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arxiv: 2402.17148 · v2 · submitted 2024-02-27 · 🪐 quant-ph · cs.LG· q-fin.CP

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

Nozomu Kobayashi , Yoshiyuki Suimon , Koichi Miyamoto This is my paper

Pith reviewed 2026-05-24 03:58 UTC · model grok-4.3

classification 🪐 quant-ph cs.LGq-fin.CP
keywords matrix product statestime series generationHeston modeloption pricingquantum computingtensor networksgenerative model
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The pith

Matrix product states can be trained to generate asset price paths matching the Heston model.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proposes using a matrix product state as a generative model for time series of asset prices. This targets the preparation of quantum states that encode joint distributions over multiple time steps, which quantum algorithms need for pricing path-dependent options. The authors detail the MPS training procedure and run numerical tests on the Heston stochastic volatility model. If the approach holds, it would let tensor networks replace more expensive methods for loading financial distributions onto quantum hardware.

Core claim

The authors demonstrate that an MPS can be trained to generate paths in the Heston model. This shows the capability of the MPS to reproduce the joint distribution of asset prices at multiple time points and thereby supports its use for preparing quantum states in path-dependent option pricing on quantum computers.

What carries the argument

The matrix product state (MPS) encoded as a qubit state and trained as a generative model to capture the joint distribution across time steps.

If this is right

  • The MPS model generates paths in the Heston model.
  • This supports preparation of quantum states encoding joint distributions for path-dependent options.
  • The training procedure for the MPS generative model is provided in detail.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Classical training of the MPS could precede loading the distribution into a quantum circuit, separating the heavy computation from the quantum device.
  • The same MPS construction might apply to other stochastic processes used in derivative pricing.
  • The method could combine with quantum amplitude estimation to form end-to-end quantum workflows for complex options.

Load-bearing premise

An MPS with practical bond dimension can be trained to faithfully reproduce the joint distribution of asset prices across multiple time steps in the Heston model.

What would settle it

Numerical tests in which the statistical properties of paths produced by the trained MPS, such as volatility or correlations, deviate from those of direct Heston model simulations.

Figures

Figures reproduced from arXiv: 2402.17148 by Koichi Miyamoto, Nozomu Kobayashi, Yoshiyuki Suimon.

Figure 1
Figure 1. Figure 1: Probability distributions of the asset price at [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Same as Figure [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
read the original abstract

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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript proposes training a matrix product state (MPS) classically as a generative model to produce time series paths distributed according to the Heston stochastic volatility model. The trained MPS is then to be encoded as a quantum state for use in quantum algorithms that price path-dependent options. The training procedure is described in detail, and numerical experiments on the Heston model are presented to demonstrate that the MPS can generate such paths, with the goal of highlighting potential advantages for quantum finance applications.

Significance. If the numerical results establish that an MPS with modest bond dimension can faithfully reproduce the joint distribution over multiple time steps, the approach would supply a concrete classical pre-processing route to the state-preparation step that is otherwise costly in quantum option-pricing algorithms. This would constitute a practical bridge between tensor-network methods and quantum finance, provided the bond-dimension scaling and training cost remain manageable.

major comments (2)
  1. [Numerical Experiments] Numerical Experiments section: the central claim that the MPS model demonstrates capability for Heston path generation rests on the assertion that a practical bond dimension suffices to reproduce the joint distribution. The manuscript must report the bond dimension(s) employed, the number of time steps, the volatility discretization, and at least one quantitative fidelity metric (KL divergence, total variation, or moment errors with error bars) so that the stress-test concern can be evaluated directly.
  2. [Training procedure] Training procedure (around the description of the MPS generative model): it is not shown whether the optimization converges to a distribution whose marginals and correlations match those of the Heston process to within a tolerance that would be acceptable for option pricing. Without these diagnostics the claim that the method is ready for path-dependent pricing remains unsupported.
minor comments (1)
  1. [Abstract] The abstract states that experiments 'demonstrate the capability' but supplies none of the quantitative results; moving at least the bond dimension and a representative error metric into the abstract would strengthen the summary.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive comments, which help clarify the presentation of our results. We address each major comment below and will revise the manuscript to incorporate the requested details and diagnostics.

read point-by-point responses

  1. Referee: [Numerical Experiments] Numerical Experiments section: the central claim that the MPS model demonstrates capability for Heston path generation rests on the assertion that a practical bond dimension suffices to reproduce the joint distribution. The manuscript must report the bond dimension(s) employed, the number of time steps, the volatility discretization, and at least one quantitative fidelity metric (KL divergence, total variation, or moment errors with error bars) so that the stress-test concern can be evaluated directly.

    Authors: We agree that these parameters and metrics are essential for evaluating the results. In the revised manuscript we will explicitly state the bond dimensions employed in the numerical experiments, the number of time steps, the discretization scheme used for the volatility process, and quantitative fidelity measures (including KL divergence and moment errors with error bars obtained from repeated training runs). These additions will directly address the concern about whether a modest bond dimension suffices. revision: yes

  2. Referee: [Training procedure] Training procedure (around the description of the MPS generative model): it is not shown whether the optimization converges to a distribution whose marginals and correlations match those of the Heston process to within a tolerance that would be acceptable for option pricing. Without these diagnostics the claim that the method is ready for path-dependent pricing remains unsupported.

    Authors: We acknowledge the value of explicit convergence diagnostics. The revised version will include additional figures and tables that track the evolution of marginal distributions and pairwise correlations during training, together with quantitative comparisons (e.g., total variation distance on marginals and absolute error on correlations) against the Heston reference. These will demonstrate that the trained MPS reproduces the target statistics to a level suitable for the intended path-dependent pricing application. revision: yes

Circularity Check

0 steps flagged

No significant circularity; MPS training and Heston validation are independent of definitional reduction

full rationale

The paper proposes training an MPS generative model for multi-step asset price paths in the Heston model, then validates the approach via numerical experiments on path generation. No equations, fitted parameters, or self-citations are shown that would make the reported capability equivalent to its inputs by construction. The central claim rests on external numerical checks of distribution reproduction rather than internal redefinition or renaming of known results. This is the most common honest finding for a methods-plus-numerics paper.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are stated beyond the standard assumption that an MPS can represent the target distribution.

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