Rewindable Quantum Computation and Its Equivalence to Cloning and Adaptive Postselection

TL;DR

This paper introduces rewinding operators in quantum computation, establishing their equivalence to cloning and adaptive postselection, and explores their computational power and classical simulability.

Contribution

It defines new complexity classes involving rewinding, cloning, and postselection, and proves their equivalence and relation to existing classes, revealing new computational capabilities.

Findings

Rewinding operators invert quantum measurements.

Complexity classes RwBQP, CBQP, and AdPostBQP are equivalent.

Rewindable Clifford circuits are classically simulatable.

Abstract

We define rewinding operators that invert quantum measurements. Then, we define complexity classes ${\sf RwBQP}$, ${\sf CBQP}$, and ${\sf AdPostBQP}$ as sets of decision problems solvable by polynomial-size quantum circuits with a polynomial number of rewinding operators, cloning operators, and adaptive postselections, respectively. Our main result is that ${\sf BPP}^{\sf PP}\subseteq{\sf RwBQP}={\sf CBQP}={\sf AdPostBQP}\subseteq{\sf PSPACE}$. As a byproduct of this result, we show that any problem in ${\sf PostBQP}$ can be solved with only postselections of events that occur with probabilities polynomially close to one. Under the strongly believed assumption that ${\sf BQP}\nsupseteq{\sf SZK}$, or the shortest independent vectors problem cannot be efficiently solved with quantum computers, we also show that a single rewinding operator is sufficient to achieve tasks that are intractable for quantum computation. Finally, we show that rewindable Clifford circuits remain classically simulatable, but rewindable instantaneous quantum polynomial time circuits can solve any problem in ${\sf PP}$.