Twins Percolation for Qubit Losses in Topological Color Codes

Abstract

In this Letter, we establish and explore a new connection between quantum information theory and classical statistical mechanics by studying the problem of qubit losses in 2D topological color codes. We introduce a protocol to cope with qubit losses, which is based on the identification and removal of a twin qubit from the code, and which guarantees the recovery of a valid three-colorable and trivalent reconstructed color code. Moreover, we show that determining the corresponding qubit loss error threshold is equivalent to a new generalized classical percolation problem. We numerically compute the associated qubit loss thresholds for two families of 2D color code and find that with p=0.461±0.005 these are close to satisfying the fundamental limit of 50% as imposed by the no-cloning theorem. Our findings reveal a new connection between topological color codes and percolation theory, show high robustness of color codes against qubit loss, and are directly relevant for implementations of topological quantum error correction in various physical platforms.

Analytical percolation theory for topological color codes under qubit loss

“…Moreover, the fundamental loss threshold p f of the three regular geometries of the color code has been computed numerically. Our results confirm the high robustness to qubit loss of the color code together with the protocol to correct qubit losses [52], which is of practical relevance for actual and future quantum processors. Furthermore, in this paper we have proven that the logical information still exists after correcting the qubit losses if and only if the set of lost and sacrificed qubits together does not include the support of a logical operator.…”

“…In this work we have explored a connection between statistical mechanics and QEC arising from the study of qubit loss in the topological color code. Here the problem of determining the robustness of the code to qubit loss is mapped to a novel classical percolation problem on coupled lattices as recently proposed in [52]. By exploring this connection we have determined analytically the tolerance of the color code to qubit loss.…”

“…The protocol proposed in [52] to correct the color code from qubit losses consists in choosing, for every lost qubit, a neighboring sacrificed qubit to be removed together with the loss. The steps of the protocol are depicted in Fig.…”

“…Then, in Sec. III we review the protocol to correct color codes from qubit losses that was proposed in [52], highlight the connection between the tolerance to qubit loss of the color code with this protocol and the percolation of the color code lattice, and provide detail on the computation of the number of edges erased to correct a qubit loss instance with the protocol. In Sec.…”

“…VI we provide an explicit recipe to compute r(p) up to any order in p. Then, in Sec. VII we describe in detail the algebraic technique proposed in [52] to obtain the fundamental qubit loss rate p f , and provide the necessary and sufficient conditions for the existence of the logical information under qubit loss. The values of p c and p f are summarized in Table I.…”