Quantum Circuit Overhead

TL;DR

The results suggest that, in terms of the upper bounds on the $T-QCO, the famous T gate is a highly non-optimal choice for the completion of the Clifford gate set, even among the gates of order 8.

Abstract

We introduce a measure for evaluating the efficiency of finite universal quantum gate sets $\mathcal{S}$, called the Quantum Circuit Overhead (QCO), and the related notion of $T$-Quantum Circuit Overhead ($T$-QCO). The overhead is based on the comparison between the efficiency of $\mathcal{S}$ versus the optimal efficiency among all gate sets with the same number of gates. We demonstrate the usefulness of the ($T$-)QCO by extensive numerical calculations of its upper bounds, providing insight into the efficiency of various choices of single-qubit $\mathcal{S}$, including Haar-random gate sets and the gate sets derived from finite subgroups, such as Clifford and Hurwitz groups. In particular, our results suggest that, in terms of the upper bounds on the $T$-QCO, the famous T gate is a highly non-optimal choice for the completion of the Clifford gate set, even among the gates of order 8. We identify the optimal choices of such completions for both finite subgroups.

Figures (8)

• Figure 1: The histograms of $Q_T$ probability density for an ensemble of type $\mathcal{C}_{\mu, \infty}$ with increasing $t$. The dashed line denotes the corresponding optimal value. The solid line corresponds to a Super-Golden gate set $\mathcal{C}_{T_{24}}$.
• Figure 2: The histograms of $Q_T$ probability density for an ensemble of type $\mathcal{C}_{\mu,8}$. The solid line denotes the corresponding optimal value.
• Figure 3: The histograms of $Q_T$ probability density for ensembles of type $\mathcal{C}_{\mu, r}$ (bottom) vs the histogram of $Q$ for the corresponding ensembles of type $\mathcal{S}_{\mu, 24, r}$ (top) for $t=500$. The solid line denotes the corresponding optimal value. Note that the scales on the Y-axis differ.
• Figure 4: The histograms of $Q_T$ probability density for an ensemble of type $\mathcal{H}_{\mu, \infty}$ with increasing $t$. The dashed line denotes the corresponding optimal value. The solid line corresponds to a Super-Golden gate set $\mathcal{H}_{T_{12}}$.
• Figure 5: The histograms of $Q_T$ probability density for an ensemble of type $\mathcal{H}_{\mu,2}$ with increasing $t$. The dashed line denotes the corresponding optimal value. The solid line corresponds to a Super-Golden gate set $\mathcal{H}_{T_{12}}$.

Theorems & Definitions (3)

• Example 1: single-qubit gate count
• Example 2: CNOT-count flavour
• Example 3: T-count flavour