Quantum Spin Glasses Quantum Annealing, and Probabilistic Information Processing

Saika

Okada

2009

J. Phys.: Conf. Ser.

Self Cite

Abstract. We numerically examine a quantum version of TAP (Thouless-Anderson-Palmer)-like mean-field algorithm for the problem of error-correcting codes. For a class of the so-called Sourlas error-correcting codes, we check the usefulness to retrieve the original bit-sequence (message) with a finite length. The decoding dynamics is derived explicitly and we evaluate the average-case performance through the bit-error rate (BER). IntroductionStatistical mechanics of information has been applied to a lot of problems in various research fields of information science and technology [1,2]. Among them, error-correcting code is one of the most developed subjects. In the research field of error-correcting codes, Sourlas showed that the convolutional codes can be constructed by spin glass with infinite range p-body interactions and the decoded message should be corresponded to the ground state of the Hamiltonian [3]. Ruján suggested that the bit error can be suppressed if one uses finite temperature equilibrium states as the decoding result, instead of the ground state [4], and the so-called Bayes-optimal decoding at some specific condition was proved by Nishimori [5] and Nishimori and Wong [6]. Kabashima and Saad succeeded in constructing more practical codes, namely, low density parity check (LDPC) codes by using the infinite range spin glass model with finite connectivities [7]. They used the so-called TAP (Thouless-Anderson-Palmer) equations to decode the original message for a given parity check.As we shall see later on, an essential key point to obtain the Bayes-optimal solution is controlling the 'thermal fluctuation' in order to satisfy the condition on the Nishimori line (the so-called Nishimori-Wong condition [6]). Then, a simple question is arisen, namely, is it possible to obtain the Bayes-optimal solution by means of the 'quantum fluctuation' induced by tunneling effects? or what is condition for the optimal control of the fluctuation?To answer these questions, Tanaka and Horiguchi introduced a quantum fluctuation into the mean-field annealing algorithm and showed that performance of image restoration is improved by controlling the quantum fluctuation appropriately during its annealing process [8,9]. The average-case performance is evaluated analytically by one of the present authors [10]. However,

Section: Replica Analysis For Finite P At Zero Temperaturesupporting

confidence: 52%

“…However, in the corresponding quantum system, the condition is not yet clarified. In our preliminary study [11], we considered the quantum version of the posterior by modifying the Hamiltonian aŝ…”

Section: Quantum Systemmentioning

confidence: 99%

Quantum mean-field decoding algorithm for error-correcting codes

Saika

Okada

2009

J. Phys.: Conf. Ser.

Self Cite

“…For various choices of parameters {J ij }, h and {τ }, one can model the problem of information processing appropriately. For instance, for the choice of J ij = J, h = 0, the above Hamiltonian describes image restoration [26,28,29] from a given set of degraded images τ = (τ 1 , · · · , τ N ) by means of Markov random fields. For J ij = 0, h = 0, it corresponds to the logarithm of the posterior of the Sourlas codes [32,33] sending the parity check of two-body interactions ξ i ξ j ∀(i, j) through the binary symmetric channel (BSC).…”

Section: The Model System and Formulationmentioning

confidence: 99%

“…. Disordered systems: An application for Statistical-Mechanical Informatics It is easy for us to extend the above formulation to some class of disordered spin systems, that is to say, the infinite-range random field Ising model which is often used to check the performance of image restoration analytically[26,28,29] as a bench mark test.…”

mentioning

confidence: 99%

Deterministic flows of order-parameters in stochastic processes of quantum Monte Carlo method

2010

J. Phys.: Conf. Ser.

Self Cite

In terms of the stochastic process of quantum-mechanical version of Markov chain Monte Carlo method (the MCMC), we analytically derive macroscopically deterministic flow equations of order parameters such as spontaneous magnetization in infinite-range (d(= ∞)dimensional) quantum spin systems. By means of the Trotter decomposition, we consider the transition probability of Glauber-type dynamics of microscopic states for the corresponding (d + 1)-dimensional classical system. Under the static approximation, differential equations with respect to macroscopic order parameters are explicitly obtained from the master equation that describes the microscopic-law. In the steady state, we show that the equations are identical to the saddle point equations for the equilibrium state of the same system. The equation for the dynamical Ising model is recovered in the classical limit. We also check the validity of the static approximation by making use of computer simulations for finite size systems and discuss several possible extensions of our approach to disordered spin systems for statisticalmechanical informatics. Especially, we shall use our procedure to evaluate the decoding process of Bayesian image restoration. With the assistance of the concept of dynamical replica theory (the DRT), we derive the zero-temperature flow equation of image restoration measure showing some 'non-monotonic' behaviour in its time evolution.

“…As we already pointed out, error-correcting codes of Sourlas type are described by spin glass with p-spin interactions and the model put in the transverse field was already investigated by [32,33]. However, recently it was revisited to investigate the thermodynamic properties from the computer scientific point of view [34,35,36].…”

Section: Several Case Studiesmentioning

confidence: 99%

Infinite-range transverse field Ising models and quantum computation

Eur. Phys. J. Spec. Top.

2015

We present a brief review on information processing, computing and inference via quantum fluctuation, and clarify the relationship between the probabilistic information processing and theory of quantum spin glasses through the analysis of the infinite-range model. We also argue several issues to be solved for the future direction in the research field.(k B : Boltzmann constant) to decrease to zero slowly enough during the Markov Chain Monte Carlo method for a given problem (the cost function or Hamiltonian). Then, the energy of the system does not decrease monotonically and sometimes it increases a