Practical implementation of Toffoli-based qubit rotation

Abstract

The Toffoli gate is an important universal quantum gate, and will alongside the Clifford gates be available in future fault-tolerant quantum computing hardware. Many quantum algorithms rely on performing arbitrarily small single-qubit rotations for their function, and these rotations may also be used to construct any unitary from a limited (but universal) gate set. How to carry out such rotations is then of significant interest. In this work, we evaluate the performance of a recently proposed single-qubit rotation algorithm using the Clifford plus Toffoli gate set by implementation of a one-shot version on both a real and a simulated quantum computer. We test the algorithm under various simulated noise levels using a per-qubit depolarizing error noise model and examine how the probabilities and process fidelities are affected. We then conduct live runs and find that the results reasonably match the simulated results. We also attempt to model the hardware noise by combining a number of noise models, matching the results to results of the live runs to approximate the hardware noise. Our results suggest that the algorithm will perform well for up to 1% noise, under the noise models we chose. We further posit the use of our algorithm as a benchmark for quantum processing units, given that it has a low complexity that is easy to fine-tune in small steps. We provide details for how to do this.

Figures (21)

• Figure 1: Circuits for approximate single-qubit rotation. a) Circuit that applies $R_\varphi$ to $\ket{\psi}$ ($\cos \varphi = 3/5$, $\sin\varphi=4/5$) with probability $5/8$; otherwise a $Z$ gate. b) Circuit that applies $R_{\theta^\ast}$ to $\ket{\psi}$, $\epsilon$-close to a desired rotation $R_\theta$ with high probability, where the number of ancillary controls $n$ and the comparison constant $k$ should be chosen as in Eqns. (\ref{['Eq:n']})-(\ref{['Eq:k']}).
• Figure 2: Bitwise ripple-carry comparator made of a sequence of Toffoli gates Hindlycke2024a. For indices where $k_i$=0, the modified Toffoli gate realizes a quantum OR gate. If carry-in is constant 1, the bit comparisons simplify.
• Figure 3: Gate array that approximates the $T$ gate to within $\epsilon < 2.6\times10^{-4}$, before and after simplifications, when performing manual basis switching prior to measurement; here $k = 181 = 10110101_2$.
• Figure 4: Rotation probabilities for different angles $\theta$, given $n$ = 8, for which $|\theta - \theta^*|$ = 0. The bars arise as the algorithm periodically reduces $n=8$ to a lower value, reducing the circuit complexity and thereby the errors. For each error rate the bar at $\theta=2\arctan\frac{1}{2}\approx 0.93$ rises highest above the baseline and corresponds to $n=2$, while the two next-highest bars are at $\theta=2\arctan\frac{1}{4}\approx0.49$ and $\theta=2\arctan\frac{3}{4}\approx1.29$ and correspond to $n=3$, and so on.
• Figure 5: Performance for the $T^*$ circuit, simulated runs. Since the same circuit is used for $n=2$ and $n=3$, the same data were used.