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arXiv:1603.02704 · Physical Review Letters

Thresholds for Universal Concatenated Quantum Codes

Christopher Chamberland, Tomas Jochym-O’Connor, Raymond Laflamme

Published Jun 28, 2016• 49 citations
arXivHTMLPDF

Abstract

Quantum error correction and fault tolerance make it possible to perform quantum computations in the presence of imprecision and imperfections of realistic devices. An important question is to find the noise rate at which errors can be arbitrarily suppressed. By concatenating the 7-qubit Steane and 15-qubit Reed-Muller codes, the 105-qubit code enables a universal set of fault-tolerant gates despite not all of them being transversal. Importantly, the cnot gate remains transversal in both codes, and as such has increased error protection relative to the other single qubit logical gates. We show that while the level-1 pseudothreshold for the concatenated scheme is limited by the logical Hadamard gate, the error suppression of the logical cnot gates allows for the asymptotic threshold to increase by orders of magnitude at higher levels. We establish a lower bound of 1.28×10^{-3} for the asymptotic threshold of this code, which is competitive with known concatenated models and does not rely on ancillary magic state preparation for universal computation.

DOI
10.1103/physrevlett.117.010501

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•Aug 24, 2016•0 citations
Quantum error correction and fault tolerance make it possible to perform quantum computations in the presence of imprecision and imperfections of realistic devices. An important question is to find the noise rate at which errors can be arbitrarily suppressed. By concatenating the 7-qubit Steane and 15-qubit Reed-Muller codes, the 105-qubit code enables a universal set of fault-tolerant gates despite not all of them being transversal. Importantly, the cnot gate remains transversal in both codes, and as such has increased error protection relative to the other single qubit logical gates. We show that while the level-1 pseudothreshold for the concatenated scheme is limited by the logical Hadamard gate, the error suppression of the logical cnot gates allows for the asymptotic threshold to increase by orders of magnitude at higher levels. We establish a lower bound of 1.28×10^{-3} for the asymptotic threshold of this code, which is competitive with known concatenated models and does not rely on ancillary magic state preparation for universal computation.
We analyze the resource overhead of recently proposed methods for universal fault-tolerant quantum computation using concatenated codes. Namely, we examine the concatenation of the 7-qubit Steane code with the 15-qubit Reed-Muller code, which allows for the construction of the 49- and 105-qubit codes that do not require the need for magic state distillation for universality. We compute a lower bound for the adversarial noise threshold of the 105-qubit code and find it to be $8.33\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}.$ We obtain a depolarizing noise threshold for the 49-qubit code of $9.69\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}$ which is competitive with the 105-qubit threshold result of $1.28\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$. We then provide lower bounds on the resource requirements of the 49- and 105-qubit codes and compare them with the surface code implementation of a logical TTT gate using magic state distillation. For the sampled input error rates and noise model, we find that the surface code achieves a smaller overhead compared to our concatenated schemes.
Abstract Concatenation of two quantum error-correcting codes with complementary sets of transversal gates can provide a means toward universal fault-tolerant quantum computation. We first show that it is generally preferable to choose the inner code with the higher pseudo-threshold to achieve lower logical failure rates. We then explore the threshold properties of a wide range of concatenation schemes. Notably, we demonstrate that the concatenation of complementary sets of Reed-Muller codes can increase the code capacity threshold under depolarizing noise when compared to extensions of previously proposed concatenation models. We also analyze the properties of logical errors under circuit-level noise, showing that smaller codes perform better for all sampled physical error rates. Our work provides new insights into the performance of universal concatenated quantum codes for both code capacity and circuit-level noise.
We propose a method for universal fault-tolerant quantum computation using concatenated quantum error correcting codes. The concatenation scheme exploits the transversal properties of two different codes, combining them to provide a means to protect against low-weight arbitrary errors. We give the required properties of the error correcting codes to ensure universal fault tolerance and discuss a particular example using the 7-qubit Steane and 15-qubit Reed-Muller codes. Namely, other than computational basis state preparation as required by the DiVincenzo criteria, our scheme requires no special ancillary state preparation to achieve universality, as opposed to schemes such as magic state distillation. We believe that optimizing the codes used in such a scheme could provide a useful alternative to state distillation schemes that exhibit high overhead costs.
Using transversal gates is a straightforward and efficient technique for fault-tolerant quantum computing. Since transversal gates alone cannot be computationally universal, they must be combined with other approaches such as magic state distillation, code switching, or code concatenation to achieve universality. In this paper we propose an alternative approach for universal fault-tolerant quantum computing, mainly based on the code concatenation approach proposed in [T. Jochym-O'Connor and R. Laflamme, Phys. Rev. Lett. 112, 010505 (2014)], but in a nonuniform fashion. The proposed approach is described based on nonuniform concatenation of the 7-qubit Steane code with the 15-qubit Reed-Muller code, as well as the 5-qubit code with the 15-qubit Reed-Muller code, which lead to two 49-qubit and 47-qubit codes, respectively. These codes can correct any arbitrary single physical error with the ability to perform a universal set of fault-tolerant gates, without using magic state distillation.
Abstract Code concatenation and conversion are two prominent methods to realize universal fault-tolerant quantum computation without the need for magic state distillation. In this paper, we analyze three quantum Reed–Muller (QRM) codes of different code lengths under code conversion schemes, and compare them with the 105-qubit concatenated code on optimized encoding circuits. The depolarizing error thresholds for the 7-, 15-, and 31-qubit QRM codes turn out to be 1.19 × 10 −3 , 5.40 × 10 −4 and 2.41 × 10 −6 respectively, all of three compared unfavorably with the improved result of 1.52 × 10 −3 for the 105-qubit code, which indicates a relatively small error-suppression capability and strict precision requirements for hardware equipment when using the code conversion scheme. While for resource requirements, code conversion schemes show obvious advantages. For the seven-qubit QRM code, the required qubit and gate overheads for level-3 gates to achieve the target logical error rate of 10 −15 are roughly 4 orders of magnitude smaller than those for the 105-qubit code.
Quantum error correction is the backbone of fault-tolerant quantum computation, a necessary requirement for any large scale quantum computer. The fault-tolerance threshold theorem has long been a target for experimental precision, allowing for the possibility of reducing logical error rates to arbitrarily low levels without excessive overhead. While there are many promising fault-tolerant architectures, the path towards the most practical fault-tolerant scheme is far from decided and may vary for differing physical models. This thesis proposes new schemes for universal fault-tolerant quantum computation in both the concatenated and topological code settings. \n \nThrough the concatenation of two different error correcting codes, a set of universal fault-tolerant gates can be obtained without requiring the need for magic state distillation. A lower bound of 1.28times10−31.28 \\times 10^{-3}1.28times10−3 for the fault-tolerance threshold under circuit level depolarizing noise is obtained. Additionally, stacked codes are proposed as a means to simulate the action of a 3D topological code in 2D, allowing for the application of a universal set of transversal operations. While fault-tolerant, unfortunately the scheme does not exhibit a threshold due to the decreasing pseudo-threshold with growing code distance, yet points to potential interesting avenues for fault-tolerant computation in 2D without distillation. \n \nOne of the primary avenues to constructing fault-tolerant logical operations is through transversal operations. In this thesis, the set of single qubit logical gates that can be implemented transversally are characterized and determined to all belong to the Clifford hierarchy. Moreover, any diagonal two-qubit operation that can be applied transversally must belong to the same level of the Clifford hierarchy as the set of gates that can be implemented in the single-qubit case. \n \nThe opposite to quantum error correction is privacy, where the output of a channel is disguised from its input. The two are fundamentally related through the complementary channel construction. This thesis presents a new class of private quantum channels, expanding the existing class beyond a seemingly fundamental restriction. This yields interesting insights into the structure of quantum information and the leaking of information to external environments. Additionally, the duality is only recovered when extending the complementary channel to sufficiently high environmental dimension. \n \nFinally, the error properties of bucket brigade quantum Random Access Memory (qRAM) are assessed. It is determined that using the bucket brigade qRAM architecture for the running of Grover's algorithm necessitates reducing the error rate of the individual components to exponentially small levels for an exponential sized memory. As such, fault-tolerant architectures will likely play an essential role in the construction of such computing primitives.