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<p>The ability to implement arbitrary unitary operations is essential for quantum information processing, since the evolution of closed quantum systems is governed by unitary dynamics. Conventional methods typically approximate the target evolution by optimizing sequences of fixed single- and two-qubit gates, which often leads to deep circuits and the accumulation of approximation errors. Here, we propose an exact and analytical method for implementing arbitrary unitary operations using discrete-time quantum walks. Specifically, an arbitrary <i>N</i>-dimensional unitary operation <i>U</i>(<i>N</i>) is constructed through a sequence of position-dependent <i>U</i>(2) coin operations and conditional shift operators, requiring only <i>N</i> steps (or <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11128_2025_4959_Article_IEq1.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="45" /> </InlineMediaObject> <EquationSource Format="TEX">\(N+1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo>+</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> for odd dimensions) for exact realization. We further present a feasible photonic implementation, where coin and position states are encoded in the polarization and spatial modes of single photons. Numerical simulations confirm the scalability and noise resilience of the approach, demonstrating its advantages for realization of arbitrary unitary operations. Our results thus provide a practical route to realizing arbitrary unitary transformations with reduced resource requirements and enhanced robustness against optical losses and phase instability.</p>

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Realization of arbitrary unitary operations using discrete-time quantum walks

  • 期刊论文

  • 发表日期：

    2025年10月15日

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  • Yongqi Han,
  • Ying Zhou,
  • Kunkun Wang

摘要

The ability to implement arbitrary unitary operations is essential for quantum information processing, since the evolution of closed quantum systems is governed by unitary dynamics. Conventional methods typically approximate the target evolution by optimizing sequences of fixed single- and two-qubit gates, which often leads to deep circuits and the accumulation of approximation errors. Here, we propose an exact and analytical method for implementing arbitrary unitary operations using discrete-time quantum walks. Specifically, an arbitrary N-dimensional unitary operation U(N) is constructed through a sequence of position-dependent U(2) coin operations and conditional shift operators, requiring only N steps (or \(N+1\) N + 1 for odd dimensions) for exact realization. We further present a feasible photonic implementation, where coin and position states are encoded in the polarization and spatial modes of single photons. Numerical simulations confirm the scalability and noise resilience of the approach, demonstrating its advantages for realization of arbitrary unitary operations. Our results thus provide a practical route to realizing arbitrary unitary transformations with reduced resource requirements and enhanced robustness against optical losses and phase instability.