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<p>Quantum signal processing (QSP) provides a systematic framework for implementing a polynomial transformation of a linear operator, and unifies nearly all known quantum algorithms. In parallel, recent works have developed <i>randomized compiling</i>, a technique that promotes a unitary gate to a quantum channel and enables a quadratic suppression of error (i.e., <i>ϵ</i> → <i>O</i>(<i>ϵ</i><sup>2</sup>)) at little to no overhead. Here we integrate randomized compiling into QSP through <i>Stochastic Quantum Signal Processing</i>. Our algorithm implements a probabilistic mixture of polynomials, strategically chosen so that the average evolution converges to that of a target function, with an error quadratically smaller than that of an equivalent individual polynomial. Because nearly all QSP-based algorithms exhibit query complexities scaling as <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41534_2025_1003_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="83" /> </InlineMediaObject> <EquationSource Format="TEX">\(O(\log (1/\epsilon ))\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>O</mi> <mrow> <mo>(</mo> <mrow> <mi>log</mi> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>/</mo> <mi>ϵ</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>—stemming from a result in functional analysis—this error suppression reduces their query complexity by a factor that asymptotically approaches 1/2. By the unifying capabilities of QSP, this reduction extends broadly to quantum algorithms, which we demonstrate on algorithms for real and imaginary time evolution, phase estimation, ground state preparation, and matrix inversion.</p>

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Halving the cost of quantum algorithms with randomization

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年3月15日

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  • John M. Martyn,
  • Patrick Rall

摘要

Quantum signal processing (QSP) provides a systematic framework for implementing a polynomial transformation of a linear operator, and unifies nearly all known quantum algorithms. In parallel, recent works have developed randomized compiling, a technique that promotes a unitary gate to a quantum channel and enables a quadratic suppression of error (i.e., ϵ → O(ϵ2)) at little to no overhead. Here we integrate randomized compiling into QSP through Stochastic Quantum Signal Processing. Our algorithm implements a probabilistic mixture of polynomials, strategically chosen so that the average evolution converges to that of a target function, with an error quadratically smaller than that of an equivalent individual polynomial. Because nearly all QSP-based algorithms exhibit query complexities scaling as \(O(\log (1/\epsilon ))\) O ( log ( 1 / ϵ ) ) —stemming from a result in functional analysis—this error suppression reduces their query complexity by a factor that asymptotically approaches 1/2. By the unifying capabilities of QSP, this reduction extends broadly to quantum algorithms, which we demonstrate on algorithms for real and imaginary time evolution, phase estimation, ground state preparation, and matrix inversion.