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ACMD Seminar: Advances in Number-Theoretic Methods for Characterizing and Optimizing Fault-Tolerant Quantum Circuits

Andrew Glaudell
Booz Allen Hamilton


Thursday, March 26, 2020, 3:00–4:00 PM
Building 101, Lecture Room A

Thursday, March 26, 2020, 1:00–2:00 PM
Building 1, Room 3304

This talk will be broadcast on-line using BlueJeans.  Contact acmdseminar [at] (acmdseminar[at]nist[dot]gov) for details.

Host: Paulina Kuo

Abstract: Kliuchnikov, Maslov, and Mosca proved in 2012 that a 2×2 unitary matrix U can be exactly represented by a single-qubit Clifford+T circuit if and only if the entries of U belong to the ring ℤ[1/\sqrt{2},i] by developing a T-optimal circuit synthesis algorithm for these circuits. Later that year, Giles and Selinger showed that the same ring 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 this talk, I will highlight two recent results which further these lines of reasoning. First, I establish that the unitaries over the rings ℤ[1/2], ℤ[1/\sqrt{2}], ℤ[1/\sqrt{-2}], and ℤ[1/2,i] are in exact correspondence to familiar universal gate sets. Second, I restrict to a slight extension of two-qubit circuits over ℤ[1/2,i], corresponding to the Clifford+CS gate set. I develop an efficient exact synthesis algorithm for such circuits using the Spin(6) ~ SU(4) isomorphism and show that this algorithm is optimal in CS-count. This is the first example of an optimal synthesis algorithm for a universal fault-tolerant multi-qubit gate set.

Bio: Andrew Glaudell earned a B.S. in Engineering Physics / Physics from the University of Wisconsin - Madison in 2013. Under the advising of Dr. Jacob M. Taylor, he received his Ph.D from the University of Maryland - College Park in December 2019, where he was a Booz Allen Hamilton Quantum Information fellow. While his doctoral work initially focused on quantum repeaters, the majority was in the development of number-theoretic methods to both characterize and optimize quantum circuits for fault-tolerant gate sets in the Clifford hierarchy for both qubits and qudits. Since graduating, Andrew has joined the BASIQ Research Team at Booz Allen Hamilton, where he and a small team continue to research quantum circuit synthesis methods, quantum algorithms, and other theoretical efforts under the umbrella of quantum information theory.

Note: Visitors from outside NIST must contact Cathy Graham; (301) 975-3800; at least 24 hours in advance.

Created March 10, 2020, Updated March 13, 2020