Quantum state matching of qubits via measurement-induced nonlinear transformations

TL;DR

This paper introduces a measurement-induced nonlinear transformation scheme for quantum state matching of qubits, enabling the discrimination of states near a reference through iterative orthogonalization maps.

Contribution

It develops a class of nonlinear maps capable of orthogonalizing nonorthogonal qubit states and analyzes their physical implementation using specific two-qubit gates.

Findings

Nonlinear maps can orthogonalize qubit states after few iterations.

The scheme can decide if an unknown state is within a neighborhood of a reference.

Single physical realization of gates suffices due to iterative process.

Abstract

We consider the task of deciding whether an unknown qubit state falls in a prescribed neighborhood of a reference state. We assume that several copies of the unknown state are given and apply a unitary operation pairwise on them combined with a post-selection scheme conditioned on the measurement result obtained on one of the qubits of the pair. The resulting transformation is a deterministic, nonlinear, chaotic map in the Hilbert space. We derive a class of these transformations capable of orthogonalizing nonorthogonal qubit states after a few iterations. These nonlinear maps orthogonalize states which correspond to the two different convergence regions of the nonlinear map. Based on the analysis of the border (the so-called Julia set) between the two regions of convergence, we show that it is always possible to find a map capable of deciding whether an unknown state is within a neighborhood of fixed radius around a desired quantum state. We analyze which one- and two-qubit operations would physically realize the scheme. It is possible to find a single two-qubit unitary gate for each map or, alternatively, a universal special two-qubit gate together with single-qubit gates in order to carry out the task. We note that it is enough to have a single physical realization of the required gates due to the iterative nature of the scheme.