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International Conference on Parallel and Distributed Computing, Applications and Technologies (PDCAT)

Fault-Tolerant Logical Hadamard Gates Implementation in Reed-Muller Quantum Codes

Dongxiao Quan, Li Niu, LiLi Zhu, Changxing Pei

Published Dec 1, 2019• 2 citations
Publisher

Abstract

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.

DOI
10.1109/pdcat46702.2019.00040

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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.
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.
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 describe a method to use measurements and correction operations in order to implement the Clifford group in a stabilizer code, generalising a result from [Bombin,2011] for topological subsystem colour codes. In subsystem stabilizer codes of distance at least 333 the process can be implemented fault-tolerantly. In particular this provides a method to implement a logical Hadamard-type gate within the 15-qubit Reed-Muller quantum code by measuring and correcting only three observables. This is an alternative to the method proposed by [Paetznick and Reichardt, 2013] to generate a set of gates which is universal for quantum computing for this code. The construction is inspired by the description of code rewiring from [Colladay and Mueller, 2018]. Inspired by the code rewiring strategy of [Colladay and Mueller, 2018], we describe a method to use measurements and correction operations in order to implement the Clifford group in the code space of any stabilizer code, and we specify a sufficient set of conditions under which the distance of the code is preserved throughout. In particular this provides a method to implement a logical Hadamard-type gate within the 15-qubit Reed-Muller quantum code by measuring and correcting only two observables, providing the only non-transversal gate required for universality. Furthermore this approach is applicable to the toric code and quantum LDPC code
We present a set of efficiently implementable logical multi-qubit gates in concatenated quantum error correction codes using parity qubits. In particular, we show how fault-tolerant high-weight rotation gates of arbitrary angle can be implemented on single physical qubits of a classical stabilizer code, or on localized regions of full quantum error correction codes. Similarly, we show how transversal CNOT gates can implement logical parity-controlled-NOT operations between arbitrarily many logical qubits. Both operation types can be implemented and in many cases parallelized without the use of lattice surgery or the need for complicated routing operations.
Steane's 7-qubit quantum error-correcting code admits a set of fault-tolerant gates that generate the Clifford group, which in itself is not universal for quantum computation. The 15-qubit Reed-Muller code also does not admit a universal fault-tolerant gate set but possesses fault-tolerant T and control-control-Z gates. Combined with the Clifford group, either of these two gates generates a universal set. Here, we combine these two features by demonstrating how to fault-tolerantly convert between these two codes, providing a new method to realize universal fault-tolerant quantum computation. One interpretation of our result is that both codes correspond to the same subsystem code in different gauges. Our scheme extends to the entire family of quantum Reed-Muller codes.
Abstract In this paper, we present the fault-tolerant conversion between quantum Reed–Muller (QRM)(2, 5) and QRM(2, 7), and also the conversion between QBCH(15, 7) and QRM(2, 7). Either of the two schemes provides a method to realize universal fault-tolerant quantum computation. In particular, the gate overhead and logical error rate of a logical T gate are provided, as well as the comparison with magic state distillation scheme. In addition, we propose two other fault-tolerant conversion schemes based on ( u ∣ u + v ) and ( a + x ∣ b + x ∣ a + b − x ) constructions.