Time-optimal CNOT between indirectly coupled qubits in a linear Ising chain

TL;DR

This paper derives analytical solutions for the fastest possible implementation of a CNOT gate between indirectly coupled qubits in a linear Ising chain, considering realistic constraints on local operations and fixed energy resources.

Contribution

It provides the first analytical time-optimal control solutions for entangling gates between non-adjacent qubits in a linear spin chain with fixed energy and non-instantaneous local operations.

Findings

Time-optimal CNOT between qubits 1 and 3 is achieved in time T=√(3/2)/J.

The same minimal time applies to a class of CNOT-like gates with local rotations.

Results improve previous estimates assuming zero-time local unitaries.

Abstract

We give analytical solutions for the time-optimal synthesis of entangling gates between indirectly coupled qubits 1 and 3 in a linear spin chain of three qubits subject to an Ising Hamiltonian interaction with equal coupling $J$ plus a local magnetic field acting on the intermediate qubit. The energy available is fixed, but we relax the standard assumption of instantaneous unitary operations acting on single qubits. The time required for performing an entangling gate which is equivalent, modulo local unitary operations, to the $\mathrm{CNOT}(1, 3)$ between the indirectly coupled qubits 1 and 3 is $T=\sqrt{3/2} J^{-1}$, i.e. faster than a previous estimate based on a similar Hamiltonian and the assumption of local unitaries with zero time cost. Furthermore, performing a simple Walsh-Hadamard rotation in the Hlibert space of qubit 3 shows that the time-optimal synthesis of the $\mathrm{CNOT}^{\pm}(1, 3)$ (which acts as the identity when the control qubit 1 is in the state $\ket{0}$, while if the control qubit is in the state $\ket{1}$ the target qubit 3 is flipped as $\ket{\pm}\rightarrow \ket{\mp}$) also requires the same time $T$.