Quantum Signal Processing and Quantum Singular Value Transformation on $U(N)$

TL;DR

The paper addresses the limitation of the traditional $U(2)$ quantum signal processing framework by developing a comprehensive theory for $U(N)$-QSP and $U(N)$-QSVT, enabling multi-output block-encoded polynomial transformations. It provides a complete characterization of achievable polynomial matrices $\bm{P}(z)$ and a constructive, recursive circuit design using parameterized $U(N)$ elements, with forward and backward formulations. Three applications demonstrate substantial advantages: bi-variate QSP via a product-rule with PCA-based low-rank decompositions, multi-interval decision achieving $\mathcal{O}(d)$ query complexity and a $\log_2 N$ speedup over $U(2)$ approaches, and QAE attaining Heisenberg-limited precision in a single non-adaptive measurement. These results extend the quantum signal processing toolkit to higher-dimensional ancilla spaces, offering new avenues for Hamiltonian simulation, quantum phase estimation, and metrology while highlighting practical challenges in optimization and implementation.

Abstract

Quantum signal processing and quantum singular value transformation are powerful tools to implement polynomial transformations of block-encoded matrices on quantum computers, and has achieved asymptotically optimal complexity in many prominent quantum algorithms. We propose a framework of quantum signal processing and quantum singular value transformation on $U(N)$, which realizes multiple polynomials simultaneously from a block-encoded input, as a generalization of those on $U(2)$ in the original frameworks. We provide a comprehensive characterization of achievable polynomial matrices and give recursive algorithms to construct the quantum circuits that realize desired polynomial transformations. As three example applications, we propose a framework to realize bi-variate polynomial functions, demonstrate $N$-interval decision achieving $O(d)$ query complexity with a $\log_2 N$ improvement over iterative $U(2)$-QSP requiring $O(d\log_2 N)$ queries, and present a quantum amplitude estimation algorithm achieving the Heisenberg limit without adaptive measurements.

Figures (8)

• Figure 1: A summary of our contributions in the paper. The orange quantum gates are for parameterized unitaries or projectors, while the blue gates are for fixed input variables.
• Figure 2: The quantum circuit for $U(N)$-QSP, parameterized by arbitrary unitaries $R_0, R_1, \ldots, R_d \in U(N)$ acting on an $N$-dimensional ancilla register. The circuit uses $d$ applications of controlled-$U$ operations, where $C_{\Pi_{\ell}}(U) := \sum_{j=0}^{\ell-1} \dyad{j}{j} \otimes U + \sum_{j=\ell}^{N-1} \dyad{j}{j} \otimes I$ applies $U$ when the ancilla is in computational basis states $\ket{0}$ through $\ket{\ell-1}$.
• Figure 3: The $U(N)$-QSVT unit, in which $\Pi=\dyad{\bm{0}}$ and $\tilde{\Pi}=\dyad{\tilde{\bm{0}}}$. The $U$ and $U^\dagger$ gates applied to the second register alternate, and it depends on the parity of $d$ whether the last two gates in the second register are $U$ and $\Pi$, or $U^\dagger$ and $\tilde{\Pi}$.
• Figure 4: Block encoding of bi-variate polynomials via uni-variate $U(N)$-QSP. The product of block-encoded matrices is obtained by treating either the blue or orange box as a single block.
• Figure 5: Comparison of PCA reconstruction error for different coefficient decay patterns. Polynomial decay significantly reduces the number of principal components needed to achieve the same error. Larger decay exponent $s$ yields better low-rank approximation. When $s=4$, only $r/N \approx 5\%$ is needed to achieve $\varepsilon < 0.1$, while random matrices require retaining over $90\%$ of principal components.

Theorems & Definitions (17)

• Theorem 1: Generalized QSP motlagh2024generalized
• Theorem 2: QSVT gilyen2018quantum
• Lemma 1: $U(N)$-QSP, forward
• Theorem 3: $U(N)$-QSP, backward
• Lemma 2: $U(N)$-QSP for full unitary
• Corollary 1: $U(N)$-QSP for Laurent polynomials
• Corollary 2: Approximate $U(N)$-QSP
• Lemma 3: $U(N)$-QSVT for one singular value
• Lemma 4: $U(N)$-QSVT, forward
• Theorem 4: $U(N)$-QSVT, backward