Applications

Abstract

It has been well known that there is an intriguing analogy between statistical mechanics of Ising spin glasses and information processing such as the associative memory, the image restoration, the pattern recognition, and etc. As a natural extension of the statistical mechanical model of information processing, one can consider a quantum mechanical model of information processing using transverse Ising models. Chapter 9 discusses such an application of transverse Ising models to information processing. The first part focuses on the Hopfield model in a transverse field for the associative memory, where the extraction of embedded patterns using quantum fluctuations is discussed. The second part of this chapter is devoted to the investigation of image restorations and error correcting codes by Bayesian statistics with quantum fluctuations. It mentions the MPM estimate as well the MAP estimate which respectively correspond to the Bayesian estimation schemes by tuning the amount of quantum fluctuations and by quantum annealing. Appendix includes details of the Hopfield model in a transverse field.

Appendix 9.A: Derivation of Saddle Point Equations for the Quantum Hopfield Model

In this appendix, according to Nishimori and Nonomura [298], we derive the saddle point equations (9.1.9)–(9.1.15) for the Hopfield model at zero temperature. We starts our argument from the Suzuki-Trotter decomposition:

$$ Z = \lim_{M \to\infty} \operatorname{Tr} \bigl( \mathrm{e}^{-\beta H_{0}/M} \, \mathrm{e}^{-\beta H_{1}/M} \bigr)^{M}=\lim_{M \to\infty} Z_{M} $$

(9.A.1)

where we defined the two parts of the Hamiltonian as

$$ H_{0}= -\frac{1}{N} \sum_{ij} \sum_{\mu=1}^{P} \xi_{i}^{\mu} \xi_{j}^{\mu} S_{i}^{z}S_{j}^{z}, \qquad H_{1} = -\varGamma\sum_{i}S_{i}^{x}. $$

(9.A.2)

Z _M is given by

(9.A.3)

with

$$ B = \frac{1}{2} \log\mathrm{cosec} \biggl( \frac{\beta\varGamma}{M} \biggr). $$

(9.A.4)

To average out the pattern-dependence in the self-averaging quantity such as free energy, we use the replica trick:

$$ {\ll}\log Z_{M} {\gg}= \lim_{n \to0} \frac{{\ll} Z_{M} {\gg}-1}{n}. $$

(9.A.5)

Then, we have

(9.A.6)

Here we assume that for the first s patterns, namely, for the so-called ‘condensed patterns’, the quantity $(\beta/M) \sum_{K\rho} \sum_{\mu=1}^{s} m_{K \mu\rho} \sum_{i}\xi_{i}^{\mu} S_{i\sigma K}$ is of order 1 object, whereas for the un-condensed patterns μ=s,…,p, the quantity $(\beta/M) \sum_{K \rho} \sum_{\mu=s+1}^{P} \sum_{i}\xi_{i}^{\mu} S_{i\rho K}$ is of order $1/\sqrt{N}$. Hence, we can rewrite (9.A.6) as

(9.A.7)

Collecting the quadratic part in (9.A.6), we obtain

$$ \prod_{\mu} \exp \biggl[ -\frac{N\beta}{2M^{2}} \sum _{K \rho} \sum_{L \sigma} m_{K \mu\rho}^{2} + \frac{\beta^{2}}{2M^{2}} \sum _{K\rho} \sum_{L \sigma} m_{K \mu\rho} m_{L \mu\sigma} \sum_{i} S_{i \rho K} S_{i\sigma L} \biggr] = \prod_{\mu}E_{\mu} $$

(9.A.8)

where we defined

$$ E_{\mu} = \exp \biggl[ -\frac{N \beta}{2M} \sum _{\mu K \rho} {\varLambda}_{K \rho,L\sigma} m_{K \mu\rho} m_{L \mu\sigma} \biggr] $$

(9.A.9)

with (Mn×Mn)-matrix:

(9.A.10)

and

$$ q_{\rho\sigma}(KL) = \frac{1}{N} \sum_{i}S_{i \rho K} S_{i \sigma L}, \qquad \overline{S}_{q} (KL) = \frac{1}{N} \sum_{i} S_{i \rho K} S_{i \sigma L}. $$

(9.A.11)

Thus, we immediately obtain

(9.A.12)

where λ _Kρ are the eigenvalues of the Λ _Kρ,Lσ . Inserting the definitions (9.A.11) by using the identities:

(9.A.13)

(9.A.14)

We write the factor of unity G{λ _Kρ }=1 as

(9.A.15)

where we used the Fourier transform of the delta function. Multiplying the G{λ _Kρ }=1 by ≪Z ⁿ≫, one obtains

(9.A.16)

where we used self-averaging properties in the limit of N→∞ and dropped the site i dependence. f denotes the free energy per spin, which is given by

(9.A.17)

where T is temperature T=β ⁻¹. Extremum conditions for the free energy give the following saddle points:

(9.A.18)

(9.A.19)

(9.A.20)

(9.A.21)

(9.A.22)

where we should notice that m _Kμρ , q _σρ (KL) and r _ρσ (KL) are overlap, spin glass order parameter and noise from un-condensed patterns, respectively. These are the same as in the classical Hopfield model. On the other hand, $\overline{S}_{\rho}(KL)$ and t _ρ (KL) represent quantum effects. The former represents the degree of quantum fluctuation by the deviation from unity and the latter denotes the effect of un-correlated patterns in the same replica.

9.1.1 9.A.1 Replica Symmetric and Static Approximation

To proceed the calculations, we assume the replica symmetric and static approximations:

(9.A.23)

(9.A.24)

(9.A.25)

(9.A.26)

(9.A.27)

namely, we consider the case in which these order parameters are independent on the replica and Trotter indexes. Then, we have the free energy per replica as follows

(9.A.28)

By using the Hubbard-Stratonovich transformation, we obtain in the limit of n→0 as

(9.A.29)

We next consider the term (1/n)logλ _Kρ in (9.A.28). The matrix elements {Λ _Kρ,Lσ } are −β/M (ρ≠σ), $-\beta\overline{S}/M$ (ρ=σ, K≠L) and 1−β/M (ρ=σ, K=L). Then, the eigenvalues are given by

(9.A.30)

(9.A.31)

(9.A.32)

Thus, one can evaluate

(9.A.33)

Collecting therms in (9.A.28), (9.A.29) and (9.A.33), we obtain

(9.A.34)

For simplicity, we now set m _μ =δ _μ,1 m,ξ _μ =δ _μ,1, namely, we consider the case in which a single pattern is recalled. The extremum conditions gives the following saddle point equations.

(9.A.35)

(9.A.36)

(9.A.37)

(9.A.38)

(9.A.39)

where we defined

(9.A.40)

(9.A.41)

9.1.2 9.A.2 Zero Temperature Limit

At the ground state T→0, we naturally expect that $\overline{S}=q$. For convenience, we introduce the parameter C as

$$ C \equiv \lim_{\beta\to\infty} \beta(\overline{S}-q) = \varGamma^{2} \int \frac{Dz}{ [(m + \sqrt{\alpha r}\,z)^{2} + \varGamma^{2}]^{3/2}}. $$

(9.A.42)

Then, we can rewrite the saddle point equations in terms of C as

(9.A.43)

(9.A.44)

(9.A.45)