Entanglement scaling in matrix product state representation of smooth functions and their shallow quantum circuit approximations

TL;DR

This paper analyzes how the entanglement in matrix product states (MPS) encoding of smooth functions decays with function properties, leading to improved shallow quantum circuits validated on IBM devices.

Contribution

It provides rigorous asymptotic expansions for entanglement decay in MPS based on function smoothness, and introduces an enhanced MPS algorithm for efficient quantum encoding.

Findings

Entanglement decay depends on the smoothness of the input function.

The improved MPS algorithm produces shallow, accurate quantum circuits.

Quantum circuits successfully loaded heavy-tailed distributions and were tested on IBM devices.

Abstract

Encoding classical data in a quantum state is a key prerequisite of many quantum algorithms. Recently matrix product state (MPS) methods emerged as the most promising approach for constructing shallow quantum circuits approximating input functions, including probability distributions, with only linear number of gates. We derive rigorous asymptotic expansions for the decay of entanglement across bonds in the MPS representation depending on the smoothness of the input function, real or complex. We also consider the dependence of the entanglement on localization properties and function support. Based on these analytical results we construct an improved MPS-based algorithm yielding shallow and accurate encoding quantum circuits. By using Tensor Cross Interpolation we are able to construct utility-scale quantum circuits in a compute- and memory-efficient way. We validate our methods by loading heavy-tailed distributions, including Levy, important in finance, but they apply to any smooth function inputs. We test the performance of the resulting quantum circuits by executing and sampling from them on IBM quantum devices, for up to 156 qubits.