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Exact and Efficient Circuit Construction for Block Encoding Matrix Polynomials
Quantum Physics · Submitted 15 Aug 2026
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim The resulting algorithm achieves a computational cost of $\mathcal{O}(d\log d)$ for explicitly constructing block encodings of matrix polynomials, improving upon the best-known theoretical bounds of previous methods.
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The abstract, sentence by sentence
A recent interpolation-based Quantum Signal Processing (QSP) framework by Alase bypasses the phase-finding procedures required in conventional QSP, allowing for a direct encoding of the target polynomial into a quantum circuit. However, this approach assumes access to a diagonal block encoding of function values without providing an explicit circuit construction. In this work, we address this gap by developing an explicit circuit construction method for diagonal block encodings. The resulting algorithm achieves a computational cost of $\mathcal{O}(d\log d)$ for explicitly constructing block encodings of matrix polynomials, improving upon the best-known theoretical bounds of previous methods. Numerical results confirm this scaling, demonstrating that circuit parameters for polynomial degrees up to $10^7$ can be computed in about a minute on a standard CPU.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | A recent interpolation-based Quantum Signal Processing (QSP) framework by Alase bypasses the phase-finding procedures required in conventional QSP, allowing for a direct encoding of the target polynomial into a quantum circuit. | background 1.00 | 0.04 | 0.05 | asserted 1.00 |
| s1 | However, this approach assumes access to a diagonal block encoding of function values without providing an explicit circuit construction. | gap 0.97 | 0.06 | 0.03 | asserted 1.00 |
| s2 | In this work, we address this gap by developing an explicit circuit construction method for diagonal block encodings. | contribution 1.00 | 0.94 | 0.17 | asserted 1.00 |
| s3 | The resulting algorithm achieves a computational cost of $\mathcal{O}(d\log d)$ for explicitly constructing block encodings of matrix polynomials, improving upon the best-known theoretical bounds of previous methods. | result 0.88 | 0.93 | 0.68 | asserted 1.00 |
| s4 | Numerical results confirm this scaling, demonstrating that circuit parameters for polynomial degrees up to $10^7$ can be computed in about a minute on a standard CPU. | result 1.00 | 0.94 | 0.16 | asserted 1.00 |
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The paper
Original abstract
A recent interpolation-based Quantum Signal Processing (QSP) framework by Alase bypasses the phase-finding procedures required in conventional QSP, allowing for a direct encoding of the target polynomial into a quantum circuit. However, this approach assumes access to a diagonal block encoding of function values without providing an explicit circuit construction. In this work, we address this gap by developing an explicit circuit construction method for diagonal block encodings. The resulting algorithm achieves a computational cost of $\mathcal{O}(d\log d)$ for explicitly constructing block encodings of matrix polynomials, improving upon the best-known theoretical bounds of previous methods. Numerical results confirm this scaling, demonstrating that circuit parameters for polynomial degrees up to $10^7$ can be computed in about a minute on a standard CPU.
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References
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What this paper does with the arXiv papers it cites
What it cites · 5 sentences without a confident label
Builds on 1 paper
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The following corollary specializes [ref] to block encodings of matrix polynomials and specifies the classical construction time using our explicit diagonal block encoding in <ref>.
Quantum signal processing without angle finding · Main Result · 1.00
Uses the method or tool of 1 paper
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We employ the recent QSP framework by Alase [ref] , which bypasses the task of finding phase factors by assuming a diagonal block encoding of function values at interpolation points.
Quantum signal processing without angle finding · Introduction · 1.00
Compares with 1 paper
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In this section, we introduce a method for constructing a diagonal block encoding for an arbitrary complex diagonal matrix using unitary controlled rotations (UCRs), compared to previous approaches [ref] .
Non-Linear Transformations of Quantum Amplitudes: Exponential Improvement, Generalization, and Applications · Diagonal encoding from unitary controlled rotations · 1.00
Cites as background 3 papers
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An interpolation-based formulation of QSP introduced by Alase [ref] avoids finding phase factors and resembles LCU in that it directly encodes information about the target function into a quantum circuit.
Quantum signal processing without angle finding · Review of QSP without angle finding · 0.97
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To tackle this, several algorithms have been developed from diverse numerical perspectives, using root-finding procedures [ref] , optimization subroutines [ref] , fixed-point iterations [ref] , and the nonlinear Fourier transform [ref] .
Finding Angles for Quantum Signal Processing with Machine Precision · Introduction · 0.86
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Despite its relatively short history, QSP has already become a central component of modern quantum algorithms across diverse areas of scientific computation, including quantum many-body problems, partial differential equations [ref] , and machine learning [ref] .
A Quantum Path to Partial Differential Equations · Introduction · 0.83
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