Implementing the quantum fanout operation with simple pairwise interactions

Abstract

It has been shown that, for even $n$, evolving $n$ qubits according to a Hamiltonian that is the sum of pairwise interactions between the particles, can be used to exactly implement an $(n+1)$-qubit fanout gate using a particular constant-depth circuit~[\href{https://arXiv.org/abs/quant-ph/0309163}{arXiv:quant-ph/0309163}]. However, the coupling coefficients in the Hamiltonian considered in that paper are assumed to be all equal. In this paper, we generalize these results and show that for all $n$, including odd $n$, one can exactly implement an $(n+1)$-qubit parity gate and hence, equivalently in constant depth an $(n+1)$-qubit fanout gate, using a similar Hamiltonian but with unequal couplings, and we give an exact characterization of the constraints that the couplings must satisfy in order for them to be adequate to implement fanout via the same circuit.}{In particular, we show the following:Letting $J_{ij}$ be the coupling strength between the $\ordth{i}$ and $\ordth{j}$ qubits, the set of couplings $\{J_{ij}\}$ is adequate to implement fanout via the circuit above if and only if there exists $J>0$ such that 1. each $J_{ij}$ is an odd integer multiple of $J$, and 2. for each $i$, there are an even number of $j\ne i$ such that $J_{ij}/J \equiv 3\pmod{4}$.