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International Conference on Parallel and Distributed Computing, Applications and Technologies (PDCAT)

Efficient Fault-Tolerant Syndrome Measurement of Quantum Error-Correcting Codes Based on "Flag"

QiFei Wei, Dongxiao Quan, Jing Liu, Changxing Pei

Published Dec 1, 2019• 0 citations
Publisher

Abstract

Fault-tolerant syndrome measurement plays an important role in the process of quantum error correction, and considerable effort had been taken for reducing the physical overhead of syndrome measurement which including the ancilla qubits, the CNOT gates and the time-slots. The two extra qubits syndrome measurement technique, known as "flag"-style syndrome measurement, cuts down the number of extra qubits to the utmost. However, it works slowly because it measures only one syndrome at a time. We extend the technique to extract all syndromes of the distance-3 quantum error-correcting code at once. We propose a new method that increases the parallelism of the syndrome measurement circuit and reduces the time overhead by allocating and adjusting the order of CNOT gates for measuring data block reasonably, which we call dynamic time-slot allocation scheme, and which is applicable to both Hamming codes and color codes. For a CSS quantum error-correcting code with the number of stabilizers m and the maximum weight w, we only need 2m extra qubits and 2 × (w+2) time-slots for one-shot measurement of all syndromes.

DOI
10.1109/pdcat46702.2019.00041

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Fault-tolerant syndrome measurement plays an important role in the process of quantum error correction, and considerable effort had been taken for reducing the physical overhead of syndrome measurement which including the ancilla qubits, the CNOT gates and the time-slots. The two extra qubits syndrome measurement technique, known as "flag"-style syndrome measurement, cuts down the number of extra qubits to the utmost. However, it works slowly because it measures only one syndrome at a time. We extend the technique to extract all syndromes of the distance-3 quantum error-correcting code at once. We propose a new method that increases the parallelism of the syndrome measurement circuit and reduces the time overhead by allocating and adjusting the order of CNOT gates for measuring data block reasonably, which we call dynamic time-slot allocation scheme, and which is applicable to both Hamming codes and color codes. For a CSS quantum error-correcting code with the number of stabilizers m and the maximum weight w, we only need 2m extra qubits and 2 × (w+2) time-slots for one-shot measurement of all syndromes.
Reliable quantum computation requires fault-tolerant protocols to prevent errors from propagating during syndrome extraction in quantum error correction. We present a novel fault-tolerant syndrome extraction technique for CSS codes, which we refer to as the cut-cat state scheme. While each ancilla qubit interacts non-fault-tolerantly with a pair of data qubits, we introduce additional cat stabilizer measurements to identify and correct the resulting hook errors. Our approach maintains the key benefit of cat-based extraction, i.e., parallelized data qubit interactions, while reducing the number of simultaneous qubits required by more than half. Compared to flag-based state-of-the-art protocols, the cut-cat scheme offers a notable advantage in terms of two-qubit gate count as the code distance increases.
Abstract Steane’s seven-qubit quantum code is a natural choice for fault-tolerance experiments because it is small and just two extra qubits are enough to correct errors. However, the two-qubit error-correction technique, known as ‘flagged’ syndrome extraction, works slowly, measuring only one syndrome at a time. This is a disadvantage in experiments with high qubit rest error rates. We extend the technique to extract multiple syndromes at once, without needing more qubits. Qubits for different syndromes can flag errors in each other. This gives equally fast and more qubit-efficient alternatives to Steane’s error-correction method, and also conforms to planar geometry constraints. We further show that Steane’s code and some others can be error-corrected with no extra qubits, provided there are at least two code blocks. The rough idea is that two seven-qubit codewords can be temporarily joined into a twelve-qubit code, freeing two qubits for flagged syndrome measurement.
Given that quantum error correction processes are unreliable, an efficient error syndrome extraction circuit should use fewer ancillary qubits, quantum gates, and measurements, while maintaining low circuit depth, to minimizing the circuit area, roughly defined as the product of circuit depth and the number of physical qubits. We propose to design parallel flagged syndrome extraction with shared flag qubits for quantum stabilizer codes. Versatile parallelization techniques are employed to minimize the required circuit area, thereby improving the error threshold and overall performance. Specifically, measurement outcomes across multiple rounds of syndrome extraction are integrated into a lookup table decoder, enabling parallelization of multiple stabilizer measurements with shared flag qubits. Additionally, we introduce an adaptive technique to reduce the overhead from excessive syndrome extraction. We present flag-sharing and fully parallel schemes for the[ ⁣[17,1,5] ⁣][\![17,1,5]\!][[17,1,5]],[ ⁣[19,1,5] ⁣][\![19,1,5]\!][[19,1,5]]Calderbank-Shor-Steane (CSS) codes and the[ ⁣[5,1,3] ⁣][\![5,1,3]\!][[5,1,3]]non-CSS code, where the[ ⁣[5,1,3] ⁣][\![5,1,3]\!][[5,1,3]]implementation achieves the minimum known circuit area. Numerical simulations have demonstrated improved pseudo-thresholds for these codes by up to an order of magnitude compared to previous schemes in the literature.
We reduce the extra qubits needed for two fault-tolerant quantum computing protocols: error correction, specifically syndrome bit measurement, and cat state preparation. For distance-three fault-tolerant syndrome extraction, we show an exponential reduction in qubit overhead over the previous best protocol. For a weight- w stabilizer, we demonstrate that stabilizer measurement tolerating one fault needs at most ⌈log2⁡w⌉+1 ancilla qubits. If qubits reset quickly, four ancillas suffice. We also study the preparation of entangled cat states, and prove that the overhead for distance-three fault tolerance is logarithmic in the cat state size. These results apply both to near-term experiments with a few qubits, and to the general study of the asymptotic resource requirements of syndrome measurement and state preparation.With a flag qubits, previous methods use O(a) flag patterns to identify faults. In order to use the same flag qubits more efficiently, we show how to use nearly all 2a possible flag patterns, by constructing maximal-length paths through the a -dimensional hypercube.
Fault-tolerant quantum error correction (FTEC) protocol is one of the most important components for a fault-tolerant simulation of a quantum circuit. It helps prevent error accumulation during the process which may cause the computation to fail. In a physical implementation of fault-tolerant circuits, the number of required ancilla qubits for the FTEC protocol can be a major concern because it grows exponentially as the level of concatenation increases whereas the resources are limited. Recently, the idea of using flag qubits to detect high-weight errors caused by fewer faults in syndrome extraction circuits has been proposed. The use of flag qubits allows the construction of fault-tolerant protocols with the fewest number of ancillas known to-date. In this work, some critical properties of CSS codes constructed from classical cyclic codes that enable the construction of a flag fault-tolerant error correction scheme are proved, and a fault-tolerant protocol along with a family of circuits for flag fault-tolerant error correction are given. The flag-FTEC protocol proposed in this work requires only four ancilla qubits and applicable to CSS codes of distance 3 constructed from classical cyclic codes.
We optimize fault-tolerant quantum error correction to reduce the number of syndrome bit measurements. Speeding up error correction will also speed up an encoded quantum computation, and should reduce its effective error rate. We give both code-specific and general methods, using a variety of techniques and in a variety of settings. We design new quantum error-correcting codes specifically for efficient error correction, e.g., allowing single-shot error correction. For codes with multiple logical qubits, we give methods for combining error correction with partial logical measurements. There are tradeoffs in choosing a code and error-correction technique. While to date most work has concentrated on optimizing the syndrome-extraction procedure, we show that there are also substantial benefits to optimizing how the measured syndromes are chosen and used. As an example, we design single-shot measurement sequences for fault-tolerant quantum error correction with the 16-qubit extended Hamming code. Our scheme uses 10 syndrome bit measurements, compared to 40 measurements with the Shor scheme. We design single-shot logical measurements as well: any logical Z measurement can be made together with fault-tolerant error correction using only 11 measurements. For comparison, using the Shor scheme a basic implementation of such a non-destructive logical measurement uses 63 measurements. We also offer ten open problems, the solutions of which could lead to substantial improvements of fault-tolerant error correction.