[Paper Review] Hexagonal matching codes with 2-body measurements

James R. Wootton|arXiv (Cornell University)|Sep 27, 2021

Semiconductor materials and devices4 citations

TL;DR

This paper presents a method to measure stabilizers in hexagonal matching codes—designed for IBM Quantum's heavy-hexagonal hardware—using only native two-body measurements, enabling scalable error detection. Experiments on 27- and 65-qubit devices show results consistent with a noise model of 1.5%–2% error probability, validating hardware performance and demonstrating compatibility with surface-code-style error correction principles.

ABSTRACT

Matching codes are stabilizer codes based on Kitaev's honeycomb lattice model. The hexagonal form of these codes are particularly well-suited to the heavy-hexagon device layouts currently pursued in the hardware of IBM Quantum. Here we show how the stabilizers of the code can be measured solely through the 2-body measurements that are native to the architecture. The process is then run on 27 and 65 qubit devices, to compare results with simulations for a standard error model. It is found that the results correspond well to simulations where the noise strength is similar to that found in the benchmarking of the devices. The best devices show results consistent with a noise model with an error probability of around $1.5\%-2\%$.

Motivation & Objective

• To implement stabilizer measurements in hexagonal matching codes using only two-body measurements native to IBM Quantum's heavy-hexagonal architecture.
• To benchmark current quantum hardware by comparing experimental stabilizer measurement outcomes with simulations under a standard error model.
• To assess the effective noise level in real devices by comparing observed logical syndrome probabilities with simulated results across varying noise strengths.
• To evaluate the scalability and reliability of stabilizer measurement protocols in large-scale quantum error correction experiments.
• To lay the groundwork for future logical qubit implementation by validating measurement fidelity and identifying hardware noise characteristics.

Proposed method

• Uses link operators (X, Y, Z-type) on edges of a hexagonal lattice to define stabilizer generators, with each link measured via a controlled-Z entangling operation between two qubits and an auxiliary ancilla qubit.
• Employs a two-group decomposition of plaquette operators into V_a and V_b, where each group’s product is measured sequentially using individual link measurements summed modulo 2.
• Leverages commutativity of V_a and V_b with each other and with the full plaquette operator to ensure measurement consistency and error-free deduction of the plaquette outcome.
• Applies a syndrome measurement protocol that splits the 6-body plaquette measurement into two 3-body measurements (V_a and V_b), each realizable via two-body interactions.
• Uses a standard depolarizing noise model in simulations and compares simulated logical syndrome probabilities (⟨p_W⟩ and ⟨p_Z⟩) with experimental results from IBM Quantum devices.
• Analyzes results using two figures of merit: ⟨p_W⟩ for plaquette measurements and ⟨p_Z⟩ for Z-link measurements, both normalized to converge at 0.5 under high noise.

Experimental results

Research questions

• RQ1 Can stabilizer measurements in hexagonal matching codes be fully implemented using only two-body measurements native to IBM Quantum’s heavy-hexagonal architecture?
• RQ2 How do experimental stabilizer measurement outcomes on real 27- and 65-qubit devices compare to simulations under a standard depolarizing noise model?
• RQ3 What is the effective noise strength in current IBM Quantum devices as inferred from the agreement between experimental and simulated syndrome probabilities?
• RQ4 To what extent do measurement errors in the hardware affect the logical syndrome readout, and how does this vary across different devices?
• RQ5 Can the observed convergence of ⟨p_W⟩ and ⟨p_Z⟩ to 0.5 in high-noise devices be used as a diagnostic for effective error rates in real hardware?

Key findings

• The stabilizer measurement protocol successfully implements all required measurements using only native two-body operations, enabling compatibility with IBM Quantum’s current hardware architecture.
• For the best-performing device, ibm_cairo, the experimental results align with simulations at a noise strength of approximately 1.5%, consistent with its reported average error rate of 1.47%.
• For ibm_hanoi, the results best match simulations at a noise strength of 2%, slightly higher than its average error rate of 1.55%, likely due to high error rate variation across qubits.
• The noisiest devices—ibmq_manhattan, ibmq_toronto, ibmq_brooklyn, and ibmq_montreal—show syndrome probabilities ⟨p_W⟩ ≈ ⟨p_Z⟩ ≈ 0.5, indicating convergence consistent with noise strengths above 4%.
• The results for ⟨p_W⟩ consistently exceed those for ⟨p_Z⟩, reflecting longer time intervals between plaquette and Z-link measurements, leading to higher error accumulation in plaquette readouts.
• The study demonstrates that large-scale stabilizer measurement protocols (up to 16 measurements per round on Hummingbird devices) are feasible on current hardware, marking one of the largest quantum error correction experiments to date.