Noisy Qudit vs Multiple Qubits : Conditions on Gate Efficiency for Enhancing Fidelity

Abstract

Today, multiple new platforms are implementing qudits, $d$-level quantum bases of information, for Quantum Information Processing (QIP). It is therefore crucial to study their efficiencies for QIP compared to more traditional qubit platforms. We present a comparative study of the infidelity scalings of a qudit and $n$-qubit systems, both with identical Hilbert space dimensions and noisy environments. The first-order response of the Average Gate Infidelity (AGI) to the noise in the Lindblad formalism, which was found to be gate-independent, was calculated analytically in the two systems being compared. This yielded a critical curve $O(d^2/\log_2(d))$ of the ratio of their respective gate times in units of decoherence time. This quantity indicates how time-efficient operations on these systems are. The curve delineates regions where each system has a higher rate of increase of the AGI than the other. This condition on gate efficiency was applied to different existing platforms. It was found that specific qudit platforms possess gate efficiencies competitive with state-of-the-art qubit platforms. Numerical simulations complemented this work and allowed for discussion of the applicability and limits of the linear response formalism.

Figures (9)

• Figure 1: Summary diagram illustrating the selected collapse operators and the associated analytically derived expected infidelity scalings as functions of the Hilbert Space dimension, as derived from (\ref{['eq:avEqubits']}) and (\ref{['eq:avEqudits']}). This is depicted for two distinct systems: multiple qubits (left) and a single qudit (right). The term 'infidelity scaling' here refers to the slopes of the first-order-in-$\gamma t$ AGIs, denoted as $c$ in (\ref{['eq:ratioE']}).
• Figure 2: Rate of increase of $\overline{\mathscr{E}_d}(\mathcal{E}_z) = c_d(J_z)\gamma t$ as a function of qudit dimension for $H=\mathbb{0}_d$ and $\gamma t \in [0,10^{-4}]$. The circled dots show the numerical results. The solid curve presents the expected analytical result given by (\ref{['eq:avEqudits']}).
• Figure 3: Average gate infidelities as a function of $\gamma t$. The data points show the computed values. The solid lines represent the linear theoretical behaviour from (\ref{['eq:Fid112']}). Each colour/marker pair corresponds to a different value of $d$.
• Figure 4: Relative deviation $1-\frac{\overline{\mathscr{E}_d}^\text{sim}}{\overline{\mathscr{E}_d}^\text{th}}$ as a function of $\gamma t$ for $H=\mathbb{0}_d$. $\overline{\mathscr{E}_d}^\text{sim}$ and $\overline{\mathscr{E}_d}^\text{th}$ were obtained from numerical computations and (\ref{['eq:avEqudits']}) respectively. Each marker corresponds to a different value of $d$.
• Figure 5: Statistical distributions of the relative deviation from the linear behaviour in (\ref{['eq:avEqudits']}) of the numerically obtained infidelity gradients $c_d$ for $N_g=5000$ gates for $\gamma t \in [10^{-5},10^{-3}]$, as a function of the dimension $d \in \llbracket 3,8 \rrbracket$. The candlestick bar chart should be interpreted as indicated in the upper right, with $\sigma$ denoting the standard deviation. The lower right inset shows the same results for $d \in \llbracket 2,4 \rrbracket$.