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, which exhibit computational hardness depending on the chosen distance-like measure (either QSZK-complete or BQP-complete), our results reveal that the space-bounded state testing problems, considering all three measures, are computationally as easy as preparing quantum states. 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：受探索有限量子比特量子计算能力的驱动，我们从量子态测试的角度出发，对空间有界量子计算提出了全新的完整刻画，涵盖了单侧误差（幺正coRQL）和双侧误差（BQL）场景：- 针对幺正coRQL的首个自然完全问题族，即迹距离与希尔伯特-施密特距离下的空间有界量子态认证；- 针对BQL的新自然完全问题族，即迹距离、希尔伯特-施密特距离及量子熵差下的空间有界量子态测试。在空间有界量子态测试问题中，我们考虑两个对数量子比特量子电路（器件）$Q_0$和$Q_1$，它们分别制备量子态$\rho_0$和$\rho_1$，并允许访问其“源代码”。我们的目标是判定在指定的距离度量下，$\rho_0$是否与$\rho_1$满足$\epsilon_1$-接近或$\epsilon_2$-远离。有趣的是，与时间有界量子态测试问题（其计算复杂度取决于所选距离度量，表现为QSZK完全或BQP完全）不同，我们的结果表明：针对上述三种度量的空间有界量子态测试问题，其计算复杂度与制备量子态相当。我们的结果主要基于Gilyén、Su、Low与Wiebe（STOC 2019）提出的空间高效量子奇异值变换（QSVT）变体，该变体本身具有独立研究价值。我们的技术为设计空间有界量子算法提供了统一框架：具体而言，对于任何逼近分段光滑函数的有界多项式，实现其QSVT仅需在投影幺正编码特殊形式的空间开销中引入常数因子。