Abstract
We present several different codes and protocols to distillT, controlled-S, and Toffoli (orCCZ) gates. One construction is based on codes that generalize the triorthogonal codes, allowing any of these gates to be induced at the logical level by transversalT. We present a randomized construction of generalized triorthogonal codes obtaining an asymptotic distillation efficiencyγ→1. We also present a Reed-Muller based construction of these codes which obtains a worseγbut performs well at small sizes. Additionally, we present protocols based on checking the stabilizers ofCCZmagic states at the logical level by transversal gates applied to codes; these protocols generalize the protocols of. Several examples, including a Reed-Muller code forT-to-Toffoli distillation, punctured Reed-Muller codes forT-gate distillation, and some of the check based protocols, require a lower ratio of input gates to output gates than other known protocols at the given order of error correction for the given code size. In particular, we find a512T-gate to10Toffoli gate code with distance8as well as triorthogonal codes with parameters[[887,137,5]],[[912,112,6]],[[937,87,7]]with very low prefactors in front of the leading order error terms in those codes.
Highlights
Magic state distillation [3,4,5] is a standard proposed approach to implementing a universal quantum computer. This approach begins by implementing the Clifford group to high accuracy using either stabilizer codes [6, 7] or using Majorana fermions [8]
There are many proposed protocols to distill magic states: for T gates from T gates [1,2,3, 5, 9, 10], for Toffoli gates from T -gates [2, 11,12,13,14,15], for Fourier states from Toffoli gates [16], CCZ(Toffoli) states from CCZ gates [17]
We present approaches to distilling Toffoli states which are not based on a single triorthogonal but rather on implementing a protocol using a sequence of checks, similar to Ref. [2]
Summary
Magic state distillation [3,4,5] is a standard proposed approach to implementing a universal quantum computer. To obtain universality, some non-Clifford operation is necessary, such as the π/4-rotation (T-gate) or the Toffoli gate (or CCZ which is equivalent to Toffoli up to conjugation by Cliffords) These non-Clifford operations are implemented using a resource, called a magic state, which is injected into a circuit that uses only Clifford operations. We present approaches to distilling Toffoli states which are not based on a single triorthogonal (or generalized triorthogonal code) but rather on implementing a protocol using a sequence of checks, similar to Ref. In 4.5 we study punctured Reed-Muller codes and find some protocols with a better ratio of input T -gates to output T -gates than any other known protocol for certain orders of error reduction Another result in 2.4 is a method of reducing the space required for any protocol based on triorthogonal codes at the cost of increased depth. Any subscript T denotes connection to the magic state for T gate
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