arXiv subjects

Koichi Miyamoto

Publications and source records attributed to Koichi Miyamoto.

At least 19 recordsLinked to original sources

arXiv2026

Calculating the squeezed bispectrum in stochastic inflation by Monte Carlo simulation

The bispectrum of cosmological perturbations is an important probe of inflation and the underlying particle physics. However, calculating it becomes challenging in inflation models where inflaton dynamics is dominated by quantum diffusion rather than slow-roll, especially in the multi-field case. In this paper, employing the stochastic-$δ\mathcal{N}$ formalism, we propose a Monte Carlo-based method to calculate the bispectrum in the squeezed limit, as an extension of the method for the power spectrum that we previously proposed. Our method involves generating paths of inflatons' time evolution that branch only several times, avoiding nested path generation, which incurs a prohibitive computational cost. As numerical demonstrations, we apply the proposed method to single- and double-field chaotic inflation for validation, and to hybrid inflation with mild waterfall, for which, to the best of our knowledge, the bispectrum is calculated for the first time in this work.

astro-ph.CO↗

arXiv2026

Quantum analog-encoding for correlated Gaussian vectors and their exponentiation with application to rough volatility

Quantum computing may speed up numerical problems involving large matrices that are demanding for classical computers, and active research on this possibility is ongoing. In this work, we propose quantum algorithms for the exact simulation of a normalised correlated Gaussian random vector $|x\rangle=\vec{x}/\lVert\vec{x}\rVert$, $\vec{x}\sim\mathcal{N}(0,Σ)$, and its exponentiation $|e^{\vec{x}} \rangle= e^{\vec{x}}/\lVert e^{\vec{x}}\rVert$. When an $O(\mathrm{polylog} N)$-gate-depth quantum data loader for the covariance matrix $Σ\in\mathbb{R}^{N\times N}$ is available, preparing $|x\rangle$ and $|e^{\vec{x}}\rangle$ require $\widetilde{O}\left(\frac{\lVertΣ\rVert_F}{λ_{\max}}κ^{1.5}\right)$ and $\widetilde{O}\left(\lVert\vec{x}\rVert\frac{\lVertΣ\rVert_F}{λ_{\max}}κ^{1.5}\right)$ elementary gate depth respectively, where $\lVertΣ\rVert_F$, $λ_{\max}$, $κ$ denote the Frobenius norm, maximal eigenvalue, and condition number of $Σ$. Motivated by financial applications, we provide an end-to-end resource analysis when $\vec{x}$ represents a sample path of a Riemann-Liouville or standard fractional Brownian motion, or of a stationary fractional Ornstein-Uhlenbeck process. As a concrete example, we construct the quantum state encoding the rough Bergomi variance process and analyse the extraction of the integrated variance via quantum amplitude estimation. Under specific conditions, the dependence of $\lVertΣ\rVert_F/λ_{\max}$ and $κ$ on $N$ is small, and subcubic complexity in $N$ is achieved, indicating a quantum advantage over classical Cholesky-based sampling methods. To our knowledge, this constitutes the first quantum algorithmic framework for the amplitude encoding of exponentiated Gaussian processes, providing foundational primitives for quantum-enhanced financial modelling.

quant-ph↗

arXiv2026

Quantum algorithm for solving high-dimensional linear stochastic differential equations via amplitude encoding of the noise term

This work studies quantum algorithms to solve high-dimensional stochastic differential equations (SDEs) $\mathrm{d} \mathbf{X}_t = A(t) \mathbf{X}_t \mathrm{d} t + B(t) \mathrm{d} \mathbf{W}_t$. Aiming for a speed-up in the dimension $N$ of $\mathbf{X}_t$, we generate quantum states that encode $\mathbf{X}_t$ in the amplitudes, while most of the existing quantum methods for SDEs employ binary encoding. A key challenge is the amplitude encoding of the noise term, and we address this by utilizing the quantum circuit implementation of a pseudorandom number generator (PRNG). We propose two methods: the Dyson series-based method and the Euler-Maruyama (EM)-based method. In the former, we express the noise term via the Dyson series approximation of the time evolution operator, while in the latter, it is approximated using the EM time discretization. Both methods use the quantum linear systems solver to generate the amplitude-encoding state of $\mathbf{X}_t$, making only ${\rm polylog}(N)$ queries to the PRNG circuit and the block-encodings of $A$ and $B$. Additionally, going beyond state preparation, we present methods to estimate expectations of functions of $\mathbf{X}_t$ using the state.

quant-ph↗

arXiv2026

Time series generation for option pricing on quantum computers using tensor network

Finance, especially option pricing, is a promising industrial field that might benefit from quantum computing. While quantum algorithms for option pricing have been proposed, it is desired to devise more efficient implementations of costly operations in the algorithms, one of which is preparing a quantum state that encodes a probability distribution of the underlying asset price. In particular, in pricing a path-dependent option, we need to generate a state encoding a joint distribution of the underlying asset price at multiple time points, which is more demanding. To address these issues, we propose a novel approach that uses a Matrix Product State (MPS), which can be encoded into a state of qubits, as a generative model for time series generation. We focus on the training of such an MPS and present its procedure in detail. To validate our approach, taking the Heston model as a target, we conduct numerical experiments to generate time series in the model. Our findings demonstrate the capability of the MPS model to generate paths in the Heston model, highlighting its potential for path-dependent option pricing on quantum computers.

quant-ph↗

arXiv2026

Calculating the power spectrum in stochastic inflation by Monte Carlo simulation and least squares curve fitting

The stochastic-$δ\mathcal{N}$ formalism is widely used to study inflation models in which the quantum diffusion of inflatons dominates the background dynamics, leading to interesting phenomena such as the production of primordial black holes. Among numerical approaches to calculate the curvature perturbation spectrum $\mathcal{P}_ζ(k)$ in this formalism, the Monte Carlo simulation-based approach has been proposed as a promising choice, especially in multifield cases. In this approach, we generate many paths of inflatons from the initial points to the end of inflation, obtain statistics of $δN$ from the paths, and then estimate $\mathcal{P}_ζ(k)$. However, this method involves a nested Monte Carlo simulation, which requires generating many branch paths from each trunk path at the point corresponding to the scale $k$ of interest, resulting in a high computational cost. In this paper, we propose a new Monte Carlo-based approach that utilizes least squares fitting, introducing two novel features for reducing computational cost. First, we devise a simple estimator of a key statistic $\langle δ\mathcal{N}_{\mathbf{X}}^2\rangle$, the variance of $δ\mathcal{N}$ conditioned on the branching point, to avoid nesting path generation. Second, via least squares fitting of a parametric function to the sampled values of the estimator, we obtain not just an estimate of $\mathcal{P}_ζ(k)$ for a single value of $k$ but an approximating function of $\mathcal{P}_ζ(k)$ over a range of $k$ of interest. We also conduct numerical demonstrations for concrete inflation models, which show the usefulness of our method.

astro-ph.CO↗

arXiv2026

Quantum simulation of Burgers turbulence: Nonlinear transformation and direct evaluation of statistical quantities

Fault-tolerant quantum computing is a promising technology to solve linear partial differential equations that are classically demanding to integrate. It is still challenging to solve non-linear equations in fluid dynamics, such as the Burgers equation, using quantum computers. We propose a novel quantum algorithm to solve the Burgers equation. With the Cole-Hopf transformation that maps the fluid velocity field $u$ to a new field $ψ$, we apply a sequence of quantum gates to solve the resulting linear equation and obtain the quantum state $\vertψ\rangle$ that encodes the solution $ψ$. We also propose an efficient way to extract stochastic properties of $u$, namely the multi-point functions of $u$, from the quantum state of $\vertψ\rangle$. Our algorithm offers an exponential advantage over the classical finite difference method in terms of the number of spatial grids when a perturbativity condition in the information-extracting step is met.

quant-ph↗

arXiv2026

Quantum algorithm for solving McKean-Vlasov stochastic differential equations

Quantum Monte Carlo integration, a quantum algorithm for calculating expectations that provides a quadratic speed-up compared to its classical counterpart, is now attracting increasing interest in the context of its industrial and scientific applications. In this paper, we propose the first application of QMCI to solving McKean-Vlasov stochastic differential equations (MVSDEs), a nonlinear class of SDEs whose drift and diffusion coefficients depend on the law $μ_t$ of the solution $X_t$ -- appearing in fields such as finance and fluid mechanics. We focus on the problem setting where the coefficients depend on $μ_t$ through expectations of some functions $\mathbb{E}[φ_k(X_t)]$, and the goal is to compute the expectation of a function $\mathbb{E}[ϕ(X_T)]$ at a terminal time $T$. We devise a quantum algorithm that leverages QMCI to compute these expectations, combined with a high-order time discretization method for SDEs and extrapolation of the expectations in time. The proposed algorithm estimates $\mathbb{E}[ϕ(X_T)]$ with accuracy $ε$, making $O(1/ε^{1+2/p})$ queries to the quantum circuit for time evolution over one step, where $p\in(1,2]$ is the weak order of the SDE discretization method. This demonstrates the speed-up over the well-known classical algorithm called the particle method with complexity of $O(1/ε^3)$. We conduct a numerical demonstration of our quantum algorithm applied to an example of MVSDEs, with some parts emulated classically, and observe that the accuracy and complexity behave as expected.

quant-ph↗

arXiv2025

Tensor train representations of Greeks for Fourier-based pricing of multi-asset options

Efficient computation of Greeks for multi-asset options remains a key challenge in quantitative finance. While Monte Carlo (MC) simulation is widely used, it suffers from the large sample complexity for high accuracy. We propose a framework to compute Greeks in a single evaluation of a tensor train (TT), which is obtained by compressing the Fourier transform (FT)-based pricing function via TT learning using tensor cross interpolation. Based on this TT representation, we introduce two approaches to compute Greeks: a numerical differentiation (ND) approach that applies a numerical differential operator to one tensor core and an analytical (AN) approach that constructs the TT of closed-form differentiation expressions of FT-based pricing. Numerical experiments on a five-asset min-call option in the Black-Sholes model show significant speed-ups of up to about $10^{5} \times$ over MC while maintaining comparable accuracy. The ND approach matches or exceeds the accuracy of the AN approach and requires lower computational complexity for constructing the TT representation, making it the preferred choice.

q-fin.CP↗

arXiv2025

Improved quantum algorithm for calculating eigenvalues of differential operators and its application to estimating the decay rate of the perturbation distribution tail in stochastic inflation

Quantum algorithms for scientific computing and their applications have been studied actively. In this paper, we propose a quantum algorithm for estimating the first eigenvalue of a differential operator $\mathcal{L}$ on $\mathbb{R}^d$ and its application to cosmic inflation theory. A common approach for this eigenvalue problem involves applying the finite-difference discretization to $\mathcal{L}$ and computing the eigenvalues of the resulting matrix, but this method suffers from the curse of dimensionality, namely the exponential complexity with respect to $d$. Our first contribution is the development of a new quantum algorithm for this task, leveraging recent quantum singular value transformation-based methods. Given a trial function that overlaps well with the eigenfunction, our method runs with query complexity scaling as $\widetilde{O}(d^3/ε^2)$ with $d$ and estimation accuracy $ε$, which is polynomial in $d$ and shows an improvement over existing quantum algorithms. Then, we consider the application of our method to a problem in a theoretical framework for cosmic inflation known as stochastic inflation, specifically calculating the eigenvalue of the adjoint Fokker--Planck operator, which is related to the decay rate of the tail of the probability distribution for the primordial density perturbation. We numerically see that in some cases, simple trial functions overlap well with the first eigenfunction, indicating our method is promising for this problem.

quant-ph↗

arXiv2025

Dividing quantum circuits for time evolution of stochastic processes by orthogonal series density estimation

Quantum Monte Carlo integration (QMCI) is a quantum algorithm to estimate expectations of random variables, with applications in various industrial fields such as financial derivative pricing. When QMCI is applied to expectations concerning a stochastic process $X(t)$, e.g., an underlying asset price in derivative pricing, the quantum circuit $U_{X(t)}$ to generate the quantum state encoding the probability density of $X(t)$ can have a large depth. With time discretized into $N$ points, using state preparation oracles for the transition probabilities of $X(t)$, the state preparation for $X(t)$ results in a depth of $O(N)$, which may be problematic for large $N$. Moreover, if we estimate expectations concerning $X(t)$ at $N$ time points, the total query complexity scales on $N$ as $O(N^2)$, which is worse than the $O(N)$ complexity in the classical Monte Carlo method. In this paper, to improve this, we propose a method to divide $U_{X(t)}$ based on orthogonal series density estimation. This approach involves approximating the densities of $X(t)$ at $N$ time points with orthogonal series, where the coefficients are estimated as expectations of the orthogonal functions by QMCI. By using these approximated densities, we can estimate expectations concerning $X(t)$ by QMCI without requiring deep circuits. Our error and complexity analysis shows that to obtain the approximated densities at $N$ time points, our method achieves the circuit depth and total query complexity scaling as $O(\sqrt{N})$ and $O(N^{3/2})$, respectively.

quant-ph↗

arXiv2025

Learning parameter dependence for Fourier-based option pricing with tensor trains

A long-standing issue in mathematical finance is the speed-up of option pricing, especially for multi-asset options. A recent study has proposed to use tensor train learning algorithms to speed up Fourier transform (FT)-based option pricing, utilizing the ability of tensor trains to compress high-dimensional tensors. Another usage of the tensor train is to compress functions, including their parameter dependence. Here, we propose a pricing method, where, by a tensor train learning algorithm, we build tensor trains that approximate functions appearing in FT-based option pricing with their parameter dependence and efficiently calculate the option price for the varying input parameters. As a benchmark test, we run the proposed method to price a multi-asset option for the various values of volatilities and present asset prices. We show that, in the tested cases involving up to 11 assets, the proposed method outperforms Monte Carlo-based option pricing with $10^6$ paths in terms of computational complexity while keeping better accuracy.

q-fin.CP↗

arXiv2025

Quantum algorithm for copula-based risk aggregation using orthogonal series density estimation

Quantum Monte Carlo integration (QMCI) provides a quadratic speed-up over its classical counterpart, and its applications have been investigated in various fields, including finance. This paper considers its application to risk aggregation, one of the most important numerical tasks in financial risk management. Risk aggregation combines several risk variables and quantifies the total amount of risk, taking into account the correlation among them. For this task, there exists a useful tool called copula, with which the joint distribution can be generated from marginal distributions with a flexible correlation structure. Classically, the copula-based method utilizes sampling of risk variables. However, this procedure is not directly applicable to the quantum setting, where sampled values are not stored as classical data, and thus no efficient quantum algorithm is known. In this paper, we introduce a quantum algorithm for copula-based risk aggregation that is compatible with QMCI. In our algorithm, we first estimate each marginal distribution as a series of orthogonal functions, where the coefficients can be calculated with QMCI. Then, by plugging the marginal distributions into the copula and obtaining the joint distribution, we estimate risk measures using QMCI again. With this algorithm, nearly quadratic quantum speed-up can be obtained for sufficiently smooth marginal distributions.

quant-ph↗

arXiv2024

Quantum algorithm for collisionless Boltzmann simulation of self-gravitating systems

The collisionless Boltzmann equation (CBE) is a fundamental equation that governs the dynamics of a broad range of astrophysical systems from space plasma to star clusters and galaxies. It is computationally expensive to integrate the CBE directly in a multi-dimensional phase space, and thus the applications to realistic astrophysical problems have been limited so far. Recently, Todorova & Steijl (2020) proposed an efficient quantum algorithm to solve the CBE with significantly reduced computational complexity. We extend the algorithm to perform quantum simulations of self-gravitating systems, incorporating the method to calculate gravity with the major Fourier modes of the density distribution extracted from the solution-encoding quantum state. Our method improves the dependency of time and space complexities on Nv , the number of grid points in each velocity coordinate, compared to the classical simulation methods. We then conduct some numerical demonstrations of our method. We first run a 1+1 dimensional test calculation of free streaming motion on 64*64 grids using 13 simulated qubits and validate our method. We then perform simulations of Jeans collapse, and compare the result with analytic and linear theory calculations. It will thus allow us to perform large-scale CBE simulations on future quantum computers.

quant-ph↗

arXiv2024

On the bias in iterative quantum amplitude estimation

Quantum amplitude estimation (QAE) is a pivotal quantum algorithm to estimate the squared amplitude $a$ of the target basis state in a quantum state $|Φ\rangle$. Various improvements on the original quantum phase estimation-based QAE have been proposed for resource reduction. One of such improved versions is iterative quantum amplitude estimation (IQAE), which outputs an estimate $\hat{a}$ of $a$ through the iterated rounds of the measurements on the quantum states like $G^k|Φ\rangle$, with the number $k$ of operations of the Grover operator $G$ (the Grover number) and the shot number determined adaptively. This paper investigates the bias in IQAE. Through the numerical experiments to simulate IQAE, we reveal that the estimate by IQAE is biased and the bias is enhanced for some specific values of $a$. We see that the termination criterion in IQAE that the estimated accuracy of $\hat{a}$ falls below the threshold is a source of the bias. Besides, we observe that $k_\mathrm{fin}$, the Grover number in the final round, and $f_\mathrm{fin}$, a quantity affecting the probability distribution of measurement outcomes in the final round, are the key factors to determine the bias, and the bias enhancement for specific values of $a$ is due to the skewed distribution of $(k_\mathrm{fin},f_\mathrm{fin})$. We also present a bias mitigation method: just re-executing the final round with the Grover number and the shot number fixed.

quant-ph↗

arXiv2024

The cross-sectional stock return predictions via quantum neural network and tensor network

In this paper, we investigate the application of quantum and quantum-inspired machine learning algorithms to stock return predictions. Specifically, we evaluate the performance of quantum neural network, an algorithm suited for noisy intermediate-scale quantum computers, and tensor network, a quantum-inspired machine learning algorithm, against classical models such as linear regression and neural networks. To evaluate their abilities, we construct portfolios based on their predictions and measure investment performances. The empirical study on the Japanese stock market shows the tensor network model achieves superior performance compared to classical benchmark models, including linear and neural network models. Though the quantum neural network model attains a lowered risk-adjusted excess return than the classical neural network models over the whole period, both the quantum neural network and tensor network models have superior performances in the latest market environment, which suggests the capability of the model's capturing non-linearity between input features.

cs.LG↗

arXiv2024

Quantum algorithm for the Vlasov simulation of the large-scale structure formation with massive neutrinos

Investigating the cosmological implication of the fact that neutrino has finite mass is of importance for fundamental physics. In particular, massive neutrino affects the formation of the large-scale structure (LSS) of the universe, and, conversely, observations of the LSS can give constraints on the neutrino mass. Numerical simulations of the LSS formation including massive neutrino along with conventional cold dark matter is thus an important task. For this, calculating the neutrino distribution in the phase space by solving the Vlasov equation is a suitable approach, but it requires solving the PDE in the $(6+1)$-dimensional space and is thus computationally demanding: Configuring $n_\mathrm{gr}$ grid points in each coordinate and $n_t$ time grid points leads to $O(n_\mathrm{gr}^6)$ memory space and $O(n_tn_\mathrm{gr}^6)$ queries to the coefficients in the discretized PDE. We propose a quantum algorithm for this task. Linearizing the Vlasov equation by neglecting the relatively weak self-gravity of the neutrino, we perform the Hamiltonian simulation to produce quantum states that encode the phase space distribution of neutrino. We also propose a way to extract the power spectrum of the neutrino density perturbations as classical data from the quantum state by quantum amplitude estimation with accuracy $ε$ and query complexity of order $\widetilde{O}((n_\mathrm{gr} + n_t)/ε)$. Our method also reduces the space complexity to $O(\mathrm{polylog}(n_\mathrm{gr}/ε))$ in terms of the qubit number, while using quantum random access memories with $O(n_\mathrm{gr}^3)$ entries. As far as we know, this is the first quantum algorithm for the LSS simulation that outputs the quantity of practical interest with guaranteed accuracy.

quant-ph↗

arXiv2023

A comprehensive survey on quantum computer usage: How many qubits are employed for what purposes?

Quantum computers (QCs), which work based on the law of quantum mechanics, are expected to be faster than classical computers in several computational tasks such as prime factoring and simulation of quantum many-body systems. In the last decade, research and development of QCs have rapidly advanced. Now hundreds of physical qubits are at our disposal, and one can find several remarkable experiments actually outperforming the classical computer in a specific computational task. On the other hand, it is unclear what the typical usages of the QCs are. Here we conduct an extensive survey on the papers that are posted in the quant-ph section in arXiv and claim to have used QCs in their abstracts. To understand the current situation of the research and development of the QCs, we evaluated the descriptive statistics about the papers, including the number of qubits employed, QPU vendors, application domains and so on. Our survey shows that the annual number of publications is increasing, and the typical number of qubits employed is about six to ten, growing along with the increase in the quantum volume (QV). Most of the preprints are devoted to applications such as quantum machine learning, condensed matter physics, and quantum chemistry, while quantum error correction and quantum noise mitigation use more qubits than the other topics. These imply that the increase in QV is fundamentally relevant, and more experiments for quantum error correction, and noise mitigation using shallow circuits with more qubits will take place.

quant-ph↗

arXiv2023

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.

quant-ph↗