A 3D lattice defect and efficient computations in topological MBQC

TL;DR

This work develops a fault-tolerant MBQC framework based on the RHG 3D cluster state, enabling topological gate implementations via lattice defects and a Rudolph-Grover rebit encoding. A local Hadamard gate is realized through a lattice dislocation defect, while the rebit encoding yields a topological S gate and complete Clifford universality without distillation for those gates. Universal quantum computation is achieved by incorporating the T gate through magic-state distillation, with substantial reductions in overhead achieved via geometric/topological circuit optimization and a formal, verifiable approach to circuit equivalence and optimization. The authors also establish a rigorous connection between measurement patterns, correlation surfaces, and Clifford operations, and provide a software verifier to automate circuit validation, yielding practical gains in the resource cost of MBQC and advancing scalable fault-tolerant quantum computation. Overall, the paper demonstrates a cohesive route to scalable, fault-tolerant universal quantum computation in a 3D cluster-state setting, balancing topological protection with circuit-level optimization.

Abstract

We describe an efficient, fully fault-tolerant implementation of Measurement-Based Quantum Computation (MBQC) in the 3D cluster state. The two key novelties are (i) the introduction of a lattice defect in the underlying cluster state and (ii) the use of the Rudolph-Grover rebit encoding. Concretely, (i) allows for a topological implementation of the Hadamard gate, while (ii) does the same for the phase gate. Furthermore, we develop general ideas towards circuit compaction and algorithmic circuit verification, which we implement for the Reed-Muller code used for magic state distillation. Our performance analysis highlights the overall improvements provided by the new methods.

Figures (13)

• Figure 1: The cluster state lattice: one unit cell (a) and a $5\times 5\times 5$ lattice (b). The physical qubits are located on the red and black dots. The entanglement induced by the $CZ$ gates is represented as solid blue lines. The underlying cubic lattice is represented by dashed lines, black qubits sit on the edges, red qubits on the faces. (b) For computation purposes, the input Toric code $\mathcal{I}$ is inserted on the cluster left plane (not visible on the figure because of perspective). It is foliated in the horizontal direction, thus simulating a « time» evolution. The output Toric code $\mathcal{O}$ is made of the highlighted black qubits on the right plane.
• Figure 2: The Toric code. The physical qubits sit on the graph edges, represented by small black dots. At the top are defined two stabilizers of different types: plaquette Z and site X. Logical qubits are encoded in pairs of missing stabilizers (blue and red squares at the bottom), they can be of two types due to the duality of the surface code. A logical Pauli operator correspond to a product of Pauli operators on physical qubits along a string either extending from one hole to the other or winding around one hole.
• Figure 3: The surfaces that correspond to the identity gate. The surface $c^Z$ is primal with black qubits on its boundary, while $\bar{c}^X$ is dual and has no boundaries at all. The black qubits in $\partial c^Z\cap \mathcal{B}$ are measured in $Z$, while all the other qubits in $\mathcal{B}$ are measured in $X$. After the measurements in the bulk, the state on $\mathcal{I}\cup \mathcal{O}$ is an encoded Bell pair.
• Figure 4: The input and output qubits and the measurement lines that correspond to the $\textsc{cnot}$ gate. The gray surface sketch the 'tubes' making up the $\bar{c}_1^X$ surface, while the red line is its boundary, which is made up of dual qubits.
• Figure 5: Hadamard defect in lattice at the unit cell level. For clarity, only the qubits on the external faces of the unit cells have been represented, except for 2 unit cells at the front where all qubits are shown. Every cell that is represented by a white cube (potentially distorted) is full: it has a qubit on every face and on every edge. On the output Toric code plane, the red qubits inside the plaquettes that are measured in X are not shown, only the one measure in X and that implements the plaquette hole is displayed. Nearest neighbor connections are represented by blue lines. Some qubits and connections are missing in the middle of the cluster to allow for the dislocation defect.

Theorems & Definitions (2)

• Theorem 1: Fundamental Theorem of Topological MBQC
• proof