spin glass model with nonlinear interactions of polynomial order $p$. This model is solved exactly for arbitrary $p$ and is shown to have first-order phase transitions between the paramagnetic and spin glass or ferromagnetic phases for $p \geq 5$. In the parameter region around the first-order transitions, the solutions give different results depending on the ensemble used for the analysis. In particular, we observe that the microcanonical specific heat can be negative and the phase may not be uniquely determined by the temperature.

cond-mat.stat-mech↗

arXiv2012

Many-body transverse interactions in the quantum annealing of the p-spin ferromagnet

We study the performance of quantum annealing for the simple $p$-body infinite-range ferromagnetic Ising model. In particular, we generalize the transverse antiferromagnetic interactions proposed by Seki and Nishimori as a quantum driver to many-body transverse interactions to understand if the two-body interactions are essential to allow the system to avoid troublesome first-order quantum phase transitions. We conclude that the general many-body interactions are effective to let the system evolve only through second-order transitions as long as a few minor conditions are satisfied. It is also discussed whether the overlap of the ground-state wave function of the new driver term with the target ground state is an essential factor for the success.

quant-ph↗

arXiv2012

Quantum annealing with antiferromagnetic fluctuations

We introduce antiferromagnetic quantum fluctuations into quantum annealing in addition to the conventional transverse-field term. We apply this method to the infinite-range ferromagnetic p-spin model, for which the conventional quantum annealing has been shown to have difficulties to find the ground state efficiently due to a first-order transition. We study the phase diagram of this system both analytically and numerically. Using the static approximation, we find that there exists a quantum path to reach the final ground state from the trivial initial state that avoids first-order transitions for intermediate values of p. We also study numerically the energy gap between the ground state and the first excited state and find evidence for intermediate values of p that the time complexity scales polynomially with the system size at a second-order transition point along the quantum path that avoids first-order transitions. These results suggest that quantum annealing would be able to solve this problem with intermediate values of p efficiently in contrast to the case with only simple transverse-field fluctuations.

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