Shorter gate sequences for quantum computing by mixing unitaries

Fault-tolerant quantum computers compose elements of a discrete gate set in order to approximate a target unitary. The problem of minimizing the number of gates is known as gate synthesis. The approximation error is a form of coherent noise, which can be significantly more damaging than comparable incoherent noise. We show how mixing over different gate sequences can convert this coherent noise into an incoherent form. As measured by diamond distance, the postmixing noise is quadratically smaller than before mixing, without increasing resource cost upper bounds. Equivalently, we can look for shorter gate sequences that achieve the same precision as unitary gate synthesis. For a broad class of problems this gives a factor $1/2$ reduction in worst-case resource costs.

Figure 1

The resource savings of our approach over unitary gate synthesis is $C_{\alpha ,\epsilon }^{\gamma }$, and here we show $C_{\alpha ,\epsilon }$ [see Eq. (13)] for $\alpha =5,10$ and a range of postmixing error rates. The different $\alpha$ correspond to different constant factors in Eqs. (7) and (11).

Figure 2

The geometric intuition of the convex hull finding algorithm. The cross marks the origin corresponding to $V$. (a) We find a $U_1=V{e}^{i{H}_1}$ so that $H_1$ is near the origin. (b) We extrapolate from ${\mu }_2=H_1$ through the origin to a point ${\tau }_2$. (c) We find a $U_2=V{e}^{i{H}_2}$ close to $V{e}^{i{\tau }_2}$, so that $H_2$ is near to ${\tau }_2$. (d) We form the convex hull of $H_1$ and $H_2$ and find the point ${\mu }_3$, which is closest to the origin. From here we extrapolate out through the origin to the point ${\tau }_3$. (e) We find a $U_3=V{e}^{i{H}_3}$ close to $V{e}^{i{\tau }_3}$, so that $H_3$ is near ${\tau }_3$. (f) We form the convex hull of $H_1,H_2$, and $H_3$ and find the origin lies inside the hull, and so the algorithm terminates. Note that none of the $H_j$ can stray far from the origin.