Reducing the number of non-Clifford gates in quantum circuits

We present a method for reducing the number of non-Clifford quantum gates, in particularly T-gates, in a circuit, an important task for efficiently implementing fault-tolerant quantum computations. This method matches or beats previous approaches to ancillae-free T-count reduction on the majority of our benchmark circuits, in some cases yielding up to 50% improvement. Our method begins by representing the quantum circuit as a ZX-diagram, a tensor networklike structure that can be transformed and simplified according to the rules of the ZX-calculus. We then extend a recent simplification strategy with a different ingredient, phase gadgetization, which we use to propagate non-Clifford phases through a ZX-diagram to find nonlocal cancellations. Our procedure extends unmodified to arbitrary phase angles and to parameter elimination for variational circuits. Finally, our optimization is self-checking, in the sense that the simplification strategy we propose is powerful enough to independently validate equality of the input circuit and the optimized output circuit. We have implemented the routines of this paper in the open-source library pyzx.

Figure 1

Overview of the steps in our phase-teleportation scheme. (a) Original circuit. (b) The circuit expanded as a ZX diagram, with 21 $T$ gates. (c) Simplified ZX-diagram. (d) 15 t gates remain after phase teleportation.

Figure 2

A convenient presentation for the ZX-calculus. These rules hold for all $\alpha ,\beta \in [0,2\pi )$, and due to $\left(\mathbf{h}\right)$ and $\left(\mathbf{i}\mathbf{2}\right)$ all rules also hold with the colors interchanged.

Figure 3

A ZX-diagram that comes from a circuit, and its equivalent graph-like ZX-diagram.