Fault-Tolerant Quantum Computation with Local Gates

TL;DR

The paper addresses achieving fault-tolerant quantum computation under locality constraints by using concatenated quantum codes and a universal gate set. It shows that a fault-tolerance threshold persists in 3D, 2D (with nearest-neighbor interactions), and 1D architectures by incorporating local interaction schemes and qubit-swapping techniques, with overhead that remains polylogarithmic in the target error rate. The analysis extends the standard threshold model to local-gate scenarios, deriving a modified threshold $1/(C r^2)$ when error locations scale as $P_{k+1} = C r^{k+1} P_k^2$, and demonstrates how ancilla verification and parallelism sustain scalability. Overall, the work provides concrete constructions and architectural considerations that enable scalable, fault-tolerant quantum computation using only local gates across dimensionalities.

Abstract

I discuss how to perform fault-tolerant quantum computation with concatenated codes using local gates in small numbers of dimensions. I show that a threshold result still exists in three, two, or one dimensions when next-to-nearest-neighbor gates are available, and present explicit constructions. In two or three dimensions, I also show how nearest-neighbor gates can give a threshold result. In all cases, I simply demonstrate that a threshold exists, and do not attempt to optimize the error correction circuit or determine the exact value of the threshold. The additional overhead due to the fault-tolerance in both space and time is polylogarithmic in the error rate per logical gate.

Figures (5)

• Figure 1: Error correction circuit for the seven qubit code. Each line represents a block of seven qubits, and each gate represents the same gate applied transversally on the block.
• Figure 2: Network to fault-tolerantly swap two computational qubits
• Figure 3: a) Computational qubits (x) arranged on a square lattice, interspersed by auxiliary qubits (o). b) adds cul-de-sacs (c) for moving qubits out of the way. c) Moving a data qubit (A) two positions.
• Figure 4: a) The logical qubits of the computer lie on separate planes. b) Each plane has the data on one line, adjacent to ancillas at various levels. The letter d represents a data qubit, 0 is a qubit from a level 0 ancilla, 1 is from a level 1 ancilla, and so on.
• Figure 5: The logical qubits of the computer lie on separate lines. Within each line, ancillas are interspersed with the data qubits. Again, d represents a data qubit, 0 is from a level 0 ancilla, and 1 is from a level 1 ancilla.