Parallel Quantum Signal Processing Via Polynomial Factorization

Quantum signal processing (QSP) is a methodology for constructing polynomial transformations of a linear operator encoded in a unitary. Applied to an encoding of a state ρ , QSP enables the evaluation of nonlinear functions of the form tr ( P ( ρ ) ) for a polynomial P ( x ) , which encompasses rele...

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Vydáno v:	Quantum (Vienna, Austria) Ročník 9; s. 1834
Hlavní autoři:	Martyn, John M., Rossi, Zane M., Cheng, Kevin Z., Liu, Yuan, Chuang, Isaac L.
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften 27.08.2025
ISSN:	2521-327X, 2521-327X
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Popis
Shrnutí:	Quantum signal processing (QSP) is a methodology for constructing polynomial transformations of a linear operator encoded in a unitary. Applied to an encoding of a state ρ , QSP enables the evaluation of nonlinear functions of the form tr ( P ( ρ ) ) for a polynomial P ( x ) , which encompasses relevant properties like entropies and fidelity. However, QSP is a sequential algorithm: implementing a degree- d polynomial necessitates d queries to the encoding, equating to a query depth d . Here, we reduce the depth of these property estimation algorithms by developing Parallel Quantum Signal Processing. Our algorithm parallelizes the computation of tr ( P ( ρ ) ) over k systems and reduces the query depth to d / k , thus enabling a family of time-space tradeoffs for QSP. This furnishes a property estimation algorithm suitable for distributed quantum computers, and is realized at the expense of increasing the number of measurements by a factor O ( poly ( d ) 2 O ( k ) ) . We achieve this result by factorizing P ( x ) into a product of k smaller polynomials of degree O ( d / k ) , which are each implemented in parallel with QSP, and subsequently multiplied together with a swap test to reconstruct P ( x ) . We characterize the achievable class of polynomials by appealing to the fundamental theorem of algebra, and demonstrate application to canonical problems including entropy estimation and partition function evaluation.
ISSN:	2521-327X 2521-327X
DOI:	10.22331/q-2025-08-27-1834