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Finding Angles for Quantum Signal Processing with Machine Precision

Rui Chao, Dawei Ding, Andras Gilyen, Cupjin Huang, Mario Szegedy

TL;DR

The paper addresses the classical challenge of synthesizing angle sequences for quantum signal processing (QSP) used in QSVT-based algorithms, including Hamiltonian simulation. It introduces two key innovations: a halving-based decomposition grounded in a uniqueness theorem within the Haah/Cayley–Dickson algebra framework, and a capitalization preprocessing that stabilizes high-degree Laurent coefficients for double-precision computation. Empirical results show that these ideas enable finding sequences with thousands of angles in a few minutes, vastly extending what is feasible under machine precision, and outperforming the prior carving approach in numerical stability and scalability. The proposed hybrid approach holds practical impact for efficient quantum algorithms on NISQ-era hardware by enabling longer-time Hamiltonian simulations and robust angle synthesis.

Abstract

We describe an algorithm for finding angle sequences in quantum signal processing, with a novel component we call halving based on a new algebraic uniqueness theorem, and another we call capitalization. We present both theoretical and experimental results that demonstrate the performance of the new algorithm. In particular, these two algorithmic ideas allow us to find sequences of more than 3000 angles within 5 minutes for important applications such as Hamiltonian simulation, all in standard double precision arithmetic. This is native to almost all hardware.

Finding Angles for Quantum Signal Processing with Machine Precision

TL;DR

The paper addresses the classical challenge of synthesizing angle sequences for quantum signal processing (QSP) used in QSVT-based algorithms, including Hamiltonian simulation. It introduces two key innovations: a halving-based decomposition grounded in a uniqueness theorem within the Haah/Cayley–Dickson algebra framework, and a capitalization preprocessing that stabilizes high-degree Laurent coefficients for double-precision computation. Empirical results show that these ideas enable finding sequences with thousands of angles in a few minutes, vastly extending what is feasible under machine precision, and outperforming the prior carving approach in numerical stability and scalability. The proposed hybrid approach holds practical impact for efficient quantum algorithms on NISQ-era hardware by enabling longer-time Hamiltonian simulations and robust angle synthesis.

Abstract

We describe an algorithm for finding angle sequences in quantum signal processing, with a novel component we call halving based on a new algebraic uniqueness theorem, and another we call capitalization. We present both theoretical and experimental results that demonstrate the performance of the new algorithm. In particular, these two algorithmic ideas allow us to find sequences of more than 3000 angles within 5 minutes for important applications such as Hamiltonian simulation, all in standard double precision arithmetic. This is native to almost all hardware.

Paper Structure

This paper contains 26 sections, 8 theorems, 42 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Quantum signal processing
  3. Post-selection Operator.
  4. Laurent Polynomials.
  5. An algebra emerges.
  6. Algebras associated with quantum signal processing
  7. The Low algebra.
  8. The Haah algebra.
  9. The Haah algebra.
  10. The syntactic versus semantic view
  11. Star operation, unitary and Hermitian elements, and degree
  12. Hermitian elements.
  13. Degree.
  14. Parity Subgroups
  15. The main lemma
  16. ...and 11 more sections

Key Result

Lemma 1

If $C$ has the effect $\sum_{-n}^{n} c_{i} W^{i}$ on all one dimensional unitary operators $W\in U(1)$, then it has the same Laurent polynomial as effect on all unitary operators.

Figures (3)

  • Figure 1: A diagrammatic illustration of the relationship of various Laurent polynomial algebras, where the algebras are identified under $\mathbb{R}$-algebra homomorphisms. The Low and Haah algebras are denoted as $L$ and $H$ and are marked red. Each line indicates a possible generator to add in the algebra below to generate the algebra above. Note that such choices might not be unique.
  • Figure 2: The running time for angle finding using the halving algorithm. Here we show the results for $\epsilon = 10^{-3}, \, \eta = 0.999$ and $\epsilon = 10^{-4}, \, \eta = 0.9999$. The running time scales as a cubic function with respect to the degree of the Laurent polynomial and hence also cubic with respect to the evolution time parameter $\tau$. Note that an instance with $\tau=1200$, corresponding to a Laurent polynomial with degree 3261, can be efficiently solved within $5$ minutes.
  • Figure 3: Comparison between the halving and carving algorithms' achievable parameter regions for the Hamiltonian simulation problem with machine precision. Note that the $y$-axis is log scaled.

Theorems & Definitions (13)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • proof
  • Lemma 4
  • Lemma 5
  • proof
  • Definition 6: Parity subgroup of real Laurent polynomials
  • Theorem 7: low2016CompositeQuantGatesgilyen2018QSingValTransfhaah2018ProdDecPerFuncQSignPRoc
  • Lemma 8: Main Lemma
  • ...and 3 more