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arXiv:2207.13356 · Physical Review A

Realizing a class of stabilizer quantum error correction codes using a single ancilla and circular connectivity

A. V. Antipov, Evgeniy O. Kiktenko, Aleksey K. Fedorov

Published Mar 3, 2023• 7 citations
arXivHTMLPDF

Abstract

We describe a class of neighboring-block stabilizer quantum error correction codes and demonstrate that such a class of codes can be implemented in a resource-efficient manner using a single ancilla and circular near-neighbor qubit connectivity. We propose an implementation for syndrome-measurement circuits for codes from the class and illustrate its workings for cases of three-qubit repetition code, Laflamme's five-qubit code, and Shor's nine-qubit code. For three-qubit repetition code and Laflamme's five-qubit code, the suggested scheme has the property that it uses only native two-qubit controlled-not-swap gates, which potentially reduces the amount of noncorrectable errors due to the shorter gate time. Elements of the scheme can be used to implement surface code with near-neighbor connectivity using a single ancilla, as demonstrated in an example. We developed efficient decoding procedures for repetition codes and the Laflamme's five-qubit code using a minimum-weight perfect-matching approach to account for the specific order of measurements in our scheme. The analysis of noise levels for which the scheme could show improvements in the fidelity of a stored logical qubit in the three-qubit repetition code and Laflamme's five-qubit code cases is provided. We complement our results by realizing the developed scheme for a three-qubit code using an IBM quantum processor and the Laflamme's five-qubit code using the state-vector simulator.

DOI
10.1103/physreva.107.032403

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We describe a class of neighboring-block stabilizer quantum error correction codes and demonstrate that such a class of codes can be implemented in a resource-efficient manner using a single ancilla and circular near-neighbor qubit connectivity. We propose an implementation for syndrome-measurement circuits for codes from the class and illustrate its workings for cases of three-qubit repetition code, Laflamme's five-qubit code, and Shor's nine-qubit code. For three-qubit repetition code and Laflamme's five-qubit code, the suggested scheme has the property that it uses only native two-qubit controlled-not-swap gates, which potentially reduces the amount of noncorrectable errors due to the shorter gate time. Elements of the scheme can be used to implement surface code with near-neighbor connectivity using a single ancilla, as demonstrated in an example. We developed efficient decoding procedures for repetition codes and the Laflamme's five-qubit code using a minimum-weight perfect-matching approach to account for the specific order of measurements in our scheme. The analysis of noise levels for which the scheme could show improvements in the fidelity of a stored logical qubit in the three-qubit repetition code and Laflamme's five-qubit code cases is provided. We complement our results by realizing the developed scheme for a three-qubit code using an IBM quantum processor and the Laflamme's five-qubit code using the state-vector simulator.
Recently, the circular connectivity pattern has been presented for a class of stabilizer quantum error correction codes. The circular connectivity pattern for such a class of stabilizer codes can be implemented in a resource-efficient manner using a single ancilla and native two-qubit Controlled-Not-Swap gates (CNS) gates, which may be interesting for demonstrating error-correction codes with superconducting quantum processors. However, one concern is that this scheme is not fault-tolerant. And it might not apply to the Calderbank-Shor-Steane (CSS) codes. In this paper, we present a fault-tolerant version of the circular connectivity pattern, named the double-circular connectivity pattern. This pattern is an implementation for syndrome-measurement circuits with a flagged error correction scheme for stabilizer codes. We illustrate that this pattern is available for Steane code (a CSS code), Laflamme’s five-qubit code, and Shor’s nine-qubit code. For Laflamme’s five-qubit code and Shor’s nine-qubit code, the pattern has the property that it uses only native two-qubit CNS gates, which are more efficient in the superconducting quantum platform.
Experimental realization of stabilizer-based quantum error correction (QEC) codes that would yield superior logical qubit performance is one of the formidable task for state-of-the-art quantum processors. A major obstacle toward realizing this goal is the large footprint of QEC codes, even those with a small distance. We propose a circuit based on the minimal distance-3 QEC code, which requires only five data qubits and five ancilla qubits, connected in a ring with iswap gates implemented between neighboring qubits. Using a density-matrix simulation, we show that, thanks to its smaller footprint, the proposed code has a lower logical error rate than surface-17 for similar physical error rates. We also estimate the performance of a neural-network-based error decoder, which can be trained to accommodate the error statistics of a specific quantum processor by training on experimental data.
The five-qubit quantum error correcting code encodes one logical qubit to five physical qubits, and protects the code from a single error. It was one of the first quantum codes to be invented, and various encoding circuits have been proposed for it. In this paper, we propose a systematic procedure for optimization of encoder circuits for stabilizer codes. We start with the systematic construction of an encoder for a five-qubit code, and optimize the circuit in terms of the number of quantum gates. Our method is also applicable to larger stabilizer codes. We further propose nearest neighbor compliant (NNC) circuits for the proposed encoder using a single swap gate, as compared to three swap gates in a prior design.
Building reliable quantum computers requires protecting fragile quantum states from inevitable environmental noise and operational errors. While quantum error correction codes like the Steane [ ⁣[7,1,3] ⁣][\![7,1,3]\!][[7,1,3]] code provide elegant theoretical solutions, their practical success hinges critically on how we measure errors - a process called syndrome extraction. The challenge lies in the ancilla qubits used for measurement: when they fail, errors can cascade across the entire quantum system, destroying the very information we're trying to protect. We address this fundamental problem by implementing and comparing three sophisticated syndrome measurement strategies: Shor's cat-state approach, which distributes measurements across multiple entangled ancillas achieving 85-92% preparation success; Steane's encoded-ancilla method using complete error-corrected logical qubits reaching 97.8% syndrome fidelity; and a flexible unified framework that adapts strategies based on hardware capabilities. Through extensive simulations using IBM's Qiskit platform spanning randomized benchmarking and T-heavy circuits, we demonstrate that intelligent ancilla management improves error suppression by up to 2.4×\times× compared to standard approaches. Our implementations achieve logical error rates as low as 5.1×10−55.1 \times 10^{-5}5.1×10−5 under realistic noise conditions with physical error rates of 10−310^{-3}10−3, while maintaining near-unity logical fidelity (0.99997) even for deep circuits. The threshold analysis reveals robust performance across distance-3 to distance-13 codes with characteristic threshold curves showing exponential error suppression below the critical physical error rate. These results provide immediately deployable tools for near-term quantum devices and establish practical design principles for scaling toward fault-tolerant quantum computers.
We realize Surface Code quantum memories for nearest-neighbor qubits with always-on Ising interactions. This is done by utilizing multi-qubit gates that mimic the functionality of several gates. The previously proposed Surface Code memories rely on error syndrome detection circuits based on CNOT gates. In a two-dimensional planar architecture, to realize a two-qubit CNOT gate in the presence of couplings to other neighboring qubits, the interaction of the target qubit with its three other neighbors must cancel out. Here we present a new error syndrome detection circuit utilizing multi-qubit parity gates. In addition to speed up in the error correction cycles, in our approach, the depth of the error syndrome detection circuit does not grow by increasing the number of qubits in the logical qubit layout. We analytically design the system parameters to realize new five-qubit gates suitable for error syndrome detection in nearest-neighbor two-dimensional array of qubits. The five-qubit gates are designed such that the middle qubit is the target qubit and all four coupled neighbors are the control qubits. In our scheme, only one control parameter of the target qubits must be adjusted to realize controlled-unitary operations. The gate operations are confirmed with a fidelity of >99.9% in a simulated system consists of nine nearest-neighbor qubits.
Quantum error correction is an important ingredient for scalable quantum computing. Stabilizer codes are one of the most promising and straightforward ways to correct quantum errors, are convenient for logical operations, and improve performance with increasing the number of qubits involved. Here, we propose a resource-efficient scaling of a five-qubit perfect code with increasing-weight cyclic stabilizers for small distances on the ring architecture, which takes into account the topological features of the superconducting platform. We show an approach to construct the quantum circuit of a correction code with ancillas entangled with non-neighboring data qubits. Furthermore, we introduce a neural network-based decoding algorithm supported by an improved lookup table decoder and provide a numerical simulation of the proposed code, which demonstrates the exponential suppression of the logical error rate.