Quantum

Rational quantum spectral filtering by repeat-until-success protocol

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Briefings

A new quantum algorithm introduces a repeat-until-success framework for rational spectral filtering, extending beyond existing polynomial methods. The protocol uses block encoding of a Hermitian operator with spectrum in [-1,1] and mid-circuit measurements to accept, reject, or continue computations. Terminal statistics are determined by a finite-dimensional Lyapunov equation. Under uniform termination and phase-inversion symmetry, any real rational function with no poles in [-1,1] and values in [0,1] can be realized as the spectral filter R(X). A constructive compiler provides bounds on memory and query cost, offering advantages over fixed-depth QSVT polynomial-postselection baseline.

Key points

• The protocol uses a finite-memory repeat-until-success framework to implement rational spectral filters on quantum computers.
• It extends beyond polynomial transformations by interleaving block encoding queries with mid-circuit measurements that accept, reject, or continue without resetting the state.

Context

This work presents a new approach to quantum spectral filtering, addressing limitations of existing polynomial-based methods like QSP and QSVT. The framework allows for exact low-degree realizations of rational functions, offering query-cost advantages over fixed-depth polynomial-postselection baselines. Applications to regularized resolvents and Lorentzian spectroscopy demonstrate practical utility.

Limitations

The protocol's terminal statistics are determined by a finite-dimensional Lyapunov equation, which may limit scalability for arbitrarily large systems. The expected query cost bounds are derived under uniform termination and phase-inversion symmetry assumptions. The source is an arXiv preprint (2611.13154v1) with no peer review or experimental validation.

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