Nonlinearity of the Fidelity in Open Qudit Systems: Gate and Noise Dependence in High-dimensional Quantum Computing

TL;DR

The paper addresses how the fidelity of quantum gates on a high-dimensional qudit is affected by Markovian environmental noise. It develops a general perturbative framework that expands the Average Infidelity $\mathcal{I}$ in powers of the dimensionless coupling $\gamma t$, yielding a gate-independent first-order term and gate-dependent higher-order corrections, with the latter arising from the interplay between the control Hamiltonian $\mathcal{S}$ and the noise operator $\mathcal{L}$. Through analytical expressions and comprehensive numerical simulations, it reveals a transition from linear to nonlinear AGI behavior in the strong-coupling regime, identifies universal bounds on the plateau infidelity $\mathcal{I}^*$ that depend only on the qudit dimension $d$, and shows how different gates (e.g., identity, X, QFT) saturate these bounds in characteristic ways. The results have practical implications for gate design, error correction, and benchmarking of near-term qudit architectures, and set the stage for extensions to multi-qudit systems and more realistic noise models, with potential experimental verification. In particular, the second-order corrections significantly improve fidelity predictions (reducing relative error to around 1% for $d=4$ and 0.01% for $d=2$ in the tested regimes).

Abstract

High-dimensional quantum computing has generated significant interest due to its potential to address scalability and error correction challenges faced by traditional qubit-based systems. This paper investigates the Average Gate Fidelity (AGF) of single qudit systems under Markovian noise in the Lindblad formalism, extending previous work by developing a comprehensive theoretical framework for the calculation of higher-order correction terms. We derive general expressions for the perturbative expansion of the Average Gate Infidelity (AGI) in terms of the environmental coupling coefficient and validate these with extensive numerical simulations, emphasizing the transition from linear to nonlinear behaviour in the strong coupling regime. Our findings highlight the dependence of AGI on qudit dimensionality, quantum gate choice, and noise strength, providing critical insights for optimising quantum gate design and error correction protocols. Additionally, we utilise our framework to identify universal bounds for the AGI in the strong coupling regime and explore the practical implications for enhancing the performance of near-term qudit architectures. This study offers a robust foundation for future research and development in high-dimensional quantum computing, contributing to the advancement of robust, high-fidelity quantum operations.

Figures (18)

• Figure 1: Comparison of the first four correction terms of the AGI to numerical simulations. AGIs for the QFT gate applied to a $d=4$ qudit undergoing pure dephasing ($L = J_z$) at couplings $\gamma t \in \left[ 0, 0.5 \right]$. The discrete points (red) represent the simulated AGI values, while the successive dashed-dotted lines represent the correction terms from first to fourth order.
• Figure 2: Deviation from linearity of the AGIs for qudits under strong dephasing. Simulations of the AGI against the strength of pure dephasing ($L = J_z$) are plotted on log-log axes over $\gamma t \in \left[10^{-2}, 10^{3} \right]$. The simulations were performed on a set of qudit dimensions, $d \in \lbrace 2, 4, 8, 16 \rbrace$ each for 100 Haar-random quantum gates. The dashed lines represent the linear regime given by the first-order correction to the AGI at each dimension. The inset shows the linear regime for smaller values of $\gamma t \in \left[10^{-5}, 10^{-2} \right]$. For stronger noise the AGIs exhibit a transition from the linear to nonlinear regime that then saturates at a stable plateau value. These plateau values vary for qudit dimension and also for different quantum gates, highlighting the gate-dependence. The saturation points $(\gamma t)^*$ of the stable regime are also dependent on both the qudit dimension and gate type, with higher-dimensional qudits deviating from linearity and saturating earlier.
• Figure 3: Large-$\gamma t$ behaviour of the AGIs for Haar-random qubit gates under pure dephasing. Simulations of the AGI were performed for a sample of one million (50 displayed) Haar-random qubit ($d=2$) gates under pure dephasing ($L = J_z$) for $\gamma t \in \left[ 10^{-2}, 10^{4} \right]$. For each gate, the AGI curves fall into one of two groups, exhibiting different behaviour near their respective saturation points ($(\gamma t)^*$). The curves in red ($\mathcal{I}_{\mathrm{mono}}$) approach their plateau values monotonically from below. The blue curves ($\mathcal{I}_{\mathrm{over}}$) rapidly approach their plateau values, before overshooting once with a single turning point, and then converging monotonically from above. The degree of blue shading for each curve indicates the degree of overshoot above the plateau value. All sampled gates converged to their $\mathcal{I}^*$ plateau values within the range $\left[\mathcal{I}^*_{\mathrm{min}}, \mathcal{I}^*_{\mathrm{max}}\right]$ indicated by the black and red horizontal lines. In the figure and insets, the lighter and darker shaded regions indicate the vertical extent of the sampled gates and their standard deviations, with the solid and dashed lines indicating the mean ($\bar{\mathcal{I}}^*$) and median ($\tilde{\mathcal{I}}^*$) AGI values, respectively. The inset figures show each group of gates in isolation. The red group span the full bounding range, but are weighted upwards by outliers near the upper limit. The blue group do not span the full range, but are weighted towards the lower limit. Only the average AGI taken over both groups converges towards the expected mean value of $0.5$.
• Figure 4: Gate-dependence of the AGIs and qudit state evolution as functions of noise strength. (Data points) AGI simulations of a $d=4$ qudit under pure dephasing were time-evolved for each of the listed quantum gates: the (blue circles) QFT gate, and (other points) interpolated $X^{\eta}$ gates for $\eta \in \left[ 0, 1 \right]$ as shown. Additionally shown for each gate: the (solid horizontal lines) plateau value $\mathcal{I}^*$ of the AGI, (dashed curves) state purity $\left( \Tr (\rho^2) \right)$, and (dashed-dotted curves) mean coherences $\mathbb{E}[\rho_{ij}]= \frac{2}{d(d-1)}\sum_{i>j} \abs{\rho_{i,j}}$ for $i>j$. Note that: (i) the AGI data points of the QFT and $X^{\eta}$ gate for $\eta = \frac{3d - 2}{4d}$ overlap and converge to the same plateau value (for any $d$), (ii) the $\eta$ values were chosen as they produce equally spaced AGI plateau values (for any $d$), (iii) the final state purity and coherences for each gate were averaged over a set of 2400 random pure Hermitian initial states, and (iv) their curves converge to $\frac{1}{d-1}$ and $0$ respectively, following the behaviour of the AGIs as they transition from the linear to nonlinear to plateau-like regimes.
• Figure 5: Saturation points ($(\gamma t)^*$) of the AGIs for interpolated $X^{\eta}$ gates as functions of qudit dimension. AGIs were calculated for a set of $12$ interpolated $X^{\eta}$ gates for $0\leq \eta \leq 1$, at dimensions $d\in\left[ 2, 12\right]$. For each dimension and gate, the AGI curve over $\gamma t$ was interpolated by a cubic spline. The saturation points $(\gamma t)^*$ were identified by root-finding algorithm of the points at which they converged (within $\varepsilon < 10^{-8}$) to their AGI plateau values. The plotted data points, on log-linear axes, indicate these root values as functions of $d$ for each gate, and the dashed lines represent their associated power-law fits. This analysis was repeated for a set of $n=4800$ Haar-random gates at each dimension. The light shaded areas are bounded by the maximum and minimum saturation points found at each dimension. The darker shaded areas represent a $1\sigma$ deviation about the mean (solid gray line).

Theorems & Definitions (11)

• Theorem A.1
• proof
• Theorem A.2
• proof
• Lemma A.1
• proof
• Theorem A.3
• proof
• Lemma C.1
• proof