Quantum Quantile Mechanics: Solving Stochastic Differential Equations for Generating Time-Series

Abstract

We propose a quantum algorithm for sampling from a solution of stochastic differential equations (SDEs). Using differentiable quantum circuits (DQCs) with a feature map encoding of latent variables, we represent the quantile function for an underlying probability distribution and extract samples as DQC expectation values. Using quantile mechanics we propagate the system in time, thereby allowing for time-series generation. We test the method by simulating the Ornstein-Uhlenbeck process and sampling at times different from the initial point, as required in financial analysis and dataset augmentation. Additionally, we analyse continuous quantum generative adversarial networks (qGANs), and show that they represent quantile functions with a modified (reordered) shape that impedes their efficient time-propagation. Our results shed light on the connection between quantum quantile mechanics (QQM) and qGANs for SDE-based distributions, and point the importance of differential constraints for model training, analogously with the recent success of physics informed neural networks.

Protocols for Trainable and Differentiable Quantum Generative Modelling

“…One option is to use a t-dependent feature map for parameterizing the model. For instance, we employed it successfully in DQC-based quantum function propagation [28]. In this case, it is convenient to use an identity-valued feature map at t 0 , and learn to adjust angles as t deviates from t 0 .…”

“…In Ref. [28] we have shown that initialization with lowdegree polynomial (truncated Chebyshev series) can vastly reduce number of optimization epochs. Here, we propose to use the structure of the quantum model in Eq.…”

“…where Q j (Z j ) are marginal quantile functions (inverted CDFs) for distribution of j-th stochastic variable. Here, we stress that copula produces correlations at the level of latent variables, as used in the inverse sampling [28]. It represents a modified PDF that deviates from a uniform multivariate distribution, and thus correlates the outcomes for multivariate PDF sampling.…”

“…First, the described generators are difficult to train as they require matching all amplitudes for N -qubit registers and finding the corresponding state for some vector θ. Second, QCBM architecture is not automatically differentiable with respect to variable x, and QGAN differentiation leads to an ill-defined loss landscape [28]. Thus, both have limited application for SDE solving.…”

“…We first note that the ability of differentiating generative models can be restored when using feature map encoding of continuous distributions [53], at the expense of multi-shot measurement to get a sample from QNNs. Second, the differential constraints at the sampling stage can be implemented using quantum quantile mechanics (QQM) [28], where a quantum circuit is trained to generate samples from SDEs and can be evolved in time, albeit with expectationbased sampling. Here, merging differentiability with fast sampling will offer both potential expressivity advantage and sampling advantage of QC.…”