Accuracy vs Memory Advantage in the Quantum Simulation of Stochastic Processes

TL;DR

This work asks whether quantum memory advantages in simulating classical stochastic processes survive when exact finite-memory $\varepsilon$-machine modelling is relaxed. It introduces tensor-network–inspired compression methods for both quantum and classical simulators and defines fidelity- and entropy-based metrics to quantify accuracy under memory constraints, validating the approach on discrete renewal processes and real data. The results show that quantum simulators can achieve the same predictive accuracy with reduced memory or higher accuracy with the same memory, with only modest entropy losses, while classical approaches lag in memory efficiency. These findings point to potential data-efficient quantum learning for stochastic processes and motivate experimental exploration with near-term quantum hardware.

Abstract

Many inference scenarios rely on extracting relevant information from known data in order to make future predictions. When the underlying stochastic process satisfies certain assumptions, there is a direct mapping between its exact classical and quantum simulators, with the latter asymptotically using less memory. Here we focus on studying whether such quantum advantage persists when those assumptions are not satisfied, and the model is doomed to have imperfect accuracy. By studying the trade-off between accuracy and memory requirements, we show that quantum models can reach the same accuracy with less memory, or alternatively, better accuracy with the same memory. Finally, we discuss the implications of this result for learning tasks.

Figures (9)

• Figure 1: Pictorial representation of the evolution for the first two time steps. Both the classical and the quantum simulators use two registers: an outcome register for the emitted symbols $x_t$, and a memory register that keeps a compressed representation of the history of previous interactions and outcomes. At each time step, the outcome register is first reset to a reference value, then it is let to interact with the memory (orange box), and finally it is measured to observe the classical outcomes $x_t$. For classical simulators, the orange box is mathematically modelled as a transition probability, Eq. \ref{['eq:P def']}, while for quantum simulators the orange box models a unitary operation, as in Eq. \ref{['eq:U']}, followed by a projective measurement to extract $x_t$.
• Figure 2: Distribution of the distance between consecutive ones in the exact and compressed quantum simulation of of a discrete renewal process of period $N$, for $N=32$ (a) or $N=64$ (b).
• Figure 3: Memory entropy vs accuracy between the exact and compressed simulators. For each $N$, the compressed simulators are computed with truncated memory dimension $M=2,\dots,N$. (a) Entropy of compressed memory vs Eq. \ref{['eq:divergence q']}. (b) Divergence $D_{1/2}$ for $L=1$ vs $M/N$. (c) Entropy vs $-\log_2(M/N)$.
• Figure 4: Compressed memory dimension $M$ vs. Bhattacharyya coefficient (a) or vs. the similarity decay rate for $L\to\infty$ (b) in the simulation of a discrete renewal process with $N=32$ and $M\leq N$. (c) Memory entropy vs. similarity decay rate in the same setting of (a-b).
• Figure 5: Distribution of the distance between consecutive ones in the exact and fitted classical (c-fit) and quantum (q-fit) simulators, and in the compressed classical (c-comp) and quantum (q-comp) simulators. The exact simulation uses $N=4$, while the fitted ones either $N=4$ (a) or $N=2$ (b). Fitting is done using a training sequence of $L=10^4$ observations, while the histogram is generated using 100 million samples. Table (c) shows the Bhattacharyya coefficients of the resulting quantum and classical simulators, either fitted from data or compressed from the exact ones, in predicting the next 10 future observations.