Modular quantum signal processing in many variables

Zane M. Rossi; Jack L. Ceroni; Isaac L. Chuang

doi:10.22331/q-2025-06-18-1776

Modular quantum signal processing in many variables

Zane M. Rossi¹, Jack L. Ceroni², and Isaac L. Chuang¹

¹Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
²Department of Mathematics, University of Toronto, Toronto, Ontario M5S1A1, Canada

Published:	2025-06-18, volume 9, page 1776
Editor:	Di Fang
Eprint:	arXiv:2309.16665v2
Doi:	https://doi.org/10.22331/q-2025-06-18-1776
Citation:	Quantum 9, 1776 (2025).

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Abstract

Featured image: A gadget 'Rosetta Stone' giving three depictions of the same self-composition of a $(2,2)$ gadget. Arrows indicate direct substitution, of (a) quantum circuits (time going left to right), (b) polynomials, and (c) gadget legs, where $A, B$ are slots (places where an oracle unitary can be substituted) and $\Phi$ corresponds to a list of programmable, generally distinct $\sigma_z$ rotation angles. The (Laurent) polynomial transforms achieved by the two protocols, $P_0, P_1$, are real-valued with real arguments, and the correction protocol of Thm. II.1 is implicitly interspersed between substitutions (discussed in the main text).

Popular summary

Despite significant advances in quantum algorithms, quantum programs in practice are often expressed at the circuit level, forgoing helpful structural abstractions common to their classical counterparts. Consequently, as many quantum algorithms have been unified with the advent of quantum signal processing (QSP) and quantum singular value transformation (QSVT), an opportunity has appeared to cast these algorithms as modules that can be combined to constitute complex programs. Specifically, QSP and QSVT are quantum algorithms that can efficiently transform the spectra of linear operators 'block encoded' (i.e., existing as a sub-block) in a unitary process; surprisingly, these algorithms can be shown to unify, simplify, and improve the presentation of most quantum algorithms by basic techniques in linear algebra. Complicating this success, however, is that while QSP/QSVT are often described by the polynomial transforms they apply to the singular values of large linear operators, and the algebraic manipulation of polynomials is simple, the QSP/QSVT protocols realizing analogous manipulations of their embedded polynomials are non-obvious to construct.

Here we provide a theory of modular multi-input-output QSP-based superoperators to address this problem, the basic unit of which we call a gadget. We show these gadgets can be snapped together with LEGO-like ease at the level of the functions they apply to the singular values of linear operators embedded in unitaries; in this way complex quantum algorithms for manipulating block encoded linear operators can be iteratively built from simpler parts, and their resource complexity analyzed. To demonstrate this ease, we also provide a Python package for assembling gadgets and compiling them to circuits. Viewed alternately, gadgets both enable families of useful multivariable functions to be applied to sub-blocks of unitaries with improved space/query complexity, and substantiate a functional-programming approach to quantum algorithm design, recasting QSP/QSVT as monadic types.

Our work's central construction are protocols which, given a black-box QSP or QSVT protocol, efficiently produce a modified protocol that can be then used as a black-box quantum algorithmic subroutine. We show that this 'correction' step is in general necessary, and analyze its query and space complexity within a variety of input models. Once this correction protocol has been established, we enumerate a basic library of achievable functions (with explicit examples), demonstrating that our methods easily allow one to bootstrap to complex, multivariable functions of commuting block-encoded linear operators. This greatly expands the techniques by which linear operators can be manipulated within quantum computers, and allows the simultaneous application of previously incompatible techniques (each with their own strengths) for constructing and manipulating block encodings.

► BibTeX data

@article{Rossi2025modularquantum,
  doi = {10.22331/q-2025-06-18-1776},
  url = {https://doi.org/10.22331/q-2025-06-18-1776},
  title = {Modular quantum signal processing in many variables},
  author = {Rossi, Zane M. and Ceroni, Jack L. and Chuang, Isaac L.},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1776},
  month = jun,
  year = {2025}
}

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[1] Yuan Liu, Shraddha Singh, Kevin C. Smith, Eleanor Crane, John M. Martyn, Alec Eickbusch, Alexander Schuckert, Richard D. Li, Jasmine Sinanan-Singh, Micheline B. Soley, Takahiro Tsunoda, Isaac L. Chuang, Nathan Wiebe, and Steven M. Girvin, "Hybrid Oscillator-Qubit Quantum Processors: Instruction Set Architectures, Abstract Machine Models, and Applications", PRX Quantum 7 1, 010201 (2026).

[2] Yuki Ito, Hitomi Mori, Kazuki Sakamoto, and Keisuke Fujii, "Polynomial time constructive decision algorithm for multivariable quantum signal processing", Quantum 10, 2102 (2026).

[3] Lorenzo Laneve, "An adversary bound for quantum signal processing", Quantum 10, 2025 (2026).

[4] D. O. Shendryk, O. V. Ivakhnenko, S. N. Shevchenko, and Franco Nori, "Efficient implementation of quantum signal processing via the adiabatic-impulse model", Physical Review A 112 4, 042437 (2025).

[5] Michel Alexis, Lin Lin, Gevorg Mnatsakanyan, Christoph Thiele, and Jiasu Wang, "Infinite quantum signal processing for arbitrary Szegő functions", arXiv:2407.05634, (2024).

[6] Jasmine Sinanan-Singh, Gabriel L. Mintzer, Isaac L. Chuang, and Yuan Liu, "Single-shot Quantum Signal Processing Interferometry", Quantum 8, 1427 (2024).

[7] Lorenzo Laneve and Stefan Wolf, "On multivariate polynomials achievable with quantum signal processing", Quantum 9, 1641 (2025).

[8] Yuki Ito, Hitomi Mori, Kazuki Sakamoto, and Keisuke Fujii, "Polynomial time constructive decision algorithm for multivariable quantum signal processing", arXiv:2410.02332, (2024).

[9] Francisca Vasconcelos and András Gilyén, "Methods for Reducing Ancilla-Overhead in Block Encodings", arXiv:2507.07900, (2025).

[10] Andrew Patterson and Leigh Lapworth, "Measurement schemes for quantum linear equation solvers", Quantum Science and Technology 10 2, 025037 (2025).

[11] Zane M. Rossi, "A Solovay-Kitaev theorem for quantum signal processing", arXiv:2505.05468, (2025).

[12] S. E. Skelton, "The Hitchhiker's Guide to QSP pre-processing", arXiv:2501.05977, (2025).

[13] Xi Lu, Yuan Liu, and Hongwei Lin, "Quantum Signal Processing and Quantum Singular Value Transformation on U(N)", Quantum 10, 2048 (2026).

[14] Kevin J. Joven, Elin Ranjan Das, Joel Bierman, Aishwarya Majumdar, Masoud Hakimi Heris, and Yuan Liu, "Scalable Quantum Computational Science: A Perspective from Block-Encodings and Polynomial Transformations", arXiv:2511.16738, (2025).

[15] Pierre-Antoine Bernard and Nathan Wiebe, "Analytical Angle-Finding and Series Expansions for Quantum Signal Processing via Orthogonal Polynomial Theory", arXiv:2605.05321, (2026).

[16] Yuxin Zhang and Changpeng Shao, "Low-ancilla block encodings via Hamiltonian simulation", arXiv:2607.01843, (2026).

The above citations are from Crossref's cited-by service (last updated successfully 2026-08-17 20:57:19) and SAO/NASA ADS (last updated successfully 2026-08-17 20:57:20). The list may be incomplete as not all publishers provide suitable and complete citation data.

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