Abstract We design forward and backward fault-tolerant conversion circuits, which convert between the Steane code and the 15-qubit Reed–Muller quantum code so as to provide a universal transversal gate set. In our method, only seven out of a total 14 code stabilizers need to be measured, and we further enhance the circuit by simplifying some stabilizers; thus, we need only to measure eight weight-4 stabilizers for one round of forward conversion and seven weight-4 stabilizers for one round of backward conversion. For conversion, we treat random single-qubit errors and their influence on syndromes of gauge operators, and our novel single-step process enables more efficient fault-tolerant conversion between these two codes. We make our method quite general by showing how to convert between any two adjacent Reed–Muller quantum codes RM ¯ ( 1 , m ) and RM ¯ ( 1 , m + 1 ) , for which we need only measure stabilizers whose number scales linearly with m rather than exponentially with m obtained in previous work. We provide the explicit mathematical expression for the necessary stabilizers and the concomitant resources required.
We investigate how to implement fault-tolerant logical Hadamard gates in Reed-Muller quantum codes(RMQCs) using the gauge-fixing method. During the realization, we consider the influence of random single-qubit errors by performing the error-detecting measurements. Moreover, some error-detecting stabilizers are simplified by the existing syndromes. Then we identify the errors and modify the syndromes, and refer to the modified syndromes to select the fix operations, and finally perform the error-correcting and fix operations together. Further, we establish a graph model for the RMQCs and exhibit a progress of how to find the fix operations for the unsatisfied stabilizers. We simulate the progress of finding corresponding fix operations for 31-quibt and 63-qubit RMQCs and the whole process of realizing logical Hadamard gate with random single-qubit errors for 15-qubit and 31-quibt RMQCs. Results show that correct fix operations can be obtained and fault-tolerant logical Hadamard gates can be realized as expected. With the implementation of the logical Hadamard gate, a universal fault-tolerant gate set is achieved in single RMQC.
The [[7,1,3]] Steane code and [[23,1,7]] quantum Golay code have been identified as good candidates for fault-tolerant quantum computing via code concatenation. These two codes have transversal implementations of all Clifford gates but require some other scheme for fault-tolerant
T gates. Using magic states, Clifford operations, and measurements is one common scheme, but magic-state distillation can have a large overhead. Code conversion is one avenue for implementing a universal gate set fault tolerantly without the use of magic-state distillation. Analogously to how the [[7,1,3]] Steane code can be fault tolerantly converted to and from the [[15,1,3]] Reed-Muller code which has a transversal
T gate, the [[23,1,7]] Golay code can be converted to a [[95,1,7]] triorthogonal code with a transversal
T gate. A crucial ingredient of this procedure is the [[49,1,5]] triorthogonal code, which can itself be seen as being related to the self-dual [[17,1,5]] two-dimensional color code. Additionally, a method for code conversion based on a transversal cnot between the codes, rather than stabilizer measurements, is described.
Designing efficient fault-tolerant circuits for converting between two quantum codes is an effective way to realize universal fault-tolerant quantum computing. In this work, we adopt the round-robin circuit construction to design a conversion circuit between CSS 7-qubit error-correcting code and Reed-Muller 15-qubit code. We take the pieceable fault-tolerant protocol to guarantee the fault tolerance of our conversion circuit. A neural network decoding scheme is proposed to suppress the propagation of quantum errors. Our decoding scheme can effectively reduce the damaging effects of imperfect syndrome extraction operations and predict possible logical errors that occurred after the error correction process. We provide a detailed description of our syndrome data model and analyze the depolarizing noise threshold of our conversion circuit through numerical simulations of the stabilizer circuit.
We propose a scheme that converts a stabilizer code into another stabilizer code in a fault tolerant manner. The scheme first puts both codes in specific forms, and proceeds the conversion from a source code to a target code by applying Clifford gates. The Clifford gates are chosen from the comparisons between both codes. The fault tolerance of the conversion is guaranteed by quantum error correction in every step during the entire conversion process. As examples, we show three conversions: the
[[5,1,3]] stabilizer code and Steane code, Steane code and
[[15,1,3]] Reed-Muller code, and Steane code and
(3,4)-QPC code.
Abstract Code concatenation and conversion are two prominent methods to realize universal fault-tolerant quantum computation without the need for magic state distillation. In this paper, we analyze three quantum Reed–Muller (QRM) codes of different code lengths under code conversion schemes, and compare them with the 105-qubit concatenated code on optimized encoding circuits. The depolarizing error thresholds for the 7-, 15-, and 31-qubit QRM codes turn out to be 1.19 × 10 −3 , 5.40 × 10 −4 and 2.41 × 10 −6 respectively, all of three compared unfavorably with the improved result of 1.52 × 10 −3 for the 105-qubit code, which indicates a relatively small error-suppression capability and strict precision requirements for hardware equipment when using the code conversion scheme. While for resource requirements, code conversion schemes show obvious advantages. For the seven-qubit QRM code, the required qubit and gate overheads for level-3 gates to achieve the target logical error rate of 10 −15 are roughly 4 orders of magnitude smaller than those for the 105-qubit code.
Abstract We investigate the implementation of fault‐tolerant logical Hadamard gates in Reed–Muller quantum codes (RMQCs). During the realization, we consider the influences of random single‐qubit errors and some error‐detecting stabilizers are simplified by the existing syndromes. We first identify the errors and modify the gauge‐fixing syndromes, then refer to the modified syndromes to select the fix operations, and finally perform the error‐correcting and fix operations together. Furthermore, we establish a graph model for the RMQCs and exhibit a progress of finding the fix operations for the unsatisfied stabilizers. For the circuit design, we optimize the choice of gauge operators, the positions of the ancillary qubits and design a parallel circuit for implementing a fault‐tolerant logical Hadamard gate in 15‐qubit RMQC. We simulate the progress of finding corresponding fix operations for 31‐qubit and 63‐qubit RMQCs and the whole process of realizing logical Hadamard gate with random single‐qubit errors for 15‐qubit and 31‐qubit RMQCs. Results show that correct fix operations can be obtained and fault‐tolerant logical Hadamard gates can be realized as expected. The performance comparisons are also given and the results show that our method can achieve a higher success rate with a reasonable higher cost. With the implementation of the logical Hadamard gate, a universal fault‐tolerant gate set is achieved in single RMQC.