Efficient Synthesis of Universal Probabilistic Quantum Circuits†

Techniques to efficiently compile higher-level quantum algorithms into lower-level fault-tolerant circuits are needed for the implementation of a scalable, general purpose quantum computer. Several universal gate sets arise from augmenting the set of Clifford gates by additional gates that arise naturally from the underlying fault-tolerance scheme. Examples are the Clifford+T basis which arises, e.g., from the surface code and the concatenated Steane code, in which the gate T = 1 0 0 e iπ/4 is added, and the Clifford+π/12 basis which arises, e.g., from quantum computing with metaplectic anyons [1], in which the gate K = 1 0 0 e iπ/6 is added. While in principle the Solovay-Kitaev algorithm [2, 3] can be used to solve the synthesis problem for any universal gate set, and therefore also the above-mentioned special cases, there are certain disadvantages to this approach, in particular the large depth of the resulting circuits. The best-known upper bound on the circuit depth is O(log 3.97 (1/ε)), where ε is the precision of the target approximation. In addition, the compilation time of the Solovay-Kitaev method, i.e., the time it takes to execute the classical algorithm that produces the output circuit, is quite high, namely almost cubic in log(1/ε). Fortunately, it was shown recently [4–7] that for the Clifford+T basis, elementary number theory can be leveraged to obtain much more efficient algorithms for approximating a single-qubit gate. The number of T gates in the resulting circuits scales close to 3 log 2 (1/ε) for single-qubit rotations around the Z-axis and the compilation time for these algorithms has essentially the same scaling. The point of the present work is twofold. The first purpose (i) is to show that the constant in the above estimates can be further reduced; this may come as a surprise as there is an information-theoretic lower bound that establishes that there are Z-rotations that require 3 log 2 (1/ε) many T gates to reach an approximation precision ε. However, this bound makes two assumptions: that the underlying circuits are unitary and that they act only on a single qubit. By relaxing both assumptions to (1) allow measurements and adaptive decisions on earlier results and (2) allow to operate on more than one qubit through the use of an ancilla, we show that this bound can be surpassed. Indeed, our best schemes lead to an expected T-gate count of 1.149 log 2 (1/ε) for arbitrary Z-rotations. The …