High-fidelity hot gates for generic spin-resonator systems

We propose and analyze a high-fidelity hot gate for generic spin-resonator systems which allows for coherent spin-spin coupling, in the presence of a thermally populated resonator mode. Our scheme is nonperturbative in the spin-resonator coupling strength, applies to a broad class of physical systems, including, for example, spins coupled to circuit-QED and surface acoustic wave resonators as well as nanomechanical oscillators, and can be implemented readily with state-of-the-art experimental setups. We provide and numerically verify simple expressions for the fidelity of creating maximally entangled states under realistic conditions.

Figure 1

Schematic illustration for a generic spin-resonator system, comprising a set of spins $\left\{{\vec{\sigma }}_i\right\}$ coupled to a common resonator mode [as provided by, e.g., (a) a transmission line or (b) nanomechanical oscillators], with a nonvanishing thermal occupation.

Figure 2

Fidelity $\mathcal{F}$ with the maximally entangled target state $\left|{\mathrm{\Psi }}_{\mathrm{tar}}\right.〉=\left(\left|\Uparrow \Downarrow \right.〉+i\left|\Downarrow \Uparrow \right.〉\right)/\sqrt{2}$ for transversal coupling $\left(\mathcal{S}={\sigma }_1^x+{\sigma }_2^x\right)$, the initial product state $\rho \left(0\right)=\left|\Uparrow \Downarrow \right.〉\left.〈\Uparrow \Downarrow \right|\otimes {\rho }_{\mathrm{th}}\left(T\right)$ and different temperatures $k_{B}T/{\omega }_c=0,1,2,3,4,5$. Independently of the temperature $T$, the spins periodically disentangle from the (hot) resonator mode and systematically build up entanglement among themselves. While the peaks are merely independent of temperature, the amplitude of the precursory oscillations do increase with temperature. Inset: occupation of the resonator ${〈\widehat{n}〉}_t$ showing small oscillations due to weak entanglement between the qubits and the cavity mode [10]. Other numerical parameters: ${\omega }_q/{\omega }_c=\mathrm{\Gamma }=0,g/{\omega }_c=\frac{1}{16},\kappa /{\omega }_c=Q^{-1}={10}^{-5}$.

Figure 3

Fidelity $\mathcal{F}$ (left) in the presence of noise, with a zoom-in around $t_{\max }$ (right). As a benchmark, the solid (topmost) black line refers to the quasi-ideal limit ($\mathrm{\Gamma }=0,\kappa /{\omega }_c=Q^{-1}={10}^{-5}$, and $k_{B}T/{\omega }_c=0$), while (only) the red dashed curve accounts for a nonzero qubit-level splitting ${\omega }_q/{\omega }_c=0.1$. The solid blue line also accounts for dephasing of the qubits with a (rather large) dephasing rate $\mathrm{\Gamma }/{\omega }_c=1\%$ and finite thermal occupation of the resonator mode with $k_{B}T/{\omega }_c=5$ (${\overline{n}}_{\mathrm{th}}\approx 4.5$). The results are relatively insensitive to the quality factor of the cavity, provided that ${\kappa }_{\mathrm{eff}}\ll \mathrm{\Gamma }$; the orange dashed line (where $Q={10}^3$) is basically identical to the $Q={10}^5$ scenario, whereas the green dashed-dotted (lowest) one with $Q={10}^2$ (that is, $\kappa /{\omega }_c=\mathrm{\Gamma }/{\omega }_c=1\%$) shows a clear reduction in $\mathcal{F}$. This result can be traced back to the hot-gate requirement given in Eq. (8). Ideally, maximum entanglement is reached for $f_{c}t=4$, with several precursory oscillation peaks at $f_{c}t=1,2,3$. Other numerical parameters: $g/{\omega }_c=\frac{1}{8},{\omega }_q/{\omega }_c=0$ (except for the red dashed curve where ${\omega }_q/{\omega }_c=0.1$).

Figure 4

Errors ($\xi =1-{\mathcal{F}}_{\max }$) due to rethermalization of the cavity mode (a) and qubit dephasing (b). (a) Rethermalization-induced error for $k_{B}T/{\omega }_c=2$ (blue) and $k_{B}T/{\omega }_c=4$ (red), and $\mathrm{\Gamma }=0$. The error ${\xi }_{\kappa }$ is found to be independent of $\mu =g/{\omega }_c$: $\mu =\frac{1}{16}$ (squares) and $\mu =\frac{1}{8}$ (blue circles and red diamonds). (b) Dephasing-induced errors for $\mu =\frac{1}{4}$ (squares), $\mu =\frac{1}{8}$ (circles), and $\mu =\frac{1}{16}$ (diamonds); here, $\kappa /{\omega }_c={10}^{-6}$ and $k_{B}T/{\omega }_c=0.01$. In both cases, the linear error scaling is verified. Other numerical parameters: ${\omega }_q/{\omega }_c=0$. (c) Total error $\xi$ as a function of both the effective rethermalization rate $\sim \kappa /{\omega }_c{\overline{n}}_{\mathrm{th}}\sim {\overline{n}}_{\mathrm{th}}/Q$ and the spin dephasing rate $\sim \mathrm{\Gamma }/{\omega }_c$ for $g/{\omega }_c=\frac{1}{16}$, $k_{B}T/{\omega }_c=2$, and ${\omega }_q=0$.

Figure 5

Thermal occupation ${\overline{n}}_{\mathrm{th}}={(\exp \left[\hslash {\omega }_c/k_{B}T\right]-1)}^{-1}$ (black solid line) and high-temperature approximate result ${\overline{n}}_{\mathrm{th}}\approx k_{B}T/\hslash {\omega }_c$ (red dashed line). For $T=4\mathrm{K}$ and ${\omega }_c/2\pi =1\mathrm{GHz}$ (${\omega }_c/2\pi =10\mathrm{GHz}$), we have $k_{B}T/\hslash {\omega }_c\approx 80$ ($k_{B}T/\hslash {\omega }_c\approx 8$). For $T=1\mathrm{K}$ and ${\omega }_c/2\pi =1\mathrm{GHz}$ (${\omega }_c/2\pi =10\mathrm{GHz}$), we have ${\overline{n}}_{\mathrm{th}}\approx 20$ (${\overline{n}}_{\mathrm{th}}\approx 2$).

Figure 6

Spectrum of the DQD Hamiltonian in the single-electron regime, $H_{\mathrm{ch}}=\frac{\epsilon }{2}{\tau }^z+t_c{\tau }^x$, as a function of the interdot detuning parameter $\epsilon$. Inset: mixing parameters $sin\theta$ (black solid) and $cos\theta$ (gray dashed) as a function of the interdot detuning parameter $\epsilon$.

Figure 7

Effective spin-resonator coupling $g^x$ (solid) and $g^z$ (dashed) as a function of the interdot detuning parameter $\epsilon$.

Figure 8

(a) Spectrum of the DQD Hamiltonian in the two-electron regime, as given in Eq. (D14), as a function of the interdot detuning parameter $\epsilon$ for $\mathrm{\Delta }=0$. Tunnel coupling between the singlet states $\left|S_{11}\right.〉$ with (1,1) charge occupation and $\left|S_{02}\right.〉$ with (0,2) charge occupation yields the hybridized singlet states $\left|S_{\pm }\right.〉$. The ellipse refers to the qubit subspace, spanned by $\left|T_0\right.〉$ and $\left|S_{-}\right.〉$, while the dotted line (red) refers to the effective exchange coupling $J\left(\epsilon \right)=t_c^2/4\epsilon$. The arrows indicate schematically how to turn on and off the effective spin-resonator coupling, by changing the effective dipole moment associated with the qubit. Inset (b): relevant level diagram in the subspace $\left\{\left|T_0\right.〉,\left|S_{11}\right.〉,\left|S_{02}\right.〉\right\}$.

Figure 9

Effective spin-resonator coupling $g_{\mathrm{sp}}/g_0={cos}^2\theta$ (solid blue line) and qubit-level splitting ${\omega }_q\approx \left|J\right|$ relative to ${\epsilon }_{+}$ (dashed red line) as a function of the interdot detuning parameter $\epsilon$. The spin-resonator coupling may reach a few percent of the bare charge-resonator coupling $g_0$, with a qubit frequency ${\omega }_q$ that is much smaller than the energy of the level $\left|S_{+}\right.〉$.

Figure 10

The error ${\xi }_q$ induced by a nonzero qubit splitting ${\omega }_q>0$, for $\mathcal{S}=S^x={\sum }_i{\sigma }_i^x$ (transversal coupling), and for different initial qubit states $\left|\mathrm{\Psi }\left(0\right)\right.〉=\left|{\uparrow }_z{\downarrow }_z\right.〉$ (red circles) and $\left|\mathrm{\Psi }\left(0\right)\right.〉=\left|{\downarrow }_z{\downarrow }_z\right.〉$ (black squares); here, $g/{\omega }_c=\frac{1}{8}$ and $k_{B}T/{\omega }_c=1$. Quadratic fits (cyan, dashed-dotted lines) verify a quadratic error scaling $\sim {\omega }_q^2$, with the numerical pre-factor ${\alpha }_q$ depending on both the spin-resonator coupling $g$ and temperature $T$. Inset: the error ${\xi }_q$ as a function of the thermal occupation number ${\overline{n}}_{\mathrm{th}}$ for $g/{\omega }_c=\frac{1}{8}$ (black squares) and $g/{\omega }_c=1/\left(8\sqrt{2}\right)$ (blue circles) for $\left|\mathrm{\Psi }\left(0\right)\right.〉=\left|{\uparrow }_z{\downarrow }_z\right.〉$ and ${\omega }_q/{\omega }_c=0.5\%$. Other numerical parameters: $\mathrm{\Gamma }=\kappa =0$.

Figure 11

Schematic illustration of the hierarchy of frequency scales assumed for the derivation of the quantum master equation. Following the standard treatment [70], the reservoir spectral density $\kappa \left(\omega \right)/2\pi$ is taken to be a flat function of $\omega$ within the frequency range of interest $\left[{\omega }_c-{\mathrm{\Delta }}_B,{\omega }_c+{\mathrm{\Delta }}_B\right]$.

Figure 12

Fidelity $\mathcal{F}$ for the two-qubit state ${\rho }_{\mathrm{qubits}}$ with the target state $\left|{\mathrm{\Psi }}_{\mathrm{tar}}\right.〉=\left(\left|\Uparrow \Downarrow \right.〉+i\left|\Downarrow \Uparrow \right.〉\right)/\sqrt{2}$ for $\mathrm{\Gamma }/{\omega }_c=0$ (blue solid line) and $\mathrm{\Gamma }/{\omega }_c=1\%$ (red dashed line). For sufficiently low noise, at ${\omega }_{c}t=2\pi$ and ${\omega }_{c}t=5\times 2\pi$ the fidelity with the maximally entangled state $\left|{\mathrm{\Psi }}_{\mathrm{tar}}\right.〉$ reaches the maximal value $\mathcal{F}=1$. Numerical parameters: ${\omega }_q/{\omega }_c=0, k_{B}T/{\omega }_c=2\left({\overline{n}}_{\mathrm{th}}\approx 1.54\right), g/{\omega }_c=\frac{1}{4},\kappa /{\omega }_c=Q^{-1}={10}^{-5}$.

Figure 13

Fidelity $\mathcal{F}=〈{\mathrm{\Psi }}_{\mathrm{tar}}\left|{\rho }_{\mathrm{qubits}}\right|{\mathrm{\Psi }}_{\mathrm{tar}}〉$ for the two-qubit state ${\rho }_{\mathrm{qubits}}={\mathrm{Tr}}_{\mathrm{cav}}\left[\rho \right]$ with the target state $\left|{\mathrm{\Psi }}_{\mathrm{tar}}\right.〉=\left(\left|\Uparrow \Downarrow \right.〉+i\left|\Downarrow \Uparrow \right.〉\right)/\sqrt{2}$ for both ${\omega }_q/{\omega }_c=0$ (solid blue line) and ${\omega }_q/{\omega }_c=0.1$ (dashed red line); here, $g/{\omega }_c=\frac{1}{16}<0.1$. Other numerical parameters: $k_{B}T/{\omega }_c=2({\overline{n}}_{\mathrm{th}}\approx 1.54), Q={10}^5$, and $\mathrm{\Gamma }/{\omega }_c=0$.

Figure 14

Logarithmic negativity $E_{\mathcal{N}}$ for $k_{B}T/{\omega }_c=1$ and different cavity quality factors: $Q={10}^5$ (solid blue), $Q={10}^2$ (dashed-dotted blue), and $Q=10$ (dashed magenta). A clear reduction of the maximum entanglement is observed, if the quality factor $Q$ is too low to satisfy the hot-gate requirement given in Eq. (8). Here, we have $g/{\omega }_c\times g/k_{B}T=\frac{1}{16}=6.25\times {10}^{-2}$. The red (dotted) curve refers to $Q={10}^2$ and ${\omega }_q/{\omega }_c=0.2$. Other numerical parameters: $g/{\omega }_c=\frac{1}{4}$ and $\mathrm{\Gamma }/{\omega }_c=0$.

Figure 15

Error as a function of the effective rethermalization rate $\kappa {\overline{n}}_{\mathrm{th}}$ for $g/{\omega }_c=\frac{1}{16}$ (red squares), $g/{\omega }_c=1/\left(8\sqrt{2}\right)$ (blue stars) and $g/{\omega }_c=\frac{1}{8}$ (green triangles), and $k_{B}T/{\omega }_c=2\left({\overline{n}}_{\mathrm{th}}\approx 1.54\right)$, within the relevant small-error regime (${\kappa }_{\mathrm{eff}}/g_{\mathrm{eff}}\ll 1$). The dashed-dotted lines in cyan refer to linear fits, demonstrating a linear error scaling in the small error regime (${\kappa }_{\mathrm{eff}}/g_{\mathrm{eff}}\ll 1$), which is independent of $\mu =g/{\omega }_c$. Accordingly, the error is larger for higher temperatures, but all temperature-related effects are approximately captured by the thermal occupation number ${\overline{n}}_{\mathrm{th}}$. Other numerical parameters: $\mathrm{\Gamma }=0$ and ${\omega }_q=0$.

Figure 16

Total error $\xi$ as a function of both the effective rethermalization rate $\sim \kappa /{\omega }_c{\overline{n}}_{\mathrm{th}}\sim {\overline{n}}_{\mathrm{th}}/Q$ and the spin dephasing rate $\sim \mathrm{\Gamma }/{\omega }_c$ for $g/{\omega }_c=\frac{1}{16}, k_{B}T/{\omega }_c=4$, and ${\omega }_q=0$.

Figure 17

Timing errors. Left: total average error $\overline{\xi }$ as a function of the thermal occupation number ${\overline{n}}_{\mathrm{th}}$ for timing windows $\left({\omega }_c/2\pi \right)\mathrm{\Delta }t=5\%$ (circles) and $\left({\omega }_c/2\pi \right)\mathrm{\Delta }t=10\%$ (squares); here, $g/{\omega }_c=\frac{1}{16}$ (red symbols) and $g/{\omega }_c=\frac{1}{8}$ (blue symbols, upper curve). All curves can be fit very well to linear error models (see black dashed lines). Center: set of underlying (temperature-dependent) simulations for both $g/{\omega }_c=\frac{1}{16}$ (terminating at ${\omega }_{c}t/2\pi =16.5$) and $g/{\omega }_c=\frac{1}{8}$ (terminating at ${\omega }_{c}t/2\pi =4.5$). Note that larger amplitudes are observed for larger values of $\mu =g/{\omega }_c$. Other numerical parameters: $Q={10}^5,\mathrm{\Gamma }=0$, and ${\omega }_q=0$. Right: same analysis as done in Fig. 15 for $g/{\omega }_c=\frac{1}{8}$ (triangles) and $g/{\omega }_c=\frac{1}{16}$ (squares). The black curves account for a finite timing accuracy $\left({\omega }_c/2\pi \right)\mathrm{\Delta }t=5\%$, showing that the detrimental effects of time jitter are less pronounced for smaller values of $\mu =g/{\omega }_c$.

Figure 18

Fidelity $\mathcal{F}$ close to the ideal time $t_{\max }$ for $g/{\omega }_c=\frac{1}{16}$. The different curves refer to $Q={10}^5,k_{B}T/{\omega }_c=2$, i.e., ${\overline{n}}_{\mathrm{th}}\approx 1.54$ (blue solid, top curve), $Q={10}^5,k_{B}T/{\omega }_c=4$, i.e., ${\overline{n}}_{\mathrm{th}}\approx 3.52$ (red solid), and $Q={10}^4,k_{B}T/{\omega }_c=4$ (red dashed dotted). The error $\xi =1-\mathcal{F}$ can be estimated well with the formula ${\xi }_k\approx 4{\overline{n}}_{\mathrm{th}}/Q$, giving (for example) $\mathcal{F}\approx 1–4\times 3.52/{10}^4\approx 0.9986$. Other numerical parameters: $\mathrm{\Gamma }=0$ and ${\omega }_q=0$.

Figure 19

Relaxation-induced error ${\xi }_{\gamma }$ for $g/{\omega }_c=\frac{1}{8}$ (blue circles), $g/{\omega }_c=1/\left(8\sqrt{2}\right)$ (black squares), and $g/{\omega }_c=\frac{1}{16}$ (red diamonds). Other numerical parameters: $k_{B}T/{\omega }_c=0.01,\kappa =0,\mathrm{\Gamma }=0$, and ${\omega }_q=0$.

Figure 20

Total average gate error $\overline{E}$ (in percent) as a function of both the effective rethermalization rate $\sim \kappa /{\omega }_c{\overline{n}}_{\mathrm{th}}\sim {\overline{n}}_{\mathrm{th}}/Q$ and the spin dephasing rate $\sim \mathrm{\Gamma }/{\omega }_c$ for $g/{\omega }_c=\frac{1}{4}$ (top) and $g/{\omega }_c=\frac{1}{8}$ (bottom). Other numerical parameters: $k_{B}T/{\omega }_c=2$ and ${\omega }_q=0$.