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

TL;DR

The paper tackles the resource burden of post-selection in Quantum Signal Processing by introducing DB-QSP, a post-selection-free framework that deterministically implements QSP transformations for Hermitian matrices. It builds a unitary synthesis for linear polynomials via U_Ψ = e^{s_Ψ[Ψ,H]} and extends to arbitrary polynomials with complex roots by incorporating state-dependent reflections, forming a recursive DB-QSP construction. Implemented through Double-Bracket Quantum Algorithms, this approach yields a deterministic QSP variant at the cost of circuit depth that grows super-exponentially with polynomial degree and requires live estimation of energy and variance at each step. The work also analyzes error propagation, sampling overhead, and depth limits, and discusses hybrid strategies to integrate DB-QSP with existing methods to warm-start and accelerate QSP-driven tasks like ground-state preparation and matrix inversion.

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.

Figures (1)

• Figure 1: Quantum Signal Processing (QSP) without auxiliary qubits and post-selection. We introduce a new formula for implementing QSP of Hermitian matrices (Thm. \ref{['thm complex QSP']}). (a) To realize a degree-$K$ polynomial of a Hermitian matrix $H$, original QSP performs measurement on auxiliary qubits so that the desired transformation is realized, as shown in Eq. \ref{['eq:qsp_conv_uni_qubitize']}. (b) In contrast, our formula does not require auxiliary qubits and accordingly the post-selection. Instead, we recursively apply the state-dependent unitary operators $e^{i\theta_k \Psi_{k}}e^{s_{k}[\Psi_{k},H]}$ with $\ket{\Psi_{k+1}}=e^{i\theta_{k} \Psi_{k}}e^{s_{k}[\Psi_{k},H]}\ket{\Psi_{k}}$, resulting in the circuit depth that grow significantly in the degree of polynomials $K$. Furthermore, to determine the time duration $s_{k}$ and phase $\theta_{k}$, energy $E_{k}=\braket{\Psi_{k}|H|\Psi_{k}}$ and variance in energy $V_{k}=\braket{\Psi_{k}|H^2|\Psi_{k}}-E_{k}^2$ must be known at each step.

Theorems & Definitions (43)

• Lemma 1: Unitary synthesis for linear polynomials without post-selection
• Theorem 2: Unitary synthesis for QSP without post-selection
• Theorem 3: DB-QSP circuit depth
• Proposition 4: Stability of Thm. \ref{['thm complex QSP']} under erroneous estimation
• Lemma B.1: Unitary synthesis for linear polynomials without post-selection
• proof : Proof of Lem. \ref{['app thm zero overhead QSP unitary single step']}
• Proposition B.2: Effective idempotence
• proof
• Lemma B.3
• proof