Quantum computing by color-code lattice surgery

TL;DR

This work develops a complete framework for universal fault-tolerant quantum computation using color-code lattice surgery on triangular 4.8.8 color codes. It shows how to implement a universal gate set, including transversal Hadamard and phase gates, with CNOT realized via lattice-surgery sequences and T|+> states via injection and distillation. Resource analyses compare color-code lattice surgery to surface-code counterparts, revealing that color codes can achieve comparable error suppression with roughly half the qubits per code distance and similar or faster operation times, particularly at small distances where explicit counts favor color codes (e.g., d=3 CNOT uses about 30 qubits vs 39 for surface codes). The study also discusses decoding options, error-scaling models, and the potential of color codes to outperform surface codes as error rates drop and code distances grow, while noting that decoder efficiency remains an active area of development.

Abstract

We demonstrate how to use lattice surgery to enact a universal set of fault-tolerant quantum operations with color codes. Along the way, we also improve existing surface-code lattice-surgery methods. Lattice-surgery methods use fewer qubits and the same time or less than associated defect-braiding methods. Furthermore, per code distance, color-code lattice surgery uses approximately half the qubits and the same time or less than surface-code lattice surgery. Color-code lattice surgery can also implement the Hadamard and phase gates in a single transversal step---much faster than surface-code lattice surgery can. Against uncorrelated circuit-level depolarizing noise, color-code lattice surgery uses fewer qubits to achieve the same degree of fault-tolerant error suppression as surface-code lattice surgery when the noise rate is low enough and the error suppression demand is high enough.

Figures (20)

• Figure 1: Triangular 4.8.8 color codes of distances 3, 5, and 7. The number of data qubits for distance $d$ is $(d^2-1)/2+d$. The number of faces (which is half the number of checks) is $(d^2+2d-3)/4$.
• Figure 2: Measurement-based $\textit{CNOT}$ circuit.
• Figure 3: Alternative measurement-based $\textit{CNOT}$ circuit.
• Figure 4: To measure $M_{XX}$ ($M_{ZZ}$) between the central logical qubit and a logical qubit adjacent to one of its sides, measure only the $X$ ($Z$) checks on the lighter-colored faces on the interface and the $X$ and $Z$ checks on the full octagons shared across the interface. (The figure compresses three separate scenarios into one.) The outcome is the product of the lighter-colored check outcomes. (color online.)
• Figure 5: The same scenario as Fig. \ref{['fig:MXX-MZZ-d3']}, except with distance-five codes. (color online)