Approximating Unitary Preparations of Orthogonal Black Box States

TL;DR

This paper explores how to efficiently approximate unitaries that prepare orthogonal quantum states from black box circuits, focusing on the case of two states and addressing constraints on auxiliary qubits.

Contribution

It introduces a polynomial-size approximation method for unitaries preparing orthogonal states with auxiliary qubits reset, specifically for the case of two states.

Findings

Polynomial size approximation for m=2 case

Techniques for auxiliary qubits to return to zero

Addressing constraints in black box state preparation

Abstract

In this paper, I take a step toward answering the following question: for m different small circuits that compute m orthogonal n qubit states, is there a small circuit that will map m computational basis states to these m states without any input leaving any auxiliary bits changed. While this may seem simple, the constraint that auxiliary bits always be returned to 0 on any input (even ones besides the m we care about) led me to use sophisticated techniques. I give an approximation of such a unitary in the m = 2 case that has size polynomial in the approximation error, and the number of qubits n.