1. INTRODUCTION

When a nondegenerate quantum system makes a cyclic evolution in the Hilbert space, it will pick up a phase, which, in general, is contributed by both the dynamical and geometric effects. The dynamical part is the time integral of the energy, while the geometric one depends upon the area enclosed by the loop that the quantum state traverses in the Hilbert space. This effect, discovered by Berry in cyclic and adiabatic evolutions [1], has been generalized to nonadiabatic [2] and degenerate [3] cases. If a system has degenerate energy levels, the cyclic evolution of the corresponding degenerate subspaces will produce a matrix-valued quantum state transformation that is path dependent and referred to as non-Abelian geometric phase or holonomy [3]. The Berry phase and holonony depend upon the global geometry of the associated loops and have intrinsic resistance to certain kinds of small errors, suggesting that quantum gates based on geometric operations have practical advantages as compared to dynamical gates [4–7]. In particular, it was shown that all of the elementary one- and two-qubit gates needed for accomplishing any quantum computation task could be achieved with Berry phase and holonomic transformations, offering a possibility for implementations of geometric quantum computation [8,9].

The conditional Berry phase was first observed in nuclear magnetic resonance systems [10]. However, the relatively long operation time associated with an adiabatic evolution represents an unfavorable condition for the implementation of geometric quantum computation with such controlled phase gates. As such, geometric effects without the adiabatic restriction are highly desirable for the implementation of quantum logic gates that are robust against noises [11–16]. So far, nonadiabatic geometric controlled-phase gates have been realized in ion traps [17–20] and superconducting circuits [21–23]. On the other hand, Sjöqvist et al. have proposed an approach for realizing a universal set of elementary gates based on nonadiabatic holonomies [24], whose robustness against noises has been analyzed [25,26]. Following this approach, a universal gate set involving two non-commutable single-qubit gates and a two-qubit controlled-NOT (CNOT) gate have been experimentally realized with nuclear magnetic resonance [27] and solid-state spins [28,29]. Several groups have demonstrated holonomic single-qubit gates in superconducting circuits [30–33], which represent a promising platform for quantum computation [34]. Recently, Egger et al. reported a holonomic operation for producing entangled states in a superconducting circuit [35]. However, a non-Abelian geometric entangling gate necessary for constructing a universal holonomic gate set has not been implemented in such scalable systems. More recently, Han et al. reported a universal set of time-optimal geometric gates with superconducting qubits [36], where single-qubit gates were realized using non-Abelian geometric phase, but the two-qubit gate was based on Abelian geometric phase.

In this paper, we propose and experimentally demonstrate a scheme for realizing a non-adiabatic, non-Abelian geometric CNOT gate for two qubits, one acting as the control qubit and the other as the target qubit. The two qubits are strongly coupled to a resonator, so that the energy levels of the target qubit depend on the state of the control qubit. This conditional energy-level shift enables the target qubit to be resonantly driven by classical fields, conditional on the state of the control qubit. With a suitable setting of the parameters, these classical fields can drive the degenerate subspace spanned by the two basis states of the target qubit to undergo a conditional cyclic evolution, realizing a CNOT gate between these two qubits. We realize this holonomic gate in a superconducting multi-qubit processor, where any two qubits can be selectively coupled to a common resonator but effectively decoupled from other qubits through frequency tuning. This flexibility enables direct implementation of holonomic gates between any pair of qubits on the chip, without the restriction of nearest-neighbor couplings. The measured process fidelity for the CNOT gate is above 0.9. With further improvements in the device design and fabrication, as confirmed by our numerical simulations, the gate fidelity can be significantly increased. Our scheme is applicable to other spin-boson systems, such as cavity QED and ion traps [37].

2. THEORETICAL MODEL

The system under consideration is composed of two qutrits coupled to a resonator. Each qutrit has three basis states, as shown in Fig. 1(a), with $| g \rangle$ and $| f \rangle$ serving as two logic states of a qubit, and $| e \rangle$, lying between $| g \rangle$ and $| f \rangle$, used as an auxiliary state for realizing the controlled logic operation. For simplicity, we will refer to the qutrits as qubits. As will be shown, the control qubit (${ Q}_1$) remains in its computational space, while the target qubit (${ Q}_2$) has a probability of being populated in the auxiliary level $| e \rangle$ during the gate operation. The transition $| g \rangle \leftrightarrow | e \rangle$ of each qubit resonantly interacts with the resonator, while $| f \rangle$ state is effectively decoupled from the resonator. In the interaction picture, the Hamiltonian describing the qubit–resonator interaction is given by

 

Fig. 1. Energy level configuration and excitation scheme for the two-qubit CNOT gate. The control and target qubits are denoted as ${Q}_1$ and ${Q}_2$, respectively. (a) Bare energy levels of ${Q}_2$ and frequencies of the drives. The quantum information of each qubit is encoded in states $| g \rangle$ and $| f \rangle$, with the auxiliary state $| e \rangle$ used for realizing the controlled-NOT gate. The transitions $| {{g_2}} \rangle \leftrightarrow | {{e_2}} \rangle$ and $| {{e_2}} \rangle \leftrightarrow | {{f_2}} \rangle$ of ${Q}_2$ are driven by classical fields of angular frequencies $({{\omega _r} - {\lambda _2}})$ and $({{\omega _{f,2}} - {\omega _r} + {\lambda _2}})$, respectively. Here ${\omega _r}$ is the angular frequency of the resonator that is strongly coupled to $| {{g_2}} \rangle \leftrightarrow | {{e_2}} \rangle$ with the coupling strength ${\lambda _2}$, and $\hbar {\omega _{f,2}}$ is the energy spacing between $| {{f_2}} \rangle$ and $| {{g_2}} \rangle$. (b) Dressed states and energy levels with ${Q}_1$ initially in $| {{f_1}} \rangle$. When being initially in $| {{f_1}} \rangle$, ${Q}_1$ is effectively decoupled from the resonator due to the large detuning. The strong coupling between ${Q}_2$ and the resonator results in dressed states $| {\psi _n^ \pm} \rangle$, whose energy levels are nonlinearly dependent on the coupling strength. The two driving fields are on-resonance with the transitions $| {{g_2}0} \rangle \leftrightarrow | {\psi _1^ -} \rangle$ and $| {\psi _1^ -} \rangle \leftrightarrow | {{f_2}0} \rangle$, respectively, but highly detuned from other transitions. (c) Dressed states and energy levels with ${Q}_1$ initially in $| {{g_1}} \rangle$. If initially in $| {{g_1}} \rangle$, ${Q}_1$, together with ${Q}_2$, is strongly coupled to the resonator, resulting in three dressed states $| {\Phi _1^ \pm} \rangle$ and $| {\Phi _1^0} \rangle$ in the single-excitation subspace. The driving fields are highly detuned from transitions of $| {{g_1}{g_2}0} \rangle$ and $| {{g_1}{f_2}0} \rangle$ to these dressed states.

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The strong couplings between the qubits and the resonator produce dressed states, whose energy levels depend on the total excitation number as well as on the number of qubits being initially populated in $| g \rangle$. When the control qubit is in the state $| {{f_1}} \rangle$, it does not interact with the resonator, and the coupling between the target qubit and the resonator is described by the Jaynes–Cummings model, whose eigenstates are given by

When ${Q}_1$ is initially in the state $| {{g_1}} \rangle$, it is also strongly coupled to the resonator, and there are three dressed states in the single-excitation subspace:

3. EXPERIMENTAL IMPLEMENTATION

The experiment is performed in a superconducting circuit involving five frequency-tunable qubits, labeled from ${Q}_1$ to ${Q}_5$, coupled to a resonator with a fixed frequency ${\omega _r}/2\pi = 5.584\;{\rm GHz}$ [21,38,39]. In our experiment, ${Q}_1$ and ${Q}_2$, whose anharmonicities are $2\pi \times 242\;{\rm MHz}$ and $2\pi \times 249\;{\rm MHz}$, are used as the control and target qubits, respectively. The on-resonance coupling strengths of the $g\! -\! e$ transitions of ${Q}_1$ and ${Q}_2$ to the resonator are respectively ${\lambda _1}= 2\pi \times 20.8\;{\rm MHz}$ and ${\lambda _2} = 2\pi \times 19.9\; {\rm MHz}$. The energy relaxation time ${T}_1$ and pure Gaussian dephasing time ${T}_2^ *$ for the basis state $| f \rangle$ of ${ Q}_1$ (${Q}_2$) are $13.0\, (10.7)\;\unicode{x00B5} {\rm s}$ and $2.1\,(1.5)\;\unicode{x00B5} {\rm s}$, while those for the intermediate state $| e \rangle$ are $23.9\,(15.9)\;\unicode{x00B5} {\rm s}$ and $2.7\,(2.1)\;\unicode{x00B5} {\rm s}$, respectively. The other qubits are on far off-resonance with the resonator so that their interactions with the resonator are effectively switched off throughout the gate operation. We note that during the gate operation, the two qubits have a probability of being populated in $| {{f_1}{e_2}} \rangle$, which is significantly coupled to $| {{e_1}{f_2}} \rangle$ via virtual photon exchange, as the two qubits almost have the same anharmonicity $\alpha \simeq 2\pi \times 240\; {\rm MHz}$. To suppress this coupling, ${Q}_1$ should be detuned from ${Q}_2$ by an amount much larger than ${\lambda _1}{\lambda _2}/\alpha$. This detuning slightly changes the energy level configuration of the dressed states associated with ${Q}_1$’s initial state $| {{g_1}} \rangle$, but does not affect the gate dynamics.

As shown in Fig. 2, the experiment starts with the initialization of ${Q}_1$ and ${Q}_2$ to the ground state $| g \rangle$ at their idle frequencies 5.47 GHz and 5.43 GHz, respectively, which is followed by the preparation of each qubit in one of the six states $\{| g \rangle$, $({| g \rangle - i| f \rangle})/\sqrt 2$, $({| g \rangle + i| f \rangle})/\sqrt 2$, $({| g \rangle + | f \rangle})/\sqrt 2$, $({| g \rangle - | f \rangle})/\sqrt 2$, $| f \rangle \}$. Except $| g \rangle$, each of the other single-qubit states is produced by a $g\! -\! e\;\pi /2$- or $\pi$-pulse followed by a $e\! -\! f\;\pi$-pulse.

 

Fig. 2. Pulse sequence. Before the gate operation, both qubits are initialized to their ground states at the corresponding idle frequencies, where single-qubit rotations are performed to prepare them in a product state. Then a ${Z}$ pulse is applied to ${Q}_1$, tuning $| {{g_1}} \rangle \leftrightarrow | {{e_1}} \rangle$ close to the resonator’s frequency; ${Q}_2$ is subjected to a ${Z}$ pulse, which brings $| {{g_2}} \rangle \leftrightarrow | {{e_2}} \rangle$ to the resonator’s frequency, and a driving pulse involving two frequency components on-resonance with the transitions $| {{g_2}0} \rangle \leftrightarrow | {\psi _1^ -} \rangle$ and $| {\psi _1^ -} \rangle \leftrightarrow | {{f_2}0} \rangle$. After the CNOT gate, realized with these pulses, both qubits are tuned back to their idle frequencies for quantum state tomography.

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After these effective single-qubit rotations, these two qubits are prepared in a product state. We then apply square Z pulses to both qubits, tuning their $| g \rangle \leftrightarrow | e \rangle$ transition frequencies to 5.58 GHz and 5.584 GHz and thus switching on their interactions with the resonator. Accompanying these ${Z}$ pulses, a driving pulse composed of two components with frequencies of 5.565 GHz and 5.369 GHz is applied to ${Q}_2$, resonantly connecting the computational states $| {{g_2}0} \rangle \;$ and $| {{f_2}0} \rangle$ to the dressed state $| {\psi _1^ -} \rangle$. The Rabi frequencies of these driving fields are ${\Omega _{\textit{ge}}} = {\Omega _{\textit{ef}}} = 2\pi \times 2.2\;{\rm MHz}$. Since the resonator is initially in the vacuum state, the system dynamics is governed by the effective Hamiltonian (5) and the time evolution given by Eq. (10). After a duration of $\tau = 205\; {\rm ns}$, the CNOT gate is realized.

One of the most important features of the CNOT gate is that it can convert a two-qubit product state into an entangled state. In particular, when the control qubit is initially in the superposition state $({| {{g_1}} \rangle + | {{f_1}} \rangle})/\sqrt 2$ and the target state in $| {{g_2}} \rangle$, they will evolve to the maximally entangled state $({| {{g_1}{g_2}} \rangle + | {{f_1}{f_2}} \rangle})/\sqrt 2$ after this gate. We measure this output state by quantum state tomography. This is realized by subsequently biasing each of the two qubits back to its idle frequency right after the gate operation, applying an $e\! -\! f\;\pi$-pulse to each qubit, and measuring its state along one of the three orthogonal ($x$, $y$, and $z$) axes of the corresponding Bloch sphere with respect to the basis $\{{| g \rangle ,| e \rangle} \}$. The $z$ measurement is directly realized by state readout, while the $x\;(y)$ measurement is realized by the combination of a $g\! -\! e\;\pi /2$-pulse and state readout. The reconstructed output two-qubit density matrix is displayed in Fig. 3, which has a fidelity of $0.935 \pm 0.016$ to the ideal maximally entangled state, and a concurrence of $0.888 \pm 0.029$.

 

Fig. 3. Measured density matrix of the output state with the input state $({| {{g_1}{g_2}} \rangle + | {{f_1}{g_2}} \rangle})/\sqrt 2$. Each matrix element is characterized by two colorbars, one for the real part and the other for the imaginary part. The black wire frames denote the matrix elements of the ideal output states.

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To fully characterize the performance of the implemented CNOT gate, we prepare a full set of 36 distinct two-qubit input states before the two-qubit gates, and measure these states and the corresponding output states. With these measured results, the process matrix for the gate operation is reconstructed. The measured process matrix, ${\chi _{{\rm meas}}}$, is presented in Fig. 4. The gate fidelity, defined as $F = tr({{\chi _{{\rm id}}}{\chi _{{\rm meas}}}})$, is $0.905 \pm 0.008$, where ${\chi _{{\rm id}}}$ is the ideal process matrix. The measured fidelity is in good agreement with the numerical simulation based on the Lindblad master equation, which yields a fidelity of 0.908. One of the error sources is the transitions from $| {{g_1}{g_2}0} \rangle$ and $| {{g_1}{f_2}0} \rangle$ to $| {\Phi _1^0} \rangle$ and $| {\Phi _1^ \pm} \rangle$ and the transition from $| {\psi _1^ -} \rangle$ to $| {\psi _2^ -} \rangle$ induced by the drive, which causes quantum information leakage to the noncomputational space. Such a leakage error can be mitigated through improvement of the qubit’s nonlinearity or by balancing the drive amplitude and the gate operation time provided the qubits’ coherence is bettered, which allows the gate fidelity to be increased by about 6.5% (see Supplement 1). On the other hand, the qubits’ energy relaxation and their dephasings contribute about 1.8% and 1.6% of the error, respectively. Our further numerical simulations show that the CNOT gate with a fidelity above 99% can be obtained with sufficiently large qubit nonlinearity ${\alpha _j}$ and qubit-resonator coupling strength ${\lambda _j}$. For instance, with the parameters ${\lambda _j}/2\pi = 110\; {\rm MHz}$, ${\alpha _j}/2\pi \; = - 3.69\; {\rm GHz}$ [40,41], ${\Omega _{\textit{ge}}}/2\pi = {\Omega _{\textit{ef}}}/2\pi = 5.9\; {\rm MHz}$, ${T_1} = 60\;\unicode{x00B5} {\rm s}$, and $T_{2,j}^ * = 86\;\unicode{x00B5} {\rm s}$, we find a CNOT gate with the operation time of about 87 ns and fidelity of 0.991, which is at the surface code threshold for fault tolerance [42–44]. We note this gate is robust against the frequency fluctuations of the driving fields. Suppose that the angular frequencies of these drives deviate from the desired values by an amount of $\delta \omega = 2\pi \times 100\; {\rm kHz}$. The infidelity incurred by this deviation is about ${[\pi {(\delta \omega)^2}/(8\Omega _{ge/ef}^2)]^2} \simeq 0.1\%$.

 

Fig. 4. Measured process matrix for the realized CNOT gate. The process matrix is measured by preparing a set of 36 distinct input product states in the computational basis $\{{| {{g_1}{g_2}} \rangle ,| {{g_1}{f_2}} \rangle ,| {{f_1}{g_2}} \rangle ,| {{f_1}{f_2}} \rangle} \}$ and reconstructing the density matrices for these states and for the output states produced by the CNOT gate. The $| e \rangle$-state populations of ${Q}_1$ and ${Q}_2$, averaged over the 36 output states, are 2.2% and 2.8%, respectively.

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4. CONCLUSION

In conclusion, we have proposed and demonstrated a scheme for implementing a non-Abelian geometric gate between two superconducting qubits, whose ground and second excited states act as the computational basis states. The conditional dynamics is realized by resonantly driving the transitions between the basis states of the target qubit to the single-excitation dressed states formed by this qubit and the resonator. This entangling gate, together with the previously demonstrated non-Abelian geometric single-qubit gates [30–33], constitutes a universal set of holonomic gates for realizing quantum computation with superconducting qubits. The method can be directly applied to other systems composed of qubits coupled to a bosonic mode, including cavity QED and ion traps.