Flag fault-tolerant error correction, measurement, and quantum computation for cyclic Calderbank-Shor-Steane codes

Abstract

Flag qubits have recently been proposed in syndrome extraction circuits to detect high-weight errors arising from fewer faults. The use of flag qubits allows the construction of fault-tolerant protocols with the fewest number of ancillas known to-date. In this work, we prove some critical properties of CSS codes constructed from classical cyclic codes that enable the construction of a flag fault-tolerant error correction scheme. We then develop fault-tolerant protocols as well as a family of circuits for flag fault-tolerant error correction and operator measurement, requiring only four ancilla qubits and applicable to cyclic CSS codes of distance 3. The measurement protocol can be further used for logical Clifford gate implementation via quantum gate teleportation. We also provide examples of cyclic CSS codes with large encoding rates.

Flag Fault-Tolerant Error Correction for any Stabilizer Code

“…In Refs. [5], [7], and [12], flag patterns can only detect or reveal partial information about the correlated data errors; adaptive syndrome measurement is further needed for full correction. In Refs.…”

“…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].…”

“…Unlike Shor's and similar schemes [1][2][3], our procedure does not verify the ancillas; no postselection is needed. Unlike previous flag schemes [5,7,12], ours is nonadaptive, in the sense that the flag-measurement outcomes alone provide enough information to correct correlated errors; no extra syndrome measurement is required. Prabhu and Reichardt [22] have recently Couple the data qubits with an unverified cat state.…”

Triangular color codes on trivalent graphs with flag qubits

“…Consequently, each qubit has degree three connectivity. Further, we show how the syndrome measurement circuits take advantage of additional ancilla qubits used as flag qubits [37][38][39][40][41][42][43]33]. In an error correction scheme, the role of flag qubits 5 is to detect errors of weight greater than v arising from < v d eff faults (where d eff if the effective distance of the code with a given decoder 6 ) and to provide additional information allowing such errors to be identified and corrected.…”

Fault-tolerant magic state preparation with flag qubits

“…Recently, new schemes using flag qubits have been introduced to implement error correction protocols using the minimal number of ancilla qubits to measure the codes stabilizer generators [20][21][22][23][24][25]. The idea behind flag error correction is to use extra ancilla qubits which flag when v ≤ t faults result in a data qubit error of weight greater then v (here t = (d − 1)/2 where d is the distance of the code).…”