Quantum Metropolis-Hastings algorithm with the target distribution calculated by quantum Monte Carlo integration

The Markov chain Monte Carlo method (MCMC), especially the Metropolis-Hastings (MH) algorithm, is a widely used technique for sampling from a target probability distribution $P$ on a state space $\Omega$ and applied to various problems such as estimation of parameters in statistical models in the Bayesian approach. Quantum algorithms for MCMC have been proposed, yielding the quadratic speedup with respect to the spectral gap $\Delta$ compered to classical counterparts. In this paper, we consider the quantum version of the MH algorithm in the case that calculating $P$ is costly because the log-likelihood $L$ for a state $x\in\Omega$ is obtained via computing the sum of many terms $\frac{1}{M}\sum_{i=0}^{M-1} \ell(i,x)$. We propose calculating $L$ by quantum Monte Carlo integration and combine it with the existing method called quantum simulated annealing (QSA) to generate the quantum state that encodes $P$ in amplitudes. We consider not only state generation but also finding a credible interval for a parameter, a common task in Bayesian inference. In the proposed method for credible interval calculation, the number of queries to the quantum circuit to compute $\ell$ scales on $\Delta$, the required accuracy $\epsilon$ and the standard deviation $\sigma$ of $\ell$ as $\tilde{O}(\sigma/\epsilon^2\Delta^{3/2})$, in contrast to $\tilde{O}(M/\epsilon\Delta^{1/2})$ for QSA with $L$ calculated exactly. Therefore, the proposed method is advantageous if $\sigma$ scales on $M$ sublinearly. As one such example, we consider parameter estimation in a gravitational wave experiment, where $\sigma=O(M^{1/2})$

Extracting a function encoded in amplitudes of a quantum state by tensor network and orthogonal function expansion

There are quantum algorithms for finding a function $f$ satisfying a set of conditions, such as solving partial differential equations, and these achieve exponential quantum speedup compared to existing classical methods, especially when the number $d$ of the variables of $f$ is large. In general, however, these algorithms output the quantum state which encodes $f$ in the amplitudes, and reading out the values of $f$ as classical data from such a state can be so time-consuming that the quantum speedup is ruined. In this study, we propose a general method for this function readout task. Based on the function approximation by a combination of tensor network and orthogonal function expansion, we present a quantum circuit and its optimization procedure to obtain an approximating function of $f$ that has a polynomial number of degrees of freedom with respect to $d$ and is efficiently evaluable on a classical computer. We also conducted a numerical experiment to approximate a finance-motivated function to demonstrate that our method works.Comment: 16 pages, 8 figure

B-mode polarization induced by gravitational waves from kinks on infinite cosmic strings

We investigate the effect of the stochastic gravitational wave (GW) background produced by kinks on infinite cosmic strings, whose spectrum was derived in our previous work, on the B-mode power spectrum of the cosmic microwave background (CMB) anisotropy. We find that the B-mode polarization due to kinks is comparable to that induced by the motion of the string network and hence the contribution of GWs from kinks is important for estimating the B-mode power spectrum originating from cosmic strings. If the tension of cosmic strings \mu is large enough i.e., G\mu >~ 10^{-8}, B-mode polarization induced by cosmic strings can be detected by future CMB experiments.Comment: 13 pages, 1 figur

Gravitational waves from kinks on infinite cosmic strings

Gravitational waves emitted by kinks on infinite strings are investigated using detailed estimations of the kink distribution on infinite strings. We find that gravitational waves from kinks can be detected by future pulsar timing experiments such as SKA for an appropriate value of the the string tension, if the typical size of string loops is much smaller than the horizon at their formation. Moreover, the gravitational wave spectrum depends on the thermal history of the Universe and hence it can be used as a probe into the early evolution of the Universe.Comment: 29 pages, 4figure