Double-bracket algorithm for quantum signal processing without post-selection

Yudai Suzuki; Bi Hong Tiang; Jeongrak Son; Nelly H. Y. Ng; Zoe Holmes; Marek Gluza

doi:10.22331/q-2025-12-23-1954

Double-bracket algorithm for quantum signal processing without post-selection

Yudai Suzuki^1,2, Bi Hong Tiang³, Jeongrak Son³, Nelly H. Y. Ng^3,4, Zoe Holmes¹, and Marek Gluza³

¹Institute of Physics, École Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
²Quantum Computing Center, Keio University, Hiyoshi 3-14-1, Kohoku-ku, Yokohama 223-8522, Japan
³School of Physical and Mathematical Sciences, Nanyang Technological University, 637371, Singapore
⁴Centre for Quantum Technologies, Nanyang Technological University, 637371, Singapore

Published:	2025-12-23, volume 9, page 1954
Editor:	Tongyang Li
Eprint:	arXiv:2504.01077v3
Doi:	https://doi.org/10.22331/q-2025-12-23-1954
Citation:	Quantum 9, 1954 (2025).

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Abstract

Quantum Signal Processing (QSP), a framework for implementing matrix-valued polynomials, is a fundamental primitive in various quantum algorithms. Despite its versatility, a potentially underappreciated challenge is that all systematic protocols for implementing QSP rely on post-selection. This can impose prohibitive costs for tasks when amplitude amplification cannot sufficiently improve the success probability. For example, in the context of ground-state preparation, this occurs when using a too poor initial state. In this work, we introduce a new formula for implementing QSP transformations of Hermitian matrices, which requires neither auxiliary qubits nor post-selection. Rather, using approximation to the exact unitary synthesis, we leverage the theory of the double-bracket quantum algorithms to provide a new quantum algorithm for QSP, termed Double-Bracket QSP (DB-QSP). The algorithm requires the energy and energetic variance of the state to be measured at each step and has a recursive structure, which leads to circuit depths that can grow super exponentially with the degree of the polynomial. With these strengths and caveats in mind, DB-QSP should be viewed as complementing the established QSP toolkit. In particular, DB-QSP can deterministically implement low-degree polynomials to ``warm start" QSP methods involving post-selection.

Featured image: Our proposal for implementing a quantum variant of signal processing proceeds recursively, in each step statistical information of the state informs how to increment the signal processing further without resorting to post-selection.

Slides: Double-bracket quantum signal processing by Marek Gluza

Popular summary

While quantum computing transformations are inherently reversible, more general irreversible operations can be achieved by embedding the system into a larger one and using post-selection, i.e. discarding data unless outcomes of measurements on additional qubits meet certain criteria. This approach is formalized by the block-encoding framework, which enables quantum signal processing and underlies many of the most efficient quantum algorithms for optimization and matrix algebra. Notably, however, in instances when the measurement outcomes required for post-selection occur with low probability, the efficiency of the algorithm deteriorates. In this work, we formulated a new quantum algorithm that implements signal processing on quantum computers without post-selection. The main difficulty is that the irreversible signal processing operations require a normalization factor which is a nonlinear function of the input state. We show that this non-linear factor can be implemented recursively, with each step performing an appropriate reflection about the input state. These reflections are reversible operations closely related to those used in the paradigmatic Grover search algorithm. In the near term, the proposed double-bracket quantum signal processing (DB-QSP) framework faces limitations similar to those of signal processing implemented using block-encodings, in that only simple signal processing is possible within the runtime constraints on existing prototypes of quantum computers. For future advanced quantum computers, we expect that combining DB-QSP with block-encoding protocols will be useful as the two approaches have complementary strengths. In particular, applying DB-QSP as an initial stage followed by block-encoding-based techniques is expected to provide an effective "warm-start" strategy for quantum signal processing in fault-tolerant quantum computing architectures.

► BibTeX data

@article{Suzuki2025doublebracket,
  doi = {10.22331/q-2025-12-23-1954},
  url = {https://doi.org/10.22331/q-2025-12-23-1954},
  title = {Double-bracket algorithm for quantum signal processing without post-selection},
  author = {Suzuki, Yudai and Tiang, Bi Hong and Son, Jeongrak and Ng, Nelly H. Y. and Holmes, Zoe and Gluza, Marek},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1954},
  month = dec,
  year = {2025}
}

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[1] Lei Zhang, Jizhe Lai, Xian Wu, and Xin Wang, "Quantum imaginary-time evolution with polynomial resources in evolution time", Quantum Science and Technology 11 3, 035041 (2026).

[2] Matteo Robbiati, Edoardo Pedicillo, Andrea Pasquale, Xiaoyue Li, Oriel Kiss, Andrew Wright, Renato M. S. Farias, Khanh Uyen Giang, Jeongrak Son, Johannes Knörzer, Siong Thye Goh, Jun Yong Khoo, Nelly H. Y. Ng, Zoë Holmes, Stefano Carrazza, and Marek Gluza, "Double-bracket quantum algorithms for high-fidelity ground-state preparation", Physical Review Research 8 3, 033078 (2026).

[3] Aarón Villanueva and Luis Pedro García-Pintos, "Hamiltonian and double-bracket flow formulations of quantum measurements", Reports on Progress in Physics 89 6, 067602 (2026).

[4] Aaron Miller, Joachim Favre, Zoë Holmes, Özlem Salehi, Rahul Chakraborty, Anton Nykänen, Zoltán Zimborás, Adam Glos, and Guillermo García-Pérez, "Simulation of Fermionic circuits using Majorana Propagation", arXiv:2503.18939, (2025).

[5] Jeongrak Son, Marek Gluza, Ryuji Takagi, and Nelly H. Y. Ng, "Quantum Dynamic Programming", Physical Review Letters 134 18, 180602 (2025).

[6] Yudai Suzuki, Marek Gluza, Jeongrak Son, Bi Hong Tiang, Nelly H. Y. Ng, and Zoë Holmes, "Grover's algorithm is an approximation of imaginary-time evolution", arXiv:2507.15065, (2025).

[7] Mohammed Alghadeer, Khanh Uyen Giang, Shuxiang Cao, Simone D. Fasciati, Michele Piscitelli, Nelly Ng, Peter J. Leek, Marek Gluza, and Mustafa Bakr, "Double-Bracket Algorithmic Cooling", arXiv:2510.00302, (2025).

[8] Ben Zindorf and Sougato Bose, "Multi-Controlled Quantum Gates in Linear Nearest Neighbor", arXiv:2506.00695, (2025).

[9] Jeongrak Son, "Quantum Qomrades: Catalysts in Resource Theories and Memories in Dynamic Programming", arXiv:2511.00454, (2025).

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[13] Tal Schwartzman, Torsten V. Zache, Hannes Pichler, and H. R. Sadeghpour, "Unitary imaginary time evolution and ground state preparation using multi-copy protocols", arXiv:2603.11208, (2026).

[14] Zhijian Lai, Hantao Nie, Jiayuan Wu, and Dong An, "Quantum circuit design from a retraction-based Riemannian optimization framework", arXiv:2602.20605, (2026).

[15] Gustav J L Jäger, Krzysztof Bieniasz, Martin B Plenio, and Hans-Martin Rieser, "Entanglement is Half the Story: Post-Selection vs. Partial Traces", arXiv:2605.02385, (2026).

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