Constructing General Unitary Maps from State Preparations

TL;DR

This paper introduces an efficient algorithm for constructing arbitrary unitary maps in high-dimensional quantum systems using a combination of eigen-decomposition, stochastic searches, and geometric methods, enabling scalable quantum control.

Contribution

The authors develop a scalable method to generate unitary maps from state preparations with polynomial resources, extending to subspace control and quantum gate implementation in atomic spins.

Findings

Efficient polynomial-scaling algorithm for unitary map construction.

Extension of the method to subspace control with fewer stochastic searches.

Application to atomic spin control for qudit gates and error correction.

Abstract

We present an efficient algorithm for generating unitary maps on a $d$-dimensional Hilbert space from a time-dependent Hamiltonian through a combination of stochastic searches and geometric construction. The protocol is based on the eigen-decomposition of the map. A unitary matrix can be implemented by sequentially mapping each eigenvector to a fiducial state, imprinting the eigenphase on that state, and mapping it back to the eigenvector. This requires the design of only $d$ state-to-state maps generated by control waveforms that are efficiently found by a gradient search with computational resources that scale polynomially in $d$. In contrast, the complexity of a stochastic search for a single waveform that simultaneously acts as desired on all eigenvectors scales exponentially in $d$. We extend this construction to design maps on an $n$-dimensional subspace of the Hilbert space using only $n$ stochastic searches. Additionally, we show how these techniques can be used to control atomic spins in the ground electronic hyperfine manifold of alkali atoms in order to implement general qudit logic gates as well to perform a simple form of error correction on an embedded qubit.