First qLDPC code constructions with transversal non-Clifford phase gates, obtained by embedding a local code with the desired transversal gate into a Tanner-based hypergraph or balanced product code, at the cost of O(1) Z-distance.
A Framework for Approximating Qubit Unitaries
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abstract
We present an algorithm for efficiently approximating of qubit unitaries over gate sets derived from totally definite quaternion algebras. It achieves $\varepsilon$-approximations using circuits of length $O(\log(1/\varepsilon))$, which is asymptotically optimal. The algorithm achieves the same quality of approximation as previously-known algorithms for Clifford+T [arXiv:1212.6253], V-basis [arXiv:1303.1411] and Clifford+$\pi/12$ [arXiv:1409.3552], running on average in time polynomial in $O(\log(1/\varepsilon))$ (conditional on a number-theoretic conjecture). Ours is the first such algorithm that works for a wide range of gate sets and provides insight into what should constitute a "good" gate set for a fault-tolerant quantum computer.
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Transversal Gates for Highly Asymmetric qLDPC Codes
First qLDPC code constructions with transversal non-Clifford phase gates, obtained by embedding a local code with the desired transversal gate into a Tanner-based hypergraph or balanced product code, at the cost of O(1) Z-distance.