quantum-computing

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

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

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Researchers introduced Double-Bracket Quantum Signal Processing (DB-QSP), a novel algorithm eliminating post-selection in quantum signal processing, addressing a key bottleneck in current QSP implementations. The method avoids auxiliary qubits by using recursive energy measurements and state variance tracking, enabling deterministic polynomial transformations of Hermitian matrices without probabilistic overhead. While circuit depth grows super-exponentially with polynomial degree, DB-QSP excels at low-degree implementations, offering a "warm start" for traditional QSP methods in ground-state preparation tasks. Near-term limitations mirror block-encoding approaches, but future fault-tolerant systems could combine DB-QSP with existing protocols for hybrid efficiency gains. The work complements established QSP frameworks, particularly where post-selection fails due to poor initial states or insufficient amplitude amplification.

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AbstractQuantum 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 summaryWhile 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} }► References [1] Guang Hao Low, Theodore J. Yoder, and Isaac L. Chuang. ``Methodology of resonant equiangular composite quantum gates''. Phys. Rev. X 6, 041067 (2016). https://doi.org/10.1103/PhysRevX.6.041067 [2] John M. Martyn, Zane M. Rossi, Andrew K. Tan, and Isaac L. Chuang. ``Grand unification of quantum algorithms''. PRX Quantum 2, 040203 (2021). https://doi.org/10.1103/PRXQuantum.2.040203 [3] András Gilyén, Yuan Su, Guang Hao Low, and Nathan Wiebe. ``Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics''. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing. Pages 193–204. (2019). https://doi.org/10.1145/3313276.3316366 [4] Aram W Harrow, Avinatan Hassidim, and Seth Lloyd. ``Quantum algorithm for linear systems of equations''.

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Physical Review Letters 102, 130503 (2009). https://doi.org/10.1103/PhysRevLett.102.130503 [65] Oumarou Oumarou, Pauline J Ollitrault, Cristian L Cortes, Maximilian Scheurer, Robert M Parrish, and Christian Gogolin. ``Molecular Properties from Quantum Krylov Subspace Diagonalization'' (2025). arXiv:2501.05286. https://doi.org/10.1021/acs.jctc.5c00194 arXiv:2501.05286 [66] Long Gui-Lu. ``General quantum interference principle and duality computer''. Communications in Theoretical Physics 45, 825 (2006). https://doi.org/10.1088/0253-6102/45/5/013 [67] Andrew M Childs, Robin Kothari, and Rolando D Somma. ``Quantum algorithm for systems of linear equations with exponentially improved dependence on precision''. SIAM Journal on Computing 46, 1920–1950 (2017). https://doi.org/10.1137/16M1087072 [68] Yulong Dong, Lin Lin, and Yu Tong. ``Ground-state preparation and energy estimation on early fault-tolerant quantum computers via quantum eigenvalue transformation of unitary matrices''. PRX Quantum 3, 040305 (2022). https://doi.org/10.1103/PRXQuantum.3.040305 [69] Ruizhe Zhang, Guoming Wang, and Peter Johnson. ``Computing ground state properties with early fault-tolerant quantum computers''. Quantum 6, 761 (2022). https://doi.org/10.22331/q-2022-07-11-761 [70] Lin Lin and Yu Tong. ``Heisenberg-Limited Ground-State Energy Estimation for Early Fault-Tolerant Quantum Computers''. PRX Quantum 3, 010318 (2022). https://doi.org/10.1103/PRXQuantum.3.010318 [71] Seth Lloyd. ``Almost any quantum logic gate is universal''.

Physical Review Letters 75, 346 (1995). https://doi.org/10.1103/PhysRevLett.75.346 [72] Nicholas C Rubin, Ryan Babbush, and Jarrod McClean. ``Application of fermionic marginal constraints to hybrid quantum algorithms''. New Journal of Physics 20, 053020 (2018). https://doi.org/10.1088/1367-2630/aab919Cited byOn Crossref's cited-by service no data on citing works was found (last attempt 2025-12-28 09:02:46). Could not fetch ADS cited-by data during last attempt 2025-12-28 09:02:46: No response from ADS or unable to decode the received json data when getting the list of citing works.This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions. AbstractQuantum 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 summaryWhile 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} }► References [1] Guang Hao Low, Theodore J. Yoder, and Isaac L. Chuang. ``Methodology of resonant equiangular composite quantum gates''. Phys. Rev. X 6, 041067 (2016). https://doi.org/10.1103/PhysRevX.6.041067 [2] John M. Martyn, Zane M. Rossi, Andrew K. Tan, and Isaac L. Chuang. ``Grand unification of quantum algorithms''. PRX Quantum 2, 040203 (2021). https://doi.org/10.1103/PRXQuantum.2.040203 [3] András Gilyén, Yuan Su, Guang Hao Low, and Nathan Wiebe. ``Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics''. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing. Pages 193–204. (2019). https://doi.org/10.1145/3313276.3316366 [4] Aram W Harrow, Avinatan Hassidim, and Seth Lloyd. ``Quantum algorithm for linear systems of equations''.

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