Fault-tolerant conversion between adjacent Reed–Muller quantum codes based on gauge fixing

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 7 out of 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.Quantum information technology, with applications such as secure quantum communication or universal quantum computing, is extremely powerful but challenging due to the fragility of quantum information in the presence of noise, loss and decoherence. Fault-tolerant quantum error correction [1][2][3] ameliorates this problem of fragility by encoding plain-text quantum information into cipher text, processing this encoded information while also measuring error syndromes and correcting, and finally turning back to plain text. For fault-tolerant error correction, transversal gates are especially valuable as qubits in each code block act bitwise between corresponding qubits in each code block, thereby naturally preventing error propagation [4]. Unfortunately, no code can simply enable a universal set of transversal gates [4-6] without invoking a technique involving ancillary qubits; thus, complicated strategies are employed to produce universal transversal gate sets. One strategy to circumvent the non-universal gate set problem is to employ magic-state distillation [7][8][9][10][11][12]. Although magic-state distillation has the advantage of a higher fault-tolerant threshold, the overhead in preparation and distillation is a major bottleneck for scalable quantum computing [13][14][15][16].Alternative techniques can provide a universal fault-tolerant gate set. The Steane code [17] provides transversal Clifford gates but not the transversal T := diag(1, exp{iπ/4}) gate whereas the 15-qubit Reed-Muller quantum code (RMQC) [18] is transversal for T, controlled-NOT (CNOT), controlled-S (CS), controlled-Z (CZ), and controlledcontrolled-Z (CCZ) gates, but not for the Hadamard (H) gate. One approach to achieving a universal set of gates employs just one code such as the Steane code or the 15-qubit RMQC and does not invoke code conversion. For example, for the Steane code, the fault-tolerant T gate can be realized with the help of an ancil...

Rewiring stabilizer codes

“…Bombin also made similar observations [32]. Quan et al later extended this idea [11]. Our approach is in the same category as these gauge fixing methods, but has the added benefit that it automatically finds the minimal set of operators which need to be measured.…”

“…The [[7, 1, 1]] Steane code is a 7-qubit code that supports transversal Clifford gates, and the 15-qubit [ [15,1,3]] quantum Reed-Muller code supports several transversal non-Clifford gates which complement those of the Steane code. Thus [9][10][11] suggested sequentially mapping between these codes to produce a universal set of transversal gates. The generators for these codes are given in table 1, where we have appended extra bits to the Steane code.…”

“…Thus we have four new elements for block C: ¢ { } g g g , 13 12 0 = ¢ { } g g g , 11 13 1 = ¢ { } g g g , 9 13 2 = ¢ { } g g , 6 3 =0. The remaining generators either belong to block A (meaning they are equal to stabilizers of the other code) or block B (in which case they are logical operators for the other code).…”

“…The former admits transversal Clifford gates, while the latter admits transversal T and control-control-Z gates. Subsequently, several authors proposed other circuits for this mapping [10,11]. Our algorithm provides a systematic approach to this, and related, problems.…”

Universal fault-tolerant quantum computation using fault-tolerant conversion schemes

“…In that case, magic state distillation can be orders of magnitude more costly than the transversal implementation [9,10]. Another idea for universal fault-tolerant quantum computation is to combine the transversal gates in two different codes using fault-tolerant conversion scheme [11,12]. Under the subsystem stabilizer formalism [13], this conversion scheme can be interpreted as different gauge fixing procedures [14][15][16] on the same subsystem code.…”