A quantum system moving cyclically in its quantum-state space acquires a geometric phase that is proportional to the area enclosed by the cyclic path.^[1,2] Since the magnitude of the geometric phase is only determined by the total area enclosed, local deformations of the path due to noise perturbations have minimal impact on the overall geometric phase acquired. This unique property offers a favorable choice for noise-resilient coherent manipulation of quantum states and for realization of high-fidelity quantum gates.^[3–11] The geometric phase has been experimentally observed in various quantum systems,^[12–21] and its noise resilience has been investigated.^[17,18]

In a recent experiment,^[22] a continuous-variable geometric phase has been utilized to implement a highly efficient quantum gate protocol in a superconducting circuit, where the resonator frequency and the qutrit |e⟩ ↔ |f⟩ transition frequency ω_e are quasi-resonant. Here |g⟩, |e⟩, and |f⟩ denote the ground, the first-excited, and the second-excited states of the qutrit, respectively. The geometric phase is due to the resonator motion in its phase space under an off-resonant microwave drive, which is present only when the qutrit is in |g⟩. However, the quasi-resonance requirement is quite stringent since it restricts the spectrum width within which the qutrits can be placed, i.e., the qutrit |e⟩ ↔ |f⟩ transition must be placed very close to the resonator in frequency. For non-tunable qutrits where the qutrit frequencies are fixed during the device design and fabrication process,^[23] the protocol in Ref. [22] is impossible to be implemented.

Here we present an alternative experiment, with less strict requirements, for observing the geometric phase while the resonator can be dispersively coupled to the qutrit |g⟩ ↔ |e⟩ and |e⟩ ↔ |f⟩ transitions. The resonator’s state components associated with |g⟩ and |e⟩ are moved along different loops in phase space, each accumulating a geometric phase proportional to the area enclosed by the corresponding loop. The geometric phase is determined by the difference of the areas enclosed by the two paths. We identify the resonator’s phase-space trajectories by Wigner tomography using an ancilla qubit, following which we demonstrate the associated geometric phase with Ramsey interferometry. The experimental results are compared with the numerical simulations with good agreement. We further use this geometric phase to construct the single-qubit π-phase gate with a quantum process fidelity of 0.851 ± 0.001.

Our device, shown in Fig. 1(a), consists of a half-wavelength coplanar waveguide resonator coupled to two Xmon qutrits,^[24,25] one serving as the test in Ramsey interferometry to read out the geometric phase and the second one used as an ancilla for Wigner tomography of the resonator. The two Xmon qutrits have weak anharmonicities η = ω_g − ω_e ≈ 2 π × 230 MHz, and their |g⟩ ↔ |e⟩ transitions are coupled to the resonator with coupling strengths λ/2π ≈ 23 MHz, where ω_g is the |g⟩ ↔ |e⟩ transition frequency.^[26] The resonator has a fixed bare resonant frequency ω_c/2π, which is estimated to be 5.042 GHz if the resonator is ideally decoupled from the test qutrit. Each qutrit can be dispersively or resonantly coupled to the resonator by tuning its frequency, and the coupling is effectively switched off when the qutrit is far off-resonant with the resonator. The Xmon qutrits can be treated as two-level qubits if their |f⟩ levels play negligible roles at certain experimental stages. For example, the ancilla qutrit is set at ω_e/2π = 4.799 GHz and is used as a qubit for Wigner tomography. In Ref. [22], the detuning between the bare resonator and the test qutrit is set to |ω_e − ω_c| ≈ λ, while here we set |ω_g − ω_c| ≫ λ. At the noted frequency, the |e⟩-state lifetime of the test qutrit is measured to be around 9 μs, while the Gaussian dephasing time is around 1 μs.^[27]

Fig. 1.

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Fig. 1. (color online) Experimental layout. (a) Device schematic showing two Xmon qutrits coupled to a resonator in the middle (left), and the corresponding potential-well landscapes and energy levels (right). (b) Energy level arrangement of the resonator and the test qutrit ignoring their interaction (left), and the resonator’s conditional frequency shifts due to the qutrit’s dispersion (right). The microwave drive with a tone ωd is detuned from ωc by an amount δd. The dispersive coupling to the qutrit shifts the resonator frequency by an amount δg (δe), and hence the effective drive-resonator detuning is Δg ≡ ωd − ωc,g = δd − δg (Δe ≡ ωd − ωc,e = δd − δe) for the qutrit state |g⟩ (|e⟩). (c) Ideal resonator’s phase-space trajectories. The resonator, initially in its vacuum state, is displaced by the microwave drive along a circular loop in its phase space with the radium Ω/Δg (Ω/Δe) and the angular velocity Δg (Δe) for the qutrit state |g⟩ (|e⟩), where Ω is the drive amplitude. When the resonator traverses the loop once, it will acquire a geometric phase that is proportional to the enclosed phase-space area. When Δg/Δe = kg/ke, with kg and ke being integers, at time T = 2π (kg − ke)/(Δg − Δe) the resonator makes a cyclic evolution and returns to the vacuum state no matter whether the qutrit is in the state |g⟩ or |e⟩; however, it acquires different geometric phases for these two qutrit states. (d) Ramsey sequence. The geometric operation, which is achieved by the combination of the microwave driving pulse (red sinusoid) and the dispersive resonator-qubit coupling, is sandwiched in between two π/2 qubit rotations (blue sinusoids) whose rotation axes differ by an angle Φ. The geometric phase difference β is manifested in the test qubit’s Ramsey interference, obtained by a dispersive measurement using a 700 ns-long microwave pulse (light brown sinusoid).

The dispersive resonator–qutrit coupling results in a frequency shift of the resonator that is dependent upon the state of the test qutrit, so that the resonator-drive detuning is also qutrit-state dependent, as shown in Fig. 1(b). The Hamiltonian describing the resonator’s conditional frequency shift in the framework rotating at the drive frequency ω_d is

.

For the resonator that is initialized in the vacuum state |0⟩ and subject to the drive Ω, it is displaced along a circular loop in phase space with the radium Ω/Δ_j and the angular velocity Δ_j depending upon the qutrit state |j⟩, as shown in Fig. 1(c). When the qutrit is in the state |j⟩, at time T = 2k_jπ/Δ_j with k_j being an integer, the resonator returns back to |0⟩ after k_j cycles in evolution, acquiring an Aharonov–Anandan phase Θ_j = −2k_jA_j, where A_j = π (Ω/Δ_j)² is the area enclosed by the resonator’s phase-space loop associated with the qutrit state |j⟩. When T = 2k_gπ/Δ_g = 2k_eπ/Δ_e with k_g and k_e both being integers, the resonator returns to |0⟩ after the cyclic evolution no matter whether the qutrit is in the state |g⟩ or |e⟩; the two subsystems are disentangled even for an initial qutrit superposition state. As a result, the geometric phase acquired by the resonator depends on the qutrit state.

Figure 2(a) displays the exemplary resonator’s phase-space trajectories for the test qutrit in |g⟩ and |e⟩ as mapped out by Wigner tomography using the ancilla,^[28] with the relevant experimental parameters listed in Table 1. In Wigner tomography, the resonator state is described by a complex number α, where |α|² represents the average photon number stored in the resonator.^[29] The apparent agreement between the experimental data in Fig. 2(a) and the theory in Fig. 1(c) indicates the existence of the geometric phases when the resonator, dressed by the test qutrit states, is off-resonantly driven. We further use the test qutrit’s |g⟩ ↔ |e⟩ Ramsey interference to detect the geometric phase,^[19,30,31] where the |f⟩ state is not participating so that the test qutrit can now be treated as a two-level qubit. The pulse sequence for the Ramsey experiment shown in Fig. 1(d) starts by applying a π/2 pulse that rotates the qubit around the x-axis on the Bloch sphere, transforming the state of the qubit from |g⟩ to . Then the resonator is driven by a microwave pulse with a tone ω_d and an amplitude Ω for a duration T ≈ 170 ns. ω_d and hence δ_d can be adjusted so that Δ_g/Δ_e ≈ k_g/k_e, as shown in Table 1. We first choose δ_d/2π = 10.3 MHz, so that k_g ≃ 2 and k_e ≃ 3, and the qubit is approximately disentangled with the resonator at the end of this pulse, yielding the qubit state , where β = Θ_e − Θ_g is the geometric phase difference associated with the resonator’s cyclic evolutions. A second π/2 rotation, around the axis with an angle Φ to the x-axis in the x–y plane, is then applied to the qubit, leading to the state

Fig. 2.

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Fig. 2. (color online) Geometric phase. (a) Exemplary resonator’s phase-space trajectories for the test qutrit in |g⟩ (blue squares) and |e⟩ (red squares) by sampling at a few selected time intervals during the evolutions with Wigner tomography using the ancilla.[28] Some relevant experimental parameters are MHz, δd/2π = 10.3 MHz, Δg/2π = 11.9 MHz, and Δe/2π = 18.0 MHz. Dashed lines are visual guides connecting the experimental data points. (b)–(d) Ramsey interferences: probabilities Pe of the test qubit in the excited state |e⟩, with color bar in (b), as functions of Φ and Ω2 for δd/2π = 10.3 MHz (b), δd/2π = 11.6 MHz (c), and δd/2π = 22.2 MHz (d), with T ≈ 170 ns. Other relevant parameters can be found in Table 1. Solid lines are numerical simulations for the dispersively coupled resonator–test qutrit system with the assumption of an extra driving strength of 0.3 Ω to the test qutrit in the full Hamiltonian due to the on-chip microwave crosstalk. For a fixed Pe, the linear scaling of Φ with Ω2 is clearly observed, which, according to Eq. (4), implies that the geometric phase difference β is also linear in Ω2.

Table 1.

Table 1.

Table 1. Choices of the drive frequency for realizing the resonator’s cyclic evolution while maintaining kg and ke both integers. . (δd/2π)/MHz (Δg/2π)/MHz (Δe/2π)/MHz kg/ke 10.3 11.9 18.0 2/3 16.2 17.8 23.9 3/4 22.2 23.8 29.9 4/5

Table 1. Choices of the drive frequency for realizing the resonator’s cyclic evolution while maintaining kg and ke both integers. .

Equation (4) shows that P_e remains unchanged when β and Φ are varied by the same amount. In Fig. 2(b), we present the measured P_e as a function of Φ and Ω², which shows that, for a fixed P_e, Φ and hence β are linear in Ω² that is proportional to the phase-space area enclosed by each loop. Equation (4) shows that if and only if β = Φ, we have P_e = 1, based on which we can directly obtain β as a function of Ω² by tracing the contour of P_e maximum in Fig. 2(b). Excellent agreement with the numerical solution (solid line) for the data in Fig. 2(b) verifies the linear scaling of β with the difference between the areas swept by the coherent states |α_j(t)⟩ associated with j = |g⟩ and |e⟩. As also verified by the numerical simulation, the total phase observed in Fig. 2(b) contains a dynamical contribution which is proportional to Ω² as well, since the test qutrit is driven through the microwave crosstalk during the evolutions.

Figures 2(c) and 2(d) display the measured P_e versus Φ and Ω² for another two choices of the drive frequency δ_d (Table 1), all for the pulse duration of T ≈ 170 ns. As expected, in each case the measured geometric phase difference β, which is equal to Φ for P_e = 1, shows the similar linear scaling with Ω². When Φ and Ω are fixed, the observed geometric phase difference decreases with increasing δ_d. This can be explained as follows. The acquired geometric phase is proportional to the product of the area enclosed by the phase-space circle and the number of turns in the pulse duration. The area decreases quadratically with the increase of the drive-resonator detuning as A_j = π (Ω/Δ_j)², while for a fixed pulse duration, the cyclic number scales linearly with the detuning.

As shown above, the geometric phase of the resonator is encoded in the final state of the test qubit, i.e., the |g⟩ ↔ |e⟩ transition of the test qutrit, producing a geometric phase gate on the qubit. In order to characterize this gate, we carry out quantum process tomography (QPT) in a separate experiment on a similar device, where the same geometric phase has been observed. In this device, the test qutrit’s |g⟩ ↔ |e⟩ transition frequency is 4.945 GHz, with a lifetime of around 17 μs and a Gaussian dephasing time of around 2 μs. The resonator frequency is 5.041 GHz, and the qutrit–resonator coupling strength is around 20 MHz. The π-phase gate with the choice of MHz and δ_d/2π = 17.1 MHz is characterized by QPT, and the QPT matrix is displayed in Fig. 3, with a process fidelity of 0.851 ± 0.001. This result demonstrates that the geometric phase can be used to realize quantum logic gates in superconducting circuits.

Fig. 3.

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	Fig. 3. (color online) Geometric π-phase gate, which is performed in a separate experiment on a similar device, with MHz, δd/2π = 17.1 MHz, Δg/2π = 12.5 MHz, Δe/2π = 18.8 MHz, and T ≈ 160 ns. Real part of the QPT matrix, χ, for the π-phase geometric gate is expressed in the Pauli basis {I, X, Y, Z}, with a process fidelity of about 0.851 ± 0.001.

We note that our experimentally achieved fidelity value of the π-phase gate, which agrees well with the numerical simulation, is mostly determined by the device and measurement parameters including the qubit-resonator detuning ω_g − ω_c, the qubit-resonator coupling λ, and the qubit anharmonicity η. Further improvement is possible by increasing |ω_g − ω_c|, λ, and η appropriately, so that the difference between the qubit-state dependent resonator frequency shifts δ_e and δ_g can be improved, and the resonator’s evolutions associated with the qubit states |g⟩ and |e⟩ can be simultaneously re-synchronized with the drive after a shorter duration. For example, setting (ω_g − ω_c)/2π = −800 MHz, λ/2π = 80 MHz, and η/2π = 500 MHz, the numerical simulation indicates that the π-phase gate fidelity can be well above 0.99 ignoring decoherence. In addition, the geometric phase is advantageous for implementing a two-qubit controlled phase gate, as investigated in Refs. [23] and [32], where the geometric phase due to the resonator’s cyclic motion in the phase space, though not directly observed via Ramsey interference, has been used to construct the resonator-induced two-qubit phase gate with fidelity up to 0.98.

In conclusion, we have carried out an alternative experiment for observation of the geometric phase of a harmonic oscillator in a superconducting circuit, which does not require the qutrit |e⟩ ↔ |f⟩ transition to be close to the resonator in frequency. The difference between the geometric phases associated with the two traversed phase-space loops, which correlate with the qutrit states |g⟩ and |e⟩, is measured with a qubit-based Ramsey interferometer. Finally, the acquired geometric phase of the resonator can be used to produce the π-phase gate on the qubit with a gate fidelity of 0.851 ± 0.001 as characterized by quantum process tomography.