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 $\rho_0$ and $\rho_1$, respectively, with access to their ``source code''. Our goal is to decide whether $\rho_0$ is $\epsilon_1$-close to or $\epsilon_2$-far from $\rho_1$ 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 Gily\'en, 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.

翻译：在探索有限量子比特数下量子计算能力的驱动下，我们提出了一种对空间受限量子计算的全新完备性刻画，该框架涵盖单侧错误（unitary coRQL）与双侧错误（BQL）两种设置，并从量子态测试的视角进行研究：- 首次给出unitary coRQL的首类自然完备问题，即基于迹距离与希尔伯特-施密特距离的空间受限量子态认证；- 提出BQL的首类自然完备问题，即基于迹距离、希尔伯特-施密特距离及量子熵差的空间受限量子态测试。在空间受限量子态测试问题中，我们考虑两个对数量子比特规模的量子电路（装置）$Q_0$ 与 $Q_1$，它们分别制备量子态 $\rho_0$ 与 $\rho_1$，且可访问其"源代码"。我们的目标是判断 $\rho_0$ 相对于 $\rho_1$ 在指定类距离度量下是 $\epsilon_1$-接近还是 $\epsilon_2$-远离。值得注意的是，与时间受限态测试问题不同，我们的结果表明所有空间受限态测试问题均对应于同一复杂度类。此外，我们针对迹距离设计的算法启发了一种算法化的Holevo-Helstrom测量，这暗示着QSZK可包含于具有量子线性空间诚实证明者的QIP(2)中。我们的结果主要建立在Gilyén、Su、Low与Wiebe（STOC 2019）提出的量子奇异值变换（QSVT）的空间高效变体之上，该变体本身具有独立的研究价值。我们的技术为设计空间受限量子算法提供了统一框架。具体而言，我们证明了对于任何逼近分段光滑函数的有界多项式，在特殊形式的投影酉编码下实现QSVT仅需常数级别的空间开销。