Analytical percolation theory for topological color codes under qubit loss

Abstract

Quantum information theory has shown strong connections with classical statistical physics. For example, quantum error correcting codes like the surface and the color code present a tolerance to qubit loss that is related to the classical percolation threshold of the lattices where the codes are defined. Here we explore such connection to study analytically the tolerance of the color code when the protocol introduced in [Phys. Rev. Lett. 121, 060501 (2018)] to correct qubit losses is applied. This protocol is based on the removal of the lost qubit from the code, a neighboring qubit, and the lattice edges where these two qubits reside. We first obtain analytically the average fraction of edges r(p) that the protocol erases from the lattice to correct a fraction p of qubit losses. Then, the threshold pc below which the logical information is protected corresponds to the value of p at which r(p) equals the bond-percolation threshold of the lattice. Moreover, we prove that the logical information is protected if and only if the set of lost qubits does not include the entire support of any logical operator. The results presented here open a route to an analytical understanding of the effects of qubit losses in topological quantum error codes.

Characterization and implementation of robust quantum information processing

“…This chapter of the thesis has presented the results that we achieved in Refs. [1,2], as well as an analytical mean-field approximation of the fraction of edges erased r(p) as a function of the qubit loss rate p that is not included in the previous references.…”

“…where the sum is performed over all of the 2 n possible bit strings e. For depolarizing noise, errors can be characterised by a string e with n elements e q ∈ {0, 1, 2, 3} corresponding to no error σ (0) = I and σ (1) = X, σ (2) = Y , σ (3) = Z errors, respectively. Each error is a Pauli operator σ e = ⊗ n q=1 σ (eq) q , and can happen with probability (p/3) [e] (1 − p) n− [e] , where [e] is the number of elements e q ∈ {1, 2, 3} in e:…”

“…In this chapter we explain the work about qubit losses on color codes that we realised and presented in Refs. [1,2]. In Ref.…”

“…This approximation does not appear in Refs. [1,2]. Provided that qubit losses happen according to a uniform distribution, when they are corrected, the qubits that are sacrificed, the edges that are erased from the original shrunk lattice, and the new edges that are added, are approximately uniformly distributed as well.…”

“…where R (2) is an invertible binary matrix, and U LUE = U S H t is constituted of local unitary operations U S representing a phase gate on a subset of the control qubits that is specified later, and H t representing Hadamard operations on the set of target qubits t. In the subsequent discussion, we explicitly calculate the l.h.s. of Eq.…”