We study two-qubit circuits over the Clifford+CS gate set which consists of Clifford gates together with the controlled-phase gate CS=diag(1,1,1,i). The Clifford+CS gate set is universal for quantum computation and its elements can be implemented fault-tolerantly in most error-correcting schemes with magic state distillation. However, since non-Clifford gates are typically more expensive to perform in a fault-tolerant manner, it is desirable to construct circuits that use few CS gates. In the present paper, we introduce an algorithm to construct optimal circuits for two-qubit Clifford+CS operators. Our algorithm inputs a Clifford+CS operator U and efficiently produces a Clifford+CS circuit for U using the least possible number of CS gates. Because our algorithm is deterministic, the circuit it associates to a Clifford+CS operator can be viewed as a normal form for the operator. We give a formal description of these normal forms as walks over certain graphs and use this description to derive an asymptotic lower bound of 5log(1/epsilon)+O(1) on the number CS gates required to epsilon-approximate any 4x4 unitary matrix.\

}, url = {https://arxiv.org/abs/2001.05997}, author = {Andrew N. Glaudell and Neil J. Ross and J. M. Taylor} } @article {2271, title = {Canonical forms for single-qutrit Clifford+T operators}, journal = {Annals of Physics}, volume = {406}, year = {2019}, month = {8/19/2019}, pages = {54-70}, abstract = {We introduce canonical forms for single qutrit Clifford+T circuits and prove that every single-qutrit Clifford+T operator admits a unique such canonical form. We show that our canonical forms are T-optimal in the sense that among all the single-qutrit Clifford+T circuits implementing a given operator our canonical form uses the least number of T gates. Finally, we provide an algorithm which inputs the description of an operator (as a matrix or a circuit) and constructs the canonical form for this operator. The algorithm runs in time linear in the number of T gates. Our results provide a higher-dimensional generalization of prior work by Matsumoto and Amano who introduced similar canonical forms for single-qubit Clifford+T circuits.\

}, doi = {https://doi.org/10.1016/j.aop.2019.04.001}, url = {https://arxiv.org/abs/1803.05047}, author = {Andrew N. Glaudell and Neil J. Ross and J. M. Taylor} } @article {2464, title = {Number-Theoretic Characterizations of Some Restricted Clifford+T Circuits}, year = {2019}, month = {8/16/2019}, abstract = {Kliuchnikov, Maslov, and Mosca proved in 2012 that a 2\×2 unitary matrix V can be exactly represented by a single-qubit Clifford+T circuit if and only if the entries of V belong to the ring Z[1/2\–\√,i]. Later that year, Giles and Selinger showed that the same restriction applies to matrices that can be exactly represented by a multi-qubit Clifford+T circuit. These number-theoretic characterizations shed new light upon the structure of Clifford+T circuits and led to remarkable developments in the field of quantum compiling. In the present paper, we provide number-theoretic characterizations for certain restricted Clifford+T circuits by considering unitary matrices over subrings of Z[1/2\–\√,i]. We focus on the subrings Z[1/2], Z[1/2\–\√], Z[1/-2\−\−\√], and Z[1/2,i], and we prove that unitary matrices with entries in these rings correspond to circuits over well-known universal gate sets. In each case, the desired gate set is obtained by extending the set of classical reversible gates {X,CX,CCX} with an analogue of the Hadamard gate and an optional phase gate.

}, url = {https://arxiv.org/abs/1908.06076}, author = {Matthew Amy and Andrew N. Glaudell and Neil J. Ross} } @article {1603, title = {Serialized Quantum Error Correction Protocol for High-Bandwidth Quantum Repeaters}, journal = {New Journal of Physics}, volume = {18}, year = {2016}, month = {2016/09/02}, pages = {093008}, abstract = {Advances in single photon creation, transmission, and detection suggest that sending quantum information over optical fibers may have losses low enough to be correctable using a quantum error correcting code. Such error-corrected communication is equivalent to a novel quantum repeater scheme, but crucial questions regarding implementation and system requirements remain open. Here we show that long range entangled bit generation with rates approaching $10^8$ ebits/s may be possible using a completely serialized protocol, in which photons are generated, entangled, and error corrected via sequential, one-way interactions with a minimal number of matter qubits. Provided loss and error rates of the required elements are below the threshold for quantum error correction, this scheme demonstrates improved performance over transmission of single photons. We find improvement in ebit rates at large distances using this serial protocol and various quantum error correcting codes.

}, doi = {10.1088/1367-2630/18/9/093008}, url = {http://iopscience.iop.org/article/10.1088/1367-2630/18/9/093008/meta}, author = {Andrew N. Glaudell and Edo Waks and J. M. Taylor} }