Triangular color codes on trivalent graphs with flag qubits

Abstract

The color code is a topological quantum error-correcting code supporting a variety of valuable faulttolerant logical gates. Its two-dimensional version, the triangular color code, may soon be realized with currently available superconducting hardware despite constrained qubit connectivity. To guide this experimental effort, we study the storage threshold of the triangular color code against circuitlevel depolarizing noise. First, we adapt the Restriction Decoder to the setting of the triangular color code and to phenomenological noise. Then, we propose a fault-tolerant implementation of the stabilizer measurement circuits, which incorporates flag qubits. We show how information from flag qubits can be used in an efficient and scalable way with the Restriction Decoder to maintain the effective distance of the code. We numerically estimate the threshold of the triangular color code to be 0.2%, which is competitive with the thresholds of other topological quantum codes. We also prove that 1-flag stabilizer measurement circuits are sufficient to preserve the full code distance, which may be used to find simpler syndrome extraction circuits of the color code. OPEN ACCESS RECEIVED

Efficient classical simulation of noisy random quantum circuits in one dimension

“…In the context of fault-tolerant quantum computing, two-qubit depolarization channels are used to model errors that happen during twoqubit gates, such as CNOT and CZ gates (i.e., to perform a detailed circuit-level noise analysis) [8,9,16,[19][20][21][22][23][24]. For instance, the fault-tolerance threshold of the surface-code p 0.01 [8] is obtained by applying the two-qubit depolarization channel N 2 [p] after each two-qubit gate.…”

“…While currently available quantum devices are not large and reliable enough to factor a large integer or simulate the dynamics of a large quantum system, it has been established over the past two decades that fault-tolerant quantum computing is in principle possible via quantum error correction [3][4][5][6][7][8]. However, despite the recent progress in reducing high resource overhead associated with the use of fault-tolerant quantum computing schemes [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24], large-scale and fault-tolerant quantum computing is not yet within reach of near-term quantum technologies.…”

On maximum-likelihood decoding with circuit-level errors

“…1 with the constant c = 1/2 κ in Eq. (11) and the input (or output) code encoding k = n 0 − r 0 qubits, where r 0 is the rank of the input-code stabilizer group. Here κ is an integer [22].…”

“…In some cases, e.g., in the case of surface codes with minimum-weight perfect matching (MWPM) decoder, some leading-order correlations can be included in the edge weights of the graph used for decoding [4,[6][7][8]. However, such a scheme is limited to codes and measurement circuits where MWPM can be done efficiently, e.g., surface codes with single-ancilla measurement circuits [9], and certain other classes or topological codes [10,11]. Further, there is necessarily a decoding accuracy loss when measurement circuit is not simply repeated, e.g., with code deformation or lattice surgery [12].…”

Flag Fault-Tolerant Error Correction for any Stabilizer Code

“…Flag error-correction schemes have been given for several code families: Hamming codes [5], rotated surface and Reed-Muller codes [7], color codes [7][8][9][10][11], cyclic CSS codes [12,13], and heavy hexagon and square codes [14]. Flag-based schemes have been proposed also for error detection [15], logical state preparation [16][17][18] and lattice surgery [19,20].…”