Surface codes: Towards practical large-scale quantum computation

Abstract

This article provides an introduction to surface code quantum computing. We first estimate the size and speed of a surface code quantum computer. We then introduce the concept of the stabilizer, using two qubits, and extend this concept to stabilizers acting on a two-dimensional array of physical qubits, on which we implement the surface code. We next describe how logical qubits are formed in the surface code array and give numerical estimates of their fault-tolerance. We outline how logical qubits are physically moved on the array, how qubit braid transformations are constructed, and how a braid between two logical qubits is equivalent to a controlled-NOT. We then describe the single-qubit Hadamard, S and T operators, completing the set of required gates for a universal quantum computer. We conclude by briefly discussing physical implementations of the surface code. We include a number of appendices in which we provide supplementary information to the main text.

54 pages, 35 figures, version incorporating referee and community feedback. Additional comments welcome

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• Proof of finite surface code threshold for matching
• Distilling one-qubit magic states into Toffoli states
• Practicality of spin chain 'wiring' in diamond quantum technologies
• Simulating the Transverse Ising Model on a Quantum Computer: Error Correction with the Surface Code
• Quantum circuit optimization by topological compaction in the surface code
• Distillation protocols for Fourier states in quantum computing
• Linked-Cluster Technique for Finding the Distance of a Quantum LDPC Code