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arXiv:1609.07497 · Physical Review A

Overhead analysis of universal concatenated quantum codes

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

Published Feb 8, 2017• 50 citations
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Abstract

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 $T$ 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.

DOI
10.1103/physreva.95.022313

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arXiv:1610.03309•Oct 11, 2016•1 citation
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×\else\texttimes\fi10\ensuremath−6.8.33\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}.8.33\ifmmode×\else\texttimes\fi10\ensuremath−6. We obtain a depolarizing noise threshold for the 49-qubit code of 9.69\ifmmode×\else\texttimes\fi10\ensuremath−49.69\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}9.69\ifmmode×\else\texttimes\fi10\ensuremath−4 which is competitive with the 105-qubit threshold result of 1.28\ifmmode×\else\texttimes\fi10\ensuremath−31.28\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}1.28\ifmmode×\else\texttimes\fi10\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.
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.
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.
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.
As existing approaches to get around the restriction of the no-go theorem generally exhibit high ancillary qubit overhead costs, we propose a scheme for universal fault-tolerant quantum computation by using the pieceable fault-tolerant protocol and code switching techniques. In additional, by utilizing the construction of the pieceable fault-tolerant circuit, we adopt a decoding strategy based on a deep-neural-network algorithm to improve the error threshold of the non-Clifford logical gate circuit. We describe our universal construction in detail with a two-level nonuniform concatenated 25-qubit code and perform numerical simulations to analyze the depolarizing noise threshold of a given universal gate set with this code. The resources required to implement this universal gate set are also estimated to further demonstrate the efficiency of our scheme. We compare these results with the existing universal concatenation methods and conclude that our method outperforms them in terms of the lower bound of the error threshold and qubit resource overhead.
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.
As there is no quantum error correction code with universal set of transversal gates, several approaches have been proposed which, in combination of transversal gates, make universal fault-tolerant quantum computation possible. Magic state distillation, code switching, code concatenation and pieceable fault-tolerance are well-known examples of such approaches. However, the overhead of these approaches is one of the main bottlenecks for large-scale quantum computation. In this paper, a hybrid approach is proposed which combines the code concatenation technique with the other mentioned approaches. The proposed approach outperforms code concatenation in terms of both number of qubits and error threshold and also significantly reduces the resource overhead of code switching, magic state distillation and pieceable fault-tolerance at the cost of reducing the effective distance of the concatenated code for implementing non-transversal gates.