Space-Bounded Unitary Quantum Computation with Postselection

TL;DR

This paper demonstrates that in space-bounded quantum computation with postselection, intermediate measurements can be eliminated without changing computational power, extending recent results and linking DQC1 models to classical probabilistic computation.

Contribution

It proves the equivalence of space-bounded quantum computations with postselection and those without, strengthening understanding of quantum computational power with space constraints.

Findings

Intermediate postselections can be eliminated in space-bounded quantum computation.

Bounded-error space-bounded DQC1 with postselection equals unbounded-error probabilistic computation.

Results connect quantum space-bounded models to classical complexity classes.

Abstract

Space-bounded computation has been a central topic in classical and quantum complexity theory. In the quantum case, every elementary gate must be unitary. This restriction makes it unclear whether the power of space-bounded computation changes by allowing intermediate measurement. In the bounded error case, Fefferman and Remscrim [STOC 2021, pp.1343--1356] and Girish, Raz and Zhan~[ICALP 2021, pp.73:1--73:20] recently provided the break-through results that the power does not change. This paper shows that a similar result holds for space-bounded quantum computation with postselection. Namely, it is proved possible to eliminate intermediate postselections and measurements in the space-bounded quantum computation in the bounded-error setting. Our result strengthens the recent result by Le Gall, Nishimura and Yakaryilmaz~[TQC 2021, pp.10:1--10:17] that logarithmic-space bounded-error quantum computation with intermediate postselections and measurements is equivalent in computational power to logarithmic-space unbounded-error probabilistic computation. As an application, it is shown that bounded-error space-bounded one-clean qubit computation (DQC1) with postselection is equivalent in computational power to unbounded-error space-bounded probabilistic computation, and the computational supremacy of the bounded-error space-bounded DQC1 is interpreted in complexity-theoretic terms.