Reversing Unknown Qubit-Unitary Operation, Deterministically and Exactly

TL;DR

This paper introduces a deterministic, exact protocol to reverse any unknown qubit-unitary operation using four calls, with potential reuse of the auxiliary state as a catalyst, advancing quantum inversion techniques.

Contribution

It presents a novel protocol for universal qubit-unitary inversion that is deterministic and exact, overcoming previous no-go limitations, and offers a simplified semidefinite programming approach.

Findings

Achieves inversion with four calls to the unknown unitary

Auxiliary state can be reused as a catalyst in multiple runs

Provides a reduced search space for optimal protocols

Abstract

We report a deterministic and exact protocol to reverse any unknown qubit-unitary operation, which simulates the time inversion of a closed qubit system. To avoid known no-go results on universal deterministic exact unitary inversion, we consider the most general class of protocols transforming unknown unitary operations within the quantum circuit model, where the input unitary operation is called multiple times in sequence and fixed quantum circuits are inserted between the calls. In the proposed protocol, the input qubit-unitary operation is called 4 times to achieve the inverse operation, and the output state in an auxiliary system can be reused as a catalyst state in another run of the unitary inversion. We also present the simplification of the semidefinite programming for searching the optimal deterministic unitary inversion protocol for an arbitrary dimension presented by M. T. Quintino and D. Ebler [Quantum $\textbf{6}$, 679 (2022)]. We show a method to reduce the large search space representing all possible protocols, which provides a useful tool for analyzing higher-order quantum transformations for unitary operations.