Scaling and logic in the color code on a superconducting quantum processor

1. Problem Statement

Quantum error correction (QEC) is essential for bridging the gap between the high error rates of physical qubits and the low logical error rates required for fault-tolerant quantum algorithms. While recent demonstrations on superconducting processors have focused on the surface code due to its high error threshold and compatibility with planar architectures, the surface code has significant limitations:

• High Qubit Overhead: It requires a large number of physical qubits to encode a single logical qubit.
• Inefficient Logical Operations: Implementing logical gates often requires complex lattice surgery or code deformation, which increases circuit depth and error accumulation.

The color code offers a promising alternative. It enables more efficient logical operations (specifically transversal Clifford gates) and requires fewer qubits for a fixed code distance. However, color codes have historically been difficult to implement on superconducting devices because:

• They require higher-weight stabilizer measurements (weight-6 vs. weight-4 in surface codes).
• They traditionally require non-local connectivity (beyond nearest neighbors), which is incompatible with standard planar superconducting architectures.
• No prior experiment had demonstrated error suppression (scaling) with the color code on any platform.

2. Methodology

The authors implemented a comprehensive demonstration of the color code on Google's 72-qubit "Willow" superconducting processor.

A. Hardware and Architecture

• Device: A 72-qubit superconducting processor with nearest-neighbor connectivity on a square grid.
• Code Design: They utilized a superdense color code based on a hexagonal lattice embedded in a square grid. This design allows for nearest-neighbor interactions only.
• Syndrome Extraction: They employed a "superdense" syndrome extraction circuit. Instead of separate rounds for X and Z stabilizers, they used two auxiliary qubits (X and Z) per tile prepared in a Bell pair. These interacted with three data qubits via CNOT gates to simultaneously measure both X-type and Z-type stabilizers in a single cycle.
• Layout Optimization: To accommodate the processor's readout constraints (where qubits on the same line are measured simultaneously), they swapped the roles of specific data and auxiliary qubits. This ensured that each readout line contained only data qubits or only auxiliary qubits, avoiding measurement-induced dephasing of mixed types.

B. Experimental Protocols

The study focused on four key pillars of fault-tolerant quantum computation:

1. Distance Scaling (Memory): Comparing logical error rates between a distance-3 ($d=3$) and distance-5 ($d=5$) code to demonstrate error suppression.
2. Logical Randomized Benchmarking (LRB): Characterizing the error rates of transversal logical Clifford gates interleaved with error correction cycles.
3. Magic State Injection: Injecting non-Clifford resource states (required for universal computation) into the logical code with high fidelity.
4. State Teleportation: Using lattice surgery to perform multi-qubit operations (specifically an MXX parity measurement) to teleport logical states between two $d=3$ logical qubits.

C. Decoding Strategies

Due to the complex connectivity of the color code (where a single error can flip three syndrome elements, creating "hyper-edges"), standard matching decoders are insufficient. The team employed three advanced decoders:

• Chromobius (Möbius Decoder): A fast, matching-based decoder using a Möbius strip topology.
• MLE (Most Likely Error): A slower, optimization-based decoder using mixed-integer programming.
• AlphaQubit (Neural Network): A recurrent neural network decoder trained on synthetic and experimental data, which provided the best performance.

3. Key Contributions

1. First Demonstration of Scaling: This is the first experimental demonstration of error suppression by increasing the code distance for the color code on any platform.
2. Superdense Syndrome Extraction: Successfully implemented a circuit that measures X and Z stabilizers simultaneously using Bell pairs, reducing the time overhead compared to sequential measurements.
3. Transversal Gate Efficiency: Demonstrated that transversal logical Clifford gates add negligible error compared to idle error correction cycles.
4. High-Fidelity Magic States: Achieved magic state injection fidelities exceeding 99% (with post-selection), surpassing the distillation threshold.
5. Lattice Surgery on Color Codes: Successfully demonstrated logical state teleportation between two logical qubits using lattice surgery, proving the feasibility of multi-qubit logical operations.

4. Key Results

• Error Suppression (Scaling):

  • Scaling the code distance from $d=3$ to $d=5$ suppressed logical errors by a factor of $\Lambda_{3/5} = 1.56(4)$.
  • The logical error per cycle for $d=5$ was $\epsilon_5 = 0.0110(2)$.
  • Simulations indicate the system is operating below the color code's noise threshold. With modest improvements in physical error rates (approx. 4x), the color code is predicted to become more qubit-efficient than the surface code.

• Logical Gate Performance:

  • Using Logical Randomized Benchmarking, the average error added by a transversal logical Clifford gate was found to be 0.0027(3).
  • This is significantly lower than the logical error per cycle ($\approx 0.017$), confirming that transversal gates are highly efficient and do not spread errors.

• Magic State Injection:

  • Injected arbitrary states and specific magic states ($|A\rangle, |H\rangle, |T\rangle$).
  • Achieved post-selection infidelities as low as 0.0008 (for $|A\rangle$) while retaining ~75% of the data.
  • These fidelities exceed the threshold required for magic state distillation and outperform previous superconducting implementations.

• State Teleportation:

  • Teleported logical states between two $d=3$ codes using lattice surgery.
  • Achieved state fidelities between 86.5% and 90.7% for Pauli eigenstates.
  • This establishes the capability to perform multi-qubit logical operations without breaking fault tolerance.

5. Significance and Outlook

This work marks a pivotal shift in the landscape of quantum error correction on superconducting processors:

• Beyond the Surface Code: It proves that the color code is a viable, and potentially superior, alternative to the surface code for superconducting hardware. While the surface code currently offers better raw error suppression on this specific device, the color code's superior qubit efficiency and gate capabilities make it a strong candidate for the future.
• Resource Efficiency: The color code requires fewer physical qubits to encode a logical qubit and allows for transversal implementation of the entire Clifford group, drastically simplifying algorithm implementation.
• Path to Universality: By combining efficient transversal Clifford gates with high-fidelity magic state injection, the authors have demonstrated the necessary building blocks for a universal, fault-tolerant quantum computer.
• Future Directions: The paper suggests that with further improvements in physical qubit coherence and gate fidelities, the color code could outperform the surface code in large-scale resource estimates, accelerating the timeline for practical fault-tolerant quantum computation.