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.
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.
Quantum computing performance simulators are needed to provide practical metrics for the effectiveness of executing theoretical quantum information processing protocols on physical hardware. In this work, we present a tool to simulate the execution of fault-tolerant quantum computation by automating the tracking of common fault paths for error propagation through an encoded circuit block and quantifying the failure probability of each encoded qubit throughout the circuit. Our simulator runs a fault path counter on encoded circuit blocks to determine the probability that two or more errors remain on the encoded qubits after each block is executed, and it combines errors from all the encoded blocks to estimate performance metrics such as the logical qubit failure probability, the overall circuit failure probability, the number of qubits used, and the time required to run the overall circuit. Our technique efficiently estimates the upper bound of the error probability and provides a useful measure of the error threshold at low error probabilities where conventional Monte Carlo methods are ineffective. We describe a way of simplifying the fault-tolerant measurement process in the Steane code to reduce the number of error correction steps necessary. We present simulation results comparing the execution of quantum adders, which constitute a major part of Shor’s algorithm.
Correcting errors in real time is essential for reliable large-scale quantum computations. Realizing this high-level function requires a system capable of several low-level primitives, including single-qubit and two-qubit operations, mid-circuit measurements of subsets of qubits, real-time processing of measurement outcomes, and the ability to condition subsequent gate operations on those measurements. In this work, we use a ten qubit QCCD trapped-ion quantum computer to encode a single logical qubit using the [[7,1,3]] color code, first proposed by Steane~\cite{steane1996error}. The logical qubit is initialized into the eigenstates of three mutually unbiased bases using an encoding circuit, and we measure an average logical SPAM error of 1.7(6)×10−3, compared to the average physical SPAM error 2.4(8)×10−3 of our qubits. We then perform multiple syndrome measurements on the encoded qubit, using a real-time decoder to determine any necessary corrections that are done either as software updates to the Pauli frame or as physically applied gates. Moreover, these procedures are done repeatedly while maintaining coherence, demonstrating a dynamically protected logical qubit memory. Additionally, we demonstrate non-Clifford qubit operations by encoding a logical magic state with an error rate below the threshold required for magic state distillation. Finally, we present system-level simulations that allow us to identify key hardware upgrades that may enable the system to reach the pseudo-threshold.
Quantum error correction will be integral in developing full scale quantum computers, but as of yet beneficial quantum error correction has not been demonstrated experimentally. An important question is therefore what prerequisites need to be met to achieve this. Results of density matrix simulations of the performance of the seven qubit Steane code in a quantum computing setting are presented. The full density matrix was kept throughout the whole simulations, which means that all errors can be accounted for. In particular, the importance of the circuit depth, i.e. the number of gates in series before error correction is applied, for the overall performance was investigated. It was found that the depth of the circuit has a large impact on the threshold error rate for which error correction becomes beneficial. A gain parameter was defined, which describes the largest constant factor by which errors can be suppressed. It was shown that there is a trade-off between the threshold error and the gain; The highest threshold value was found to be around p_th~10^-4, which is in line with other estimates, but the maximum gain for this value was only 3. To achieve a gain of 100, an error rate of p_err~2x10^-9 is required. In addition, performance statistics such as run-time as a function of circuit depth and width for the matlab code used for simulations are presented.
The ultimate goal of quantum error correction is to create logical qubits with very low error rates (e.g. 1e-12) and assemble them into large-scale quantum computers capable of performing many (e.g. billions) of logical gates on many (e.g. thousands) of logical qubits. However, it is necessarily difficult to directly assess the performance of such high-quality logical qubits using standard Monte Carlo sampling because logical failure events become very rare. Building on existing approaches to this problem, we develop three complementary techniques to characterize the rare-event regime for general quantum low-density parity-check (qLDPC) codes under circuit noise. (I) We propose a well-motivated, low-parameter ansatz for the failure spectrum (the fraction of fault sets of each size that fail) that empirically fits all the QEC systems we studied and predicts logical error rates at all physical error rates. (II) We find min-weight logical operators of syndrome measurement circuits and exactly compute the number of min-weight failing configurations. (III) We generalize the splitting method to qLDPC codes using multi-seeded Metropolis sampling to improve convergence for systems with many inequivalent logical operators. We apply these tools to distance-6, -12, and -18 bivariate bicycle codes under circuit noise, observing strong low-error-rate performance with the recently proposed Relay decoder but also considerable scope for further improvement.
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.
Correcting errors in real time is essential for reliable large-scale quantum computations. Realizing this high-level function requires a system capable of several low-level primitives, including single-qubit and two-qubit operations, midcircuit measurements of subsets of qubits, real-time processing of measurement outcomes, and the ability to condition subsequent gate operations on those measurements. In this work, we use a 10-qubit quantum charge-coupled device trapped-ion quantum computer to encode a single logical qubit using the [[7,1,3]] color code, first proposed by Steane [Phys. Rev. Lett. 77, 793 (1996)PRLTAO0031-900710.1103/PhysRevLett.77.793]. The logical qubit is initialized into the eigenstates of three mutually unbiased bases using an encoding circuit, and we measure an average logical state preparation and measurement (SPAM) error of 1.7(2)×10^{-3}, compared to the average physical SPAM error 2.4(4)×10^{-3} of our qubits. We then perform multiple syndrome measurements on the encoded qubit, using a real-time decoder to determine any necessary corrections that are done either as software updates to the Pauli frame or as physically applied gates. Moreover, these procedures are done repeatedly while maintaining coherence, demonstrating a dynamically protected logical qubit memory. Additionally, we demonstrate non-Clifford qubit operations by encoding a T[over ¯]|+⟩_{L} magic state with an error rate below the threshold required for magic state distillation. Finally, we present system-level simulations that allow us to identify key hardware upgrades that may enable the system to reach the pseudothreshold.