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An adversary bound for quantum signal processing

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An adversary bound for quantum signal processing

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AbstractQuantum signal processing (QSP) and quantum singular value transformation (QSVT), have emerged as unifying frameworks in the context of quantum algorithm design. These techniques allow to carry out efficient polynomial transformations of matrices block-encoded in unitaries, involving a single ancilla qubit. Recent efforts try to extend QSP to the multivariate setting (M-QSP), where multiple matrices are transformed simultaneously. However, this generalization faces problems not encountered in the univariate counterpart: in particular, the class of polynomials achievable by M-QSP seems hard to characterize. In this work we borrow tools from query complexity, namely the state conversion problem and the adversary bound: we first recast QSP as a state conversion problem over the Hilbert space of square-integrable functions. We then show that the adversary bound for a state conversion problem in this space precisely identifies all and only the QSP protocols in the univariate case. Motivated by this first result, we extend the formalism to several variables: the existence of a feasible solution to the adversary bound implies the existence of a M-QSP protocol, and the computation of a protocol of minimal space is reduced to a rank minimization problem involving the feasible solution space of the adversary bound.Featured image: On the left, the process of constructing a QSP protocol (univariate or multivariate) using the adversary bound. On the right a visualization of $Q_\tau$, a convex space of feasible solutions to the adversary bound for constructing a desired parameterized quantum state $\tau(z)$. If $Q_\tau$ is non-empty, then a QSP protocol for $\tau(z)$ exists.Popular summaryQuantum signal processing is an important primitive in the quantum algorithmic literature. Roughly speaking, by alternating occurrences of a unitary that encodes some complex number $z$ (the "signal" operator) with operations $A_k$ that are independent of $z$ (the "processing" operators), the matrix product will contain a polynomial $P(z)$ in one of the entries, whose degree depends on the number of calls to the signal operator, and whose coefficients depend on the $A_k$ we choose. It turns out that we can make any possible polynomial $P(z)$ by choosing an appropriate sequence of processing operators. However, this is not clear when we have access to multiple signals $z_1, \ldots, z_m$ (the so-called multivariate quantum signal processing). This work uses the adversary bound — a convex optimization problem borrowed from quantum query complexity theory — to establish a necessary and sufficient condition for a given polynomial $P(z_1, \ldots, z_m)$ to admit a sequence of processing operators.► BibTeX data@article{Laneve2026adversarybound, doi = {10.22331/q-2026-03-13-2025}, url = {https://doi.org/10.22331/q-2026-03-13-2025}, title = {An adversary bound for quantum signal processing}, author = {Laneve, Lorenzo}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2025}, month = mar, year = {2026} }► References [1] Gilles Brassard, Peter Hoyer, Michele Mosca, and Alain Tapp. ``Quantum Amplitude Amplification and Estimation''. Quantum Computation and Information 305, 53–74 (2002). https://doi.org/10.1090/conm/305/05215 [2] Dominic W Berry, Andrew M Childs, and Robin Kothari. ``Hamiltonian Simulation with Nearly Optimal Dependence on all Parameters''. In 2015 IEEE 56th Annual Symposium on Foundations of Computer Science. Pages 792–809. (2015). https://doi.org/10.1109/FOCS.2015.54 [3] Andrew M Childs and Nathan Wiebe. ``Hamiltonian simulation using linear combinations of unitary operations''. Quantum Information and Computation 12, 901–924 (2012). https://doi.org/10.26421/QIC12.11-12-1 [4] Dong An, Jin-Peng Liu, and Lin Lin. ``Linear Combination of Hamiltonian Simulation for Nonunitary Dynamics with Optimal State Preparation Cost''.

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AbstractQuantum signal processing (QSP) and quantum singular value transformation (QSVT), have emerged as unifying frameworks in the context of quantum algorithm design. These techniques allow to carry out efficient polynomial transformations of matrices block-encoded in unitaries, involving a single ancilla qubit. Recent efforts try to extend QSP to the multivariate setting (M-QSP), where multiple matrices are transformed simultaneously. However, this generalization faces problems not encountered in the univariate counterpart: in particular, the class of polynomials achievable by M-QSP seems hard to characterize. In this work we borrow tools from query complexity, namely the state conversion problem and the adversary bound: we first recast QSP as a state conversion problem over the Hilbert space of square-integrable functions. We then show that the adversary bound for a state conversion problem in this space precisely identifies all and only the QSP protocols in the univariate case. Motivated by this first result, we extend the formalism to several variables: the existence of a feasible solution to the adversary bound implies the existence of a M-QSP protocol, and the computation of a protocol of minimal space is reduced to a rank minimization problem involving the feasible solution space of the adversary bound.Featured image: On the left, the process of constructing a QSP protocol (univariate or multivariate) using the adversary bound. On the right a visualization of $Q_\tau$, a convex space of feasible solutions to the adversary bound for constructing a desired parameterized quantum state $\tau(z)$. If $Q_\tau$ is non-empty, then a QSP protocol for $\tau(z)$ exists.Popular summaryQuantum signal processing is an important primitive in the quantum algorithmic literature. Roughly speaking, by alternating occurrences of a unitary that encodes some complex number $z$ (the "signal" operator) with operations $A_k$ that are independent of $z$ (the "processing" operators), the matrix product will contain a polynomial $P(z)$ in one of the entries, whose degree depends on the number of calls to the signal operator, and whose coefficients depend on the $A_k$ we choose. It turns out that we can make any possible polynomial $P(z)$ by choosing an appropriate sequence of processing operators. However, this is not clear when we have access to multiple signals $z_1, \ldots, z_m$ (the so-called multivariate quantum signal processing). This work uses the adversary bound — a convex optimization problem borrowed from quantum query complexity theory — to establish a necessary and sufficient condition for a given polynomial $P(z_1, \ldots, z_m)$ to admit a sequence of processing operators.► BibTeX data@article{Laneve2026adversarybound, doi = {10.22331/q-2026-03-13-2025}, url = {https://doi.org/10.22331/q-2026-03-13-2025}, title = {An adversary bound for quantum signal processing}, author = {Laneve, Lorenzo}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2025}, month = mar, year = {2026} }► References [1] Gilles Brassard, Peter Hoyer, Michele Mosca, and Alain Tapp. ``Quantum Amplitude Amplification and Estimation''. Quantum Computation and Information 305, 53–74 (2002). https://doi.org/10.1090/conm/305/05215 [2] Dominic W Berry, Andrew M Childs, and Robin Kothari. ``Hamiltonian Simulation with Nearly Optimal Dependence on all Parameters''. In 2015 IEEE 56th Annual Symposium on Foundations of Computer Science. Pages 792–809. (2015). https://doi.org/10.1109/FOCS.2015.54 [3] Andrew M Childs and Nathan Wiebe. ``Hamiltonian simulation using linear combinations of unitary operations''. Quantum Information and Computation 12, 901–924 (2012). https://doi.org/10.26421/QIC12.11-12-1 [4] Dong An, Jin-Peng Liu, and Lin Lin. ``Linear Combination of Hamiltonian Simulation for Nonunitary Dynamics with Optimal State Preparation Cost''.

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An adversary bound for quantum signal processing

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