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Driven by exploring the power of quantum computation with a limited number ofqubits, we present a novel complete characterization for space-bounded quantumcomputation, 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 andHilbert-Schmidt distance;  - A new family of natural complete problems for BQL, i.e., space-boundedquantum state testing for trace distance, Hilbert-Schmidt distance, and quantumentropy difference.  In the space-bounded quantum state testing problem, we consider twologarithmic-qubit quantum circuits (devices) denoted as $Q_0$ and $Q_1$, whichprepare quantum states $\rho_0$ and $\rho_1$, respectively, with access totheir ``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 aspecified distance-like measure. Interestingly, unlike time-bounded statetesting problems, which exhibit computational hardness depending on the chosendistance-like measure (either QSZK-complete or BQP-complete), our resultsreveal that the space-bounded state testing problems, considering all threemeasures, are computationally as easy as preparing quantum states.  Our results primarily build upon a space-efficient variant of the quantumsingular value transformation (QSVT) introduced by Gily\'en, Su, Low, and Wiebe(STOC 2019), which is of independent interest. Our technique provides a unifiedapproach for designing space-bounded quantum algorithms. Specifically, we showthat implementing QSVT for any bounded polynomial that approximates apiecewise-smooth function incurs only a constant overhead in terms of the spacerequired for special forms of the projected unitary encoding.

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