Fault-tolerant quantum computation with high threshold in two dimensions

TL;DR

The paper addresses scalable, fault-tolerant universal quantum computation in a strictly two-dimensional, locally connected architecture. It combines topological error correction on a 2D cluster state with surface codes, a defect-based CNOT, and magic-state distillation to realize non-Clifford gates, then maps the 3D construction into a slice-by-slice 2D scheme. It derives a per-source error threshold of $7.5\times 10^{-3}$, analyzes a concrete error model with equal-strength depolarizing channels, and shows poly-logarithmic overhead with encoded size $S'\sim S\log^3 S$. This work demonstrates a practical, high-threshold route for scalable quantum computation using only local interactions, with potential implementations in optical lattices, segmented ion traps, quantum dots, or superconducting qubits.

Abstract

We present a scheme of fault-tolerant quantum computation for a local architecture in two spatial dimensions. The error threshold is 0.75% for each source in an error model with preparation, gate, storage and measurement errors.

Figures (4)

• Figure 1: (Color online.) The CNOT gate $\Lambda(X)_{c,t}$ ($c$: control, $t$: target) formed by topologically entangled lattice defects. Each pair of defects carries an encoded qubit. Defects exist as primal (blue) and dual (black), and are created by local measurement. The primal correlation surface (light blue) shown here converts an incoming Pauli operator $Z_t$ into an outgoing $Z_t\otimes Z_c$, as required for a CNOT gate.
• Figure 2: (Color online.) Lattice definitions. a) Elementary cell of the cluster lattice ${\cal{L}}$. 1-chains of ${\cal{L}}$ (dashed lines), and graph edges (solid lines). b) A surface code obtained from a 2D cluster state by local $X$-measurements. c) A pair of electric ("e") or magnetic ("m") holes in the code plane each support an encoded qubit. $\overline{Z}^{e/m}$ and $\overline{X}^{e/m}$ denote the encoded Pauli operators $Z$ and $X$, respectively.
• Figure 3: (Color online.) Remaining gates for universal fault-tolerant computation. The relevant correlation surfaces are shown in light blue and gray. Replace Out(put) by In(put) for a measurement. a) Preparation of a $\overline{Z}$-eigenstate for an electric qubit. b) Preparation of an $\overline{X}$-eigenstate for an electric qubit. c) Creation of a Bell pair among a bare $S$-qubit and an encoded qubit.
• Figure 4: Elementary cell of the 2D lattice. Temporal order of operations in $V$: The labels on the edges denote the time steps at which the corresponding $\Lambda(Z)$ gate is performed. The labels at the syndrome vertices ("$\circ$") denote measurement and (re-)preparation times $[t_M, t_P]$, and the labels at the code vertices ("$\bullet$") denote times for Hadamard gates $(t_H,t_H^\prime)$. The pattern is periodic in space, and in time with period six.