Mitigation of Decoherence-Induced Quantum-Bit Errors and Quantum-Gate Errors Using Steane’s Code

Abstract

Quantum processors require Quantum Error Correction Codes (QECC's) for improving the fidelity of quantum logic gates. Fault tolerant QECC's are capable of providing error rate improvements in quantum processors as long as the components are operating below a certain gate error probability. In this contribution, we quantify the depolarization probability bound, below which transversal QECC's would give a better error probability than an uncoded gate. Both a low-complexity repetition code and Steane's 7-bit QECC are characterized. INDEX TERMS Fault tolerance, quantum error correction codes, quantum stabilizer codes, quantum gates.

Gate-Error-Resilient Quantum Steane Codes

“…Let us first briefly introduce the Steane Code, whilst mentioning that its tutorial description be found in [3], [25]. The Steane code belongs to the family of Calderbank-Shor-Steane (CSS) code, which is a general construction using a pair of classical linear block codes C 1 and C 2 where C 2 ⊂ C 1 .…”

“…More explicitly, a fault-tolerant circuit limits the effects of a single gate error to a correctable number of qubit errors [1]. However, unfortunately many traditional encoding circuits are not fault-tolerant [1]- [3]. This is because these circuits have twoqubit controlled-NOT (CNOT) gate connections which have the property that a single qubit error propagates to many qubits, hence proliferating the errors [4].…”

“…This overwhelms the error correction capability of the [n, k, d] QECC, hence more errors are inflicted by the circuit than are corrected, where n is the number of encoded physical qubits, k is the number of original information qubits, d is the minimum distance and t is the error correction capability where we have t = (d − 1)/2 for the family of maximum-minimum distance codes. Therefore, rather than satisfying our original objective of improving the error rates, the QECC failing to rely on a fault-tolerant architecture prepares encoded states that have a higher error rate than the original uncoded information [3].…”

“…However, the prerequisite for using this scheme is that it needs clean all-zero ancilla qubits in order to achieve fault-tolerance. In addition, this scheme relies on a full stabilizer measurement, which is costly compared to the non-fault tolerant unitary encoding circuit in terms of qubit overheads [3], [15]. Nevertheless, if the architectural assumptions of the stabilizer circuit are met by the processor, the encoderless scheme imposes no further connectivity constraints on the device.…”

“…Shor's code motivated the design of Calderbank-Shor-Steane (CSS) codes [18], [19], which exploit the properties of classical linear block codes, providing a more general framework to correct both bit and phase-errors than Shor's code. As a further development, using the [7,4,3] Hamming code the 1 7 -rate Steane code was devised, which can correct a single arbitrary qubit error [19]. This code rate was then further improved in [20], [21], showing that a 1 5 -rate code was the shortest possible codeword length capable fo correcting a single qubit error.…”