Optimal ancilla-free Clifford+T approximation of z-rotations

We consider the problem of approximating arbitrary single-qubit z-rotations by ancilla-free Clifford+T circuits, up to given epsilon. We present a fast new probabilistic algorithm for solving this problem optimally, i.e., for finding the shortest possible circuit whatsoever for the given problem instance. The algorithm requires a factoring oracle (such as a quantum computer). Even in the absence of a factoring oracle, the algorithm is still near-optimal under a mild number-theoretic hypothesis. In this case, the algorithm finds a solution of T-count m + O(log(log(1/epsilon))), where m is the T-count of the second-to-optimal solution. In the typical case, this yields circuit approximations of T-count 3log_2(1/epsilon) + O(log(log(1/epsilon))). Our algorithm is efficient in practice, and provably efficient under the above-mentioned number-theoretic hypothesis, in the sense that its expected runtime is O(polylog(1/epsilon)).Comment: 40 pages. New in v3: added a section on worst-case behavio

Graphical Methods in Device-Independent Quantum Cryptography

We introduce a framework for graphical security proofs in device-independent quantum cryptography using the methods of categorical quantum mechanics. We are optimistic that this approach will make some of the highly complex proofs in quantum cryptography more accessible, facilitate the discovery of new proofs, and enable automated proof verification. As an example of our framework, we reprove a previous result from device-independent quantum cryptography: any linear randomness expansion protocol can be converted into an unbounded randomness expansion protocol. We give a graphical proof of this result, and implement part of it in the Globular proof assistant.Comment: Publishable version. Diagrams have been polished, minor revisions to the text, and an appendix added with supplementary proof