[Paper Review] Efficient Quantum Circuits for Accurate State Preparation of Smooth, Differentiable Functions

Adam Holmes, A. Y. Matsuura|arXiv (Cornell University)|May 9, 2020

Quantum Computing Algorithms and Architecture42 references11 citations

TL;DR

This paper presents a linear-time algorithm to generate linear-depth quantum circuits that prepare smooth, differentiable, real-valued functions—such as Gaussian, lognormal, and Lorentzian distributions—with high fidelity. By encoding piecewise polynomial approximations into matrix product states (MPS) and leveraging low-rank MPS compression and gate extraction, the method achieves over 99% fidelity even for highly squeezed states, enabling efficient initialization for quantum algorithms.

ABSTRACT

Effective quantum computation relies upon making good use of the exponential information capacity of a quantum machine. A large barrier to designing quantum algorithms for execution on real quantum machines is that, in general, it is intractably difficult to construct an arbitrary quantum state to high precision. Many quantum algorithms rely instead upon initializing the machine in a simple state, and evolving the state through an efficient (i.e. at most polynomial-depth) quantum algorithm. In this work, we show that there exist families of quantum states that can be prepared to high precision with circuits of linear size and depth. We focus on real-valued, smooth, differentiable functions with bounded derivatives on a domain of interest, exemplified by commonly used probability distributions. We further develop an algorithm that requires only linear classical computation time to generate accurate linear-depth circuits to prepare these states, and apply this to well-known and heavily-utilized functions including Gaussian and lognormal distributions. Our procedure rests upon the quantum state representation tool known as the matrix product state (MPS). By efficiently and scalably encoding an explicit amplitude function into an MPS, a high fidelity, linear-depth circuit can directly be generated. These results enable the execution of many quantum algorithms that, aside from initialization, are otherwise depth-efficient.

Motivation & Objective

• To address the intractable challenge of initializing arbitrary quantum states with high precision on near-term quantum hardware.
• To enable practical execution of quantum algorithms—especially Monte Carlo and quantum machine learning—by providing efficient, scalable state preparation.
• To develop a method that constructs accurate quantum circuits for smooth, differentiable, real-valued functions with bounded derivatives.
• To ensure the algorithm scales linearly in classical computation time and maintains high fidelity as system size increases.

Proposed method

• The method uses piecewise polynomial approximation to represent target smooth functions with bounded derivatives.
• Each polynomial segment is encoded into a matrix product state (MPS) with controlled bond dimension.
• Variational compression is applied to reduce the MPS to low, bounded bond dimension while preserving fidelity.
• A known algorithm from ran2020encoding is used to extract a linear-depth quantum circuit from the compressed MPS.
• The entire procedure is optimized for linear classical runtime, with the bottleneck being variational MPS compression.
• The approach is tunable, allowing adjustments to polynomial order and approximation strategy to balance accuracy and resource use.

Experimental results

Research questions

• RQ1 Can smooth, differentiable, real-valued functions with bounded derivatives be prepared with high fidelity using quantum circuits of linear depth and size?
• RQ2 Does the matrix product state representation of these functions exhibit exponentially decreasing von Neumann entropy with increasing qubit count, enabling efficient compression?
• RQ3 Can piecewise polynomial approximation combined with variational MPS compression yield high-fidelity state preparation with linear classical computation time?
• RQ4 How does the fidelity of the constructed state scale with system size and function parameterization, especially for highly squeezed distributions?
• RQ5 To what extent can the algorithm be generalized to other analytical functions or classical data sets that are well-approximated by low-order polynomials?

Key findings

• The algorithm achieves over 99% state preparation fidelity for Gaussian, lognormal, and Lorentzian distributions using only cubic polynomials.
• Fidelity increases to 99.8%, 99.91%, and 99.95% respectively when using fifth-order polynomials, demonstrating high accuracy across diverse distributions.
• The optimality ratio between the constructed circuit and the theoretical optimal state remains constant or improves with increasing system size, indicating scalability.
• The SVD-based χ=2 MPS approximation maintains high fidelity across larger system sizes, supporting the conjecture that low-bond-dimension representations scale effectively.
• The method enables linear-time classical construction of linear-depth quantum circuits, with the variational MPS compression being the primary computational bottleneck.
• Empirical results show no decay in accuracy with increasing discretization, supporting the potential for scaling to large quantum systems.