We show that a simple modification of the surface code can exhibit an enormous gain in the error correction threshold for a noise model in which Pauli Z errors occur more frequently than X or Y errors. Such biased noise, where dephasing dominates, is ubiquitous in many quantum architectures. In the limit of pure dephasing noise we find a threshold of 43.7(1)% using a tensor network decoder proposed by Bravyi, Suchara and Vargo. The threshold remains surprisingly large in the regime of realistic noise bias ratios, for example 28.2(2)% at a bias of 10. The performance is, in fact, at or near the hashing bound for all values of the bias. The modified surface code still uses only weight-4 stabilizers on a square lattice, but merely requires measuring products of Y instead of Z around the faces, as this doubles the number of useful syndrome bits associated with the dominant Z errors. Our results demonstrate that large efficiency gains can be found by appropriately tailoring codes and decoders to realistic noise models, even under the locality constraints of topological codes.For quantum computing to be possible, fragile quantum information must be protected from errors by encoding it in a suitable quantum error correcting code. The surface code [1] (and related topological stabilizer codes [2]) are quite remarkable among the diverse range of quantum error correcting codes in their ability to protect quantum information against local noise. Topological codes can have surprisingly large error thresholds-the break-even error rate below which errors can be corrected with arbitrarily high probability-despite using stabilizers that act on only a small number of neighboring qubits [3]. It is the combination of these high error thresholds and local stabilizers that make topological codes, and the surface code in particular, popular choices for many quantum computing architectures.Here we demonstrate a significant increase in the error threshold for a surface code when the noise is biased, i.e., when one Pauli error occurs at a higher rate than others. For qubits defined by nondegenerate energy levels with a Hamiltonian proportional to Z, the noise model is typically described by a dephasing (Z-error) rate that is much greater than the rates for relaxation and other energy-nonpreserving errors. Such biased noise is common in many quantum architectures, including superconducting qubits [4], quantum dots [5], and trapped ions [6], among others. The increased error threshold is achieved by tailoring the standard surface code stabilizers to the noise in an extremely simple way and by employing a decoder that accounts for correlations in the error syndrome. In particular, using the tensor network decoder of Bravyi, Suchara and Vargo (BSV) [7], we give evidence that the error correction threshold of this tailored surface code with pure Z noise is p c = 43.7(1)%, a fourfold increase over the optimal surface code threshold for pure Z noise of 10.9% [7]. These gains result from the following simple observations. For a Z error in the s...

The surface code, with a simple modification, exhibits ultrahigh error-correction thresholds when the noise is biased toward dephasing. Here, we identify features of the surface code responsible for these ultrahigh thresholds. We provide strong evidence that the threshold error rate of the surface code tracks the hashing bound exactly for all biases and show how to exploit these features to achieve significant improvement in logical failure rate. First, we consider the infinite bias limit, meaning pure dephasing. We prove that the error threshold of the modified surface code for pure dephasing noise is 50%, i.e., that all qubits are fully dephased, and this threshold can be achieved by a polynomial-time decoding algorithm. We demonstrate that the subthreshold behavior of the code depends critically on the precise shape and boundary conditions of the code. That is, for rectangular surface codes with standard rough and smooth open boundaries, it is controlled by the parameter g = gcd(j, k), where j and k are dimensions of the surface code lattice. We demonstrate a significant improvement in logical failure rate with pure dephasing for coprime codes that have g = 1, and closely-related rotated codes, which have a modified boundary. The effect is dramatic: The same logical failure rate achievable with a square surface code and n physical qubits can be obtained with a coprime or rotated surface code using only O( √ n) physical qubits. Finally, we use approximate maximum-likelihood decoding to demonstrate that this improvement persists for a general Pauli noise biased toward dephasing. In particular, comparing with a square surface code, we observe a significant improvement in logical failure rate against biased noise using a rotated surface code with approximately half the number of physical qubits.

Performing large calculations with a quantum computer will likely require a fault-tolerant architecture based on quantum error-correcting codes. The challenge is to design practical quantum error-correcting codes that perform well against realistic noise using modest resources. Here we show that a variant of the surface code—the XZZX code—offers remarkable performance for fault-tolerant quantum computation. The error threshold of this code matches what can be achieved with random codes (hashing) for every single-qubit Pauli noise channel; it is the first explicit code shown to have this universal property. We present numerical evidence that the threshold even exceeds this hashing bound for an experimentally relevant range of noise parameters. Focusing on the common situation where qubit dephasing is the dominant noise, we show that this code has a practical, high-performance decoder and surpasses all previously known thresholds in the realistic setting where syndrome measurements are unreliable. We go on to demonstrate the favourable sub-threshold resource scaling that can be obtained by specialising a code to exploit structure in the noise. We show that it is possible to maintain all of these advantages when we perform fault-tolerant quantum computation.

Noise in quantum computing is countered with quantum error correction. Achieving optimal performance will require tailoring codes and decoding algorithms to account for features of realistic noise, such as the common situation where the noise is biased towards dephasing. Here we introduce an efficient high-threshold decoder for a noise-tailored surface code based on minimum-weight perfect matching. The decoder exploits the symmetries of its syndrome under the action of biased noise and generalises to the fault-tolerant regime where measurements are unreliable. Using this decoder, we obtain fault-tolerant thresholds in excess of 6% for a phenomenological noise model in the limit where dephasing dominates. These gains persist even for modest noise biases: we find a threshold of ∼ 5% in an experimentally relevant regime where dephasing errors occur at a rate one hundred times greater than bit-flip errors.The surface code [1,2] is among the most promising quantum error-correcting codes to realise the first generation of scalable quantum computers [3][4][5]. This is due to its two-dimensional layout and low-weight stabilizers that help give it its high threshold [2,6,7], and its universal set of fault-tolerant logical gates [2,[8][9][10][11]. Ongoing experimental work [12][13][14][15] is steadily improving the surface code error rates. Concurrent work on improved decoding algorithms [6,7,[16][17][18] is leading to higher thresholds and lower logical failure rates, reducing the exquisite control demanded of experimentalists to realise such a system.Identifying the best decoder for the surface code depends critically on the noise model. Minimum-weight perfect matching (MWPM) [19,20] is near-optimal in the case of a bit-flip error model [2] and for a phenomenological error model with unreliable measurements [6]; see [21,22]. More recently, attention has turned to tailoring the decoder to perform under more realistic types of noise, such as depolarising noise [16,18,23,24] and correlated errors [25][26][27]. Of particular note is noise that is biased towards dephasing: a common feature of many architectures. With biased noise and reliable measurements, it is known that the surface code can be tailored to accentuate commonly occurring errors and that an appropriate decoder will give substantially increased thresholds [28,29]. However, these high thresholds were obtained using decoders with no known efficient implementation in the realistic setting where measurements are unreliable.Here we propose an efficient decoder for the surface code that is tailored to correct for local noise biased towards dephasing, demonstrating exceptional fault-tolerant thresholds. Our decoder uses the MWPM algorithm together with a recent technique to exploit symmetries of a given quantum errorcorrecting code [30]. Rather than using the symmetries of the code, we generalize this idea and use the symmetries of the entire system. Specifically, we exploit the symmetries of the syndrome with respect to its incident error model. Applied to pure depha...

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We show that a variant of the surface code—the XZZX code—offers remarkable performance for fault-tolerant quantum computation. The error threshold of this code matches what can be achieved with random codes (hashing) for every single-qubit Pauli noise channel; it is the first explicit code shown to have this universal property. We present numerical evidence that the threshold even exceeds this hashing bound for an experimentally relevant range of noise parameters. Focusing on the common situation where qubit dephasing is the dominant noise, we show that this code has a practical, high-performance decoder and surpasses all previously known thresholds in the realistic setting where syndrome measurements are unreliable. We go on to demonstrate the favorable sub-threshold resource scaling that can be obtained by specializing a code to exploit structure in the noise. We show that it is possible to maintain all of these advantages when we perform fault-tolerant quantum computation. We finally suggest some small-scale experiments that could exploit noise bias to reduce qubit overhead in two-dimensional architectures

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Tensor-network codes enable the construction of large stabilizer codes out of tensors describing smaller stabilizer codes. An application of tensor-network codes was an efficient and exact decoder for holographic codes. Here, we show how to write some topological codes, including the surface code and colour code, as simple tensor-network codes. We also show how to calculate distances of stabilizer codes by contracting a tensor network. The algorithm actually gives more information, including a histogram of all logical coset weights. We prove that this method is efficient in the case of stabilizer codes encoded via local log-depth circuits in one dimension and holographic codes. Using our tensor-network distance calculator, we find a modification of the rotated surface code that has the same distance but fewer minimum-weight logical operators by “doping” the tensor network, i.e., we break the homogeneity of the tensor network by locally replacing tensors. For this example, this corresponds to an improvement in successful error correction of almost 2% against depolarizing noise (in the perfect-measurement setting), but comes at the cost of introducing three higher-weight stabilizers. Our general construction lets us pick a network geometry (e.g., a Euclidean lattice in the case of the surface code), and, using only a small set of seed codes (constituent tensors), build extensive codes with the potential for optimisation.