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arXiv:1708.00236 · Physical Review A

Quantum annealing with a nonvanishing final value of the transverse field

Kohji Nishimura, Hidetoshi Nishimori

Published Oct 10, 2017• 6 citations
arXivHTMLPDF

Abstract

We study the problem to infer the original ground state of a spin-glass Hamiltonian out of the information from the Hamiltonian with interactions deviated from the original ones. Our motivation comes from quantum annealing on a real device in which the values of interactions are degraded by noise. We show numerically for quasi-one-dimensional systems that the Hamming distance between the original ground state and the inferred spin state is minimized when we stop the process of quantum annealing before the amplitude of the transverse field reaches zero in contrast to the conventional prescription. This result means that finite quantum fluctuations compensate for the effects of noise, at least, to some extent. Analytical calculations using the infinite-range mean-field model support our conclusion qualitatively.

DOI
10.1103/physreva.96.042310

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•Mar 24, 2014•0 citations
We study the problem to infer the original ground state of a spin-glass Hamiltonian out of the information from the Hamiltonian with interactions deviated from the original ones. Our motivation comes from quantum annealing on a real device in which the values of interactions are degraded by noise. We show numerically for quasi-one-dimensional systems that the Hamming distance between the original ground state and the inferred spin state is minimized when we stop the process of quantum annealing before the amplitude of the transverse field reaches zero in contrast to the conventional prescription. This result means that finite quantum fluctuations compensate for the effects of noise, at least, to some extent. Analytical calculations using the infinite-range mean-field model support our conclusion qualitatively.
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.
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 in finding 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 for which 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.
Although quantum annealing is usually considered as a method for locating the ground states of difficult spin-glass and optimization problems, its use in approximate optimization -- finding low- but not zero-energy states in a reasonably short amount of time -- is no less important. Here we investigate the behavior of quantum annealing at approximate optimization in the canonical mean-field spin-glass models, the spherical ppp-spin models, and find that it performs surprisingly well. Whereas it had long been assumed that infinite-range spin glasses have a unique ``threshold'' energy at which all quench and annealing dynamics become trapped until exponential timescales, recent work has shown that two-stage quenches can in fact reach states below the naive threshold in more generic situations. We demonstrate that quantum annealing is also capable of exploiting this effect to locate sub-threshold states in O(1)O(1)O(1) time. Not only can it attain energies as far below the threshold as classical annealing algorithms, but it can do so significantly faster: for an annealing schedule taking time τττ, the residual energy under quantum annealing decays as τ−ατ^{-α}τ−α with an exponent up to twice as large as that of simulated annealing in the cases considered. Importantly, by deriving and numerically solving closed integro-differential equations that hold in the thermodynamic limit, our results are free from finite-size effects and hold for annealing times that are unambiguously independent of system size.
Quantum annealing represents an essential milestone towards the goal of adiabatic quantum computing. In quantum annealing, the computation involves finding the ground state of a classical Ising-like Hamiltonian realized as interactions between qubits. Quantum fluctuations are introduced to allow the wavefunction of the qubits to explore the energy landscape, the hope being that the wavefunction finds a minimum energy configuration and possibly giving the result of the computation. While quantum annealing likely may not be as powerful as adiabatic quantum computing, it is possible that it may be better at optimization compared to analogous classical algorithms. In physical realizations of quantum annealing, there are still questions as to the role of quantum fluctuations in the operation of a device given the short coherence times of the individual qubits. These questions have consistently posed a serious theoretical challenge making it difficult to verify experimental results. Here we simplify the problem by considering a system of qubits with ferromagnetic interactions, modeling the decoherence effects as classical noise in the transverse-field of each qubit. We compare the calculations to data collected from a system of manufactured qubits produced by D-wave Systems by performing a finite-size scaling analysis that captures the competition between quantum fluctuations of the transverse-field and bit-flip errors from the noise. We argue that on time-scales larger than the single-qubit decoherence time, the device produces the expected quantum fluctuations for the many-body system. Using this finite-size scaling, one can diagnose sources of noise in the system. Hopefully, in the near future, these devices will not only be realizing coherent quantum annealing but will likely be useful as another example of synthetic quantum matter.
This thesis provides theoretical studies on a probabilistic information processing incorporated quantum fluctuations through statistical mechanical approaches.Quantum fluctuations cause transitions between states, which play same role as thermal fluctuations.Such an idea has been used in optimization problems for searching a ground state of a cost function.The algorithm is called quantum annealing (QA) in a manner analogous to simulated annealing.Simulated annealing is the conventional algorithm for finding the ground state by using the thermal fluctuations.QA has been applied to various optimization problems by solving the Schrödinger equation or carrying out Quantum Monte Carlo simulations on classical computers.However, what we call a quantum annealer with current superconducting devices has been launched by D-wave systems.For this situation, the algorithms by using the quantum fluctuations have received a lot of attention in recent years.With the developments in the research fields of the algorithms by making use of the quantum fluctuations, it is also interesting for us to consider the possible application of the quantum fluctuations to probabilistic information processing.Main problem of the probabilistic information processing in the context of this thesis is to recover the original information from damaged information, e.g., image restoration, error correcting codes, and CDMA multiuser demodulation.Such problems can be denoted as the infiniterange spin glass model, adopting the framework of Bayesian statistics.Then, we can apply statistical mechanical approaches to it in order to figure out the average-case performance of those systems.We examine the average-case performance of the probabilistic information processing in which quantum fluctuations is utilized to recover the original information in the context of the Bayesian inference.The quantum fluctuations are built into the system as transverse fields put in the infinite-range Ising spin glass model.We evaluate the performance measurement by using statistical mechanics which mean mean-field theory.Accordingly, we find the following properties on the quantum fluctuations in the probabilistic information processing