[Paper Review] Space-bounded quantum state testing via space-efficient quantum singular value transformation

François Le Gall, Yupan Liu+1 more|arXiv (Cornell University)|Aug 9, 2023

Quantum Computing Algorithms and Architecture4 citations

TL;DR

This paper presents a novel characterization of space-bounded quantum computation by introducing natural, complete problems for both one-sided (coRQL) and two-sided (BQL) error settings through quantum state testing. It introduces space-efficient quantum singular value transformation (QSVT) to show that space-bounded state testing for trace distance, Hilbert-Schmidt distance, and entropy difference are as easy as state preparation, establishing completeness and tight complexity bounds via logspace Karp reductions.

ABSTRACT

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 $ρ_0$ 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, 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 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.

Motivation & Objective

• To characterize the computational power of space-bounded quantum computation with limited qubits.
• To identify natural, complete problems for coRQL (one-sided error) and BQL (two-sided error) in the quantum state testing framework.
• To unify and simplify the complexity characterization of space-bounded quantum computation using quantum state testing.
• To establish tight completeness results via logspace Karp reductions for trace distance, Hilbert-Schmidt distance, and entropy difference.

Proposed method

• Develops a space-efficient variant of quantum singular value transformation (QSVT) using averaged Chebyshev truncation for bounded and piecewise-smooth functions.
• Applies the space-efficient QSVT to projected unitary encodings with only constant space overhead.
• Uses polynomial approximation via averaged Chebyshev truncation to implement QSVT for functions like sign and normalized logarithm.
• Constructs quantum circuits that prepare purified states and simulate target functions with bounded error using logarithmic space.
• Employs logspace Karp reductions to prove BQL- and coRQL-hardness for state testing problems.
• Integrates error reduction techniques and norm bounds leveraging function smoothness to ensure space efficiency.

Experimental results

Research questions

• RQ1 What are the natural complete problems for space-bounded quantum computation with one-sided error (coRQL)?
• RQ2 How do space-bounded quantum state testing problems for trace distance, Hilbert-Schmidt distance, and entropy difference relate to BQL?
• RQ3 Can space-efficient QSVT be constructed for piecewise-smooth functions with only constant space overhead?
• RQ4 Is quantum state testing under these distance measures as easy as state preparation in the space-bounded setting?
• RQ5 What is the complexity of space-bounded quantum state testing under different distance measures?

Key findings

• CertQSDlog and CertQHSlog are complete for coRQL, providing natural one-sided error problems for space-bounded quantum computation.
• GapQSDlog, GapQHSlog, and GapQEDlog are all in BQL, showing that space-bounded state testing for these measures is as easy as state preparation.
• GapQSDlog is BQUL-hard, and GapQJSlog and GapQEDlog are also BQUL-hard, establishing tight complexity bounds.
• The space-efficient QSVT construction incurs only constant overhead in space for bounded polynomial approximations of piecewise-smooth functions.
• The reduction from GapQSDlog to GapQJSlog and GapQEDlog confirms the completeness of these problems under logspace Karp reductions.
• The results show that space-bounded state testing is computationally no harder than state preparation, unlike time-bounded counterparts which are QSZK- or BQP-complete.