The experimental realization of high-fidelity 'shortcut-to-adiabaticity' quantum gates in a superconducting Xmon qubit

A single-qubit of $\{| 0\rangle ,| 1\rangle \}$ can be mapped onto a spin-1/2 particle $\{| \uparrow \rangle ,| \downarrow \rangle \}$ driven by an external field [1]. In the rotating frame, the time-dependent Hamiltonian is written as

$$\begin{eqnarray}&&{H}_{0}(t)={\rm{\hslash }}{{\boldsymbol{B}}}_{0}(t)\cdot {\boldsymbol{\sigma }}/2,\end{eqnarray} \tag{ 1 }$$

where ${{\boldsymbol{B}}}_{0}(t)={\rm{\Omega }}(t)({\rm{\sin }}\theta (t){\rm{\cos }}\phi (t),{\rm{\sin }}\theta (t){\rm{\sin }}\phi (t),{\rm{\cos }}\theta (t))$ is the vector of an external field and ${\boldsymbol{\sigma }}=({\sigma }_{x},{\sigma }_{y},{\sigma }_{z})$ is the vector of Pauli matrices. The amplitude Ω(t), the polar angle θ(t) and the azimuthal angle ϕ(t) are modulated by microwave pulse sequences in our experiment [23, 24]. At a given time t, the instantaneous eigenstates, $\{| {\psi }_{+}(t)\rangle ,| {\psi }_{-}(t)\rangle \}$, are obtained by a rotation of the reference states, $\{| \uparrow \rangle ,| \downarrow \rangle \}$, where the rotation matrix to change the frame is given by

$$\begin{eqnarray}S(t)=\left(\begin{array}{cc}{\rm{\cos }}\displaystyle \frac{\theta (t)}{2} & {\rm{\sin }}\displaystyle \frac{\theta (t)}{2}{{\rm{e}}}^{-{\rm{i}}\phi (t)}\\ -{\rm{\sin }}\displaystyle \frac{\theta (t)}{2}{{\rm{e}}}^{{\rm{i}}\phi (t)} & {\rm{\cos }}\displaystyle \frac{\theta (t)}{2}\end{array}\right).\end{eqnarray} \tag{ 2 }$$

For an extremely slow variation of the external field, the spin-1/2 particle remains at the same instantaneous eigenstate, $| {\psi }_{+/-}(t)\rangle$, if it is prepared at $| {\psi }_{+/-}(0)\rangle$ initially. During this adiabatic propagation, only the dynamic and geometric phases are accumulated. With respect to the instantaneous eigenbasis, a unitary transformation is thus defined as ${U}_{\mathrm{ad}}(t)=| {\psi }_{+}(t)\rangle {U}_{\mathrm{ad};++}(t)\langle {\psi }_{+}(0)| +| {\psi }_{-}(t)\rangle {U}_{\mathrm{ad};--}(t)\langle {\psi }_{-}(0)|$ with ${U}_{\mathrm{ad};++}(t)=\exp [{\rm{i}}{\varphi }_{d}(t)+{\rm{i}}{\gamma }_{+}(t)]$ and ${U}_{\mathrm{ad};--}(t)=\exp [-{\rm{i}}{\varphi }_{d}(t)+{\rm{i}}{\gamma }_{-}(t)]$. Here ${\varphi }_{{\rm{d}}}(t)=-(1/2){\int }_{0}^{t}{\rm{\Omega }}(\tau ){\rm{d}}\tau$ and ${\gamma }_{\pm }(t)={\rm{i}}{\int }_{0}^{t}\langle {\psi }_{\pm }(\tau )| {\partial }_{\tau }| {\psi }_{\pm }(\tau )\rangle {\rm{d}}\tau$ are the dynamic and geometric phases, respectively. In a matrix representation, this adiabatic unitary transformation is explicitly written as

$$\begin{eqnarray}{U}_{\mathrm{ad}}({t})=\left(\begin{array}{cc}{{\rm{e}}}^{{\rm{i}}{\varphi }_{d}({t})+{\rm{i}}{\gamma }_{+}({t})} & 0\\ 0 & {{\rm{e}}}^{-{\rm{i}}{\varphi }_{d}({t})+{\rm{i}}{\gamma }_{-}({t})}\end{array}\right).\end{eqnarray} \tag{ 3 }$$

In our experiment, we consider a special form of the amplitude evolution,

$$\begin{eqnarray}&&{\rm{\Omega }}(t)=A\sin \left(\displaystyle \frac{2\pi t}{T}\right),\end{eqnarray} \tag{ 4 }$$

where the parameter T is the time of our quantum operation. The accumulated dynamic phases vanish, i.e., φ_d(T) = 0. After a global phase shift, the unitary transformation is simplified to

$$\begin{eqnarray}{U}_{\mathrm{ad}}(T)=\left(\begin{array}{cc}1 & 0\\ 0 & {{\rm{e}}}^{-{\rm{i}}{\rm{\Delta }}\gamma (T)}\end{array}\right),\end{eqnarray} \tag{ 5 }$$

with ${\rm{\Delta }}\gamma (T)={\gamma }_{+}(T)-{\gamma }_{-}(T)$. If the initial preparation and final measurement are performed in the reference basis of $\{| \uparrow \rangle ,| \downarrow \rangle \}$, the combined unitary transformation is given by

$$\begin{eqnarray}&&U={S}^{+}(T){U}_{\mathrm{ad}}(T)S(0),\end{eqnarray} \tag{ 6 }$$

which leads to an arbitrary single-qubit quantum gate [1]. This adiabatic construction can be straightforwardly extended to multi-qubit gates, which will be studied in the future.

In practice, the remaining non-adiabatic transition introduces an inevitable error for an adiabatic quantum gate. In the STA protocol, an additional counter-diabatic Hamiltonian is applied to cancel this non-adiabatic error [9–17]. A general time-dependent Hamiltonian H₀(t) can be expanded in its instantaneous eigenbasis, giving ${H}_{0}(t)={\sum }_{n}{\epsilon }_{n}(t)| {\psi }_{n}(t)\rangle \langle {\psi }_{n}(t)|$ with _n(t) the nth eigenenergy and $| {\psi }_{n}(t)\rangle$ the nth eigenstate. Accordingly, the counter-diabatic Hamiltonian ${H}_{\mathrm{cd}}(t)$ is formally written as [10]

$$\begin{eqnarray}&&{H}_{\mathrm{cd}}(t)={\rm{i}}{\rm{\hslash }}\displaystyle \sum _{n}[| {\partial }_{t}{\psi }_{n}(t)\rangle \langle {\psi }_{n}(t)| -\langle {\psi }_{n}(t)| {\partial }_{t}{\psi }_{n}(t)\rangle | {\psi }_{n}(t)\rangle \langle {\psi }_{n}(t)| ],\end{eqnarray} \tag{ 7 }$$

which suppresses the non-adiabatic transition for each eigenstate $| {\psi }_{n}(t)\rangle$. The quantum system driven $H(t)={H}_{0}(t)+{H}_{\mathrm{cd}}(t)$ rigorously evolves along the instantaneous eigenstates of H₀(t). The time propagator becomes exactly diagonal in the reference instantaneous eigenbasis, i.e.,

$$\begin{eqnarray}&&{U}_{\mathrm{STA}}(t)=\displaystyle \sum _{n}| {\psi }_{n}(t)\rangle {U}_{\mathrm{STA};{nn}}(t)\langle {\psi }_{n}(0)| .\end{eqnarray} \tag{ 8 }$$

Each diagonal element of the time propagator is written as ${U}_{\mathrm{STA};{nn}}(t)=\exp \{{\rm{i}}[{\varphi }_{d;n}(t)+{\gamma }_{n}(t)]\}$ where ${\varphi }_{d;n}(t)$ and γ_n(t) are accumulated dynamic and geometric phases of the nth reference adiabatic state. The adiabatic quantum gate introduced in equation (6) is thus changed to a STA quantum gate,

$$\begin{eqnarray}&&U={S}^{+}(T){U}_{\mathrm{STA}}(T)S(0),\end{eqnarray} \tag{ 9 }$$

by replacing U_ad(T) with U_STA(T) and excluding the global phase. As the quantum operation time T is decreased, the error induced by relaxation and decoherence can be significantly reduced while the non-adiabatic error is fully suppressed in the ideal scenario. The STA protocol provides an alternative design of high-fidelity quantum gates [25].

For the spin-1/2 particle under the Hamiltonian in equation (1), the counter-diabatic Hamiltonian follows a similar form,

$$\begin{eqnarray}&&{H}_{\mathrm{cd}}(t)={\rm{\hslash }}{{\boldsymbol{B}}}_{\mathrm{cd}}(t)\cdot {\boldsymbol{\sigma }}/2.\end{eqnarray} \tag{ 10 }$$

Through a tedious but straightforward derivation from equation (7), the three elements of the counter-diabatic field ${{\boldsymbol{B}}}_{\mathrm{cd}}(t)$ are explicitly given by

$$\begin{eqnarray}&&\left\{\begin{array}{l}{B}_{\mathrm{cd};x}(t)=-\dot{\theta }(t){\rm{\sin }}\phi (t)-\dot{\phi }(t){\rm{\sin }}\theta (t){\rm{\cos }}\theta (t){\rm{\cos }}\phi (t)\\ {B}_{\mathrm{cd};y}(t)=\dot{\theta }(t){\rm{\cos }}\phi (t)-\dot{\phi }(t){\rm{\sin }}\theta (t){\rm{\cos }}\theta (t){\rm{\sin }}\phi (t)\\ {B}_{\mathrm{cd};z}(t)=\dot{\phi }(t){{\rm{\sin }}}^{2}\theta (t).\end{array}\right.\end{eqnarray} \tag{ 11 }$$

Equation (11) can be further organized into a cross product form as [10, 23, 24]

$$\begin{eqnarray}&&{{\boldsymbol{B}}}_{\mathrm{cd}}(t)=\displaystyle \frac{1}{{| {{\boldsymbol{B}}}_{0}(t)| }^{2}}{{\boldsymbol{B}}}_{0}(t)\times {\dot{{\boldsymbol{B}}}}_{0}(t),\end{eqnarray} \tag{ 12 }$$

which is always orthogonal to the reference field ${{\boldsymbol{B}}}_{0}(t)$. By applying the external field, ${\boldsymbol{B}}(t)={{\boldsymbol{B}}}_{0}(t)+{{\boldsymbol{B}}}_{\mathrm{cd}}(t)$, to a single-qubit, the STA gates will be testified experimentally in our Xmon qubit system.

In many artificial systems, the influence of higher excited states cannot be fully ignored so that the two-level qubit has to be re-modeled as a multi-level anharmonic oscillator [1, 32]. For example, the Hamiltonian of a three-level anharmonic oscillator in the rotating frame is written as [24]

$$\begin{eqnarray}&&H(t)=\displaystyle \frac{{\rm{\hslash }}}{2}{\boldsymbol{B}}(t)\cdot {\boldsymbol{S}}+{\rm{\hslash }}{{\rm{\Delta }}}_{2}| 2\rangle \langle 2| ,\end{eqnarray} \tag{ 13 }$$

where the operator vector ${\boldsymbol{S}}$ is given by

$$\begin{eqnarray}&&\left\{\begin{array}{l}{S}_{x}={\displaystyle \sum }_{n=0}^{1}\sqrt{n+1}(| n+1\rangle \langle n| +| n\rangle \langle n+1| )\\ {S}_{y}={\displaystyle \sum }_{n=0}^{1}\sqrt{n+1}({\rm{i}}| n+1\rangle \langle n| -{\rm{i}}| n\rangle \langle n+1| )\\ {S}_{z}={\displaystyle \sum }_{n=0}^{2}(1-2n)| n\rangle \langle n| \end{array}\right.\end{eqnarray} \tag{ 14 }$$

and Δ₂ is an anharmonic parameter. In the STA protocol, the external field is given by ${\boldsymbol{B}}(t)={{\boldsymbol{B}}}_{0}(t)+{{\boldsymbol{B}}}_{\mathrm{cd}}(t)$. A technical treatment is to apply the derivative removal by adiabatic gates (DRAG) method, which decouples the interaction between the lowest two levels (qubit) and higher excited states [24, 28–31]. With the increment of another field, ${{\boldsymbol{B}}}_{{\rm{d}}}(t)=({B}_{{\rm{d}};x}(t),{B}_{{\rm{d}};y}(t),{B}_{{\rm{d}};z}(t))$, the total external field is changed to ${{\boldsymbol{B}}}_{\mathrm{tot}}(t)={\boldsymbol{B}}(t)+{{\boldsymbol{B}}}_{{\rm{d}}}(t)$ and the total Hamiltonian in equation (13) is modified to be ${H}^{{\prime} }(t)=({\rm{\hslash }}/2){{\boldsymbol{B}}}_{\mathrm{tot}}(t)\cdot {\boldsymbol{S}}+{\rm{\hslash }}{{\rm{\Delta }}}_{2}| 2\rangle \langle 2|$. In addition, we introduce the DRAG frame (${ \mathcal D }$-frame), in which the total Hamiltonian is transformed into

$$\begin{eqnarray}&&{H}_{{ \mathcal D }}(t)={{ \mathcal D }}^{+}(t){H}^{{\prime} }(t){ \mathcal D }(t)+{\rm{i}}{\dot{{ \mathcal D }}}^{+}(t){ \mathcal D }(t)\end{eqnarray} \tag{ 15 }$$

where ${ \mathcal D }(t)$ is a unitary operator. The density matrix in the ${ \mathcal D }$-frame is given by ${\rho }_{{ \mathcal D }}(t)={ \mathcal D }(t)\rho (t){{ \mathcal D }}^{+}(t)$. With a delicate design of ${{\boldsymbol{B}}}_{\mathrm{tot}}(t)$ and ${ \mathcal D }(t)$, the transformed Hamiltonian ${H}_{{ \mathcal D }}(t)$ is factorized into

$$\begin{eqnarray}&&{H}_{{ \mathcal D }}(t)=\left[\varepsilon (t)+\displaystyle \frac{{\rm{\hslash }}}{2}{\boldsymbol{B}}(t)\cdot {\boldsymbol{\sigma }}\right]\oplus {\varepsilon }_{2}(t)| 2\rangle \langle 2| ,\end{eqnarray} \tag{ 16 }$$

where ε(t) and ε₂(t) are two shifted energies [24, 29]. The qubit subspace of $\{| 0\rangle ,| 1\rangle \}$ is decoupled with the second excited state $| 2\rangle$. To avoid an artifact of the ${ \mathcal D }$-frame, we would expect an requirement of

$$\begin{eqnarray}&&{ \mathcal D }(t=0)=1\,\,\,\mathrm{and}\,\,\,{ \mathcal D }(t=T)=1,\end{eqnarray} \tag{ 17 }$$

so that the density matrices at the initial and final moments of the quantum operation are unaffected, i.e., ${\rho }_{{ \mathcal D }}(0)=\rho (0)$ and ${\rho }_{{ \mathcal D }}(T)=\rho (T)$. In the DRAG method, ${{\boldsymbol{B}}}_{\mathrm{tot}}(t)$ and ${ \mathcal D }(t)$ are evaluated by a perturbation approach with the assumption of a large anharmoncity, i.e., $| {{\rm{\Delta }}}_{2}| \gg | {\boldsymbol{B}}(t)|$. On the first order correction, the DRAG field ${{\boldsymbol{B}}}_{{\rm{d}}}(t)$ is explicitly given by [24, 29]

$$\begin{eqnarray}&&\left\{\begin{array}{l}{B}_{{\rm{d}};x}(t)=\displaystyle \frac{1}{2{{\rm{\Delta }}}_{2}}[{\dot{B}}_{y}(t)-{B}_{z}(t){B}_{x}(t)]\\ {B}_{{\rm{d}};y}(t)=-\displaystyle \frac{1}{2{{\rm{\Delta }}}_{2}}[{\dot{B}}_{x}(t)+{B}_{z}(t){B}_{y}(t)]\\ {B}_{{\rm{d}};z}(t)=0,\end{array}\right.\end{eqnarray} \tag{ 18 }$$

under a presumption of ${B}_{{\rm{d}};z}(t)=0$. In our experiment, the Xmon qubit is driven by the total external field, ${{\boldsymbol{B}}}_{\mathrm{tot}}(t)={{\boldsymbol{B}}}_{0}(t)+{{\boldsymbol{B}}}_{\mathrm{cd}}(t)+{{\boldsymbol{B}}}_{{\rm{d}}}(t)$, under the STA protocol and with the DRAG correction.

The unitary matrices representing the π and π/2 rotations about the X-axis (X_π and X_π/2 rotations) are explicitly written as [1]

$$\begin{eqnarray}{U}_{{X}_{\pi }}=\left(\begin{array}{cc}0 & -i\\ -i & 0\end{array}\right)\,\,\,\mathrm{and}\,\,\,{U}_{{X}_{\pi /2}}=\displaystyle \frac{\sqrt{2}}{2}\left(\begin{array}{cc}1 & -i\\ -i & 1\end{array}\right).\end{eqnarray} \tag{ 19 }$$

To design the X_π rotation, the reference 'adiabatic' field ${{\boldsymbol{B}}}_{0}(t)$ is specified as

$$\begin{eqnarray}&&\left\{\begin{array}{l}{B}_{0;x}(t)=0\\ {B}_{0;y}(t)=-{\rm{\Omega }}(t)\sin \theta (t)\\ {B}_{0;z}(t)={\rm{\Omega }}(t)\cos \theta (t).\end{array}\right.\end{eqnarray} \tag{ 20 }$$

The drive amplitude, polar and azimuthal angles are ${\rm{\Omega }}(t)=A\sin (2\pi t/T)$, $\theta (t)=(\pi /2)[1-\cos (\pi t/T)]$, and ϕ(t) = −π/2, respectively. In our experiment, we set the pulse length (operation time) at T = 30 ns and the maximum drive amplitude at A/2π = 20 MHz. The same two parameters will be used in other STA gates. The pulse length is comparable to the typical value of a truncated Gaussian pulse. In principle, these two parameters can be modified independently under the STA protocol. The counter-diabatic field ${{\boldsymbol{B}}}_{\mathrm{cd}}(t)$ and the DRAG field ${{\boldsymbol{B}}}_{{\rm{d}}}(t)$ are calculated using equations (11) and (18). Due to the limitation of space, we will not present the analytical forms of ${{\boldsymbol{B}}}_{\mathrm{cd}}(t)$ and ${{\boldsymbol{B}}}_{{\rm{d}}}(t)$. In figures 2(a) and (b), we plot the x-, y- and z-components of the reference field ${{\boldsymbol{B}}}_{0}(t)$ and the total field ${{\boldsymbol{B}}}_{\mathrm{tot}}(t)={{\boldsymbol{B}}}_{0}(t)+{{\boldsymbol{B}}}_{\mathrm{cd}}(t)+{{\boldsymbol{B}}}_{{\rm{d}}}(t)$. As a comparison, the major difference between the two fields appears in their x-components. With the condition of $| A| \ll | {{\rm{\Delta }}}_{2}|$, the DRAG correction is a minor effect. Notice that if the amplitude of ${{\boldsymbol{B}}}_{0}(t)$ is gradually decreased to zero, the dominant counter-diabatic field ${{\boldsymbol{B}}}_{\mathrm{cd}}(t)$ recovers the standard X_π-pulse along the x-direction. Thus, other gate schemes can in principle be mapped onto their counterparts in the STA protocol. For an initial preparation at the spin-up state ($| \uparrow \rangle =| 0\rangle$), the fast 'adiabatic' trajectory of the qubit is shown in figure 2(c). In an ideal scenario, the qubit vector evolves from the north to south pole along 270°-longitude of the Bloch sphere, and the final qubit state is the spin-down state ($| \downarrow \rangle =| 1\rangle$). Figure 2(c) shows that this trajectory can be excellently generated under the STA control field ${{\boldsymbol{B}}}_{\mathrm{tot}}(t)$ [24].

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With the consideration of the errors in state preparation, STA operation and readout, the output state is obtained through a map of the input state [1], i.e.,

$$\begin{eqnarray}&&\ \ \varepsilon :\rho \mapsto \varepsilon (\rho )=\displaystyle \sum _{i=1}^{4}{E}_{i}\rho {E}_{i}^{+},\end{eqnarray} \tag{ 21 }$$

with ρ the initial density matrix of the qubit. Each linear operators E_i=1,⋯,4 can be expanded over a fixed set of operators, $\{{\tilde{E}}_{m}=I,{\sigma }_{x},{\sigma }_{y},{\sigma }_{z}\}$, giving ${E}_{i}={\sum }_{m}{e}_{{im}}{\tilde{E}}_{m}$. The output density matrix is rewritten as

$$\begin{eqnarray}&&\ \ \varepsilon (\rho )=\displaystyle \sum _{{mn}}{\chi }_{{mn}}{\tilde{E}}_{m}\rho {\tilde{E}}_{n}^{+}\end{eqnarray} \tag{ 22 }$$

with ${\chi }_{{mn}}={\sum }_{i}{e}_{{im}}{e}_{{in}}^{* }$. The χ matrix thus completely characterizes the behavior of a specific gate. To experimentally determine the χ matrix, we perform the QPT by selecting 6 different initial states, $\{| 0\rangle ,| 1\rangle ,(| 0\rangle \pm | 1\rangle )/\sqrt{2},(| 0\rangle \pm {\rm{i}}| 1\rangle )/\sqrt{2}\}$ [1, 38, 39]. Each input state is driven by ${{\boldsymbol{B}}}_{\mathrm{tot}}(t)$ and the output state is measured by the quantum state tomographymethod. The χ matrix is then numerically calculated by solving equation (22). For the STA X_π-gate, the experimental result of the χ(X_π) matrix is plotted in figures 3(a) and (b). Consistent with the theoretical prediction of an ideal X_π-gate, the dominant element of the χ(X_π) matrix is the operator of σ_x. To quantify the fidelity of the whole quantum process, we calculate the process fidelity using [1]

$$\begin{eqnarray}&&{F}_{P}=\mathrm{Tr}\{\chi {\chi }_{\mathrm{ideal}}\}.\end{eqnarray} \tag{ 23 }$$

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The experimental result is ${F}_{P}({X}_{\pi })=95.21 \%$. To exclude the errors in state preparation and readout, we perform an interleaved randomized benchmarking measurement (see section 4.4), which gives the gate fidelity of the STA X_π rotation at ${F}_{g}({X}_{\pi })=99.82 \%$. This number is very close to the current highest fidelity of a Xmon qubit [6], and the 0.1% deviation could be improved by the future optimization of our system. On the other hand, if the operation time is increased to T = 500 ns, the counter-diabatic field becomes nearly negligible and the process fidelity of the adiabatic X_π-gate is decreased to ${F}_{P}({X}_{\pi })=92.15 \%$ due to the accumulation of the dissipation-induced error.

To design the X_π/2 rotation, we take the same reference 'adiabatic' field ${{\boldsymbol{B}}}_{0}(t)$ except for that the azimuthal angle is changed to $\theta (t)=(\pi /4)[1-\cos (\pi t/T)]$. The counter-diabatic and DRAG fields, ${{\boldsymbol{B}}}_{\mathrm{cd}}(t)$ and ${{\boldsymbol{B}}}_{{\rm{d}}}(t)$, are analytically calculated accordingly. After the QPT measurement, the experimentally reconstructed $\chi ({X}_{\pi /2})$ matrix is plotted in figures 3(c) and (d), agreeing excellently with the theoretical prediction of an ideal X_π/2 gate. As compared to the χ(X_π) matrix, the χ(X_π/2) matrix includes auto and cross correlations between the operators of I and σ_x. The experimental measurement shows that the process and gate fidelities of our STA X_π/2 rotation are ${F}_{P}({X}_{\pi /2})=95.03 \%$ and ${F}_{g}({X}_{\pi /2})=99.81 \%$.

The second group of STA quantum gates we inspect are the π and π/2 rotations about Z-axis. The corresponding unitary matrices are [1]

$$\begin{eqnarray}{U}_{{Z}_{\pi }}=\left(\begin{array}{cc}-i & 0\\ 0 & i\end{array}\right),\,\,\,\mathrm{and}\,\,\,{U}_{{Z}_{\pi /2}}=\left(\begin{array}{cc}{{\rm{e}}}^{-{\rm{i}}\pi /4} & 0\\ 0 & {{\rm{e}}}^{{\rm{i}}\pi /4}\end{array}\right).\end{eqnarray} \tag{ 24 }$$

To design these two gates, the reference 'adiabatic' field ${{\boldsymbol{B}}}_{0}(t)$ is specified as

$$\begin{eqnarray}&&\left\{\begin{array}{l}{B}_{0;x}(t)={\rm{\Omega }}(t)\cos \phi (t)\\ {B}_{0;y}(t)={\rm{\Omega }}(t)\sin \phi (t)\\ {B}_{0;z}(t)=0,\end{array}\right.\end{eqnarray} \tag{ 25 }$$

where the drive amplitude is ${\rm{\Omega }}(t)=A\sin (2\pi t/T)$ and the polar angle is θ(t) = π/2. The azimuthal angles for Z_π and Z_π/2 rotations are $\phi (t)=(\pi /2)[1-\cos (\pi t/T)]$ and $\phi (t)=(\pi /4)[1-\cos (\pi t/T)]$, respectively. The control parameters, A and T, are the same as those in the X-rotation gates. The counter-diabatic and DRAG fields, ${{\boldsymbol{B}}}_{\mathrm{cd}}(t)$ and ${{\boldsymbol{B}}}_{{\rm{d}}}(t)$, are also analytically calculated for the experimental generation. The QPT measurements of χ(Z_π) and $\chi ({Z}_{\pi /2})$ matrices are presented in figures 4(a)–(d), also agreeing excellently with the results in an ideal scenario. The process fidelities of these two STA gates are ${F}_{P}({Z}_{\pi })=95.23 \%$ and ${F}_{P}({Z}_{\pi /2})=95.20 \%$. After excluding errors in state preparation and readout, the gates fidelities are ${F}_{g}({Z}_{\pi })=99.89 \%$ and ${F}_{g}({Z}_{\pi /2})\,=99.87 \%$.

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An arbitrary single-qubit quantum gate can be realized by a combination of sequential rotations about X-, Y- and Z-axes. For example, the Hadamard gate can be generated by π/2 rotation about the Y-axis followed by π rotation about the X-axis [1], i.e.,

$$\begin{eqnarray}{U}_{H}={U}_{{X}_{\pi }}{U}_{{Y}_{\pi /2}}=\displaystyle \frac{\sqrt{2}}{2}\left(\begin{array}{cc}1 & 1\\ 1 & -1\end{array}\right).\end{eqnarray} \tag{ 26 }$$

In the STA protocol, the Hadamard gate can be realized by a one-step operation, which reduces the errors accumulated through multiple steps. Our reference 'adiabatic' field ${{\boldsymbol{B}}}_{0}(t)$ is designed as

$$\begin{eqnarray}&&\left\{\begin{array}{l}{B}_{0;x}(t)=\displaystyle \frac{\sqrt{2}}{2}{\rm{\Omega }}(t){\rm{\cos }}\varphi (t)\\ {B}_{0;y}(t)={\rm{\Omega }}(t)\sin \varphi (t)\\ {B}_{0;z}(t)=-\displaystyle \frac{\sqrt{2}}{2}{\rm{\Omega }}(t){\rm{\cos }}\varphi (t)\end{array}\right.\end{eqnarray} \tag{ 27 }$$

with ${\rm{\Omega }}(t)=A\sin (2\pi t/T)$ and $\varphi (t)=(\pi /2)[1-\cos (\pi t/T)]$. After including counter-diabatic field ${{\boldsymbol{B}}}_{\mathrm{cd}}(t)$ and the DRAG field ${{\boldsymbol{B}}}_{{\rm{d}}}(t)$, we perform the same QPT measurement as above. The experimentally reconstructed χ(H) matrix is displayed in figures 5(a) and (b). The process fidelity is F_P(H) = 94.93% while the gate fidelity with the errors in state preparation and readout excluded is F_g(H) = 99.81%.

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In the QPT measurement, the errors of state preparation and readout are mixed with the error of a quantum gate operation. To extract the gate fidelity, we perform the Clifford-based randomized benchmarking measurement [6, 40–43]. For a single-qubit, the Clifford group consists of 24 rotations preserving the octahedron in the Bloch sphere. In principle, each Clifford operator can be realized by a combination from the elements of $\{I,{X}_{\pi },{X}_{\pm \pi /2},{Y}_{\pi },{Y}_{\pm \pi /2}\}$. The qubit is initially prepared at the spin-up state ($| \uparrow \rangle =| 0\rangle$), and then driven by a sequence of m randomly selected Clifford gates. The combined operation is described by a unitary matrix, ${U}_{C}={\prod }_{i=1}^{m}{U}_{i}$. Since the Clifford group is a closed set, U_C is always a Clifford operator. Subsequently, the (m + 1)th step is the reversed step of U_C and the total quantum operation is written as

$$\begin{eqnarray}&&{U}_{\mathrm{tot}}={U}_{C}^{+}\displaystyle \prod _{i=1}^{m}{U}_{i}.\end{eqnarray} \tag{ 28 }$$

The remaining population P₀(t_f) of the initial state is measured afterwards. After repeating the above random operation sequence k (=50 in our experiment) times, we calculate the average result of P₀(t_f), which represents a sequence fidelity, F_seq(m). As shown in figure 6, this sequence fidelity can be well fitted by a power-law decaying function [41],

$$\begin{eqnarray}&&{F}_{\mathrm{seq}}={A}_{0}{p}^{m}+{B}_{0},\end{eqnarray} \tag{ 29 }$$

where A₀ and B₀ absorbs the errors in state preparation and readout, and p is a depolarizing parameter. The average error over the randomized Clifford gates is given by [41]

$$\begin{eqnarray}&&r=\displaystyle \frac{d-1}{d}(1-p),\end{eqnarray} \tag{ 30 }$$

where d = 2^N is the dimension of the Hilbert space for an array of N qubits. In our experiment, the value of the average error is r = 0.0011, or equivalently the fidelity of a randomized Clifford gate is 99.89%, which serves as a reference for our next interleaved operation (see figure 6).

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To extract the fidelity of a specific gate U_g, we make an interleaved operation [41]. At each step, the qubit is driven by a combination of a randomly select Clifford operator followed U_g. With the product operator, ${U}_{C}^{{\prime} }={\prod }_{i=1}^{m}({U}_{g}{U}_{i})$, and the $(m+1)$th operator of ${({U}_{C}^{{\prime} })}^{+}$, the total quantum operation is described by ${U}_{\mathrm{tot}}^{{\prime} }={({U}_{C}^{{\prime} })}^{+}{\prod }_{i=1}^{m}({U}_{g}{U}_{i})$ [6, 41]. Similarly, we measure the sequence fidelity ${F}_{\mathrm{seq}}^{{\prime} }(m)$. As shown by the examples in figure 6, ${F}_{\mathrm{seq}}^{{\prime} }(m)$ can also be well fitted by equation (29) with a new depolarizing parameter ${p}^{{\prime} }$. Here ${p}^{{\prime} }$ can be considered as a product of the average number p of a randomized Clifford operator and the intrinsic number p_g of the specific gate U_g, i.e., ${p}_{g}={p}^{{\prime} }/p$. Substituting p_g into equation (30), we obtain the intrinsic error r_g and the gate fidelity of U_g is given by

$$\begin{eqnarray}&&{F}_{g}=1-\displaystyle \frac{d-1}{d}\left(1-\displaystyle \frac{{p}^{{\prime} }}{p}\right).\end{eqnarray} \tag{ 31 }$$

In figure 6, we list the results of 8 example STA gates, and all the values of F_g are equal or greater than 99.8%. In the STA protocol, the fidelity of the Hadamard gate (F_g(H) = 99.81%) is higher than the product of the fidelities of the Y_π/2 and X_π gates (${F}_{g}({X}_{\pi }){F}_{g}({Y}_{\pi /2})=99.65 \%$). Thus, our one-step STA gate can efficiently reduce the error accumulation in a combined operation of multiple gates.