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arXiv:2407.20976

An iterative transversal CNOT decoder

Kwok Ho Wan, Mark Webber, Austin G. Fowler, W. K. Hensinger

Published Jul 30, 2024• 0 citations
arXivHTMLPDF

Abstract

Modern platforms for potential qubit candidates, such as trapped ions or neutral atoms, allow long range connectivity between distant physical qubits through shuttling. This opens up an avenue for transversal logical CNOT gates between distant logical qubits, whereby physical CNOT gates are performed between each corresponding physical qubit on the control and target logical qubits. However, the transversal CNOT can propagate errors from one logical qubit to another, leading to correlated errors between logical qubits. We have developed a multi-pass iterative decoder that decodes each logical qubit separately to deal with this correlated error. We show that under circuit-level noise and only $\mathcal{O}(1)$ code cycles, a threshold can still persist, and the logical error rate will not be significantly degraded, matching the sub-threshold logical error rate scaling of $p^{\lfloor\frac{d}{2}\rfloor}$ for a distance $d$ rotated surface code.

DOI
10.48550/arxiv.2407.20976

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Modern platforms for potential qubit candidates, such as trapped ions or neutral atoms, allow long range connectivity between distant physical qubits through shuttling. This opens up an avenue for transversal logical CNOT gates between distant logical qubits, whereby physical CNOT gates are performed between each corresponding physical qubit on the control and target logical qubits. However, the transversal CNOT can propagate errors from one logical qubit to another, leading to correlated errors between logical qubits. We have developed a multi-pass iterative decoder that decodes each logical qubit separately to deal with this correlated error. We show that under circuit-level noise and only O(1)\mathcal{O}(1)O(1) code cycles, a threshold can still persist, and the logical error rate will not be significantly degraded, matching the sub-threshold logical error rate scaling of p⌊d2⌋p^{\lfloor\frac{d}{2}\rfloor}p⌊2d​⌋ for a distance ddd rotated surface code.
A scalable and programmable quantum computer holds the potential to solve computationally intensive tasks that classical computers cannot accomplish within a reasonable time frame, achieving quantum advantage. However, the vulnerability of the current generation of quantum processors to errors poses a significant challenge towards executing complex and deep quantum circuits required for practical problems. Quantum error-correction codes such as stabilizer codes offer a promising path forward for fault-tolerant quantum computing; however, their realization on quantum hardware is an on-going area of research. In particular, fault-tolerant quantum processing must employ logical gates on logical qubits with error suppression with realistically large size codes. This work has implemented a transversal controlled-not (cnot) gate between two logical qubits constructed using the repetition code with flag qubits, and demonstrated error suppression with increasing code size under multiple rounds of error detection and correction. Our results conducted on IBM quantum devices through cloud access show that despite the potential for error propagation among logical qubits during the transversal cnot gate operation, scaling the code distance as we increase the number of physical qubits from 21 to 39 and 57 can suppress errors, which persists over ten rounds of error detection and correction. Our work establishes the feasibility of employing logical cnot gates alongside error detection on a superconductor-based processor using current generation quantum hardware.
Recent experimental advances have made it possible to implement logical multiqubit transversal gates on surface codes in a multitude of platforms. A transversal controlled- (t) gate on two surface codes introduces correlated errors across the code blocks and thus requires modified decoding compared to established methods of decoding surface-code quantum memory (SCQM) or lattice-surgery operations. In this work, we examine and benchmark the performance of three different decoding strategies for the t for scalable fault-tolerant quantum computation. In particular, we present a low-complexity decoder based on minimum-weight perfect matching (MWPM) that achieves the same threshold as the SCQM MWPM decoder. We extend our analysis with a study of tailored decoding of a transversal-teleportation circuit, along with a comparison between the performance of lattice-surgery and transversal operations under Pauli- and erasure-noise models. Our investigation builds toward systematic estimation of the cost of implementing large-scale quantum algorithms based on transversal gates in the surface code.
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To date, the surface code has become a promising candidate for quantum error correcting codes because of its high threshold, nearest gate operations and low weight stabilizers. Here, we assume that the number of CNOT gates is restricted because of the quantum crosstalk due to residual inter-qubit coupling or max channels per pulse gate. We thus calculated the effective physical error rate in this case and determine the surface code distance accordingly. The optimal code distance was obtained based on the logical error rate performance estimation of the surface code for some T1and T2time.