Double-bracket algorithm for quantum signal processing without post-selection

• URL: http://arxiv.org/abs/2504.01077v2
• Date: Wed, 16 Apr 2025 11:49:19 GMT
• Title: Double-bracket algorithm for quantum signal processing without post-selection
• Authors: Yudai Suzuki, Bi Hong Tiang, Jeongrak Son, Nelly H. Y. Ng, Zoë Holmes, Marek Gluza,
• Abstract summary: Quantum signal processing (QSP) is a fundamental primitive in various quantum algorithms.<n>Despite its versatility, a potentially underappreciated challenge is that all systematic protocols for implementing QSP rely on post-selection.<n>We introduce a new formula for implementing QSP transformations of Hermitian matrices, which requires neither auxiliary qubits nor post-selection.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Quantum signal processing (QSP), a framework for implementing matrix-valued polynomials, is a fundamental primitive in various quantum algorithms. Despite its versatility, a potentially underappreciated challenge is that all systematic protocols for implementing QSP rely on post-selection. This can impose prohibitive costs for tasks when amplitude amplification cannot sufficiently improve the success probability. For example, in the context of ground-state preparation, this occurs when using a too poor initial state. In this work, we introduce a new formula for implementing QSP transformations of Hermitian matrices, which requires neither auxiliary qubits nor post-selection. Rather, using approximation to the exact unitary synthesis, we leverage the theory of the double-bracket quantum algorithms to provide a new quantum algorithm for QSP, termed Double-Bracket QSP (DB-QSP). The algorithm requires the energy and energetic variance of the state to be measured at each step and has a recursive structure, which leads to circuit depths that can grow super exponentially with the degree of the polynomial. With these strengths and caveats in mind, DB-QSP should be viewed as complementing the established QSP toolkit. In particular, DB-QSP can deterministically implement low-degree polynomials to "warm start" QSP methods involving post-selection.

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