Hyperbolic Cluster States for Fault-Tolerant Measurement-Based Quantum Computing

Abstract

Fault-tolerant measurement-based quantum computing (MBQC) provides a compelling framework for fault-tolerant quantum computation, in which quantum information is processed through single-qubit measurements on a three-dimensional entangled resource known as cluster state. To date, this resource has been predominantly studied on Euclidean lattices, most notably in the Raussendorf-Harrington-Goyal (RHG) construction, which underlies topological fault tolerance in MBQC. In this work, we introduce the hyperbolic cluster state, a generalization of the three-dimensional cluster state to negatively curved geometries, obtained via the foliation of periodic hyperbolic lattices. We present an explicit construction of hyperbolic cluster states and investigate their fault-tolerant properties under a realistic circuit-level depolarizing noise model. Using large-scale numerical simulations, we perform memory experiments to characterize their logical error rates and decoding performance. Our results demonstrate that hyperbolic cluster states exhibit a fault-tolerance threshold comparable to that of the Euclidean RHG cluster state, while simultaneously supporting a constant encoding rate in the thermodynamic limit. This represents a substantial improvement in qubit overhead relative to conventional cluster-state constructions. These findings establish hyperbolic geometry as a powerful and experimentally relevant resource for scalable, fault-tolerant MBQC and open new avenues for leveraging negative curvature in quantum information processing.

Figures (6)

• Figure 1: Stereographic projection of the regular hyperbolic $\{8,3\}$ lattice into the Poincaré disk with the unit cell highlighted in red.
• Figure 2: Two-layer foliated hyperbolic cluster state derived from an $\{8,3\}$ lattice before and after imposing PBCs.
• Figure 3: Representative correlation surfaces of the hyperbolic cluster state obtained by foliating the $\{8,3\}$ hyperbolic QEC code. These surfaces realize the logical degrees of freedom of the underlying QEC code.
• Figure 4: Geometric realization of parity-check operators in the hyperbolic cluster state obtained from the $\{8,3\}$ hyperbolic QEC code. The left (right) panel shows an $X$-type ($Z$-type) parity check, supported on a three-dimensional bi-pyramidal structure whose equatorial face lies in a single layer and whose apices extend to the neighboring layers. The parity-check measurement outcome is given by the sum (mod 2) of the single-qubit measurement outcomes associated with the qubits represented by the nodes of the corresponding structure.
• Figure 5: CZ-gate schedule around a primal plaquette, applied in a fixed counterclockwise order. An $X$ fault on the face ancilla occurring after the fourth CZ gate propagates through the remaining entangling operations, inducing a contiguous chain of $Z$ faults on the subsequent neighboring data qubits. This chain flips the outcomes of the two $X$-type parity checks at its endpoints, producing a correlated pair of syndrome defects (thick blue dashed triangles). The other $X$-type check operators involved are represented by thinner blue dashed triangles. Since they contain two $Z$ errors each, they do not produce a syndrome deffect.