Universal hardware-efficient topological measurement-based quantum computation via color-code-based cluster states

Abstract

Topological measurement-based quantum computation (MBQC) enables one to carry out universal fault-tolerant quantum computation via single-qubit measurements with a family of large entangled states called cluster states as resources. Raussendorf's three-dimensional cluster states (RTCSs) based on the surface codes are mainly considered for topological MBQC. In such schemes, however, the logical Hadamard, phase (Z1/2), and T (Z1/4) gates which are essential for building up arbitrary logical gates are not implemented natively without using state distillation or lattice dislocations, to the best of our knowledge. In particular, state distillation generally consumes many ancillary logical qubits; thus it is a severe obstacle against practical quantum computing. To solve this problem, we suggest an MBQC scheme via a family of cluster states called color-code-based cluster states (CCCSs) based on the two-dimensional color codes instead of the surface codes. We define logical qubits, construct elementary logical gates, and describe error correction schemes. We show that all logical Clifford gates generated by the cnot, Hadamard, and phase gates can be implemented natively in a fault-tolerant manner, although the T gate still requires state distillation to be fault-tolerant. The error thresholds of MBQC via CCCSs for logical-Z errors are calculated to be 2.7%–2.8%, which are comparable to the values for RTCSs, assuming a simple error model where physical qubits have nontrivial errors independently with the same probability. We analyze and compare the resource overheads of both the schemes. In particular, we show that the number of physical qubits required for implementing a phase gate with CCCSs is at least about 26 times smaller than with RTCSs using state distillation, for the same code distance.