[Paper Review] Infinite quantum signal processing

Yulong Dong, Lin Lin+2 more|arXiv (Cornell University)|Sep 21, 2022

Quantum Computing Algorithms and Architecture4 citations

TL;DR

This paper introduces infinite quantum signal processing (IQSP), a generalization of quantum signal processing that enables the representation of non-polynomial functions via phase factors in the ℓ¹ space. It proves that when the ℓ¹ norm of Chebyshev coefficients is bounded, the phase factors converge, and presents a numerically stable fixed-point algorithm using double-precision arithmetic with provable convergence for large-degree approximations.

ABSTRACT

Quantum signal processing (QSP) represents a real scalar polynomial of degree $d$ using a product of unitary matrices of size $2 imes 2$, parameterized by $(d+1)$ real numbers called the phase factors. This innovative representation of polynomials has a wide range of applications in quantum computation. When the polynomial of interest is obtained by truncating an infinite polynomial series, a natural question is whether the phase factors have a well defined limit as the degree $d o \infty$. While the phase factors are generally not unique, we find that there exists a consistent choice of parameterization so that the limit is well defined in the $\ell^1$ space. This generalization of QSP, called the infinite quantum signal processing, can be used to represent a large class of non-polynomial functions. Our analysis reveals a surprising connection between the regularity of the target function and the decay properties of the phase factors. Our analysis also inspires a very simple and efficient algorithm to approximately compute the phase factors in the $\ell^1$ space. The algorithm uses only double precision arithmetic operations, and provably converges when the $\ell^1$ norm of the Chebyshev coefficients of the target function is upper bounded by a constant that is independent of $d$. This is also the first numerically stable algorithm for finding phase factors with provable performance guarantees in the limit $d o \infty$.

Motivation & Objective

• To investigate whether phase factors in quantum signal processing have a well-defined limit as polynomial degree d → ∞ when approximating infinite series.
• To establish conditions under which the phase factors converge in the ℓ¹ space for non-polynomial target functions.
• To develop a numerically stable, efficient algorithm for computing phase factors in the infinite-degree limit with provable convergence guarantees.
• To explore the connection between the regularity of target functions and the decay properties of phase factors.
• To enable practical quantum algorithms for smooth, non-polynomial functions via quantum singular value transformation.

Proposed method

• Generalizes quantum signal processing to infinite sequences of phase factors in ℓ¹ space, ensuring convergence under bounded ℓ¹ norm of Chebyshev coefficients.
• Uses a fixed-point iteration algorithm (Algorithm 1) to compute phase factors, relying on double-precision arithmetic and provable convergence when ‖c‖₁ ≤ r_c.
• Analyzes the structure of the derivative matrix DF and its invertibility in ℓ¹ to establish existence and uniqueness of solutions.
• Employs Chebyshev polynomial expansions to represent smooth functions and relates their coefficient decay to phase factor decay.
• Applies Banach space theory and matrix norm estimates to prove convergence and stability of the fixed-point method.
• Validates results numerically using target functions like |x|³ and odd/even components of f_even and f_odd, comparing with existing solvers.

Experimental results

Research questions

• RQ1 Under what conditions do the phase factors in quantum signal processing converge as the polynomial degree d → ∞?
• RQ2 Can a consistent parameterization of phase factors be defined such that the limit exists in ℓ¹ space for non-polynomial functions?
• RQ3 What is the relationship between the regularity of a target function and the decay rate of its corresponding phase factors?
• RQ4 Can a numerically stable, provably convergent algorithm be constructed for computing phase factors in the infinite-degree limit?
• RQ5 How does the performance of the proposed fixed-point algorithm compare to existing methods like QSPPACK in terms of speed and accuracy?

Key findings

• The phase factors converge in ℓ¹ space when the ℓ¹ norm of the Chebyshev coefficients of the target function is bounded by a constant independent of d.
• The fixed-point iteration algorithm converges provably under the same ℓ¹ norm condition and runs in double-precision arithmetic with CPU time scaling as O(τ²), where τ is the truncation degree.
• The number of iterations required for convergence is stable and independent of τ, typically between 14 and 16 iterations for τ up to 1000.
• The decay of phase factors closely matches the decay of Chebyshev coefficients, suggesting a strong link between function smoothness and phase factor sparsity.
• Numerical experiments show that the algorithm outperforms QSPPACK in speed and maintains high accuracy (ε = 10⁻¹²) even for large τ.
• The method remains effective even when the ℓ¹ norm exceeds the theoretical bound (e.g., for τ = 3.2332), indicating potential for relaxation of the condition.