Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2205.06261 · Quantum

Multivariable quantum signal processing (M-QSP): prophecies of the two-headed oracle

Zane M. Rossi, Isaac L. Chuang

Published Sep 20, 2022• 30 citations
arXivHTMLPDF

Abstract

Recent work shows that quantum signal processing (QSP) and its multi-qubit lifted version, quantum singular value transformation (QSVT), unify and improve the presentation of most quantum algorithms. QSP/QSVT characterize the ability, by alternating ansätze, to obliviously transform the singular values of subsystems of unitary matrices by polynomial functions; these algorithms are numerically stable and analytically well-understood. That said, QSP/QSVT require consistent access to a single oracle, saying nothing about computing joint properties of two or more oracles; these can be far cheaper to determine given an ability to pit oracles against one another coherently. This work introduces a corresponding theory of QSP over multiple variables: M-QSP. Surprisingly, despite the non-existence of the fundamental theorem of algebra for multivariable polynomials, there exist necessary and sufficient conditions under which a desired stable multivariable polynomial transformation is possible. Moreover, the classical subroutines used by QSP protocols survive in the multivariable setting for non-obvious reasons, and remain numerically stable and efficient. Up to a well-defined conjecture, we give proof that the family of achievable multivariable transforms is as loosely constrained as could be expected. The unique ability of M-QSP to obliviously approximate joint functions of multiple variables coherently leads to novel speedups incommensurate with those of other quantum algorithms, and provides a bridge from quantum algorithms to algebraic geometry.

DOI
10.22331/q-2022-09-20-811

Related papers

Comment on "Multivariable quantum signal processing (M-QSP): prophecies of the two-headed oracle"

Hitomi Mori, Kaoru Mizuta, Keisuke Fujii

Multivariable Quantum Signal Processing (M-QSP) \cite{Rossi2022multivariable} is expected to provide an efficient means to handle polynomial transformations of multiple variables simultaneously. However, we identified several inconsistencies in the main theorem, where necessary and sufficient conditions for achievable polynomials are provided, and its proof in Ref. \cite{Rossi2022multivariable}. Moreover, a counterexample to the conjecture in Ref. \cite{Rossi2022multivariable}, based on which the main theorem is constructed, is presented in Ref. \cite{n^^c3^^a9meth2023variants}, meaning the requirement of the conjecture should be included as a condition in the theorem. Here we note our observations and propose the revised necessary conditions for M-QSP. We also show that these necessary conditions cannot be sufficient conditions, and thus some additional condition on top of these revisions is essentially required for the complete M-QSP theorem.

arXiv:2310.00918•Quantum•Oct 29, 2024•4 citations

Polynomial time constructive decision algorithm for multivariable quantum signal processing

Yuki Ito, Hitomi Mori, Kazuki Sakamoto +1 more

Quantum signal processing (QSP) and quantum singular value transformation (QSVT) have provided a unified framework for understanding many quantum algorithms, including factorization, matrix inversion, and Hamiltonian simulation. As a multivariable version of QSP, multivariable quantum signal processing (M-QSP) is proposed. M-QSP interleaves signal operators corresponding to each variable with signal processing operators, which provides an efficient means to perform multivariable polynomial transformations. However, the necessary and sufficient condition for what types of polynomials can be constructed by M-QSP is unknown. In this paper, we propose a classical algorithm to determine whether a given pair of multivariable Laurent polynomials can be implemented by M-QSP, which returns True or False. As one of the most important properties of this algorithm, its returning True is the necessary and sufficient condition. The proposed classical algorithm runs in polynomial time in the number of variables and signal operators. Our algorithm also provides a constructive method to select the necessary parameters for implementing M-QSP. These findings offer valuable insights for identifying practical applications of M-QSP.

arXiv:2410.02332•Quantum•May 12, 2026•0 citations

An adversary bound for quantum signal processing

Lorenzo Laneve

Quantum 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.

arXiv:2506.20484•Quantum•Mar 13, 2026•0 citations

Quantum signal processing with continuous variables

Zane M. Rossi, V. M. Bastidas, William J. Munro +1 more

Quantum singular value transformation (QSVT) enables the application of polynomial functions to the singular values of near arbitrary linear operators embedded in unitary transforms, and has been used to unify, simplify, and improve most quantum algorithms. QSVT depends on precise results in representation theory, with the desired polynomial functions acting simultaneously within invariant two-dimensional subspaces of a larger Hilbert space. These two-dimensional transformations are largely determined by the related theory of quantum signal processing (QSP). While QSP appears to rely on properties specific to the compact Lie group SU(2), many other Lie groups appear naturally in physical systems relevant to quantum information. This work considers settings in which SU(1,1) describes system dynamics and finds that, surprisingly, despite the non-compactness of SU(1,1), one can recover a QSP-type ansatz, and show its ability to approximate near arbitrary polynomial transformations. We discuss various experimental uses of this construction, as well as prospects for expanded relevance of QSP-like ansätze to other Lie groups.

arXiv:2304.14383•Apr 27, 2023•3 citations

On multivariate polynomials achievable with quantum signal processing

Lorenzo Laneve, Stefan Wolf

Quantum signal processing (QSP) is a framework which was proven to unify and simplify a large number of known quantum algorithms, as well as discovering new ones. QSP allows one to transform a signal embedded in a given unitary using polynomials. Characterizing which polynomials can be achieved with QSP protocols is an important part of the power of this technique, and while such a characterization is well-understood in the case of univariate signals, it is unclear which multivariate polynomials can be constructed when the signal is a vector, rather than a scalar. This work uses a slightly different formalism than what is found in the literature, and uses it to find simpler necessary conditions for decomposability, as well as a sufficient condition – the first, to the best of our knowledge, proven for a (generally inhomogeneous) multivariate polynomial in the context of quantum signal processing.

arXiv:2407.20823•Quantum•Feb 20, 2025•5 citations

Quantum signal processing over SU(N)

Lorenzo Laneve

Quantum signal processing (QSP) and the quantum singular value transformation (QSVT) are pivotal tools for simplifying the development of quantum algorithms. These techniques leverage polynomial transformations on the eigenvalues or singular values of block-encoded matrices, achieved with the use of just one control qubit. In contexts where the control qubit is used to extract information about the eigenvalues or singular values, the amount of extractable information is limited to one bit per protocol. In this work, we extend the original QSP ansatz by introducing multiple control qubits. We show that, much like in the single-qubit case, nearly any vector of polynomials can be implemented with a multi-qubit QSP ansatz, and the gate complexity scales polynomially with the dimension of such states. Moreover, assuming that powers of the matrix to transform are easily implementable - as in Shor's algorithm - we can achieve polynomial transformations with degrees that scale exponentially with the number of control qubits. This work aims to provide a partial characterization of the polynomials that can be implemented using this approach, with phase estimation schemes and discrete logarithm serving as illustrative examples.

arXiv:2311.03949•Nov 7, 2023•0 citations
Recent work shows that quantum signal processing (QSP) and its multi-qubit lifted version, quantum singular value transformation (QSVT), unify and improve the presentation of most quantum algorithms. QSP/QSVT characterize the ability, by alternating ansätze, to obliviously transform the singular values of subsystems of unitary matrices by polynomial functions; these algorithms are numerically stable and analytically well-understood. That said, QSP/QSVT require consistent access to a single oracle, saying nothing about computing joint properties of two or more oracles; these can be far cheaper to determine given an ability to pit oracles against one another coherently. This work introduces a corresponding theory of QSP over multiple variables: M-QSP. Surprisingly, despite the non-existence of the fundamental theorem of algebra for multivariable polynomials, there exist necessary and sufficient conditions under which a desired stable multivariable polynomial transformation is possible. Moreover, the classical subroutines used by QSP protocols survive in the multivariable setting for non-obvious reasons, and remain numerically stable and efficient. Up to a well-defined conjecture, we give proof that the family of achievable multivariable transforms is as loosely constrained as could be expected. The unique ability of M-QSP to obliviously approximate joint functions of multiple variables coherently leads to novel speedups incommensurate with those of other quantum algorithms, and provides a bridge from quantum algorithms to algebraic geometry.
Multivariable Quantum Signal Processing (M-QSP) \cite{Rossi2022multivariable} is expected to provide an efficient means to handle polynomial transformations of multiple variables simultaneously. However, we identified several inconsistencies in the main theorem, where necessary and sufficient conditions for achievable polynomials are provided, and its proof in Ref. \cite{Rossi2022multivariable}. Moreover, a counterexample to the conjecture in Ref. \cite{Rossi2022multivariable}, based on which the main theorem is constructed, is presented in Ref. \cite{n^^c3^^a9meth2023variants}, meaning the requirement of the conjecture should be included as a condition in the theorem. Here we note our observations and propose the revised necessary conditions for M-QSP. We also show that these necessary conditions cannot be sufficient conditions, and thus some additional condition on top of these revisions is essentially required for the complete M-QSP theorem.
Quantum signal processing (QSP) and quantum singular value transformation (QSVT) have provided a unified framework for understanding many quantum algorithms, including factorization, matrix inversion, and Hamiltonian simulation. As a multivariable version of QSP, multivariable quantum signal processing (M-QSP) is proposed. M-QSP interleaves signal operators corresponding to each variable with signal processing operators, which provides an efficient means to perform multivariable polynomial transformations. However, the necessary and sufficient condition for what types of polynomials can be constructed by M-QSP is unknown. In this paper, we propose a classical algorithm to determine whether a given pair of multivariable Laurent polynomials can be implemented by M-QSP, which returns True or False. As one of the most important properties of this algorithm, its returning True is the necessary and sufficient condition. The proposed classical algorithm runs in polynomial time in the number of variables and signal operators. Our algorithm also provides a constructive method to select the necessary parameters for implementing M-QSP. These findings offer valuable insights for identifying practical applications of M-QSP.
Quantum 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.
Quantum singular value transformation (QSVT) enables the application of polynomial functions to the singular values of near arbitrary linear operators embedded in unitary transforms, and has been used to unify, simplify, and improve most quantum algorithms. QSVT depends on precise results in representation theory, with the desired polynomial functions acting simultaneously within invariant two-dimensional subspaces of a larger Hilbert space. These two-dimensional transformations are largely determined by the related theory of quantum signal processing (QSP). While QSP appears to rely on properties specific to the compact Lie group SU(2), many other Lie groups appear naturally in physical systems relevant to quantum information. This work considers settings in which SU(1,1) describes system dynamics and finds that, surprisingly, despite the non-compactness of SU(1,1), one can recover a QSP-type ansatz, and show its ability to approximate near arbitrary polynomial transformations. We discuss various experimental uses of this construction, as well as prospects for expanded relevance of QSP-like ansätze to other Lie groups.
Quantum signal processing (QSP) is a framework which was proven to unify and simplify a large number of known quantum algorithms, as well as discovering new ones. QSP allows one to transform a signal embedded in a given unitary using polynomials. Characterizing which polynomials can be achieved with QSP protocols is an important part of the power of this technique, and while such a characterization is well-understood in the case of univariate signals, it is unclear which multivariate polynomials can be constructed when the signal is a vector, rather than a scalar. This work uses a slightly different formalism than what is found in the literature, and uses it to find simpler necessary conditions for decomposability, as well as a sufficient condition – the first, to the best of our knowledge, proven for a (generally inhomogeneous) multivariate polynomial in the context of quantum signal processing.
Quantum signal processing (QSP) and the quantum singular value transformation (QSVT) are pivotal tools for simplifying the development of quantum algorithms. These techniques leverage polynomial transformations on the eigenvalues or singular values of block-encoded matrices, achieved with the use of just one control qubit. In contexts where the control qubit is used to extract information about the eigenvalues or singular values, the amount of extractable information is limited to one bit per protocol. In this work, we extend the original QSP ansatz by introducing multiple control qubits. We show that, much like in the single-qubit case, nearly any vector of polynomials can be implemented with a multi-qubit QSP ansatz, and the gate complexity scales polynomially with the dimension of such states. Moreover, assuming that powers of the matrix to transform are easily implementable - as in Shor's algorithm - we can achieve polynomial transformations with degrees that scale exponentially with the number of control qubits. This work aims to provide a partial characterization of the polynomials that can be implemented using this approach, with phase estimation schemes and discrete logarithm serving as illustrative examples.