Constructing General Unitary Maps from State Preparations

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.

9 pages, 3 figures

References in corpus (6)

• Experimental demonstration of quantum memory for light
• Mapping photonic entanglement into and out of a quantum memory
• Optimal control of quantum gates and suppression of decoherence in a system of interacting two-level particles
• Optimal control of atom transport for quantum gates in optical lattices
• Quantum control of the hyperfine-coupled electron and nuclear spins in alkali atoms
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Cited by in corpus (10)

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• Search complexity and resource scaling for the quantum optimal control of unitary transformations
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• Quantum logic gates for superconducting resonator qudits
• Quantum Optimal Control of Nuclear Spin Qudecimals in ${}^{87}\text{Sr}$
• Control of inhomogeneous atomic ensembles of hyperfine qudits
• Quantum tomography of the full hyperfine manifold of atomic spins via continuous measurement on an ensemble
• Maximum information gain in weak or continuous measurements of qudits: complementarity is not enough
• Quantum Control of d-Dimensional Quantum Systems with Application to Alkali Atomic Spins
• Theory of quantum control landscapes: Overlooked hidden cracks