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IEEE VTS Asia Pacific Wireless Communications Symposium (APWCS)

Analysis of surface code under the restricted number of CNOT gates

Jonghyun Lee, Jinyoung Ha, Yujin Kang, Jun Heo

Published Aug 24, 2022• 0 citations
Publisher

Abstract

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.

DOI
10.1109/apwcs55727.2022.9906509

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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.
Abstract To date, the surface code has become a promising candidate for quantum error correcting codes because it achieves a high threshold and is composed of only the nearest gate operations and low-weight stabilizers. Here, we have exhibited that the logical failure rate can be enhanced by manipulating the lattice size of surface codes that they can show an enormous improvement in the number of physical qubits for a noise model where dephasing errors dominate over relaxation errors. We estimated the logical error rate in terms of the lattice size and physical error rate. When the physical error rate was high, the parameter estimation method was applied, and when it was low, the most frequently occurring logical error cases were considered. By using the minimum weight perfect matching decoding algorithm, we obtained the optimal lattice size by minimizing the number of qubits to achieve the required failure rates when physical error rates and bias are provided .
As current experiments already realize small quantum circuits on error corrected qubits, it is important to fully understand the effect of physical errors on the logical error channels of these fault-tolerant circuits. Here, we investigate a lattice-surgery-based CNOT operation between two surface code patches under phenomenological error models. (i) For two-qubit logical Pauli measurements – the elementary building block of the CNOT – we optimize the number of stabilizer measurement rounds, usually taken equal to d , the size (code distance) of each patch. We find that the optimal number can be greater or smaller than d , depending on the rate of physical and readout errors, and the separation between the code patches. (ii) We fully characterize the two-qubit logical error channel of the lattice-surgery-based CNOT. We find a symmetry of the CNOT protocol, that results in a symmetry of the logical error channel. We also find that correlations between X and Z errors on the logical level are suppressed under minimum weight decoding.
The surface code is one of the leading quantum error correction codes for realizing large-scale fault-tolerant quantum computing (FTQC). One major challenge in realizing surface-code-based FTQC is the extremely large number of qubits required. To mitigate this problem, fusing multiple codewords of the surface code into a densely packed configuration has been proposed. It is known that by using dense packing, the number of physical qubits required per logical qubit can be reduced to approximately three-fourths compared to simply placing surface-code patches side by side. Despite its potential, concrete deformation procedures and quantitative error-rate analyses have remained largely unexplored. In this work, we present a detailed code-deformation procedure that transforms multiple standard surface code patches into a densely packed, connected configuration, along with a conceptual microarchitecture to utilize this dense packing. We also propose a CNOT gate-scheduling for stabilizer measurement circuits that suppresses hook errors in the densely packed surface code. We performed circuit-level Monte Carlo noise simulation of densely packed surface codes using this gate scheduling. The numerical results demonstrate that as the code distance of the densely packed surface code increases and the physical error rate decreases, the logical error rate of the densely packed surface code becomes lower than that of the standard surface code. Furthermore, we find that only when employing hook-error-avoiding syndrome extraction can the densely packed surface code achieve a lower logical error rate than the standard surface code, while simultaneously reducing the space overhead.
Practical quantum computers will require resource-efficient error-correcting codes. The rotated surface code uses approximately half the number of qubits as the unrotated surface code to create a logical qubit with the same error-correcting distance. However, instead of distance, a more useful qubit-saving metric would be based on logical error rates. In this work we find the well-below-threshold scaling of logical to physical error rates under circuit-level noise for both codes at high odd and even distances and then compare the number of qubits used by each code to achieve equal logical error rates. We perform Monte Carlo sampling of memory experiment circuits with all valid CNOT orders using the stabilizer simulator Stim and the uncorrelated minimum-weight perfect matching decoder PyMatching 2. We find that the rotated code uses about 74 % the number of qubits used by the unrotated code to achieve a logical error rate of p L = 10 − 12 at the operational physical error rate of p = 10 − 3 . The ratio remains ≈ 75 % for p values within a factor of two of p = 10 − 3 for all useful p L . Our work finds the low- p L scaling of the surface code and clarifies the qubit savings provided by the rotated surface code, providing numerical justification for its use in future implementations of the surface code.
The field of quantum computation currently lacks a formal proof of experimental feasibility. Qubits are fragile and sophisticated quantum error correction is required to achieve reliable quantum computation. The surface code is a promising quantum error correction code, requiring only a physically reasonable 2D lattice of qubits with nearest neighbor interactions. However, existing proofs that reliable quantum computation is possible using this code assume the ability to measure four-body operators and, despite making this difficult to realize assumption, require that the error rate of these operator measurements is less than 10(-9), an unphysically low target. High error rates have been proved tolerable only when assuming tunable interactions of strength and error rate independent of distance, which is also unphysical. In this work, given a 2D lattice of qubits with only nearest neighbor two-qubit gates, and single-qubit measurement, initialization, and unitary gates, all of which have error rate p, we prove that arbitrarily reliable quantum computation is possible provided p < 7.4 × 10(-4), a target that many experiments have already achieved. This closes a long-standing open problem, formally proving the experimental feasibility of quantum computation under physically reasonable assumptions.
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.