Single-shot realization of nonadiabatic holonomic gates with a superconducting Xmon qutrit

Nonadiabatic holonomic quantum computation has received increasing attention due to its robustness against control errors as well as high-speed realization. The original protocol of nonadiabatic holonomic one-qubit gates has been experimentally demonstrated with superconducting transmon qutrit. However, the original protocol requires two noncommuting gates to realize an arbitrary one-qubit gate, which doubles the exposure time of gates to error sources and therefore makes the gates vulnerable to environment-induced decoherence. Single-shot protocol was subsequently proposed to realize an arbitrary one-qubit nonadiabatic holonomic gate. In this paper, we experimentally realize the single-shot protocol of nonadiabatic holonomic single qubit gates with a superconducting Xmon qutrit, where all the Clifford element gates are realized by a single-shot implementation. Characterized by quantum process tomography and randomized benchmarking, the single-shot gates reach a fidelity larger than 99%.

The original protocol [10,11] of nonadiabatic holonomic one-qubit gates has been experimentally demonstrated with a superconducting circuit [20], nuclear magnetic resonance [21], and nitrogen-vacancy centers in diamond [22,23]. However, based on the original protocol, a single step operation can only rotate the quantum state about arbitrary axes, with a fixed angle of π. An arbitrary one-qubit gate then requires two sequential steps, which doubles the exposure time of gates to error sources. A single-shot protocol of nonadiabatic holonomic gates is further proposed, in which the quantum state is rotated about arbitrary axes with variable angles [16,17]. Recently, the single-shot protocol of nonadiabatic holonomic gates were experimentally realized with nitrogen-vacancy centers in diamond [24,25] and nuclear magnetic resonance [26]. With a different approach, a single-loop protocol of holonomic gates is also proposed [18,28] and experimentally realized [27].
A superconducting circuit provides an appealing scalable platform for implementing nonadiabatic holonomic quantum computation. As a solid state system, the integrated circuit can be easily scaled up to a multi-qubit system, with each qubit controlled by individual lines. The superconducting Xmon is a high quality qubit with relative long coherence time, the design of which also balances the coherence, the connectivity, as well as the fast control [29][30][31]. In this paper, we experimentally realize the single-shot protocol of nonadiabatic holonomic one-qubit gates with a three-level superconducting Xmon qutrit.
An arbitrary single qubit gate can be realized with the single-shot protocol. Here we choose a group of single qubit Clifford gates as examples to present results. Excluding the identity operation, we classify the group to π-rotation, π/2-rotation, and 2π/3-rotation Clifford gates. The π-rotation gates are simultaneously driven by two resonant microwave pulses as in the original protocol [10,11], while the other gates are driven by two off-resonant pulses following the single-shot protocol [16,17]. By using both the quantum process tomography (QPT) [32] and randomized benchmarking (RB) [33][34][35], we demonstrate the realization of the single-shot gate with high fidelity.

Protocol
We first explain how a nonadiabatic holonomic gate arises [10,11]. Consider a quantum system described by a Ndimensional state space and exposed to the Hamiltonian H(t). Assume there is a time-dependent L-dimensional subspace S(t) = Span{|ψ k (t) } L k=1 , where |ψ k (t) satisfies the Schrödinger equation i|ψ k (t) = H(t)|ψ k (t) . We take S(0) as the computational space. If the following requirements are satisfied, the unitary transformation acting on S(0) is a nonadiabatic holonomic gate. Here, condition (i) entails that S(t) undergoes a cyclic evolution and condition (ii) ensures a parallel transport with vanishing dynamical phases. In our experiment, we realize a single-shot protocol of nonadiabatic holonomic gate [16,17]. Consider a three-level superconducting Xmon consisting of three lowest levels |0 , |e and |1 with ladder configuration, as shown in Fig. 1(a). The states |0 and |1 are taken as the qubit computational basis while the state |e acts as an auxiliary state, with ω oe and ω e1 being energy differences between neighboring states. The transitions |0 ↔ |e and |e ↔ |1 are facilitated by two microwave pulses with pump frequency ω p (t) and stocks frequency ω s (t) [36]. Here the two microwave drives are off-resonant with detunings ∆ p (t) and ∆ s (t), where ∆ p (t) = ω oe − ω p (t) and ∆ s (t) = ω 1e − ω s (t). In the two-photon resonance condition ω p (t) + ω s (t) = ω oe + ω e1 , the Hamiltonian of the Xmon, in the double-rotating frame and by using rotating wave approximation, reads where Ω p (t) and Ω s (t) are time-dependent envelopes and H.c. represents the Hermitian conjugate terms.
To realize the nonadiabatic holonomic gates, the parameters in Eq.
In the original protocol of nonadiabatic holonomic gates, the three level system is controlled by two resonant pulses with ∆ p (t) = 0, which is equivalent to a fixed rotation angle of γ = π. In the single-shot protocol, the two off-resonant pulses are applied with a variable detuning. An arbitrary rotation angle γ can be obtained. In other words, an arbitrary one-qubit gate can be realized in a single-shot implementation. The original protocol is a specific case of the current single-shot protocol.
Single Qubit Clifford Group An arbitrary single qubit gate can be implemented with the single-shot nonadiabatic holonomic protocol, by choosing specific gate parameters α, θ and ϕ. To demonstrate the arbitrariness of the protocol, we implement single qubit gates in the Clifford group (Clifford gates). For a single qubit, Clifford gates consist 24 rotations preserving quantum states along vertexes of an octahedron in the Bloch sphere. The rotation axes are lines connecting the origin of the Bloch sphere and a face center, vertex or midpoint of an edge of the octahedron, as shown in Fig. 1(c). Classified by the rotation angle, the single qubit Clifford gates can be divided into four sets: identity, π-rotation, π/2-rotation and 2π/3-rotation. For the set of π-rotation, there are nine single qubit gates in the Clifford group, red axis and blue axis in Fig. 1(c) . This set of gates can be realized by the original nonadiabatic holonomic protocol, and implemented with a single step of resonant pump and stocks drives. However, for the sets of π/2-(blue axis) and 2π/3-(green axis) rotations, the remaining 14 gates have to be realized by combing two π-rotation gates in the original nonadiabatic holonomic protocol. With the single-shot protocol, these two sets can be implemented with a single step of off-resonant pump and stocks drives.

Experimental Parameters
The superconducting Xmon used in this work is an aluminum-based circuit operated at about 10 mK in a cryogen-free dilution refrigerator. For a single Xmon in this experiment, the lowest three levels are used as |0 , |e , and |1 in the singleshot protocol. The relevant transition frequencies are ω 0e /2π = 4.849 GHz and ω e1 /2π = 4.597 GHz, and the nonlinearity η = (ω e1 − ω 0e )/2π = −252 MHz. The coherence of the qubit is characterized by energy relaxation time T e0 1 = 25.3µs, T 1e 1 = 12.8µs, and the pure dephasing time T e0 φ = 28.1µs, T 1e φ = 13.4µs measured with a Ramsey interference experiment. In the single-shot nonadiabatic holonomic protocol, the rotation angle γ is determined by the time-independent parameter α. For the single qubit Clifford gates, the set of γ = π, π/2and 2π/3-rotations are accordingly assigned with α = 0, −π/6 and − arcsin(1/3). The rotation axis for each gate is determined by two other time-independent parameters θ and ϕ. The same time-dependent pulse envelope Ω(t) is shared for all the gates regardless of different gate parameters β, θ and ϕ. π-rotation Clifford gates We initially implement holonomic gates with π-rotation in the Clifford group. In this special case, the single-shot protocol falls back to the original protocol. Accordingly, two resonant microwave pulses simultaneously drive the |0 ↔ |e and |e ↔ |1 level transitions with the pump pulse Ω p (t) = Ω(t) cos θ 2 and the drive pulse Ω s (t) = Ω(t) sin θ 2 , respectively. The gate parameter α = 0 (i.e. ∆ p (t) = 0), yielding the rotation angle γ = π. The rotation axis of the quantum gates is determined by choosing specific parameters θ and ϕ. To completely characterize the holonomic gates, we perform a quantum process tomography (QPT) involving all three basis |0 , |e , |1 [37], which is quantified by a reconstructed χ-matrix. As shown in Fig. 1(b), for the 3level QPT, we prepare 16 different initial states ρ i by sequentially applying the identity I, the π/2-rotation X/2, Y/2 and the π-rotation X pulses to |0 ↔ |e and |e ↔ |1 transitions. Initialized states are then followed by a specified holonomic quantum gate in Clifford group. Finally we perform a state measurement with full quantum state tomography (QST). The output state ρ f is extracted using the maximum likelihood estimation method. The process matrix, χ, is reconstructed from the input and output states by numerically solving the equation ρ f = m,n χ mn E m ρ i E † n [32]. The full set of nine orthogonal basis operators E m is chosen as {I 01 , X 01 , Y 01 , Z 01 , X 0e , Y 0e , X e1 , Y e1 , I e } [20,37]. The first four operators represent the operation between the computation subspace, {|0 , |1 }. For the holonomic gate within the subspace, a population leakage to the auxiliary state |e may happen due to nonideal microwave pulses, and energy relaxation/dephasing of both |e and |1 . The leakage can be determined by the trace of the reduced χ-matrix,χ, which describes the process involving the computation states |0 and |1 [20].
As an example of the π-rotation gate, a Hadamard gate (H) is experimentally realized and demonstrated, with the setting parameters α = 0, θ = π/4 and ϕ = 0. The real and imaginary parts of χ-matrix for the Hadamard gate are shown in Fig. 2(a) and Fig. 2(b), respectively. The dominant elements in the subspace {|0 , |1 } is X 01 and Z 01 , and the imaginary part of χ-matrix is close to zero, which are both the same as the theoretical expectation. Moreover, the element χ I e ,I e is close to one, which represents that the auxiliary state is nearly unaffected during the holonomic gate operation, as described in the single-shot protocol. The population leakage of Hadamard gate is described by the trace of the reduced χ-matrix, Tr(χ), which is about 0.96. The main leakage error comes from the imperfection of the microwave signal and the limited nonlinearity. Following the definition of fidelity [21,38,39], we calculate the fidelity of the process matrix using the ideal process matrix χ th as a reference. For the Hadamard gate, the fidelity of QPT F(H) reaches 99.2%, and the error of the process is less than 1%. In Fig. 2(c), we show the fidelity for all π-rotation gates in single qubit Clifford group with specific gate parameters θ and ϕ. The fidelities for all these gates are above 99%, with an average fidelity 99.3%. π/2-rotation Clifford gates To go beyond the π-rotation gates, we follow the single-shot protocol and implement holonomic gates with π/2-rotation in the Clifford group. To obtain a rotation angle γ = π/2, the off-resonant pump pulse and stocks pulse are simultaneously applied, with the parameter α set as −π/6 and the detuning ∆ p (t) = −Ω(t)/2. Different π/2-rotation Clifford gates are specified by their corresponding rotation axes, which determine the parameters θ and ϕ.
As an example of π/2-rotation gates, following we realize and demonstrate the X/2 gate by setting α = −π/6, θ = π/2 and ϕ = 0. With the same QPT measurement as described previously, we reconstruct the process χ-matrix for X/2 gate and present its real and imaginary parts in Fig. 3(a) and Fig. 3(b), respectively. The χ-matrix of X/2 includes auto and cross correlations between operators I and X. The element of I e , χ I e ,I e , is close to 1, showing neglectable effect to the auxiliary state during the gate operation. The population leakage to the auxiliary subspace is about 0.97, characterized by Tr(χ). The fidelity of the full QPT is F( X 2 ) = 99.2%, which is similar to the fidelity in π-rotation gates. In Fig. 3(c), we show the fidelities of all the gates with π/2-rotation in single qubit Clifford group. From the experimental result, all the fidelities are above 99% with an average fidelity 99.2%.
With the original protocol of holonomic gates, a π/2-rotation gate can only be realized by sequentially combining two πrotation gates. The fidelity of such a gate can be roughly estimated to be about 98%, with a similar fidelity of π-rotation gate as in our experiment. With the single-shot protocol, the arbitrary gate can be implemented within a single step. By shortening the gate operation time, we reduce the accumulated environment-induced error in a combined operation, and increase the corresponding fidelity.

2π/3-rotation Clifford gates
To complete the whole Clifford group of single-qubit gates, we realize and demonstrate eight 2π/3-rotation gates with the single-shot protocol. The axes of these gates are the lines connecting the origin and face centers of the octahedron, as shown in Fig. 1(c). For all the 2π/3-rotation gates, we set α = − arcsin(1/3), leading to a rotation angle γ = 2π/3. The detuning satisfies ∆ p (t) = −Ω(t)/3, and the other two parameters θ and ϕ are determined by the corresponding rotation axis for each gate.
For example, the gate C 8 is an operation of 2π/3-rotation about the axis n = 1 √ 3 (1, 1, 1), which determine the parameters θ = arccos(1/3) and ϕ = π/4. From Eq. (7), the gate operation in the computation subspace for C 8 reads as: In Fig. 4(a) and Fig. 4(b), we show the χ-matrix for the gate C 8 . The real part of χ-matrix shows similar value in the X, Y, Z components, agreeing with the theoretic expectation in Eq. (9). The QPT fidelity for the gate C 8 is about F(C 8 ) = 99.2%, which is comparable to the previous holonomic gates with other rotation angles. And the population leakage characterized by the quantity Tr(χ) is about 98%. In Fig. 4(c), we present the fidelities for all the Clifford gates with a 2π/3-rotation. All the fidelities are larger than 99%, with an average fidelity reaching 99.1%.

Randomized Benchmarking
Randomized benchmarking (RB) is another systematic method to extract the quantum gate fidelity. The gate fidelity in the RB measurement is separately quantified by excluding errors in state preparation and measurement [30,[33][34][35]. In this section, we perform the Clifford-based RB measurement to obtain the holonomic gate fidelity. Twenty-four Clifford holonomic gates are used in the RB experiment. Each gate is individually implemented by choosing specific gate parameters α, θ, and ϕ.
A reference RB experiment is performed first. As shown in the pulse sequence in Fig. 5(a), the qubit is initially prepared at |0 state, then a sequence of m Clifford gates are randomly chosen to drive the qubit. Since the Clifford group is a closed set, a recovery gate can be defined and finally applied to reverse the operation of m Clifford gates. The remaining population of the initial state is measured afterwards. After repeating this random operation sequence k (= 50 in our experiment) times, we obtain the average result of remaining population as a function of m, which is also called a sequence fidelity. As shown in Fig. 5(b), the sequence fidelity can be fitted using the function [34]: where F is the sequence fidelity, p is a depolarizing parameter, and the parameters A and B absorb the error in state preparation and measurement. The average error r over the randomized Clifford gates is given by r = (1 − p)/2. From the fitting of the reference RB measurement, we obtain the depolarized parameter p ref = 0.986, which yields an average error r ref = 0.007, or an average RB fidelity 99.3%.
The reference RB experiment can only give an average gate fidelity over the Clifford gates. For a specific gate, interleaved RB experiment can be applied to determine the gate fidelity [35]. In Fig. 5(a), we show the pulse sequence for the interleaved RB experiment. At each step, the qubit is driven by a combination of a randomly selected Clifford gate and the target holonomic gate. After a reversed recovery gate is applied, the remaining population or the sequence fidelity is measured as a function of the number of steps m. Similarly, the sequence fidelity is fitted by the same function as in the reference RB experiment, leading to a new depolarized parameter p gate .

III. DISCUSSION
From the QPT and RB experiments, we verify that all the single qubit Clifford gates can be implemented by single-shot protocol with high fidelity. It is confirmed that all the fidelities for single qubit gates are higher than 99.0%. We suggest that the main error comes from the decoherence and the energy relaxation, which is confirmed by the fidelity of the idle gate, F = 99.1%. We obtain similar fidelities for π-rotation gates and gates with other rotation angles, which means that the control errors are similar for resonant and off-resonant pulses.
The single qubit gates in our experiment are compatible with the previously proposed two-qubit nonadiabatic holonomic gate. By combining two-qubit gates with the current results, we can obtain a universsal nonadiabatic holonomic quantum computation in the future.

IV. METHODS
The Xmon sample is fabricated on a silicon substrate, with a standard nano-fabrication method. Four arms of the Xmon cross are connected to different lines for separate functions of coupling, control and readout. The lowest three energy levels of the Xmon are utilized in the single-shot protocol. A readout resonator couples the Xmon and a readout line for a dispersive measurement of the quantum state of the three-level qutrit. A 1 µs-long microwave signal is send to the sample, with frequency f = 6.56 GHz. After interacting with the readout resonator, the signal is amplified by a Josephson parametric amplifier [40,41] and a high electron mobility transistor (HEMT). The signal is further digitalized and demodulated by an analog to digital convertor for a high fidelity measurement. By heralding the ground state |0 [42], the readout fidelity for the lowest three levels are F 0 = 99.5%, F e = 92.3% and F 1 = 89.5%, respectively.
The operations of gates C i shown in the figures from Fig. 2 to Fig. 5 are given in the supplementary information.
V. ACKNOWLEDGEMENT TABLE I. The 23 single qubit Cliffords with the name C i and the rotation axis n excluding the identity operation I (C 0 ). Here X/2 donates a π/2 rotation over the X axis with unitary R X (π/2) = exp(−iπσ X /4).

VI. CLIFFORD GATES
We implement 24 single qubit Clifford gates through single-shot protocol in our experiment. To simplify the notation, we use the C i , (i = 0, 1, ..., 23) as the names of the Clifford gates with the rotation axes and rotation angles shown in the Tab. I.