Entanglement dynamics and performance of two-qubit gates for superconducting qubits under non-Markovian effects

TL;DR

This work investigates non-Markovian open-system effects on two-qubit gates for superconducting qubits using the nonperturbative FP-HEOM framework with tensor-train representations. Independent reservoirs with Lorentzian and broadband spectra are modeled, and the rotating wave approximation (RWA) is critically assessed by comparing full and RWA couplings at zero temperature. Key findings show that counter-rotating terms can cause substantial deviations in entanglement dynamics, including dark periods and entanglement sudden death in broadband noise, with memory effects linking gate operation to idle phases. The study also analyzes Hadamard + CNOT sequences, revealing that shorter sequences and specific initial states yield higher fidelity and concurrence, while reservoir memory can cause nonmonotonic behavior even when total fidelity decays monotonically. Overall, the results provide guidance for gate design and noise modeling in superconducting devices and demonstrate the necessity of nonperturbative, memory-aware simulations for accurate performance predictions.

Abstract

Within a numerically exact simulation technique, the dissipative dynamics of a two-qubit architecture is considered in which each qubit couples to its individual noise source (reservoir). The goal is to reveal the role of subtle qubit-reservoir correlations including non-Markovian processes as a prerequisite to guide further improvements of quantum computing devices. This paper addresses the following three topics. First, we examine the validity of the rotating wave approximation imposed previously on the qubit-reservoir coupling with respect to the disentanglement dynamics. Second, generation of the entanglement as well as destruction are analyzed by monitoring the reduced dynamics during and after application of a $\sqrt{\mbox{iSWAP}^\dagger}$ gate, also focusing on memory effects caused by reservoirs. Finally, the performance of a Hadamard + CNOT sequence is analyzed for different gate decomposition schemes. In all three cases, various types of noise sources and qubit parameters are considered.

Figures (10)

• Figure 1: Schematic of the model we consider in this study. The qubits are directly coupled with each other with the strength $J (t)$, but each qubit couples with a different reservoir (gray rectangle). Each reservoir is characterized by the spectral noise power $S_j^{\beta} (\omega)$. For single-qubit gates, external rotating fields are applied to each qubit [$\Omega_j (t)$]. The external fields and the qubit--qubit coupling can be switched on and off according to gate operation.
• Figure 2: Time traces of the concurrence $\mathcal{C}(t)$ of two-qubit systems coupled with Lorentzian noise sources for various coupling strengths $\kappa$ and initial states $\hat{\rho}_0$. (a) Results without detuning, $\Delta = 0$. Results obtained with (w/ RWA, red curves and circles) and without (w/o RWA, blue curves) the RWA are depicted. Insets of (ii) and (v): Magnifications of $\mathcal{C}(t)$ around certain time periods. (b), (c) Results with detuning $\Delta\neq 0$. Dynamics in a whole time domains (b) and magnifications of those in long-time domains (c) are displayed. The panels (i)--(vi) in (b) and (c) display the same dynamics in different time domains. For the case without RWA (w/o RWA), the values $\Delta = 0.05 \omega_q$ [($+$), green curves] and $-0.05 \omega_q$ [($-$), red curves] are taken into account. For the case with RWA (w/ RWA), both results with $\Delta = 0.05 \omega_q$ and $-0.05 \omega_q$ are exactly the same, and hence only one blue curve is depicted in each panel. The dashed curves are the results without detuning (0), corresponding to the blue curves in (a).
• Figure 3: Time traces of the concurrence $\mathcal{C}(t)$ of two-qubit systems coupled with broadband noise sources for various spectral exponents $s$ and initial states $\hat{\rho}_0$. Results obtained with (w/ RWA, red curve) and without (w/o RWA, blue curve) the RWA are depicted. The green filled circles indicate the times at which the concurrence reaches zero (ESD). The times of the ESD are indicated by the associated numbers.
• Figure 4: (a) Schematic of the sequence of the qubit--qubit coupling strength $J(t)$. (b)--(i) Linear--log plots of the time traces of the concurrence $\mathcal{C}(t)$ during and after application of a $\sqrt{\hbox{iSWAP}^\dagger}$ gate in homogeneous environments, $s_1 = s_2 = s$. The vertical dashed lines indicate the end time of the gate operation. The system--reservoir coupling strength is $2\pi\hbar \kappa = 0.04$ in (b) and (d)--(f), and $0.004$ in (c) and (g)--(i), respectively. (b), (c) Comparison between different spectral exponents with the fixed qubit--qubit coupling strength ($J / \omega_q = 1/8$). Three cases for the exponents, $s = 1$ (blue curves), $1/2$ (red curves), and $1/8$ (green curves), are depicted. Inset of (b): Time traces of $2|\rho_{23}(t)|$ (dashed curve) and $2\sqrt{\rho_{11}(t)\rho_{44}(t)}$ (solid curve) for $s = 1/8$, both of which constitute the concurrence. Inset of (c): Magnification of the main plot around the time in which the concurrence takes the maximum value (linear--linear plot). (d)--(i) Comparison between different strengths of the qubit--qubit coupling with the fixed spectral exponents: $s = 1$ in (d) and (g), $1/2$ in (e) and (h), and $1/8$ in (f) and (i). The stronger ($J/\omega_q = 1/8$, blue curves) and weaker ($J/\omega_q = 1/12$, red curves) coupling are considered. The filled circles indicate the maximum value of the concurrence.
• Figure 5: Dynamics of the off-diagonal element $\rho_{23}(t)$ in heterogeneous environments ($s_1 \neq s_2$) during and after application of a $\sqrt{\hbox{iSWAP}^\dagger}$ gate. The dotted and dashed curves represent the real and imaginary part of $\rho_{23}(t)$, respectively. The results with the exponents $(s_1, s_2) = (1, 1/8)$ (red curve) are depicted, and a homogeneous case $(s_1, s_2) = (1, 1)$ is also displayed as a reference (blue curve). The vertical dashed line indicates the end time of the gate operation. The coupling strengths between the qubits and the qubits and reservoirs are fixed to $J/\omega_q = 1/8$ and $2 \pi \hbar \kappa = 0.04$, respectively.