Noise-resistant phase gates with amplitude modulation

Constructing a quantum computer requires sequences of accurate gate operations on a large number of quantum bits (qubits) Figgatt et al. (2019). Any unitary transformation, including multiqubit gates, can be decomposed into a series of elementary two-qubit gates together with single-qubit operations in principle Barenco et al. (1995). Recently, much attention has been attracted on realizing universal two-qubit controlled-not gates and controlled-phase gates in various physical systems including trapped ions Leibfried et al. (2003); Ge et al. (2019); Palmero et al. (2017); Zheng (2006); Zou et al. (2002); Zheng (2003), neutral atoms Rauschenbeutel et al. (1999); Welte et al. (2018); Isenhower et al. (2010); Sun et al. (2020); Shen et al. (2019); Wu et al. (2020), linear optics Kiesel et al. (2005); Lemr et al. (2011), and superconducting circuit quantum electrodynamics (QED) Puri and Blais (2016); Xu et al. (2020a); Yang et al. (2019); Zhang et al. (2017); Hua et al. (2014). In addition to these atom-based (both natural and artificial atoms) gates, photon-based gates enabled by atom-mediated interactions have also been extensively studied. Recent examples include the designs of single–photon Hadamard gates Chen et al. (2019a) and two–photon controlled–phase gates Chen et al. (2019b); Hua et al. (2014) on the solid-state platforms. Among various physical architectures, circuit QED has recently become a leading platform for quantum computing as well as quantum information processing (QIP). One of the outstanding features of superconducting circuit is that the coupling strength between a superconducting qubit and a microwave field can be artificially engineered Gu et al. (2017). On this basis, tunable qubit-resonator coupling has been realized experimentally using a tunable mutual inductance between a phase qubit and a lumped-element resonator Allman et al. (2010), or using a three-island transmon qubit with a tunable dipole moment Srinivasan et al. (2011); Gambetta et al. (2011); Hoffman et al. (2011). Based on the context of circuit QED, high two-qubit gate fidelity above $99\%$ has been demonstrated in recent experiments Barends et al. (2014); Sheldon et al. (2016).

Despite rapid progress in realizing high-fidelity gate over the years, gates fidelity remains a major bottleneck for scalable QIP as well as fault-tolerant quantum computation. The gate fidelity is generally limited by two sources of noise, namely control errors in gate parameters and decoherence induced by noisy environment. The former can arise due to slow drifts in experimental parameters such as the resonator frequency, or imperfections in the control apparatus as well as imperfect operations. The latter is caused by the inevitable interaction between the system and environment Chen et al. (2018, 2017), which will collapse the quantum state and make the quantum information no longer correct, thereby compromising the power of quantum computation Duan and Guo (1997); Yang et al. (2020). Although fault-tolerant quantum error correcting is capable of counteracting a certain degree of errors or decoherence, it cannot be brought into play unless the underlying error rates are small to begin with Nazir et al. (2002); Raussendorf and Harrington (2007). Therefore, seeking for high-fidelity gate operations that are robust in the presence of noises or errors becomes an imperative.

In this article, we present a simple and experimentally feasible method to implement fast arbitrary phase gates in a qutrit-cavity coupled system, introducing amplitude modulation to make the operation more resilient to the two sources of noise described above. Specifically, the features and advantages of the present work are summarized as follows. The scheme employs a parametric drive to squeeze the cavity mode, which induces Rabi interactions between the qutrits and cavity with exponentially improved effective coupling strength. Then two-qubit arbitrary phase gates can be implemented by designing proper logical states and parameter conditions. Since the gate time is inversely proportional to the effective qubit-cavity coupling, it can be shortened significantly by employing strong parametric drive. In contrast to previous gate schemes based on the dispersive regime where the desired effective interactions are derived using perturbation theory with large-detuning conditions Puri and Blais (2016); Yang et al. (2018, 2012); Han et al. (2020, 2019), this proposal enables faster implementation of the phase gates. Besides, we show that the gate robustness against parameter errors, including gate timing error and frequency detuning error, can be increased by shaping the qutrit-cavity coupling strength with time-dependent amplitude modulation. In principle, the error suppression is achieved by means of optimizing the phase-space trajectory of the cavity mode for minimizing residual qubit-cavity entanglement in case of errors. Furthermore, the proposed amplitude-shaped gates are robust to the energy relaxations of qutrits due to the coding of quantum imformation on the qutrits ground states and in particular, exhibit significantly enhanced robustness against cavity decay compared to the unshaped gate.

The remainder of the paper is structured as follows. In Sec. II, we describe the physical model and demonstrate in detail how to implement the two-qubit arbitrary phase gate. We then take the $\pi$-phase gate as an example for numerical simulations to verify the gate performance. In Sec. III, we propose to use amplitude modulation to improve the gate robustness against parameter errors. Then, the effects of decoherence caused by the energy relaxations of qutrits and cavity decay on the shaped gates are discussed. Finally, we briefly discuss the possible experimental implementation and summarize our conclusions in Sec. IV.

We consider a system containing two qutrits coupled to a single-mode cavity that is subjected to a parametric drive. Figure 1(a) illustrates a potential implementation of the system in the context of circuit QED Johansson et al. (2009); Peropadre et al. (2010); Xiang et al. (2013). The two qutrits have identical level structure, i.e., an excited state $\lvert e\rangle$ and two ground states $\lvert g\rangle$ and $\lvert f\rangle$ [see Fig. 1(b)]. The transitions between $\lvert e\rangle$ and $\lvert g\rangle$ are coupled to the cavity mode with coupling strength $g$. A classical field (with frequency $\omega$) is imposed on the qutrits to drive the transitions between $\lvert e\rangle$ and $\lvert g\rangle$ with Rabi frequency $\Omega$. The state $\lvert f\rangle$ is not affected during the interaction. Particularly, a parametric (two-photon) driving field of frequency $\omega_{p}$ and amplitude $\Omega_{p}$ is introduced to drive the cavity, which produces a nonlinear process equivalent to degenerate parametric amplification. For the current model we assume a sufficiently narrow bandwidth of the generated photon pair centered on the cavity frequency, which is much less than the scale of amplitude $\Omega_{p}$ to ensure that the model is valid. As shown in Fig. 1(a), this parametric drive of the resonator can be implemented by modulating the flux $\Phi_{\mathrm{ext}}(t)$ threading the SQUID loop connected to the middle of the transmission line Macklin et al. (2015); Yamamoto et al. (2008); Zhong et al. (2013).

A direct effect of the parametric drive is that it will result in exponentially enhanced qutrit-cavity interactions that can be well-described by a usual Rabi Hamiltonian Leroux et al. (2018); Qin et al. (2018); Chen et al. (2019c); Wang et al. (2019a). To be more specific, the Hamiltonian of the system described above in a proper observation frame is given by ($\hbar=1$, hereinafter)

$$\begin{split}H=&\Delta_{c}a^{\dagger}a+\sum_{j=1,2}\Big{[}\Delta_{q}\lvert e\rangle_{j}\langle e\lvert+g(\lvert e\rangle_{j}\langle g\lvert a+a^{\dagger}\lvert g\rangle_{j}\langle e\lvert)\\ &-\frac{\Omega}{2}(\lvert e\rangle_{j}\langle g\lvert+\lvert g\rangle_{j}\langle e\lvert)\Big{]}-\frac{\Omega_{p}}{2}(a^{2}+a^{{\dagger}2}).\end{split}$$

(1)

Here, $a^{\dagger}(a)$ is the creation (annihilation) operator of the cavity mode with frequency $\omega_{a}$; The detunings are $\Delta_{c}=\omega_{a}-\omega_{p}/2$, $\Delta_{q}=\omega_{e}-\omega_{g}-\omega_{p}/2$, where $\omega_{z}$ is the frequency associated with level $\lvert z\rangle$ ($z=e,g$); We have also assumed $\omega=\omega_{p}/2$. By diagonalizing the cavity-only part in $H$ with unitary $U_{s}=\exp[r_{p}(a^{2}-a^{{\dagger}2})/2]$, where the squeeze parameter $r_{p}$ is defined via $r_{p}=(1/2)\mathrm{arctanh}(\Omega_{p}/\Delta_{c})$, the Hamiltonian (1) is rewritten as

$$\begin{split}H_{s}=&\Delta a^{\dagger}a+\sum_{j=1,2}\Big{[}\frac{g}{2}e^{r_{p}}(a+a^{\dagger})(\lvert e\rangle_{j}\langle g\lvert+\lvert g\rangle_{j}\langle e\lvert)\\ &-\frac{g}{2}e^{-r_{p}}(a^{\dagger}-a)(\lvert e\rangle_{j}\langle g\lvert-\lvert g\rangle_{j}\langle e\lvert)\\ &-\frac{\Omega}{2}(\lvert e\rangle_{j}\langle g\lvert+\lvert g\rangle_{j}\langle e\lvert)\Big{]},\end{split}$$

(2)

where $\Delta=\Delta_{c}\mathrm{sech}(2r_{p})$ and we have set $\Delta_{q}=0$ for simplicity. Apparently, the first term in the summation has the form of Rabi Hamiltonian. The second term $H_{\mathrm{Err}}\equiv\sum_{j}\left[-\frac{g}{2}e^{-r_{p}}(a^{\dagger}-a)(\lvert e\rangle_{j}\langle g\lvert-\lvert g\rangle_{j}\langle e\lvert)\right]$, indicating the deviation from ideal Rabi Hamiltonian, could be neglected in a large amplification limit $e^{r_{p}}\to\infty$ (which requires $r_{p}$ to be sufficiently large, see Appendix for detailed discussion), such that $H_{s}$ after moving to the interaction picture with respect to $H_{0}=\Delta a^{\dagger}a$ is transformed to

$$\begin{split}H_{s}^{\prime}=g_{s}S_{x}(ae^{-\mathrm{i}\Delta t}+a^{\dagger}e^{\mathrm{i}\Delta t})-\Omega S_{x},\end{split}$$

(3)

where $g_{s}=ge^{r_{p}}$ and $S_{x}=\sum_{j}(\lvert e\rangle_{j}\langle g\lvert+\lvert g\rangle_{j}\langle e\lvert)/2$.

The coupled system described by $H_{s}^{\prime}$ undergoes a unitary evolution with the propagator being $U(t)=e^{-\mathrm{i}F(t)S_{x}a}e^{-\mathrm{i}G(t)S_{x}a^{\dagger}}e^{-\mathrm{i}A(t)S_{x}^{2}}e^{-\mathrm{i}B(t)S_{x}}$, where we have

$$\begin{split}F(t)=&g_{s}\int_{0}^{t}e^{-\mathrm{i}\Delta t^{\prime}}dt^{\prime}=\frac{\mathrm{i}g_{s}}{\Delta}(e^{-\mathrm{i}\Delta t}-1),\\ G(t)=&g_{s}\int_{0}^{t}e^{\mathrm{i}\Delta t^{\prime}}dt^{\prime}=\frac{\mathrm{i}g_{s}}{\Delta}(1-e^{\mathrm{i}\Delta t}),\\ A(t)=&\mathrm{i}g_{s}\int_{0}^{t}F(t^{\prime})e^{\mathrm{i}\Delta t^{\prime}}dt^{\prime}=\frac{-g_{s}^{2}}{\Delta}(t-\frac{e^{\mathrm{i}\Delta t}}{\mathrm{i}\Delta}+\frac{1}{\mathrm{i\Delta}}),\\ B(t)=&-\int_{0}^{t}\Omega dt^{\prime}=-\Omega t.\end{split}$$

(4)

As can be inferred from $U(t)$, a maximally two-qubit entangled state emerges in the $S_{z}$ basis for $\Delta t=2\pi$, $A(t)=\pi/2$ and $\Omega=0$, which corresponds to the M$\o$lmer-S$\o$rensen gate operation Sørensen and Mølmer (2000); Webb et al. (2018); Shapira et al. (2018). In the $S_{x}$ basis, $U(t)$ could implement a two-qubit phase gate coded via proper logical states. To this end, we assume that the ground state $\lvert f\rangle$ of qutrits is coded to carry the logical state 0 and superposition state $\lvert+\rangle=(\lvert e\rangle+\lvert g\rangle)/\sqrt{2}$ to carry the logical state 1. By setting $\Delta\tau=2k_{1}\pi$ (where $\tau$ is the gate time and $k_{1}$ a positive integer), we have $F(\tau)=G(\tau)=0$. Then the propagator can be written as merely acting in the qubit subspace, which is reduced to $U(\tau)=e^{-\mathrm{i}A(\tau)S_{x}^{2}}e^{-\mathrm{i}B(\tau)S_{x}}$, with $A(\tau)$ and $B(\tau)$ being $A(\tau)=-2k_{1}\pi g_{s}^{2}/\Delta^{2}$ and $B(\tau)=-2k_{1}\pi\Omega/\Delta$, respectively. It can be easily found that

$$\begin{split}&\lvert f\rangle_{1}\lvert f\rangle_{2}\stackrel{{\scriptstyle U(\tau)}}{{\longrightarrow}}\lvert f\rangle_{1}\lvert f\rangle_{2},\\ &\lvert f\rangle_{1}\lvert+\rangle_{2}\stackrel{{\scriptstyle U(\tau)}}{{\longrightarrow}}e^{-\mathrm{i}\frac{A(\tau)}{4}}e^{-\mathrm{i}\frac{B(\tau)}{2}}\lvert f\rangle_{1}\lvert+\rangle_{2},\\ &\lvert+\rangle_{1}\lvert f\rangle_{2}\stackrel{{\scriptstyle U(\tau)}}{{\longrightarrow}}e^{-\mathrm{i}\frac{A(\tau)}{4}}e^{-\mathrm{i}\frac{B(\tau)}{2}}\lvert+\rangle_{1}\lvert f\rangle_{2},\\ &\lvert+\rangle_{1}\lvert+\rangle_{2}\stackrel{{\scriptstyle U(\tau)}}{{\longrightarrow}}e^{-\mathrm{i}A(\tau)}e^{-\mathrm{i}B(\tau)}\lvert+\rangle_{1}\lvert+\rangle_{2}.\end{split}$$

(5)

Further, we demand that $A(\tau)/4+B(\tau)/2=-2k_{2}\pi$ and $A(\tau)+B(\tau)=-(2k_{3}\pi+\varphi)$ where $k_{2}$ and $k_{3}$ are integers and satisfy $k_{3}\geq 2k_{2}-1/2$ (see below). In this case, Eq. (5) becomes

$$\begin{split}&\lvert f\rangle_{1}\lvert f\rangle_{2}\stackrel{{\scriptstyle U(\tau)}}{{\longrightarrow}}\lvert f\rangle_{1}\lvert f\rangle_{2},\\ &\lvert f\rangle_{1}\lvert+\rangle_{2}\stackrel{{\scriptstyle U(\tau)}}{{\longrightarrow}}\lvert f\rangle_{1}\lvert+\rangle_{2},\\ &\lvert+\rangle_{1}\lvert f\rangle_{2}\stackrel{{\scriptstyle U(\tau)}}{{\longrightarrow}}\lvert+\rangle_{1}\lvert f\rangle_{2},\\ &\lvert+\rangle_{1}\lvert+\rangle_{2}\stackrel{{\scriptstyle U(\tau)}}{{\longrightarrow}}e^{\mathrm{i}\varphi}\lvert+\rangle_{1}\lvert+\rangle_{2},\end{split}$$

(6)

which conducts a two-qubit arbitrary phase gate. In the following, we take the gate with $\varphi=\pi$ as an example to discuss detailedly the gate performance. The corresponding constraints become $A(\tau)=-(2k_{3}+1-4k_{2})\pi$ and $B(\tau)=(2k_{3}+1-8k_{2})\pi$, which result in $\Delta=g_{s}/\sqrt{(2k_{3}+1-4k_{2})/k_{1}}$ and $\Omega=-(2k_{3}+1-8k_{2})\Delta/2k_{1}$, respectively. The gate time is thus inversely proportional to the effective coupling $g_{s}$, i.e., $\tau\sim 1/g_{s}$. Since $g_{s}$ increases significantly with enhancing the parametric drive, which could be orders of magnitude larger than the original coupling, the gate operation could be attained at a very short time.

To show clearly the gate operation, we plot in Fig. 2(a) the evolutions of the four computational states’ population and the gate fidelity Johansson et al. (2012, 2013), which are obtained by numerically solving the Liouville equation with respect to the full Hamiltonian given by Eq. (2). Here we choose the initial state of the system to be $\lvert\psi_{0}\rangle=\frac{1}{2}(\lvert ff\rangle+\lvert f+\rangle+\lvert+f\rangle+\lvert++\rangle)\otimes\lvert 0\rangle$ where the latter state refers to initial cavity vacuum. The gate fidelity is defined via $F=\langle\psi_{\tau}\lvert\rho_{q}(t)\lvert\psi_{\tau}\rangle$ in which $\lvert\psi_{\tau}\rangle=U(\tau)\lvert\psi_{0}\rangle$ and $\rho_{q}(t)$ is the reduced density operator of the system after tracing out the cavity degree of freedom. Also, we plot the phase evolutions of the four computational states in Fig. 2(b). It can be found in Fig. 2 that the populations of the four computational states remain unchanged during the gate evolution, while $\lvert++0\rangle$ obtains an accumulated phase of $\pi$ at the gate time $\tau=2\pi/\Delta$ ($k_{1}$=1 and $k_{2}=k_{3}=0$). Correspondingly, the gate fidelity reaches above 0.9999 at the instant $\tau$. For an experimentally feasible coupling $g/2\pi=50$ MHz Niemczyk et al. (2010), the gate time is estimated approximately as 0.16 ns.

In the numerical simulations above, we assume a particular two-qubit superposition state to evaluate the gate fidelity. Strictly speaking, using only a specific initial state is insufficient to test the gate performance. In view of this, we introduce the average fidelity metric that takes into account all possible initial inputs, which is defined based on a trace-preserving quantum operator, asNielsen (2002)

$$\bar{F}(\varepsilon,U)=\frac{\sum_{l=1}^{4^{N+1}}\mathrm{tr}\left[Uu_{l}^{{\dagger}}U^{\dagger}\varepsilon(u_{l})\right]+d^{2}}{d^{2}(d+1)},$$

(7)

where $u_{l}$ is the tensor of Pauli matrices $II,I\sigma_{x},\dots,\sigma_{z}\sigma_{z}$ in the computational basis $\{|f\rangle,|+\rangle\}$, $U$ is the perfect phase gate, $d=2^{N+1}$ for a $(N+1)$-qubit gate, and $\varepsilon$ is the trace-preserving quantum operation obtained through solving the master equation. Based on Eq. (7), we plot the average gate fidelity using the same set of parameters as above, as depicted by the hollow squares in Fig. 2(a). One can clearly see that the average fidelity at the gate time is also close to unity (precisely, above 0.9999). For simplicity, we use the initial-state-specified gate fidelity $F$ to characterize the gate against noises in the following sections.

As discussed above, a most promising pathway for implementating our proposal could be the current circuit QED platform. The proposed amplitude-shaped gates are realized based on a unitary $U(t)$ with a specific evolution period. This requires the control of the time-dependent qutrit-resonator coupling $g(t)$, the classical field drives on the qutrits $\Omega$, and the parametric drive of the resonator $\Omega_{p}$, when considering the practical implementation in a superconducting circuit architecture. Specifically, the classical field drives could be implemented by introducing square microwave pulses with sine ramp-up and ramp-down edges and specific pulse length Xu et al. (2018, 2020b), while the parametric drive could be implemented through modulating the magnetic flux threading the SQUID loop connected to the resonator Yamamoto et al. (2008); Zhong et al. (2013). Regarding the time-dependent coupling $g(t)$, there have been both theoretical Shapiro et al. (2015); Zhukov et al. (2016) and experimental Allman et al. (2010); Srinivasan et al. (2011) research concerning coherent tunable qubit-resonator coupling. Experimentally, the desired amplitude modulation on $g(t)$ could be implemented directly by adopting controlled voltage pulses generated by an arbitrary waveform generator (AWG) to tune the flux threading the SQUID loop of each qutrits Salathé et al. (2015). All drives on qutrits could be set to respective pulse envelopes and be performed synchronously for a specific duration.

We next give a brief evaluation of the gate performance using experimentally accessible parameters. For a flux qubit coupled to a CPW resonator, it is demonstrated that the coupling strength up to several hundreds of MHz is achievable Niemczyk et al. (2010). As an example, we assume a coupling strength of $g/2\pi=50$ MHz, which gives approximately $\Delta_{1}/2\pi=556.05$ MHz and $\Omega_{1}/2\pi=-139.01$ MHz for our 1st order ($k=1$) shaped phase gate when considering a squeezing parameter of $r_{p}=2.5$, and the corresponding gate time is calculated as $\tau_{1}=3.6$ ns. In addition, for the parameters assumed above, the other system parameters could be chosen as $\omega_{a}/2\pi=43.76$ GHz, $\omega_{p}/2\pi=5$ GHz, $(\omega_{e}-\omega_{g})/2\pi=2.5$ GHz, and $\omega/2\pi=2.5$ GHz. Based on the above parameters, together with an assumption of experimentally feasible decay rates $\kappa/2\pi=\gamma/2\pi=0.5$ MHz Yan et al. (2016); Leek et al. (2010), the shaped gate fidelity could reach above 0.99 at the gate time of about 3.6 ns.

Regarding the effects of the potential errors in real experiments on the gate, we simulate the 1st order shaped gate fidelity versus varying actual values of $t$ and $\Delta_{1}$, as plotted in Fig. 7. The upper surface (labelled as “S.I”) corresponds to the case with perfect control of the pulse amplitude and the classical field amplitude, i.e., $\{g_{m},\Omega_{1}\}$ precisely equals to $2\pi\times\{50,-139.01\}$ MHz, while the lower surface (labelled as “S.II”) depicts the case with slight deviation in these two amplitudes. Apparently, in the absence of all control errors, the shaped gate fidelity could achieve 0.9906, as marked by the square dot on S.I. When different control errors exist simultaneously, e.g., $t=3.69$ ns and $\{\Delta_{1},g_{m},\Omega_{1}\}=2\pi\times\{569.95,50.5,-140.4\}$ MHz, the corresponding gate fidelity would be reduced to 0.9851.

In conclusion, we have proposed a theoretical method for implementing fast two-qubit arbitrary phase gates and introduced amplitude modulation to improve the gate robustness against control errors and decoherence. The shaped gates provide enhanced robustness to errors in gate time and frequency detuning compared to the standard gate without amplitude-shaped coupling. In addition, the scheme presents intrinsic robustness against energy relaxations of the qutrits and in particular, significantly enhanced robustness against cavity decay is achieved with the shaped gates. Further improvement of the gate robustness would be possible by designing suitable pulse shapes with modulated amplitude and phase. We anticipate that our proposal could be helpful for future studies on this field and offer a feasible method for implementing efficient gate operations in quantum information protocols.

This work was supported by National Natural Science Foundation of China (NSFC) (11675046), Program for Innovation Research of Science in Harbin Institute of Technology (A201412), and Postdoctoral Scientific Research Developmental Fund of Heilongjiang Province (LBH-Q15060).

As seen in Eq. (2), the error term $H_{\mathrm{Err}}=\frac{g}{2}e^{-r_{p}}(a^{\dagger}-a)(\lvert e\rangle_{k}\langle g\lvert-\lvert g\rangle_{k}\langle e\lvert)$ describes undesired corrections to the ideal Rabi Hamiltonian. In principle, $H_{\mathrm{Err}}$ can be neglected completely for a large $r_{p}$ satisfying $e^{r_{p}}\to\infty$. Here we demonstrate numerically that a modest $r_{p}$ is able to supress $H_{\mathrm{Err}}$ sufficiently. To be specific, we plot in Fig. 8 the dynamical evolution of the fidelity between the exact state $\psi(t)$ and the ideal Rabi-model state $\psi_{\mathrm{Rabi}}(t)$ for several values of $r_{p}$, where $\psi(t)$ and $\psi_{\mathrm{Rabi}}(t)$ are obtained by solving the Hamiltonian (2) with and without $H_{\mathrm{Err}}$, respectively. Apparently, the fidelity keeps near unity over long time scales for $r_{p}=2.5$, which implies that the system described by the exact Hamiltonian faithfully reproduces the ideal Rabi-model evolution. This allows us to choose $r_{p}=2.5$ in the simulations, as shown in Fig. 2, where a high-fidelity gate can be achieved correspondingly.