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 (mixed) state testing perspective:
- The first family of natural complete problems for unitary coRQL, namely space-bounded quantum state certification for trace distance and Hilbert-Schmidt distance;
- A new family of (arguably simpler) natural complete problems for 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 $\rho_0$ 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 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.