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

We first explain how nonadiabatic holonomic gates arise [10, 11]. A quantum system is described by a N-dimensional state space and exposed to the Hamiltonian $H(t)$. Then it is assumed that there is a time-dependent L-dimensional subspace ${ \mathcal S }(t)=\mathrm{Span}{\{| {\psi }_{k}(t)\rangle \}}_{k=1}^{L}$, where $| {\psi }_{k}(t)\rangle$ satisfies the Schrödinger equation ${\rm{i}}| {\dot{\psi }}_{k}(t)\rangle =H(t)| {\psi }_{k}(t)\rangle$. ${ \mathcal S }(0)$ is taken as the computational space. The unitary transformation acting on ${ \mathcal S }(0)$ is a nonadiabatic holonomic gate if the following requirements are satisfied:

$$\begin{eqnarray}&&\begin{array}{l}({\rm{i}})\displaystyle \sum _{k=1}^{L}| {\psi }_{k}(\tau )\rangle \langle {\psi }_{k}(\tau )| =\displaystyle \sum _{k=1}^{L}| {\psi }_{k}(0)\rangle \langle {\psi }_{k}(0)| ,\\ (\mathrm{ii})\ \langle {\psi }_{k}(t)| H(t)| {\psi }_{l}(t)\rangle =0,\,k,l=1,2,\cdot \cdot \cdot ,\,L.\end{array}\end{eqnarray} \tag{ 1 }$$

Here, condition (i) entails that ${ \mathcal S }(t)$ undergoes a cyclic evolution and condition (ii) ensures a parallel transport with vanishing dynamical phases.

In this experiment, a single-shot protocol of nonadiabatic holonomic gate [16, 17] is realized. Consider a three-level superconducting Xmon consisting of three lowest levels $| 0\rangle$, $| e\rangle$ and $| 1\rangle$ with ladder configuration, as shown in figure 1(a). The states $| 0\rangle$ and $| 1\rangle$ are taken as the qubit computational basis, while the state $| e\rangle$ acts as an auxiliary state, with ω_0e and ω_e1 being energy differences between neighboring states. The transitions $| 0\rangle \leftrightarrow | e\rangle$ and $| e\rangle \leftrightarrow | 1\rangle$ are facilitated by two microwave pulses with pump frequency ω_p(t) and stocks frequency ω_s(t) [42]. In this work, the two microwave drives are off-resonant with detunings Δ_p(t) and Δ_s(t) , where Δ_p(t) = ω_0e − ω_p(t) and Δ_s(t) = ω_1e − ω_s(t) . In the two-photon resonance condition ω_p(t) + ω_s(t) = ω_0e + ω_e1, the Hamiltonian of the Xmon in the double-rotating frame and by using rotating wave approximation, reads

$$\begin{eqnarray}&&H(t)={{\rm{\Delta }}}_{p}(t)| e\rangle \langle e| +\displaystyle \frac{1}{2}\left[{{\rm{\Omega }}}_{p}(t)| e\rangle \langle 0| +{{\rm{\Omega }}}_{s}(t)| e\rangle \langle 1| +{\rm{h}}.{\rm{c}}.\right],\end{eqnarray} \tag{ 2 }$$

where ${{\rm{\Omega }}}_{p}(t)$ and Ω_s(t) are time-dependent envelopes and h.c. represents the Hermitian conjugate terms.

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To realize the nonadiabatic holonomic gates, the parameters in equation (2) are taken as

$$\begin{eqnarray}\begin{array}{rcl}{{\rm{\Delta }}}_{p}(t) & = & {\rm{\Omega }}(t)\sin \alpha ,\\ {{\rm{\Omega }}}_{p}(t) & = & {\rm{\Omega }}(t)\cos \alpha \cos \displaystyle \frac{\theta }{2},\\ {{\rm{\Omega }}}_{s}(t) & = & {\rm{\Omega }}(t)\cos \alpha \sin \displaystyle \frac{\theta }{2}{{\rm{e}}}^{-{\rm{i}}\varphi },\end{array}\end{eqnarray} \tag{ 3 }$$

where Ω(t) is time-dependent, and α, θ and φ are time-independent constants. Accordingly, the Hamiltonian in equation (2) can be expressed as

$$\begin{eqnarray}&&{ \mathcal H }(t)={\rm{\Omega }}(t)\sin \alpha | e\rangle \langle e| +\displaystyle \frac{1}{2}{\rm{\Omega }}(t)\cos \alpha \left(| e\rangle \langle b| +| b\rangle \langle e| \right),\end{eqnarray} \tag{ 4 }$$

with the bright state $| b\rangle =\cos \tfrac{\theta }{2}| 0\rangle +\sin \tfrac{\theta }{2}{{\rm{e}}}^{{\rm{i}}\varphi }| 1\rangle$. Here, a dark state is orthogonal to the bright state, as $| d\rangle =\sin \tfrac{\theta }{2}{{\rm{e}}}^{{\rm{i}}\varphi }| 0\rangle -\cos \tfrac{\theta }{2}| 1\rangle$. If the evolution period τ is taken to satisfy

$$\begin{eqnarray}&&{\int }_{0}^{\tau }\displaystyle \frac{{\rm{\Omega }}(t)}{2}{\rm{d}}{t}=\pi ,\end{eqnarray} \tag{ 5 }$$

the evolution operator can be obtained as

$$\begin{eqnarray}&&U(\tau )=| d\rangle \langle d| +{{\rm{e}}}^{-{\rm{i}}\pi (1+\sin \alpha )}| b\rangle \langle b| +{{\rm{e}}}^{-{\rm{i}}\pi (1+\sin \alpha )}| {e}\rangle \langle {e}| .\end{eqnarray} \tag{ 6 }$$

Consequently, an arbitrary one-qubit gate can be obtained by projecting the evolution operator onto the computational subspace, denoted by

$$\begin{eqnarray}&&{U}_{L}(\tau )={{\rm{e}}}^{-{\rm{i}}\gamma {\boldsymbol{n}}\cdot {\boldsymbol{\sigma }}/2},\end{eqnarray} \tag{ 7 }$$

where $\gamma =\pi (1+\sin \alpha )$, ${\boldsymbol{n}}=(\sin \theta \cos \varphi ,\sin \theta \sin \varphi ,\cos \theta )$, and ${\boldsymbol{\sigma }}=({\sigma }_{x},{\sigma }_{y},{\sigma }_{z})$ with σ_x, σ_y, σ_z being the Pauli operators acting on computational basis $| 0\rangle$ and $| 1\rangle$. Here, an unimportant global phase is neglected.

Next, ${U}_{L}(\tau )$ is demonstrated to be a nonadiabatic holonomic gate. First, condition (i) is satisfied as $U(\tau )(| 0\rangle \langle 0| +| 1\rangle \langle 1| ){U}^{\dagger }(\tau )=| 0\rangle \langle 0| +| 1\rangle \langle 1|$. Second, with the aid of the commutation relation $[{ \mathcal H }(t),U(t)]=0$, it can be verified that condition (ii) is satisfied as $\langle {{\rm{\Phi }}}_{i}(t)| { \mathcal H }(t)| {{\rm{\Phi }}}_{j}(t)\rangle$ $=\langle {i}| {U}^{\dagger }(t){ \mathcal H }(t)U(t)| j\rangle$ $=\,\langle {i}| { \mathcal H }(t)| j\rangle =0$, where $| {{\rm{\Phi }}}_{i(j)}(t)\rangle =U(t)| i(j)\rangle$ with $U(t)={{\rm{e}}}^{-{\rm{i}}{\int }_{0}^{t}{ \mathcal H }({t}^{{\prime} }){{\rm{d}}{t}}^{{\prime} }}$ and i(j) = 0, 1. Therefore, U_L(τ) is a nonadiabatic holonomic gate.

In the original protocol of nonadiabatic holonomic gates, the three-level system is controlled by two resonant pulses with Δ_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 nonzero and time-dependent 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.

An arbitrary single-qubit gate can be implemented using the single-shot nonadiabatic holonomic protocol by choosing specific gate parameters α, θ and φ. To demonstrate the arbitrariness of the protocol, single-qubit gates in the Clifford group (Clifford gates) are implemented. For a single qubit, the Clifford group consists of 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 figure 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, corresponding to the red and blue axes in figure 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 axes) and 2π/3- (green axes) rotations, the remaining 14 gates must be realized by combining 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 by the off-resonant pump and stocks drives.

The superconducting Xmon used in this work is an aluminum-based circuit operated at approximately 10 mK in a cryogen-free dilution refrigerator. The Xmon sample was fabricated on a silicon substrate using a standard nano-fabrication method. Four arms of the Xmon cross were connected to different lines for separate functions of coupling, control and readout. The lowest three energy levels of the Xmon were then utilized in the single-shot protocol. A readout resonator coupling the Xmon and a readout line were used for a dispersive measurement of the quantum state of the three-level qutrit. A 1 μs long microwave signal was sent to the sample with a frequency of f = 6.56 GHz. After interacting with the readout resonator, the signal was amplified by a Josephson parametric amplifier [43, 44] and a high electron mobility transistor. The signal was further digitalized and demodulated using an analog to digital convertor for high fidelity measurement. By heralding the ground state $| 0\rangle$ [45], the readout fidelity for the lowest three levels was F₀ = 99.5%, F_e = 92.3% and F₁ = 89.5%, respectively. For a single Xmon in this experiment, the lowest three levels are used as $| 0\rangle$, $| e\rangle$, and $| 1\rangle$ in the single-shot protocol. The relevant transition frequencies were ω_0e/2π = 4.849 GHz and ω_e1/2π = 4.597 GHz, and the nonlinearity was η = (ω_e1 − ω_0e)/2π = −252 MHz. The coherence of the qubit was characterized by energy relaxation time ${T}_{1}^{e0}=25.3\,\mu {\rm{s}}$, ${T}_{1}^{1e}=12.8$ μs, and the pure dephasing time ${T}_{\phi }^{e0}=28.1$ μs, ${T}_{\phi }^{1e}=13.4\,\mu {\rm{s}}$ was measured using 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/2- and 2π/3- rotations can be assigned with α = 0, −π/6 and −arcsin(1/3) , respectively. 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 the different gate parameters β, θ and φ. In principle, the form of Ω(t) can be arbitrary as long as it satisfies the constraint of equation (5). Due to general experimental restrictions, a square-shaped pulse was employed in the original proposal [10]. However, this requires a precise control of the starting and ending time, and the time jitter of a nonzero ending of the pulse may induce a population leakage. In this experiment, the form of Ω(t) is designed as ${\rm{\Omega }}(t)={{\rm{\Omega }}}_{0}{\sin }^{2}(\pi t/\tau )$, with the gate time τ set at 100 ns and the constraint of equation (5) as Ω₀τ = 4π. For this specified form of Ω(t) rather than using the square-shaped pulse, the microwave drive pulse is turned on and off smoothly with Ω(0) = Ω(τ) = 0, and the population leakage is suppressed. Flexibly-shaped detunings can be induced by changing the frequency of the driving pulses, as details in appendix B. Following this process, all single-qubit Clifford gates are realized using the nonadiabatic holonomic single-shot protocol.

In the Clifford group, holonomic gates with π-rotation were firstly implemented. In this unique case, the single-shot protocol reverts back to the original protocol. Accordingly, in the experiment, two resonant microwave pulses simultaneously drove the $| 0\rangle \leftrightarrow | e\rangle$ and $| e\rangle \leftrightarrow | 1\rangle$ level transitions with the pump pulse ${{\rm{\Omega }}}_{p}(t)={\rm{\Omega }}(t)\cos \tfrac{\theta }{2}$ and the drive pulse ${{\rm{\Omega }}}_{s}(t)={\rm{\Omega }}(t)\sin \tfrac{\theta }{2}$, respectively. The gate parameter α = 0 (i.e. Δ_p(t) = 0), yielded the rotation angle γ = π. The rotation axis of the quantum gates was then determined by choosing specific parameters θ and φ. To completely characterize the holonomic gates, we performed a QPT involving all three basis $| 0\rangle ,| e\rangle ,| 1\rangle$ [46], which is quantified by a reconstructed χ-matrix. As shown in figure 1(b), for the three-level QPT, 16 different initial states ρ_i were prepared by sequentially applying the identity I, the π/2-rotation X/2, Y/2 and the π-rotation X pulses to $| 0\rangle \leftrightarrow | e\rangle$ and $| e\rangle \leftrightarrow | 1\rangle$ transitions. Initialized states were then followed by a specified holonomic quantum gate in the Clifford group. Finally, a state measurement was performed with full quantum state tomography. The output state ρ_f was then extracted using the maximum likelihood estimation method. The process matrix χ was reconstructed from the input and output states by numerically solving the equation ${\rho }_{f}={\sum }_{m,n}{\chi }_{{mn}}{E}_{m}{\rho }_{i}{E}_{n}^{\dagger }$ [38]. The full set of nine orthogonal basis operators E_m was chosen as {I₀₁, X₀₁, Y₀₁, Z₀₁, X_0e, Y_0e, X_e1, Y_e1, I_e} [21, 46], in which the first four operators represent the operation between the computation subspace, $\{| 0\rangle ,| 1\rangle \}$. For the holonomic gate within the subspace, a population leakage to the auxiliary state $| e\rangle$ may occur due to imperfect microwave pulses and the energy relaxation/dephasing of both $| e\rangle$ and $| 1\rangle$. The leakage can be determined by traces of the reduced χ-matrix, $\tilde{\chi }$, which describes the process involving the computation states $| 0\rangle$ and $| 1\rangle$ [21].

As an example of the π-rotation gate, a Hadamard gate (H) was experimentally carried out with the setting parameters $\alpha =0$, θ = π/4 and φ = 0. The real and imaginary parts of χ-matrix for the Hadamard gate are provided in figures 2(a) and (b), respectively. The dominant elements in the subspace $\{| 0\rangle ,| 1\rangle \}$ is X₀₁ and Z₀₁, and the imaginary part of χ-matrix is close to zero, which are both consistent with the theoretical expectation. Moreover, the element ${\chi }_{{I}_{e},{I}_{e}}$ is close to one, which illustrates that the auxiliary state was almost 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, $\mathrm{Tr}(\,\tilde{\chi })$, which is about 0.96. The main leakage error arises from the imperfection of the microwave signal and the limited nonlinearity. Following the definition of fidelity [22, 47, 48]

$$\begin{eqnarray}&&F=\displaystyle \frac{| \mathrm{Tr}(\,({\chi }_{\exp }{\chi }_{\mathrm{th}}^{\dagger }))| }{\sqrt{\mathrm{Tr}(\,({\chi }_{\exp }{\chi }_{\exp }^{\dagger }))\mathrm{Tr}(\,({\chi }_{\mathrm{th}}{\chi }_{\mathrm{th}}^{\dagger }))}},\end{eqnarray} \tag{ 8 }$$

the fidelity of the process matrix can be calculated 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 figure 2(c), the fidelity for all π-rotation gates in single-qubit Clifford group are provided with specific gate parameters θ and φ. The operations of gates ${{ \mathcal C }}_{i}$ shown in this and following figures are provided in appendix A. The fidelities for all gates are above 99%, with an average fidelity 99.3%.

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Advancing beyond the π-rotation gates, the single-shot protocol was followed and holonomic gates with π/2-rotation in the Clifford group were implemented. To obtain a rotation angle γ = π/2, the off-resonant pump pulse and stocks pulse were simultaneously applied with the parameter α set as −π/6 and the detuning of Δ_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, the X/2 gate was demonstrated by setting α = −π/6, θ = π/2 and φ = 0. With the same QPT measurement as described previously, the process χ-matrix was reconstructed for X/2 gate and its real and imaginary parts are presented in figures 3(a) and (b), respectively. The χ-matrix of X/2 includes auto and cross correlations between operators I and X. The element of I_e, ${\chi }_{{I}_{e},{I}_{e}}$, is close to 1, demonstrating an insignificant effect on the auxiliary state during the gate operation. The population leakage to the auxiliary subspace is about 0.97, characterized by $\mathrm{Tr}(\,\tilde{\chi })$. The fidelity of the full QPT is $F(\tfrac{X}{2})=99.2 \%$, which is similar to the fidelity in π-rotation gates. Figure 3(c) shows fidelities of all the gates with π/2-rotation in single-qubit Clifford group. The experimental result shows that all fidelities are above 99% with an average fidelity of 99.2%.

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According to the original protocol of holonomic gates, a π/2-rotation gate can only be realized by sequentially combining two π-rotation gates. The fidelity of such a gate can be roughly estimated to be about 98%, with a similar fidelity of π-rotation gate used in the current experiment. With the single-shot protocol, the arbitrary gate can be implemented within a single step. By shortening the gate operation time, the accumulated environment-induced error is reduced in a combined operation, and the corresponding fidelity is increased.

To complete the whole Clifford group of single-qubit gates, eight 2π/3-rotation gates were realized using the single-shot protocol. The axes of these gates are the lines connecting the origin and face centers of the octahedron, as shown in figure 1(c). For all 2π/3-rotation gates, $\alpha =-\arcsin (1/3)$, leading to a rotation angle γ = 2π/3. The detuning satisfies Δ_p(t) = −Ω(t)/3, and the other two parameters θ and φ were determined by the corresponding rotation axis for each gate.

For example, gate ${{\boldsymbol{ \mathcal C }}}_{8}$ is an operation of 2π/3-rotation about the axis ${\boldsymbol{n}}=\tfrac{1}{\sqrt{3}}(1,1,1)$, which determines the parameters $\theta =\arccos (1/3)$ and φ = π/4. According to equation (7), the gate operation in the computation subspace for ${{ \mathcal C }}_{8}$ reads as:

$$\begin{eqnarray}U=\displaystyle \frac{1}{2}\left[\begin{array}{cc}1-{\rm{i}} & -1-{\rm{i}}\\ 1-{\rm{i}} & 1+{\rm{i}}\end{array}\right]=\displaystyle \frac{1}{2}\left(I-{\rm{i}}{\sigma }_{x}-{\rm{i}}{\sigma }_{y}-{\rm{i}}{\sigma }_{z}\right).\end{eqnarray} \tag{ 9 }$$

The χ-matrix for the gate ${{ \mathcal C }}_{8}$ is provided in figures 4(a) and (b). The real part of χ-matrix has a similar value in the X, Y, Z components, corresponding well with the theoretic expectation in equation (9). The QPT fidelity for gate ${{ \mathcal C }}_{8}$ is approximate to $F({{ \mathcal C }}_{8})=99.2 \%$, which is comparable to previous holonomic gates with other rotation angles. The population leakage characterized by the quantity $\mathrm{Tr}(\,\tilde{\chi })$ is about 98%. In figure 4(c), the fidelities for all Clifford gates with a 2π/3-rotation are presented, in which all the fidelities are larger than 99%, with an average fidelity reaching 99.1%.

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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 [36, 39–41]. The Clifford-based RB measurement was implemented to obtain the holonomic gate fidelity. Twenty-four Clifford holonomic gates were used in the RB experiment, with each gate individually implemented by choosing specific gate parameters α, θ, and φ.

A reference RB experiment was first performed. As shown in the pulse sequence in figure 5(a), the qubit was initially prepared at $| 0\rangle$ state, then a sequence of m Clifford gates were randomly chosen to drive the qubit. As 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, the average result of the remaining population as a function of m was obtained, which is also called a sequence fidelity. As shown in figure 5(b), the sequence fidelity can be fitted using the function [40]:

$$\begin{eqnarray}&&F={{Ap}}^{m}+B,\end{eqnarray} \tag{ 10 }$$

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, the depolarized parameter p_ref = 0.986 is obtained, which yields an average error r_ref = 0.007, or an average RB fidelity 99.3%.

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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 [41]. The pulse sequence for the interleaved RB experiment is provided in figure 5(a). At each step, the qubit was driven by a combination of a randomly selected Clifford gate and the target holonomic gate. After a reversed recovery gate was applied, the remaining population or the sequence fidelity was measured as a function of the number of steps m. Similarly, the sequence fidelity was fitted by the same function as in the reference RB experiment, leading to a new depolarized parameter p_gate.

The specific gate fidelity is given by F_gate = 1 − (1 − p_gate/p_ref)/2. The experimental results for seven gates are shown in figure 5(b). The chosen gates were the idle gate I, two π-rotation gates, X and H, two π/2-rotation gates X/2 and Z/2, and two 2π/3-rotation gates ${{ \mathcal C }}_{8}$ (rotation axis ${\boldsymbol{n}}=\tfrac{1}{\sqrt{3}}(1,1,1)$) and ${{ \mathcal C }}_{4}$ (rotation axis ${\boldsymbol{n}}=\tfrac{1}{\sqrt{3}}(1,1,-1)$). According to the interleaved RB experiment, the fidelities for X, H, X/2, Z/2, ${{ \mathcal C }}_{8}$ and ${{ \mathcal C }}_{4}$ are 99.2%, 99.0%, 99.3%, 99.4%, 99.0% and 99.3%, respectively. After benchmarking all the Clifford gates, RB fidelities for all the Clifford holonomic gates are obtained, each of which is larger than 99.0%.