A family of surface codes with general lattice structures is proposed. We can control the error tolerances against bit and phase errors asymmetrically by changing the underlying lattice geometries. The surface codes on various lattices are found to be efficient in the sense that their threshold values universally approach the quantum Gilbert-Varshamov bound. We find that the error tolerance of the surface codes depends on the connectivity of the underlying lattices; the error chains on a lattice of lower connectivity are easier to correct. On the other hand, the loss tolerance of the surface codes exhibits an opposite behavior; the logical information on a lattice of higher connectivity has more robustness against qubit loss. As a result, we come upon a fundamental trade-off between error and loss tolerances in the family of surface codes with different lattice geometries.
A family of surface codes with general lattice structures is proposed. We can control the error tolerances against bit and phase errors asymmetrically by changing the underlying lattice geometries. The surface codes on various lattices are found to be efficient in the sense that their threshold values universally approach the quantum Gilbert-Varshamov bound. We find that the error tolerance of the surface codes depends on the connectivity of the underlying lattices; the error chains on a lattice of lower connectivity are easier to correct. On the other hand, the loss tolerance of the surface codes exhibits an opposite behavior; the logical information on a lattice of higher connectivity has more robustness against qubit loss. As a result, we come upon a fundamental trade-off between error and loss tolerances in the family of surface codes with different lattice geometries.
We propose a family of surface codes with general lattice structures, where the error tolerances against bit and phase errors can be controlled asymmetrically by changing the underlying lattice geometries. The surface codes on various lattices are found to be efficient in the sense that their threshold values universally approach the quantum Gilbert-Varshamov bound. We find that the error tolerance of the surface codes depends on the connectivity of the underlying lattices; the error chains on a lattice of lower connectivity are easier to correct. On the other hand, the loss tolerance of the surface codes exhibits an opposite behavior; the logical information on a lattice of higher connectivity has more robustness against qubit loss. As a result, we come upon a fundamental trade-off between error and loss tolerances in the family of surface codes with different lattice geometries. We also provide the physical aspects of the present results from the viewpoint of statistical physics, which leads to an equality that captures well both the error and loss tolerances of these surface codes.
Quantum computers are far more error-prone than their classical counterparts. Therefore, to build a quantum computer capable of running large-scale quantum algorithms, we must use the techniques of quantum error correction to ensure that the computer produces the correct output even when its components are unreliable. However, the resource requirements of building such a fault-tolerant quantum computer are currently prohibitive. Here, we examine the utility of using three-dimensional (3D) surface codes in a fault-tolerant quantum computer. This family of topological error-correcting codes is a generalization of the well-known 2D surface code to three spatial dimensions. We show that certain 3D surface codes have a transversal logical non-Clifford gate. In a quantum computing architecture, a non-Clifford gate is required to achieve computational universality. Transversal gates do not entangle qubits in different codes, so they are naturally fault tolerant because they do not spread errors. Next, we consider the problem of decoding 3D surface codes. In a quantum error-correcting code, we cannot observe the qubits directly, so we measure parity-check operators to gain information about the state of the code. Decoding is the problem of estimating what error has occurred given a list of unsatisfied parity checks. We observe that 3D surface codes offer asymmetric protection against bit-flip and phase-flip errors, but in both cases, we find that a threshold error rate exists below which we can suppress logical errors by increasing the size of the code. We use our results about logical gates and decoding to propose two fault-tolerant quantum computing architectures that utilize 3D surface codes. Finally, we compare the resource requirements of our architectures with the requirements of leading quantum computing architectures based on topological codes. We find that one of our architectures may be competitive with the leading architectures, depending on the properties of the physical systems used to build the qubits.
Abstract Surface codes are quantum error correcting codes typically defined on a 2D array of qubits. A [ d x , d z ] surface code design is being introduced, where d x ( d z ) represents the distance of the code for bit (phase) error correction, motivated by the fact that the severity of bit flip and phase flip errors in the physical quantum system is asymmetric. We present pseudo‐threshold and threshold values for the proposed surface code design for asymmetric error channels in the presence of various degrees of asymmetry of , errors in a depolarisation channel. We demonstrate that compared to symmetric surface codes, our asymmetric surface codes can provide almost double the pseudo‐threshold rates while requiring less than half the number of physical qubits in the presence of increasing asymmetry in the error channel. Our results show that for low degree of asymmetry, it is advantageous to increase d x along with d z . However, as the asymmetry of the channel increases, higher pseudo‐threshold is obtained with increasing d z when d x is kept constant at a low value. Additionally, we also show that the advantage in the pseudo‐threshold rates begins to saturate for any possible degree of asymmetry in the error channel as the surface code asymmetry is continued to be increased.
Quantum fault tolerance is a prerequisite to harness the full potential of quantum computers. However, achieving fault tolerance remains a major challenge since it requires optimizing tradeoffs between several intertwined theoretical and experimental parameters, including the encoding rate, code distance, parity-check weights, code planarity, and routing. Surface codes remain strong candidates for fault tolerance due to their simple geometric structure, established decoding methods, locality, and experimental attainability. In this work, we present finite families of hyperbolic surface codes that leverage the optimal efficiency scaling kd2/n=C(logk)2 attainable by surface codes with bounded local geometry. We show that optimizing the periodic identifications can double the code distance and thereby increase the code's efficiency by a factor of four, at fixed qubit count, encoding rate, and parity-check weight. We also address the routing problem of hyperbolic surface codes by introducing a topology-aware algorithm that partitions a hyperbolic code into bounded planar modules while reducing long-range communications demand. Circuit-level Monte Carlo simulations yield finite-size threshold estimates for both modular and monolithic layouts, showing that tripling the long-range gate error probability mildly lowers, but does not eliminate, the estimated error-correction thresholds. Together, these results strengthen the case for hyperbolic surface codes as structured and highly efficient large-scale quantum error correction codes for future fault-tolerant architectures.
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 the 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 or smooth open boundaries, it is controlled by the parameter g ¼ gcdðj; kÞ, where j and k are dimensions of the surface code lattice.We demonstrate a significant improvement in the logical failure rate with pure dephasing for coprime codes that have g ¼ 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 Oð ffiffiffi n p Þ 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 the logical failure rate against biased noise using a rotated surface code with approximately half the number of physical qubits.
For quantum computers to reach their full potential will require error correction. We study the surface code, one of the most promising quantum error correcting codes, in the context of predominantly dephasing (Z-biased) noise, as found in many quantum architectures. We find that the surface code is highly resilient to Y-biased noise, and tailor it to Z-biased noise, whilst retaining its practical features. We demonstrate ultrahigh thresholds for the tailored surface code: ~39% with a realistic bias of = 100, and ~50% with pure Z noise, far exceeding known thresholds for the standard surface code: ~11% with pure Z noise, and ~19% with depolarizing noise. Furthermore, we provide strong evidence that the threshold of the tailored surface code tracks the hashing bound for all biases. We reveal the hidden structure of the tailored surface code with pure Z noise that is responsible for these ultrahigh thresholds. As a consequence, we prove that its threshold with pure Z noise is 50%, and we show that its distance to Z errors, and the number of failure modes, can be tuned by modifying its boundary. For codes with appropriately modified boundaries, the distance to Z errors is O(n) compared to O(n1/2) for square codes, where n is the number of physical qubits. We demonstrate that these characteristics yield a significant improvement in logical error rate with pure Z and Z-biased noise. Finally, we introduce an efficient approach to decoding that exploits code symmetries with respect to a given noise model, and extends readily to the fault-tolerant context, where measurements are unreliable. We use this approach to define a decoder for the tailored surface code with Z-biased noise. Although the decoder is suboptimal, we observe exceptionally high fault-tolerant thresholds of ~5% with bias = 100 and exceeding 6% with pure Z noise. Our results open up many avenues of research and, recent developments in bias-preserving gates, highlight their direct relevance to experiment.