Time-optimal Unitary Operations in Ising Chains II: Unequal Couplings and Fixed Fidelity

TL;DR

This paper analytically derives the minimal time and optimal control strategies for implementing entangling CNOT gates in a three-qubit Ising chain with unequal couplings, considering fixed fidelity and energy constraints.

Contribution

It provides exact solutions for equal couplings and perturbative solutions for unequal couplings, demonstrating time-optimal control in quantum gate synthesis.

Findings

Exact solutions for equal Ising couplings at perfect fidelity.

Perturbative solutions for unequal couplings with tolerated errors.

Comparison shows previous numerical methods are not time-optimal.

Abstract

We analytically determine the minimal time and the optimal control laws required for the realization, up to an assigned fidelity and with a fixed energy available, of entangling quantum gates ($\mathrm{CNOT}$) between indirectly coupled qubits of a trilinear Ising chain. The control is coherent and open loop, and it is represented by a local and continuous magnetic field acting on the intermediate qubit. The time cost of this local quantum operation is not restricted to be zero. When the matching with the target gate is perfect (fidelity equal to one) we provide exact solutions for the case of equal Ising coupling. For the more general case when some error is tolerated (fidelity smaller than one) we give perturbative solutions for unequal couplings. Comparison with previous numerical solutions for the minimal time to generate the same gates with the same Ising Hamiltonian but with instantaneous local controls shows that the latter are not time-optimal.