Direct measurement of Bacon-Shor code stabilizers

Abstract

A Bacon-Shor code is a subsystem quantum error-correcting code on an L × L lattice where the 2(L − 1) weight-2L stabilizers are usually inferred from the measurements of (L − 1) 2 weight-2 gauge operators. Here we show that the stabilizers can be measured directly and fault tolerantly with bare ancillary qubits by constructing circuits that follow the pattern of gauge operators. We then examine the implications of this method for small quantum error-correcting codes by comparing distance 3 versions of the rotated surface code and the Bacon-Shor code with the standard depolarizing model and in the context of a trapped ion quantum computer. We find that for a simple circuit of prepare, error correct and measure the Bacon-Shor code outperforms the surface code by requiring fewer qubits, taking less time, and having a lower error rate.Quantum information experiments are approaching the number of qubits and operational fidelity necessary for quantum error correction to improve performance [1][2][3]. Classical error correction on quantum devices have already shown the ability to suppress introduced errors and increase memory times [4][5][6][7][8]. Two promising quantum error-correcting codes for data qubits arranged on an L × L lattice are the surface code [9-11] and the Bacon-Shor code [12][13][14]. Numerical simulation of the surface code shows a high memory threshold of 1% error per operation for increasing L and for distance 3 codes a pseudothreshold of 0.3% error per operation for a depolarizing error model [11]. The Bacon-Shor code is a subsystem code and has no threshold as L grows [15] but promising performance for small distance codes with a pseudothreshold of 0.2% for a depolarizing error model [14] and a fault-tolerant protocol for implementing universal gates without distillation [16]. The rotated surface code has L 2 − 1 check operators of weight 4 in the bulk and weight 2 on the boundary [10]. The advantage of the Bacon-Shor code comes from using weight-2 gauge operators to determine the weight 2L check operators and the lack of threshold is a result of having only 2(L−1) checks [12,14,15].For the [[9,1,3]] surface code, Tomita and Svore [11] pointed out that in the circuit model, the order in which the weight 4 check operators are measured prevents unwanted error propagation and allows for parallelization of operations. For Bacon-Shor, the idea has always been to measure gauge operators because direct measurements of the stabilizers would require long-range interaction between qubits and higher-weight stabilizers usually require more complicated ancillary qubit preparation [17][18][19][20]. Inspired by Tomita and Svore and the fact that both the surface code and the Bacon-Shor code can be thought as gauge choices on the compass model, we found that * ken.brown@duke.edu Bacon-Shor codes can be measured with bare ancillary qubits.In condensed matter physics the compass model is used to describe a family of lattice models involving interacting quantum degrees of freedom [21]. The relationship between compas...

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