Repeated measurements enable rational quantum spectral filters

What the filter changes

An operator’s eigenvalues describe its possible spectral values; for an energy operator, they are possible energies. A spectral filter selectively changes the probability of observing those values. If an input has some probability weight at eigenvalue x, acceptance multiplies that weight by a chosen response R(x). The accepted distribution is then normalized to sum to one.

Established methods called quantum signal processing and quantum singular value transformation, or QSP and QSVT, implement polynomial transformations through finite coherent circuit sequences. Rational functions—ratios of polynomials—are useful targets because they naturally describe regularized inverses and narrow spectral windows.

The new work asks how to realize rational acceptance responses directly, rather than first approximating them with polynomials.

A circuit that can continue without restarting

The proposed repeat-until-success protocol combines a signal register, finite quantum memory and a measurement flag. Each round applies a controlled quantum walk: a unitary operation that encodes information about the target operator. A fixed additional operation couples the memory and flag, which is then measured.

There are three outcomes: accept, reject or continue. Acceptance and rejection end the run. Continuation preserves the conditional joint state of the signal and memory; only the measurement flag is reset. Rejection, by contrast, requires a fresh copy of the original input when seeking another chance at acceptance.

The circuit can repeat arbitrarily many times without needing an ever-growing memory. Different stopping times correspond to distinct measurement records, so their contributions add as probabilities, not as coherently interfering amplitudes.

For the authors’ constructive scheme, a complete trial has polynomial probabilities of acceptance, rejection and continuation. Summing acceptance over every possible number of continuing trials produces a rational response. The denominator arises from the repetition itself.

An exact characterization, under stated conditions

The central theorem concerns a Hermitian operator—one with real eigenvalues—scaled so its spectrum lies between −1 and 1. Under uniform termination and phase-inversion symmetry, the realizable acceptance responses are exactly the real rational functions that remain between zero and one and have no poles on that interval. A pole is a point where a reduced rational function becomes singular.

Uniform termination supplies a common decaying bound on the probability of continuing across all relevant signal phases. Phase-inversion symmetry ensures that the two quantum-walk phases associated with the same operator eigenvalue have equal acceptance probabilities.

Although the protocol has infinitely many possible stopping times, the authors calculate its eventual statistics using a finite-dimensional linear matrix equation, called a discrete Lyapunov equation.

They also give a constructive circuit compiler. For a nonconstant response of rational degree r—the larger of its reduced numerator and denominator degrees—the construction needs at most r controlled-walk queries per complete trial and a controller memory of dimension at most 3r. These are per-trial bounds, not total runtime bounds: trials may repeat many times.

Two examples and their costs

The first example weights low energies using the regularized inverse response γ/(λ + γ), where λ is an eigenvalue of a positive operator bounded by Λ and γ sets the filtering scale. It uses one controlled-walk query per trial. Including continuation and fresh-input restarts after rejection, the expected cost per accepted output is exactly Λ/γ + 1, independent of the input spectral distribution.

For a representative broad input spectrum, the paper compares this with fixed-depth polynomial postselection that uniformly approximates the target response. Writing K = Λ/γ, the rational construction’s cost grows proportionally to K, while the stated polynomial baseline requires at least order K to the three-halves power.

The second example is a Lorentzian window, which favors eigenvalues near a chosen center. Its width parameter η is the distance from the center at which the response falls to half its maximum. A trial uses two queries. For a fixed interior center and a smooth input spectral density positive there, the rational method avoids a logarithmic factor in the baseline’s expected accepted-output cost as the window narrows.

What the results do not establish

These comparisons are not lower bounds against every quantum algorithm. They concern fixed-depth, uniformly approximating polynomial postselection with independent restarts, without amplitude amplification. Query counts exclude input re-preparation costs and do not establish practical hardware speedups.

Acceptance probabilities also do not fully specify the output quantum state. Measurement histories can retain information that reduces coherence between different eigenvalue sectors. The authors characterize this with a coherence kernel, which describes how those quantum superpositions change. Exact spectral reweighting therefore does not automatically mean fully coherent filtering.

The preprint provides mathematical proofs and numerical illustrations, not an experimental demonstration. Its exactness applies to ideal operations and the full repeat-until-termination process; imposing a finite stopping limit leaves an unresolved probability. The result is a theoretical expansion of spectral-filter design, with practical performance still dependent on implementation and measurement costs.