Quantum Fanout and GHZ states using spin-exchange interactions

TL;DR

This paper demonstrates how spin-exchange interactions can implement fanout and GHZ states for logical qubits, generalizing previous equal-coupling results to arbitrary coupling strengths with exact conditions.

Contribution

It provides exact conditions on pairwise couplings for implementing fanout and GHZ states using spin-exchange interactions with unequal couplings.

Findings

Fanout operation implemented in constant depth using spin-exchange interactions.

Conditions on coupling strengths for exact implementation of quantum gates.

Creation of GHZ states using the same physical interactions.

Abstract

We show how the fanout operation on $n$ logical qubits can be implemented via spin-exchange (Heisenberg) interactions between $2n$ physical qubits, together with a physical target qubit and $1$- and $2$-qubit gates in constant depth. We also show that the same interactions can be used to implement Mod_q gates for any $q>1$. These results allow for unequal coupling strengths between physical qubits. This work generalizes an earlier result by Fenner & Zhang [arXiv: quant-ph/0407125], wherein the authors showed similar results assuming all pairwise couplings are equal. The current results give exact conditions on the pairwise couplings that allow for this implementation. Precisely, each logical qubit is encoded into two physical qubits. Couplings between physical qubits encoding the same logical qubit are termed as internal couplings and couplings between the ones encoding different logical qubits are termed as external couplings. We show that for a suitable time $T$ of evolution, the following conditions should hold: a) every external coupling should be an odd integer multiple of $\pi/2T$; b) every internal coupling should be an integer multiple of $\pi/T$; and c) the external magnetic strength in $z$-direction should be an integer multiple of $\pi/T$. Since generalized GHZ (''cat'') states can be created in constant depth using fanout, the same interactions can be used to create these states.