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Quantum signal processing polynomials stay within physical bounds through Fourier retraction

Enforcing physical amplitude limits during polynomial design helps quantum subroutines for molecular simulation and linear systems avoid compilation failure, according to a new preprint.

By Primary News Team

30 statements added by Rahul · 25 Sep see what was added
Quantum signal processing polynomials stay within physical bounds through Fourier retraction
Published25 Sep 2026, 17:37 Last updated25 Sep 2026, 17:37 Sources
Source mode Marks each sentence drawn from a source or a contributor

Quantum algorithms promise dramatic computational advantages across science and engineering, from simulating molecules and other complex physical systems to accelerating materials discovery, solving vast linear systems, and modeling differential equations.12 Most of these modern quantum algorithms rely on quantum signal processing and the closely related quantum singular value transformation, which serve as foundational algorithmic primitives or ansätze for quantum circuit design.32 In this framework, algorithms begin with a block encoding of a linear operator, embedding a target matrix into the upper-left block of an achievable unitary quantum circuit, and then perform a polynomial transformation to obtain a block encoding of a transformed operator.12 Inside quantum signal processing, a challenging polynomial approximation problem inevitably arises: compilers must approximate a useful target function, such as 1/x for matrix inversion, using a polynomial, and then implement that transformation through an alternating sequence of single-qubit rotations and controlled operations.1

This computational translation faces a strict physical barrier rooted in the fundamental laws of quantum mechanics.contributed Because the probabilities of all possible physical measurement outcomes must sum exactly to one, every quantum operation is represented by a unitary matrix whose entries can never exceed a magnitude of one.contributed If a compiler approximates a target function with a polynomial that breaches that unit threshold by even an infinitesimal margin at a single coordinate, the mathematical instructions become physically impossible to build.contributed The synthesis algorithm cannot calculate valid rotation angles, and the sequence of quantum gates collapses before running on physical hardware.contributed

To establish a valid quantum routine, classical compilation requires a chain of strict conditions. First, a designer specifies a target response function across an active operating interval.2 Second, an optimization algorithm searches for a polynomial of matching parity, composed exclusively of even or odd powers, that tracks the target curve within an allowable error threshold.2 Third, that approximating polynomial must satisfy a continuous physical bound across the full domain from zero to one. Fourth, a classical factorization algorithm converts the valid polynomial into an explicit list of rotation angles, known as phase factors, which quantum processors execute directly as physical circuit pulses.23

In a preprint posted to the arXiv server on September 1, 2026, researchers Yulong Dong, James B. Larsen, Lin Lin, and lead author Rahul Sarkar introduced a computational framework that prevents approximating polynomials from breaching these mandatory boundaries.1 Their method pairs an active-set Remez exchange algorithm with a technique called nonlinear Fourier retraction, ensuring that quantum signal processing polynomials satisfy continuous physical constraints without inflating the polynomial degree.2

Why does standard numerical optimization fail to protect quantum limits?

Standard numerical solvers fail because quantum signal processing demands continuous constraint satisfaction across every point on a continuous interval rather than across a finite grid of test points.2 The mathematical task forms a constrained minimax approximation problem, in which an algorithm seeks a polynomial of prescribed parity that minimizes the maximum difference between itself and a target function uniformly across a fitting set. Because the polynomial must also stay bounded in magnitude by one across every point on the interval from zero to one, the system must satisfy what mathematicians call a semi-infinite constraint, meaning an infinite collection of conditions applied to a finite set of polynomial coefficients.

When an algorithm discretizes this problem by checking only a finite grid of sampled points through a linear program, the polynomial can still spike above the unit threshold between those grid points.2 Sarkar and colleagues noted that this breakdown occurs most frequently when the mathematically optimal curve presses directly against the outer boundary of the feasible set.2 In those critical regions, standard linear programming solutions report success on the sampled grid while producing a polynomial that violates physical bounds on the continuous domain.2

Sarkar, the lead author of the preprint, explained that this boundary instability stems directly from the geometry of discretized semi-infinite programs.contributed As Sarkar told Primary, "This is a generic phenomenon that happens when solving LSIP (Linear semi-infinite programs) with discretization, especially when the optimal solutions lie on or close to the feasible set boundary (this situation is known as the fully-coherent regime in the context of quantum signal processing (QSP)). If a non-constant polynomial is touching the bound constraints at two successive discretization points, the polynomial has to either have a lower or higher value in between the two points. What our paper demonstrates is that for many QSP problems, the polynomials end up having higher values in between two successive discretization points, violating the bound constraints for the QSP problem instances."contributed

How does the new mathematical workflow enforce continuous physical bounds?

Sarkar and colleagues investigated two complementary strategies to enforce physical feasibility across the complete interval.2 The first strategy combines a Remez exchange method, an iterative algorithm that continuously updates a set of reference points toward local error peaks, with active-set constraint enforcement to pull the polynomial back whenever it threatens to breach the unit bound.2 While this active-set exchange technique proved computationally efficient across many tested instances, the authors reported that its numerical stability depends heavily on the chosen target function and the geometry of the optimization problem.2

Quantum signal processing polynomials stay within physical bounds through Fourier retraction

Addressing how the algorithm manages constraint boundaries during optimization, the authors explained that the exchange procedure applies strict boundary pinning without an arbitrary tolerance buffer.contributed "For the active-set step, there is no additional feasibility tolerance: any candidate extremum in the constraint region with |f(x)| > 1 is activated and pinned to the nearest boundary, +1 or -1," the authors said in response to questions from Primary.contributed The authors added that numerical tolerance serves solely as a stopping rule, stating that "The tolerance in the Remez and active-set based algorithm of our paper is only used as a convergence stopping criterion; any residual violation is removed by the final rescaling step."contributed

When asked which problem geometries cause the active-set exchange solver to falter, the authors identified competing polynomial requirements as the central vulnerability.contributed "The main unstable example we observed is uniform singular value amplification," the authors said.contributed "Its target function lies in the low-degree approximation space on the fitting interval, while higher-degree components are still needed to enforce the global unit bound outside that interval. This competition can break equioscillation or produce substantial overshoot. We do not claim a complete classification of unstable target families."contributed

This mathematical strategy extends a lineage of research connecting nonlinear Fourier analysis to quantum algorithms developed since 2024.52 The foundation was established by Michel Alexis, Gevorg Mnatsakanyan, and Christoph Thiele in their 2024 paper, "Quantum signal processing and nonlinear Fourier analysis," which uncovered the core link between nonlinear Fourier analysis and quantum signal processing.5 That framework was advanced by Michel Alexis, Lin Lin, Gevorg Mnatsakanyan, Christoph Thiele, and Jiasu Wang in their 2024 work, "Infinite quantum signal processing for arbitrary Szego functions," which introduced the Riemann-Hilbert-Weiss algorithm.1 Building directly on these foundations, Hongkang Ni, Rahul Sarkar, Lexing Ying, and Lin Lin developed the layer-stripping and inverse nonlinear fast Fourier transform routines in their 2025 paper, "Inverse nonlinear fast Fourier transform on SU(2) with applications to quantum signal processing," providing the computational stability and fast algorithms that the current paper's retraction procedure relies upon.contributed

To resolve cases where the Remez heuristic becomes unstable, the authors introduced an approach termed nonlinear Fourier retraction.2 Nonlinear Fourier retraction takes a polynomial that is only nearly feasible and applies quantum signal processing completion and phase synthesis to generate phase factors for a strictly feasible polynomial without increasing the polynomial degree.2

Sarkar explained the algebraic mechanism that allows this transformation to maintain degree while guaranteeing physical boundaries.contributed "The key is that the Nonlinear Fourier Transform (NLFT) has the property that it takes sequences of complex numbers and outputs 2x2 matrices, which are elements of the Lie group SU(2), where the matrix entries are Laurent polynomials," Sarkar told Primary.contributed "SU(2) matrices have determinant one, and this translates into the property that the polynomial entries of the matrix must have absolute value at most one inside the unit disk."contributed Building on prior computational stability findings for layer-stripping routines established by Ni, Sarkar, Ying, and Lin, the retraction algorithm evaluates polynomials even outside the formal image of the transform.3 As Sarkar noted, layer stripping generates coefficient sequences whose transform lies strictly within SU(2), guaranteeing physical bound satisfaction while preserving the degree.contributed

What can the retraction framework deliver for quantum compilers?

Across representative test problems, Sarkar and colleagues found that nonlinear Fourier retraction largely preserves approximation accuracy and remains effective on difficult problem instances where the Remez exchange heuristic fails.2 The researchers integrated the resulting workflow into an open software package named qsppack, providing a unified pipeline for quantum algorithm development.2 Detailed documentation and executable examples are publicly accessible through the pyqsppack repository on GitHub and its dedicated Read the Docs portal.contributed

The study represents an algorithmic demonstration across representative benchmark problems and does not supply an analytical convergence guarantee for every conceivable target function.2 The Remez exchange technique remains sensitive to problem geometry, while nonlinear Fourier retraction relies on small initial boundary violations.2 As Sarkar cautioned to Primary, "We think this is a good method only when the constraint violations of the input polynomial are both small in magnitude and occur over a small region of the constraint enforcement set. If your polynomial is violating the constraints by a lot, we do not recommend using nonlinear Fourier retraction for constraint enforcement, even though the method would still work, but the result will probably be extremely suboptimal."contributed Furthermore, the research was posted as an arXiv preprint on September 1, 2026, and has not yet completed formal peer review.1

Establishing reliable mathematical pipelines from continuous target functions to discrete rotation sequences represents an essential milestone for quantum software engineering. By ensuring that approximating polynomials never breach physical amplitude thresholds, the qsppack tools help protect practical quantum subroutines in Hamiltonian simulation, quantum chemistry, and numerical linear algebra from compilation failure.21

What this rests on

52 sentences trace to 4 sources and 1 contributor.

  1. 1 L lin.caltech.edu Third-party reporting · lin.caltech.edu Lin Lin's homepage See the source
  2. 2 YD Yulong Dong, James B. Larsen, Lin Lin, Rahul Sarkar Paper · arxiv.org · 1 Sep 2026 Constrained minimax approximation for quantum signal processing Preprint · may not have been peer reviewed See the source
  3. 3 A arxiv.org Paper · arxiv.org Inverse nonlinear fast Fourier transform on SU(2) with applications to quantum signal processing Preprint · may not have been peer reviewed See the source
  4. 4 R Rahul Contributor · interview 7–8 Sep 2026 · in the article since 25 Sep 2026 Contribution — Rahul 30 statements added to this article
  5. 5 A arxiv.org Paper · arxiv.org Quantum signal processing and nonlinear Fourier analysis Preprint · may not have been peer reviewed See the source

Article history

  1. Published 25 Sep 2026, 17:37
    Assembled by the Primary desk from 4 sources · 1 contributor · 52 cited sentences
  2. Rahul, lead author 30 statements added 18 days before publication
    1. Go to the sentence: This computational translation faces a strict physical barrier rooted in the fundamental laws of quantum mechanics.
    2. Go to the sentence: Because the probabilities of all possible physical measurement outcomes must sum exactly to one, every quantum operation is represented by a unitary matrix whose entries can never exceed a magnitude of one.
    3. Go to the sentence: If a compiler approximates a target function with a polynomial that breaches that unit threshold by even an infinitesimal margin at a single coordinate, the mathematical instructions become physically impossible to build.
    4. Go to the sentence: The synthesis algorithm cannot calculate valid rotation angles, and the sequence of quantum gates collapses before running on physical hardware.
    5. Go to the sentence: When an algorithm discretizes this problem by checking only a finite grid of sampled points through a linear program, the polynomial can still spike above the unit threshold between those grid points.
    6. Go to the sentence: Sarkar and colleagues noted that this breakdown occurs most frequently when the mathematically optimal curve presses directly against the outer boundary of the feasible set.
    7. Go to the sentence: In those critical regions, standard linear programming solutions report success on the sampled grid while producing a polynomial that violates physical bounds on the continuous domain.
    8. Go to the sentence: Sarkar, the lead author of the preprint, explained that this boundary instability stems directly from the geometry of discretized semi-infinite programs.
    9. Go to the sentence: As Sarkar told Primary, "This is a generic phenomenon that happens when solving LSIP (Linear semi-infinite programs) with discretization, especially when the optimal solutions lie on or close to the feasible set boundary (this situation is known as the fully-coherent regime in the context of quantum signal processing (QSP)). If a non-constant polynomial is touching the bound constraints at two successive discretization points, the polynomial has to either have a lower or higher value in between the two points. What our paper demonstrates is that for many QSP problems, the polynomials end up having higher values in between two successive discretization points, violating the bound constraints for the QSP problem instances."
    10. Go to the sentence: The first strategy combines a Remez exchange method, an iterative algorithm that continuously updates a set of reference points toward local error peaks, with active-set constraint enforcement to pull the polynomial back whenever it threatens to breach the unit bound.
    11. Go to the sentence: While this active-set exchange technique proved computationally efficient across many tested instances, the authors reported that its numerical stability depends heavily on the chosen target function and the geometry of the optimization problem.
    12. Go to the sentence: Addressing how the algorithm manages constraint boundaries during optimization, the authors explained that the exchange procedure applies strict boundary pinning without an arbitrary tolerance buffer.
    13. Go to the sentence: "For the active-set step, there is no additional feasibility tolerance: any candidate extremum in the constraint region with |f(x)| > 1 is activated and pinned to the nearest boundary, +1 or -1," the authors said in response to questions from Primary.
    14. Go to the sentence: The authors added that numerical tolerance serves solely as a stopping rule, stating that "The tolerance in the Remez and active-set based algorithm of our paper is only used as a convergence stopping criterion; any residual violation is removed by the final rescaling step."
    15. Go to the sentence: When asked which problem geometries cause the active-set exchange solver to falter, the authors identified competing polynomial requirements as the central vulnerability.
    16. Go to the sentence: "The main unstable example we observed is uniform singular value amplification," the authors said.
    17. Go to the sentence: "Its target function lies in the low-degree approximation space on the fitting interval, while higher-degree components are still needed to enforce the global unit bound outside that interval. This competition can break equioscillation or produce substantial overshoot. We do not claim a complete classification of unstable target families."
    18. Go to the sentence: Building directly on these foundations, Hongkang Ni, Rahul Sarkar, Lexing Ying, and Lin Lin developed the layer-stripping and inverse nonlinear fast Fourier transform routines in their 2025 paper, "Inverse nonlinear fast Fourier transform on SU(2) with applications to quantum signal processing," providing the computational stability and fast algorithms that the current paper's retraction procedure relies upon.
    19. Go to the sentence: Nonlinear Fourier retraction takes a polynomial that is only nearly feasible and applies quantum signal processing completion and phase synthesis to generate phase factors for a strictly feasible polynomial without increasing the polynomial degree.
    20. Go to the sentence: Sarkar explained the algebraic mechanism that allows this transformation to maintain degree while guaranteeing physical boundaries.
    21. Go to the sentence: "The key is that the Nonlinear Fourier Transform (NLFT) has the property that it takes sequences of complex numbers and outputs 2x2 matrices, which are elements of the Lie group SU(2), where the matrix entries are Laurent polynomials," Sarkar told Primary.
    22. Go to the sentence: "SU(2) matrices have determinant one, and this translates into the property that the polynomial entries of the matrix must have absolute value at most one inside the unit disk."
    23. Go to the sentence: Building on prior computational stability findings for layer-stripping routines established by Ni, Sarkar, Ying, and Lin, the retraction algorithm evaluates polynomials even outside the formal image of the transform.
    24. Go to the sentence: As Sarkar noted, layer stripping generates coefficient sequences whose transform lies strictly within SU(2), guaranteeing physical bound satisfaction while preserving the degree.
    25. Go to the sentence: Across representative test problems, Sarkar and colleagues found that nonlinear Fourier retraction largely preserves approximation accuracy and remains effective on difficult problem instances where the Remez exchange heuristic fails.
    26. Go to the sentence: The researchers integrated the resulting workflow into an open software package named qsppack, providing a unified pipeline for quantum algorithm development.
    27. Go to the sentence: Detailed documentation and executable examples are publicly accessible through the pyqsppack repository on GitHub and its dedicated Read the Docs portal.
    28. Go to the sentence: The study represents an algorithmic demonstration across representative benchmark problems and does not supply an analytical convergence guarantee for every conceivable target function.
    29. Go to the sentence: The Remez exchange technique remains sensitive to problem geometry, while nonlinear Fourier retraction relies on small initial boundary violations.
    30. Go to the sentence: As Sarkar cautioned to Primary, "We think this is a good method only when the constraint violations of the input polynomial are both small in magnitude and occur over a small region of the constraint enforcement set. If your polynomial is violating the constraints by a lot, we do not recommend using nonlinear Fourier retraction for constraint enforcement, even though the method would still work, but the result will probably be extremely suboptimal."