Fault-tolerant gates via homological product codes

Tomas Jochym-O'Connor

Walter Burke Institute for Theoretical Physics, Institute for Quantum Information & Matter, California Institute of Technology, Pasadena, CA 91125, USA

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A method for the implementation of a universal set of fault-tolerant logical gates is presented using homological product codes. In particular, it is shown that one can fault-tolerantly map between different encoded representations of a given logical state, enabling the application of different classes of transversal gates belonging to the underlying quantum codes. This allows for the circumvention of no-go results pertaining to universal sets of transversal gates and provides a general scheme for fault-tolerant computation while keeping the stabilizer generators of the code sparse.

Quantum error correction provides a means to combat errors present in a physical realization of a quantum computer. However, one must also prevent errors from proliferating when computing on the stored quantum data. This is the central goal of fault tolerance. Unfortunately, there is no simple method to both protect and manipulate the data of a single code in a universal manner, that is exploit the full power of the quantum computer. Rather, additional tricks or resources are required.

We present a method to cross between different codes in a fault-tolerant manner, allowing one to exploit the desirable properties of both codes. The path between different codes may spread errors, but importantly it does so in a controlled manner that allows for the resulting code to clean up any noise that has propagated. Moreover, the method keeps the code simple and does not rely on preparing special resource states or measuring complicated objects. This provides a potential fruitful alternative to universal fault tolerance.

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