Fault-Tolerant Thresholds for the Surface Code in Excess of 5% Under Biased Noise

Abstract

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...

Tailoring Surface Codes for Highly Biased Noise

“…Finally, although this paper focuses on features of surface codes with Y or Y -biased noise rather than the issue of fault-tolerant decoding, our numerical results motivate the search for fast fault-tolerant decoders for the surface code with biased noise. The highly significant question of whether the high performance of surface codes with biased noise can be preserved in the context of faulttolerant quantum computing, is addressed in a forthcoming paper [32], where a fast but suboptimal decoder for tailored surface codes achieves fault-tolerant thresholds in excess of 5% with biased noise. Investigating the optimal fault-tolerant thresholds with biased noise and the performance well below threshold remain important avenues of research.…”

Fault-tolerant bosonic quantum error correction with the surface–Gottesman-Kitaev-Preskill code

“…Recently, there have been proposals for scaling up the cat codes by concatenating them with a repetition code [17] or a surface code [18] which are tailored to biased noise models [19][20][21]. These schemes take advantage of * kyungjoo.noh@yale.edu † christopher.chamberland@ibm.com the fact that the cat code can suppress bosonic dephasing (stochastic random rotation) errors exponentially in the size of the cat code, thereby yielding a qubit with a biased noise predominated either by bit-flip or phaseflip errors.…”

Towards a realistic GaAs-spin qubit device for a classical error-corrected quantum memory

“…If two-qubit gates are of very high fidelity (with negligible error) and noise is dominated by phase-flip errors during slow, noisy measurements, then, instead of using the phase-flip code as bottom code, one might be able to directly benefit from using a modified surface code in which one measures X and Y checks, considered in [67]. With an (incoming) phenomenological Z-error rate p Z ≈ p and X -and Y -error rate equal to p X = p Y ≈ p/200 (noise bias η = 100) and a measurement error rate equal to p, the reported threshold in [67] is p c = 5%. For the GaAs numbers in Tables I and II, p Z ≈ t readout /T 2 = 1.25 × 10 −3 and p X ≈ t readout /T 1 = 5 × 10 −4 , putting one safely below this 5% threshold.…”

“…However, this picture is not realistic since the two-qubit gate error rates are in fact not negligible and biased-noise or bias-preserving gates have not been the focus of previous research. For example, doing four CNOT gates in a surface code QEC cycle, each CNOT gate would have to have X -error rate at least below 5% 800 = 0.625 × 10 −4 in order to get below this 5% threshold of [67]. It might be interesting to develop such noise-bias preserving CNOT with dominant Z noise as has been done for superconducting devices in [68].…”

“…Given that the noise bias is unlikely to be a direct advantage in surface code decoding a la [67], we can further consider using the phase-flip code as the bottom code to equalize the X -and Z-error rates as we have numerically explored in Sec. IV E. Thus the phase-flip repetition code could then be concatenated, in principle, with the surface code.…”