First quantum error correcting code code#
Relating its logical code basis to Majorana dimers, we efficiently compute boundary-state properties even for the non-Gaussian case of generic logical input. Equipped with this framework, we revisit the recently proposed hyperbolic pentagon code (HyPeC). This picture allows for an efficient contraction with a simple diagrammatic interpretation and is amenable to analytical study of holographic quantum error-correcting codes. In this work, we introduce a versatile framework of Majorana dimers capturing the intersection of stabilizer and Gaussian Majorana states. Holographic quantum error-correcting codes have been proposed as toy models that describe key aspects of the anti-de Sitter/conformal field theory (AdS/CFT) correspondence. In particular, comparing with a square surface code, we observe a significant improvement in the logical failure rate against biased noise using a rotated surface code with approximately half the number of physical qubits. Finally, we use approximate maximum-likelihood decoding to demonstrate that this improvement persists for a general Pauli noise biased toward dephasing. 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. We demonstrate a significant improvement in the logical failure rate with pure dephasing for coprime codes that have g=1 and closely related rotated codes, which have a modified boundary. That is, for rectangular surface codes with standard rough or 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 that the subthreshold behavior of the code depends critically on the precise shape and boundary conditions of the code. 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. First, we consider the infinite bias limit, meaning pure dephasing.
First quantum error correcting code how to#
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 the logical failure rate. Here, we identify features of the surface code responsible for these ultrahigh thresholds. The surface code, with a simple modification, exhibits ultrahigh error-correction thresholds when the noise is biased toward dephasing. We show that it is possible to maintain all of these advantages when we perform fault-tolerant quantum computation.
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. 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 present numerical evidence that the threshold even exceeds this hashing bound for an experimentally relevant range of noise parameters. 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. Here we show that a variant of the surface code-the XZZX code-offers remarkable performance for fault-tolerant quantum computation. The challenge is to design practical quantum error-correcting codes that perform well against realistic noise using modest resources. Performing large calculations with a quantum computer will likely require a fault-tolerant architecture based on quantum error-correcting codes.
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