Space-bounded quantum state testing via space-efficient quantum singular value transformation

TL;DR

The paper addresses the computational power of quantum computers constrained to a small number of qubits by introducing space-bounded quantum state testing and certification, and showing complete problems for unitary coRQL and BQL. It develops a space-efficient quantum singular value transformation (QSVT) based on averaged Chebyshev truncation, enabling space overheads that scale with the log-space bound and yielding space-efficient implementations of polynomial approximations. A key contribution is the algorithmic Holevo-Helstrom measurement, which operationalizes optimal state discrimination in this setting and places QSZK in QIP(2) with a linear-space honest prover. The results provide a unified, scalable framework for designing space-bounded quantum algorithms and establish tight complexity characterizations for natural state-testing problems, highlighting practical implications for verifiable quantum device testing and quantum information processing under stringent resource constraints.

Abstract

Driven by exploring the power of quantum computation with a limited number of qubits, we present a novel complete characterization for space-bounded quantum computation, which encompasses settings with one-sided error (unitary coRQL) and two-sided error (BQL), approached from a quantum state testing perspective: - The first family of natural complete problems for unitary coRQL, i.e., space-bounded quantum state certification for trace distance and Hilbert-Schmidt distance; - A new family of natural complete problems for BQL, i.e., space-bounded quantum state testing for trace distance, Hilbert-Schmidt distance, and quantum entropy difference. In the space-bounded quantum state testing problem, we consider two logarithmic-qubit quantum circuits (devices) denoted as $Q_0$ and $Q_1$, which prepare quantum states $ρ_0$ and $ρ_1$, respectively, with access to their ``source code''. Our goal is to decide whether $ρ_0$ is $ε_1$-close to or $ε_2$-far from $ρ_1$ with respect to a specified distance-like measure. Interestingly, unlike time-bounded state testing problems, our results reveal that the space-bounded state testing problems all correspond to the same class. Moreover, our algorithms on the trace distance inspire an algorithmic Holevo-Helstrom measurement, implying QSZK is in QIP(2) with a quantum linear-space honest prover. Our results primarily build upon a space-efficient variant of the quantum singular value transformation (QSVT) introduced by Gilyén, Su, Low, and Wiebe (STOC 2019), which is of independent interest. Our technique provides a unified approach for designing space-bounded quantum algorithms. Specifically, we show that implementing QSVT for any bounded polynomial that approximates a piecewise-smooth function incurs only a constant overhead in terms of the space required for special forms of the projected unitary encoding.

Figures (3)

• Figure 1: General framework for quantum state testing $\mathcal{T}(Q_{\rho},U_A,P_{d'})$.
• Figure 2: Quantum tester $\mathcal{T}(Q,U_A,P_{d'},\epsilon)$: the circuit implementation.
• Figure 3: Algorithmic Holevo-Helstrom measurement.

Theorems & Definitions (129)

• theorem 1.1: Informal of \ref{['thm:space-bounded-quantum-state-certification-RQL-complete']}
• theorem 1.2: Informal of \ref{['thm:space-bounded-quantum-state-testing-BQL-complete']}
• theorem 1.3: Informal of \ref{['thm:algo-HH-meas']}
• theorem 1.4: Informal of \ref{['thm:GapQSD-in-QIP(2)-bounded-prover']}
• theorem 1.5: Space-efficient QSVT, informal of \ref{['thm:space-efficient-QSVT']}
• definition 2.1: Singular value decomposition of a projected unitary, adapted from Definition 7 in GSLW19
• definition 2.2: Singular value transformation by even or odd functions, adapted from Definition 9 in GSLW19
• definition 2.3: Quantum distances and divergences
• lemma 1: Trace distance vs. fidelity, adapted from FvdG99
• lemma 2: Joint entropy theorem, adapted from Theorem 11.8(5) in NC02