François Le Gall, Yupan Liu, Qisheng Wang
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 $\sf coRQL$) and two-sided error ($\sf BQL$), approached from a quantum state testing perspective: - The first family of natural complete problems for unitary $\sf coRQL$, namely space-bounded quantum state certification for trace distance and Hilbert-Schmidt distance; - A new family of natural complete problems for $\sf BQL$, namely space-bounded quantum state testing for trace distance, Hilbert-Schmidt distance, and (von Neumann) 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, which exhibit computational hardness depending on the chosen distance-like measure, 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 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.