  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.
