Quantum State Preparation for Probability Distributions with Reflection Symmetry Using Matrix Product States

TL;DR

The paper tackles the challenge of loading probability distributions with low-depth quantum circuits, focusing on reflection-symmetric distributions. It combines tensor-network representations (Matrix Product States) with a Matrix Product Disentangler and a reflection qubit to exploit symmetry and reduce entanglement, achieving near-linear circuit-depth scaling in the qubit count. Results show two orders-of-magnitude improvements in KL divergence over prior tensor-network methods, with accuracy governed primarily by the bond dimension $\chi$ rather than the number of qubits, demonstrated on $n=10$ and $n=20$ qubits on a real processor. The approach extends to Lorentzian and Student's $t$ distributions and has practical implications for finance and physical-system simulations as a reusable subroutine for quantum state preparation on noisy devices.

Abstract

Quantum circuits for loading probability distributions into quantum states are essential subroutines in quantum algorithms used in physics, finance engineering, and machine learning. The ability to implement these with high accuracy in low-depth quantum circuits is a critical issue. We propose a novel quantum state preparation method for probability distribution with reflection symmetry using matrix product states. By considering reflection symmetry, our method reduces the entanglement of probability distributions and improves the accuracy of approximations by matrix product states. As a result, we improved the accuracy by two orders of magnitude over existing methods using matrix product states. Our approach, characterized by linear scalability with qubit count, is highly advantageous for noisy quantum devices. Also, our demonstration results reveal that the approximation accuracy in tensor networks depends heavily on the bond dimension, with minimal reliance on the number of qubits. Our method is demonstrated for a normal distribution encoded into 10 and 20 qubits on a real quantum processor.

Figures (13)

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• Figure 4: The Kullback-Leibler divergence between the normal distribution $\mathcal{N}(\mu=0, \sigma^2 = 0.01)$$(\mathrm{min}:-0.5, \mathrm{max}:0.5)$ (or the left half of normal distribution) and the probability distributions represented by each MPS (10 sites) with limited bond dimension.
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