Fault-tolerant quantum computation on logical cluster states

Keisuke Fujii

Christopher M. Dawson, Michael A. Nielsen

Keisuke Fujii, Katsuji Yamamoto

We investigate a scheme of fault-tolerant quantum computation based on the cluster model. Logical qubits are encoded by a suitable code such as the Steane's 7-qubit code. Cluster states of logical qubits are prepared by post-selection through verification at high fidelity level, where the unsuccessful ones are discarded without recovery operation. Then, gate operations are implemented by transversal measurements on the prepared logical cluster states. The noise threshold is improved significantly by making the high fidelity preparation and transversal measurement. It is estimated to be about 3% by a numerical simulation.

Simon J. Devitt, Austin G. Fowler, Lloyd C. L. Hollenberg

Ahmed Adel Mahmoud, Gabrielle Tournaire, Sven Bachmann +1 more

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.

Keisuke Fujii, Katsuji Yamamoto, Alexander Lvovsky

A novel scheme is presented for fault‐tolerant quantum computation based on the cluster model. Some relevant logical cluster states are constructed in concatenation by post‐selection through verification, without necessity of recovery operation, where a suitable code such as the Steane’s 7‐qubit code is adopted for transversal operations. This simple concatenated construction of verified cluster states achieves a high noise threshold ∼4%, and restrains the divergence of resources.