# Efficient phase-factor evaluation in quantum signal processing

arXiv: https://arxiv.org/abs/2002.11649 · HTML: https://mlsanity.com/paper/2002.11649/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 26 Feb 2020.

## Built on by (most cited first, 6 of 6)

- [Operator-Level Quantum Acceleration of Non-Logconcave Sampling](https://mlsanity.com/paper/2505.05301/) (2025)
- [Quantum enhanced rare event discovery and sampling](https://mlsanity.com/paper/2606.06316/) (2026)
- [Beyond Heisenberg Limit Quantum Metrology through Quantum Signal Processing](https://mlsanity.com/paper/2209.11207/) (2022)
- [Statistics-Informed Parameterized Quantum Circuit via Maximum Entropy Principle for Data Science and Finance](https://mlsanity.com/paper/2406.01335/) (2024)
- [On Encoding Matrices using Quantum Circuits](https://mlsanity.com/paper/2510.20030/) (2025)
- [Block-encodings as programming abstractions: The Eclipse Qrisp BlockEncoding Interface](https://mlsanity.com/paper/2604.18276/) (2026)

## Abstract (arXiv)

Quantum signal processing (QSP) is a powerful quantum algorithm to exactly implement matrix polynomials on quantum computers. Asymptotic analysis of quantum algorithms based on QSP has shown that asymptotically optimal results can in principle be obtained for a range of tasks, such as Hamiltonian simulation and the quantum linear system problem. A further benefit of QSP is that it uses a minimal number of ancilla qubits, which facilitates its implementation on near-to-intermediate term quantum architectures. However, there is so far no classically stable algorithm allowing computation of the phase factors that are needed to build QSP circuits. Existing methods require the usage of variable precision arithmetic and can only be applied to polynomials of relatively low degree. We present here an optimization based method that can accurately compute the phase factors using standard double precision arithmetic operations. We demonstrate the performance of this approach with applications to Hamiltonian simulation, eigenvalue filtering, and the quantum linear system problems. Our numerical results show that the optimization algorithm can find phase factors to accurately approximate polynomials of degree larger than 10,000 with error below 10-12.