Multipartite entanglement and quantum error identification in $D$-dimensional cluster states

Abstract

An entangled state is said to be $m$-uniform if the reduced density matrix of any $m$ qubits is maximally mixed. This formal definition is known to be intimately linked to pure quantum error correction codes (QECCs), which allow not only to correct errors, but also to identify their precise nature and location. Here, we show how to create $m$-uniform states using local gates or interactions and elucidate several QECC applications. We first point out that $D$-dimensional cluster states, i.e. the ground states of frustration-free local cluster Hamiltonians, are $m$-uniform with $m=2D$. We discuss finite size limitations of $m$-uniformity and how to achieve larger $m$ values using quasi-$D$ dimensional cluster states. We demonstrate experimentally on a superconducting quantum computer that the 1D cluster state allows to detect and identify 1-qubit errors, distinguishing, $X$, $Y$ and $Z$ errors. Finally, we show that $m$-uniformity allows to formulate pure QECCs with a finite logical space