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arXiv:1612.08044

Error tracing in linear and concatenated quantum circuits

Ritajit Majumdar, Saikat Basu, Priyanka Mukhopadhyay, Susmita Sur‐Kolay

Published Dec 23, 2016• 7 citations
arXivHTMLPDF

Abstract

Descriptions of quantum algorithms, communication etc. protocols assume the existence of closed quantum system. However, real life quantum systems are open and are highly sensitive to errors. Hence error correction is of utmost importance if quantum computation is to be carried out in reality. Ideally, an error correction block should be placed after every gate operation in a quantum circuit. This increases the overhead and reduced the speedup of the quantum circuit. Moreover, the error correction blocks themselves may induce errors as the gates used for error correction may be noisy. In this paper, we have proposed a procedure to trace error probability due to noisy gates and decoherence in quantum circuit and place an error correcting block only when the error probability exceeds a certain threshold. This procedure shows a drastic reduction in the required number of error correcting blocks. Furthermore, we have considered concatenated codes with tile structure layout lattice architecture[25][21],[24] and SWAP gate based qubit transport mechanism. Tracing errors in higher levels of concatenation shows that, in most cases, after 1 or 2 levels of concatenation, the number of QECC blocks required become static. However, since the gate count increases with increasing concatenation, the percentage saving in gate count is considerably high.

DOI
10.48550/arxiv.1612.08044

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Descriptions of quantum algorithms, communication etc. protocols assume the existence of closed quantum system. However, real life quantum systems are open and are highly sensitive to errors. Hence error correction is of utmost importance if quantum computation is to be carried out in reality. Ideally, an error correction block should be placed after every gate operation in a quantum circuit. This increases the overhead and reduced the speedup of the quantum circuit. Moreover, the error correction blocks themselves may induce errors as the gates used for error correction may be noisy. In this paper, we have proposed a procedure to trace error probability due to noisy gates and decoherence in quantum circuit and place an error correcting block only when the error probability exceeds a certain threshold. This procedure shows a drastic reduction in the required number of error correcting blocks. Furthermore, we have considered concatenated codes with tile structure layout lattice architecture[25][21],[24] and SWAP gate based qubit transport mechanism. Tracing errors in higher levels of concatenation shows that, in most cases, after 1 or 2 levels of concatenation, the number of QECC blocks required become static. However, since the gate count increases with increasing concatenation, the percentage saving in gate count is considerably high.
One of the most important challenges in constructing large-scale quantum computers is handling the high error rates in quantum systems. Using error correction codes and frequently applying error correction on qubits is a common approach to compete with this challenge. Quantum error analysis techniques are required to compare the effectiveness of error correction codes and also to efficiently insert the error correction procedures (ECPs) in a quantum circuit. However, most existing assessment approaches are either inaccurate and inefficient or are slow and time-consuming. To address these problems, in this paper, a fast analytical approach is proposed that analyzes errors and smartly inserts ECPs into a quantum circuit and finally estimates the failure probability of the circuit. The approach is also able to estimate the pseudo-threshold of quantum error correction codes. Experimental results show that the proposed method improves the number of ECPs and the latency of the circuit encoded by the Steane’s [[7, 1, 3]] code on average by about 52.6% and 29.4% for the one-level one and by about 79.4% and 48.5% for the two-level concatenated code respectively compared with a related method in the literature. Moreover, the proposed method takes only a few seconds to run on even our relatively large benchmark circuits.
Resource overhead problem caused by concatenation in quantum error correction (QEC) is of significant importance for the realization of fault-tolerant quantum computation (FTQC). To attack this problem, we propose a novel scheme by considering integrated FTQC architecture where the concatenation level is controlled dynamically; i.e., less (or more) concatenation levels are imposed by good (or poor) performance gates-we call this scheme "dynamic concatenation" in this sense. Such a dynamic concatenation is realizable in an integrated structure of FTQC, as the information of the concatenation can be communicated between classical system elements (e.g., compiler and system organizer) and the logical qubits in real-time. We derive the effective lower and upper bounds of the length of gate decomposition in order to achieve the practical advantage, namely of reduction of the overall operation time. By considering two non-trivial examples, it is shown that the aforementioned advantage can indeed be achieved in the presented scheme. Our result also provides an important scientific message, i.e., the interplay between "classical" and "quantum" can be helpful in QEC.
In this work, the efficient quantum error-correction protocol against the general independent noise is constructed with the three-qubit codes. The rules of concatenation are summarized according to the error-correcting capability of the codes. The codes not only play the role of correcting errors, but the role of polarizing the effective channel. For any independent noise, the most suitable error-correction protocol can be constructed based on the rules of concatenation. The most significant aspect of using the concatenated three-qubit codes is to realize quantum error-correction with the non-ideal gate operations, because the error-correction schemes of the three-qubit codes are simple and modular. For example, for the amplitude damping noise with the initial channel fidelity 0.9, the effective channel fidelity can reach 0.94731 (or 0.961634 with the ideal quantum gate operations) when using the 4 levels concatenated quantum error-correction with the three-qubit codes. The protocol with the three-qubit codes needs 81 qubits and 366 quantum gate operations when the accuracy rate is 0.9995. Meanwhile, when using the 3 levels concatenated quantum error-correction with the five-qubit code, the effective channel fidelity can only reach 0.922798 (or 0.975488 with the ideal quantum gate operations). The protocol with the five-qubit code needs 125 qubits and 1147 quantum gate operations when the accuracy rate is 0.9995. The physical resources costed is multiple of the physical resources costed by realizing the three-qubit quantum error-correction, the protocol has a higher fault tolerance threshold, and no increase in complexity. So, we believe it will be helpful for realizing quantum error-correction in the physical system.
To solve problems of practical importance1,2, quantum computers probably need to incorporate quantum error correction, in which a logical qubit is redundantly encoded in many noisy physical qubits3–5. The large physical-qubit overhead associated with error correction motivates the search for more hardware-efficient approaches6–18. Here, using a superconducting quantum circuit19, we realize a logical qubit memory formed from the concatenation of encoded bosonic cat qubits with an outer repetition code of distance d = 5 (ref. 10). A stabilizing circuit passively protects cat qubits against bit flips20–24. The repetition code, using ancilla transmons for syndrome measurement, corrects cat qubit phase flips. We study the performance and scaling of the logical qubit memory, finding that the phase-flip correcting repetition code operates below the threshold. The logical bit-flip error is suppressed with increasing cat qubit mean photon number, enabled by our realization of a cat-transmon noise-biased CX gate. The minimum measured logical error per cycle is on average 1.75(2)% for the distance-3 code sections, and 1.65(3)% for the distance-5 code. Despite the increased number of fault locations of the distance-5 code, the high degree of noise bias preserved during error correction enables comparable performance. These results, where the intrinsic error suppression of the bosonic encodings enables us to use a hardware-efficient outer error-correcting code, indicate that concatenated bosonic codes can be a compelling model for reaching fault-tolerant quantum computation. Bosonic qubits can be engineered to feature intrinsic protection against certain kinds of errors, which makes quantum error correction across many bosonic qubits possible with less overhead.
Quantum error correction protocols have been developed to offset the high sensitivity to noise inherent in quantum systems. However, much is still unknown about the behavior of a quantum error-correcting code under general noise, including noisy measurements. This lack of knowledge is largely due to the computational cost of simulating quantum systems large enough to perform nontrivial encodings. In this paper, we develop general methods for incorporating noisy measurement operations into simulations of quantum error-correcting codes and show that measurement errors on readout qubits manifest as a renormalization on the effective logical noise. We also derive general methods for reducing the computational complexity of calculating the exact effective logical noise by many orders of magnitude. This reduction is achieved by determining when different recovery operations produce equivalent logical noise. These methods could also be used to better approximate soft decoding schemes for concatenated codes or to reduce the size of a lookup table to speed up the error correction step in implementations of quantum error-correcting codes. We give examples of such reductions for the three-qubit, five-qubit, Steane, concatenated, and toric codes.
Large-scale quantum computers mandate error correction and fault tolerance. Due to constraints on the number of qubits, fault tolerance is difficult to achieve in near-term quantum systems. Therefore, error correction should require minimal resources. Gates in the near-term devices are also noisy. Quantum error correction code blocks built with these noisy gates can inject further error in the circuit. The goals for error correction in near-term systems are as follows: (i) using a small number of qubits for encoding, and (ii) keeping cost of circuits for encoding and decoding low. In this paper, we propose two techniques to achieve these mutually orthogonal goals. For a binary quantum system we propose an error estimation method that can aid in reducing the number of error correcting blocks via sparse scheduling. For ternary quantum systems, we propose an approximate code that can correct errors with high probability while significantly reducing the circuit cost. These techniques are expected to be helpful for error mitigation in near-term systems in the absence of fault tolerance.