A Game of Surface Codes: Large-Scale Quantum Computing with Lattice Surgery

Daniel Litinski

Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany

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Given a quantum gate circuit, how does one execute it in a fault-tolerant architecture with as little overhead as possible? In this paper, we discuss strategies for surface-code quantum computing on small, intermediate and large scales. They are strategies for space-time trade-offs, going from slow computations using few qubits to fast computations using many qubits. Our schemes are based on surface-code patches, which not only feature a low space cost compared to other surface-code schemes, but are also conceptually simple~--~simple enough that they can be described as a tile-based game with a small set of rules. Therefore, no knowledge of quantum error correction is necessary to understand the schemes in this paper, but only the concepts of qubits and measurements.

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Useful, classically intractable quantum computations can be very long, potentially consisting of billions of quantum gates. Some of these computations use as few as 100 qubits, but these need to be logical error-corrected qubits, rather than physical qubits, as the coherence times of currently available physical qubits are many orders of magnitude shorter than the execution times of these computations. With logical qubits, the set of available operation is determined by the error-correcting code, independently of the underlying physical hardware. For full-scale quantum computations on hundreds of logical qubits (i.e., potentially hundreds of thousands of physical qubits), it is useful to have a framework that describes logical qubits and operations without keeping track of the details of the physical hardware and error-correction protocols. This paper introduces such a framework with a focus on surface codes, but can also be applied to other topological codes. It uses lattice surgery to describe all logical operations via the easy-to-understand concepts of qubits and measurements, avoiding anyons and topological braiding diagrams. Using this framework, a complete full-scale quantum computer is constructed, consisting of qubit blocks that perform magic state distillation and blocks that consume magic states to advance the computation. Finally, space-time trade-offs are discussed, i.e., how to use more qubits to compute faster.

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