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abstract: 'Schrödinger cat states are crucial for exploration of fundamental issues of quantum mechanics and have important applications in quantum information processing. Here, we propose and experimentally demonstrate a method for manipulating cat states in a cavity with the Aharonov-Anandan phase acquired by a superconducting qubit, which is dispersively coupled to the cavity. Based on this dispersive coupling, the qubit can be forced to trace out a circuit in the projective Hilbert space conditional on one coherent state. By preparing the cavity in a superposition of two coherent states, the geometric phase associated with this transport is encoded to the relative probability amplitude of these two coherent states. We demonstrate the photon-number parity of a cat state in a cavity can be controlled by adjusting this geometric phase, which offers the possibility for protecting its quantum coherence from single-photon loss. Based on this geometric effect, we realize phase gates for one and two photonic qubits whose logical basis states are encoded in two quasi-orthogonal coherent states. We further demonstrate two-cavity gates with symmetric and asymmetric Fock state encoding schemes. Our method can be directly extended to implementation of controlled-phase gates between error-correctable logical qubits.'
author:
- 'Y. Xu'
- 'W. Cai'
- 'Y. Ma'
- 'X. Mu'
- 'W. Dai'
- 'W. Wang'
- 'L. Hu'
- 'X. Li'
- 'J. Han'
- 'H. Wang'
- 'Y. P. Song'
- 'Zhen-Biao Yang'
- 'Shi-Biao Zheng'
- 'L. Sun'
title: Geometrically manipulating photonic Schrödinger cat states and realizing cavity phase gates
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When a quantum system is parallel-transported along a circuit in its quantum state space, it collects information about the geometry of this path, acquiring a “memory" of its motion in the form of a phase. This phase is referred to as the geometric phase and has close relations to many physical phenomena [@Anandan1992The; @Wilczek1989Geometric]. This effect was first discovered by Berry in the context of adiabatic passage [@berry1984quantal]. One remarkable feature of Berry phase is that it is robust against fast parameter fluctuations whose effect on the enclosed parameter-space area averages out [@Chiara2003Berry]. As such, Berry phase has been considered as a choice for fault-tolerant quantum computation [@duan2001geometric; @jones2000geometric]. So far, observation of this phase and demonstration of its noise-resilient feature have been reported in various physical systems [@jones2000geometric; @Tycko1987Adiabatic; @Leek1889Observation; @Filipp2009Experimental; @Pechal2012Geometric; @gasparinetti2016measurement]. Berry’s discovery has triggered considerable interest in quantum-mechanical geometric effects, leading to important generalizations in various directions [@aharonov1987phase; @Samuel1988General]. In particular, Aharonov and Anandan defined geometric phase in the projective Hilbert space, instead of in parameter space [@aharonov1987phase], removing the adiabatic condition. The geometric nature of Aharonov-Anandan (AA) phase lies in the fact that it is related to the area enclosed by the circuit traversed by the state vector.

When two or more quantum systems are coupled, the geometric phase acquired by one system can be employed to manipulate the quantum state of the others [@Zheng2004Unconventional; @Pechal2012Geometric]. The geometric phase of a harmonic vibrational mode of trapped ions has been utilized for implementing high-fidelity entangling gates for the ionic qubits [@Leibfried2003Experimental]. In a recent experiment [@Song2017Geometric], the geometric phase of a continuous-variable field mode was observed through Ramsey interference and used for realizing controlled phase gates with up to four qubits in a superconducting circuit. On the other hand, it has been shown that the geometric phase of a superconducting qubit can be used for realizing Selective Number-dependent Arbitrary Phase (SNAP) gates on a cavity [@Krastanov2015]. This kind of gates has been experimentally demonstrated and used to produce a single-photon state [@Heeres2015]. Recently, a quantum controlled-NOT (CNOT) gate between two cavity systems has been demonstrated by use of both the dynamical and AA phases produced by controllably coupling these cavities to a qubit  [@Rosenblum2018]. This gate requires the logic states of the control qubit to be respectively encoded on the vacuum state and a nonzero photon-number state, which renders it incompatible with quantum error correction schemes; on the occurrence of single-photon loss the control qubit will collapse to a Fock state, leading to complete loss of the stored information.

![Geometric manipulation of a photonic cat state. (a) Schematic of the nonadiabatic AA phase of a qubit. Two successive $\pi$ rotations of the qubit produce a geometric phase $\gamma = \pi + \varphi$, where $\varphi$ is the angle between the two rotation axes. (b) Experimental sequence to manipulate the cat state. A cavity is dispersively coupled to the qubit and initialized in a cat state $\left({\ensuremath{\left|0\right\rangle}}+{\ensuremath{\left|2\sqrt{2}\right\rangle}}\right)/\sqrt{2}$ with the help of an ancillary qubit $Q_2$. The AA phase produced by the rotations of $Q_1$ conditional on the cavity’s vacuum state is encoded in the probability amplitude of ${\ensuremath{\left|0\right\rangle}}$, resulting in a phase gate. (c) Measured Wigner function of the cavity state before the phase gate, corresponding to fidelity of 0.980 to the ideal cat state. (d) Wigner function of the cavity state after the gate with $\varphi=0$. The slight rotation and deformation of the Wigner function is due to the self-Kerr effect of the cavity. (e) Measured parity of the cavity state as a function of $\varphi$ after a displacement $D(-\sqrt{2} e^{i\delta} )$ for different values of $\delta$. Symbols are experimental data, in excellent agreement with numerical simulations (solid lines).[]{data-label="fig:fig1"}](Figure1_final.pdf)

We here propose and experimentally demonstrate a scheme for manipulating the parity of a cat state in a cavity with the AA phase of a qubit dispersively coupled to the cavity in a superconducting circuit. Cat states are of fundamental interest [@Deleglise] and can be used to encode error-correctable logical qubits [@LeghtasPRL2013; @Mirrahimi2014; @Ofek2016; @heeres2017implementing]. Thus, manipulating these states and protecting them from decoherence is a subject of great importance. In our experiment, the qubit is parallel-transported along a closed loop on the Bloch sphere, picking up a geometric phase, conditional on one of the two quasiclassical components forming the cat state. We demonstrate the photon-number parity of the cat state can be manipulated by this geometric operation. This manipulation technique, in combination with the parity jump tracking method [@SunNature], allows for the protection of the quantum coherence of cat states from single-photon loss. We then employ this phase to realize logic gates for a cat-encoded qubit, and generalize our method to implementation of two-cavity controlled-phase gates with different encoding schemes and two-cavity SNAP gates for entangling two cavities. Our procedure can be directly generalized to implement gates between logic qubits with inherent error correction function.

![Quantum process tomography of single-cavity geometric phase gates. (a) Experimental sequence. (b) The Pauli transfer process $R$ matrix fidelity as a function of $m$, the number of the Z gate on the cavity state. The inserts show the measured $R$ matrices after one and nine Z gates, respectively. A linear fit of the process fidelity decay gives the Z gate fidelity $F_\mathrm{Z} = 0.987\pm0.001$. (c) The measured and ideal Pauli transfer $R$ matrices of the S gate and T gate with fidelities $F_{S} = 0.968$ and $F_{T} = 0.964$.[]{data-label="fig:fig2"}](Figure2_final.pdf)

![Two-cavity geometric phase gate. (a) A 3D view of Device B. A superconducting transmon qubit $Q_3$ at the center couples to two coaxial cavities $S_1$ and $S_2$, which couple to two other individual ancillary transmon qubits $Q_1$ and $Q_2$, respectively. Each of these transmon qubits independently couples to a stripline readout resonator used to perform simultaneous single-shot readout. (b) Schematic of the experimental sequence. (c) Ideal (left) and measured Pauli transfer $R$ matrices of two-cavity CZ gates with coherent state encoding {${\ensuremath{\left|0\right\rangle}}$, ${\ensuremath{\left|2\sqrt{2}\right\rangle}}$} (middle) and Fock state encoding {${\ensuremath{\left|0\right\rangle}}$, ${\ensuremath{\left|1\right\rangle}}$} (right) for both cavities. The process fidelities, $F_\mathrm{CZ\_ED}$ ($F_\mathrm{ED}$), for these two encodings are 0.727 (0.869) and 0.862 (0.905), respectively. (d) Ideal (left) and measured (right) Pauli transfer $R$ matrices of the two-cavity CNOT gate with the encoding {${\ensuremath{\left|0\right\rangle}}$, ${\ensuremath{\left|1\right\rangle}}$} for cavity $S_1$ and {${\ensuremath{\left|0\right\rangle}}_L = \left({\ensuremath{\left|0\right\rangle}}+{\ensuremath{\left|2\right\rangle}}\right)/\sqrt{2}$, ${\ensuremath{\left|1\right\rangle}}_L = \left({\ensuremath{\left|0\right\rangle}}-{\ensuremath{\left|2\right\rangle}}\right)/\sqrt{2}$} for cavity $S_2$. The corresponding process fidelity $F_\mathrm{CNOT\_ED}$ ($F_\mathrm{ED}$) is 0.829 (0.857).[]{data-label="fig:fig3"}](Figure3_final.pdf)

The experiments presented in this work are based on two circuit quantum electrodynamics (QED) devices [@Wallraff; @Clarke2008Superconducting; @You2011Atomic; @Schoelkopf2013; @Gu2017Microwave]. Device A, on which single-cavity geometric phase gates are performed, consists of two transmon qubits simultaneously dispersively coupled to two three-dimensional cavities [@Paik; @Kirchmair; @Vlastakis; @Liu2017; @Wang2017]. The parameters and architecture setup are described in Ref. [@Xu2018]. Device B, on which two-cavity geometric phase gates are performed, consists of three transmon qubits dispersively coupled to two cylindrical cavities [@Reagor2016] and three stripline readout cavities [@axline2016an]. The device parameters are described in Ref. [@Supplement]. In Device A, the coupling between the qubit ($Q_1$) used to produce the geometric phase and the cavity used to encode this phase is described by the Hamiltonian $$H=-\hbar \chi _{\mathrm{qs}}a^{\dagger }a{\ensuremath{\left|e\right\rangle}}{\ensuremath{\left\langlee\right|}},$$ where $\chi_{\mathrm{qs}}$ denotes the qubit frequency shift induced by per photon, $a^{\dagger }$ and $a$ are the creation and annihilation operators for the particular cavity field respectively, and ${\ensuremath{\left|e\right\rangle}}$ $({\ensuremath{\left|g\right\rangle}})$ is the excited (ground) state of the qubit. In Device B, the qubit, commonly coupled to two cavities used to store the photonic qubits, undergoes a frequency shift dependent on the photon numbers of both cavities.

The geometric manipulation technique is well exemplified with the even cat state $\left({\ensuremath{\left|\alpha\right\rangle}} + {\ensuremath{\left|-\alpha\right\rangle}} \right)/\sqrt{2}$, where ${\ensuremath{\left|\alpha\right\rangle}}$ and ${\ensuremath{\left|-\alpha\right\rangle}}$ are coherent states with $\left\langle \alpha | -\alpha \right\rangle \approx 0$. To realize conditional qubit rotations, a phase-space displacement, $D(\alpha)$, is applied to the cavity, transforming its state to $\left({\ensuremath{\left|2\alpha\right\rangle}} + {\ensuremath{\left|0\right\rangle}} \right)/\sqrt{2}$. The qubit, initially in the ground state $\left| g\right\rangle $, is then driven by a classical field on resonance with the qubit frequency conditioned on the cavity’s vacuum state $\left| 0\right\rangle $. We here assume that the Rabi frequency $\varepsilon $ of the drive is much smaller than $\bar{n}\chi _\mathrm{qs}$, where $\bar{n}$ is the average photon number of the state ${\ensuremath{\left|2\alpha\right\rangle}}$. In this case, the qubit’s state is not changed by the drive when the cavity is in ${\ensuremath{\left|2\alpha\right\rangle}}$ due to the large detuning, and the system dynamics is described by the effective Hamiltonian $$H_{\mathrm{eff}}=\frac 12\hbar \varepsilon e^{i\phi }\left| e\right\rangle
\left\langle g\right| \otimes \left| 0\right\rangle \left\langle 0\right|
+h.c.,$$ where $\phi $ is the phase of the drive. This Hamiltonian produces a qubit rotation $R_{{\bf n}}^\theta $ conditional on the cavity’s vacuum state, where $R_{{\bf n}}^\theta $ represents the operation that rotates the qubit’s state by an angle $\theta =\int_0^\tau \varepsilon dt$ around the axis ${\bf n}$ with an angle $\phi $ to $x$ axis on the equatorial plane of the Bloch sphere, with $\tau $ being the pulse duration.

After two successive conditional $\pi $ rotations $R_{{\bf n}_1}^{\pi ,0}=R_{ {\bf n}_1}^\pi \otimes \left| 0\right\rangle \left\langle 0\right| $ and $R_{ {\bf n}_2}^{\pi ,0}=R_{{\bf n}_2}^\pi \otimes \left| 0\right\rangle \left\langle 0\right| $, the qubit makes a cyclic evolution, returning to the initial state $\left| g\right\rangle $ but acquiring a phase $\gamma =\pi+\Delta \phi = \Omega/2$, where $\Delta \phi =\phi _1-\phi _2$ represents the angle between the two rotation axes, and $\Omega$ is the solid angle substended by the trajectory traversed by the qubit on the Bloch sphere, as shown in Fig. \[fig:fig1\](a). This conditional phase shift is encoded in the probability amplitude of the state component $\left| 0\right\rangle $, leading to the cavity state $\left( {\ensuremath{\left|2\alpha\right\rangle}} + e^{i\gamma}{\ensuremath{\left|0\right\rangle}} \right)/\sqrt{2}$. A subsequent displacement $D(-\alpha )$ transforms the cavity to the state $(\left| \alpha \right\rangle +e^{i\gamma }\left| -\alpha \right\rangle )/\sqrt{2}$. Due to the quantum interference of the two superposed coherent state components $\left| \alpha \right\rangle $ and $\left| -\alpha \right\rangle $, the cavity parity $P$ exhibits a periodical oscillation when the geometric phase $\gamma $ is varied: $P = \cos{\gamma}$. The procedure allows for manipulation of the parity of the cat state; when $\gamma=\pi$, the parity is reversed. This procedure is equivalent to a phase gate operation for the cavity qubit with ${\ensuremath{\left|\alpha\right\rangle}}$ and ${\ensuremath{\left|-\alpha\right\rangle}}$ acting as the logic basis states, and can be used to correct for the parity jump caused by single-photon loss.

To simplify the operation, in our experiment the cavity displacement before the conditional qubit rotation is incorporated with the preparation of the initial cavity state; ${\ensuremath{\left|0\right\rangle}}$ and ${\ensuremath{\left|2\alpha\right\rangle}}$ act as the two logic basis states for the gate demonstration. The experimental sequence to manipulate a cat state with Device A is shown in Fig. \[fig:fig1\](b). The cavity is initialized in the cat state $\left({\ensuremath{\left|0\right\rangle}} + {\ensuremath{\left|2\sqrt{2}\right\rangle}}\right)/\sqrt{2}$ \[the measured Wigner function is shown in Fig. \[fig:fig1\](c)\] with the help of ancillary qubit $Q_2$ following the gradient ascent pulse engineering (GRAPE) technique [@Khaneja2005; @deFouquieres2011]. The two subsequent conditional $\pi$ rotations on $Q_1$, with the first one around the axis with an angle $\varphi$ to the $x$ axis and the second one around the $x$ axis, yield a geometric phase $\gamma = \pi + \varphi$ conditional on the cavity’s vacuum state. The Wigner function of the cavity state after this single-cavity geometric phase gate is shown in Fig. \[fig:fig1\](d) with $\varphi=0$, which demonstrates the phase-space inteference fringes are reversed by the geometric manipulation. After a displacement $D(-\sqrt{2} e^{i\delta} )$, the parity of the cavity state as a function of $\varphi$ is measured and shown in Fig. \[fig:fig1\](e), in excellent agreement with numerical simulations.

Process tomography is used to benchmark the cavity geometric phase gate performance, with the experimental sequence shown in Fig. \[fig:fig2\](a). Since trusted operations and measurements necessary for quantum process tomography are unavailable in the cat-encoded subspace, we characterize the gate by decoding the quantum information on the cavity back to the transmon qubit $Q_2$. We use the so-called Pauli transfer process $R$ matrix as a measure of our gate [@Chow2012Universal], which connects the input and output Pauli operators with $P_\mathrm{out} = R\cdot P_\mathrm{in}$. Figure \[fig:fig2\](b) shows the $R$ matrix fidelity decay as a function of $m$, the number of the $\pi$-phase (Z) gate. The fidelity at $m=0$ quantifies the “round trip" process fidelity $F_{\mathrm{ED}}=0.969$ of the encoding and decoding processes only. A linear fit of the process fidelity decay gives the Z gate fidelity $F_Z = 0.987$, also consistent with the fidelity calculated from $F_Z = 1 - \left( F_\mathrm{ED} - F_{Z\_\mathrm{ED}} \right)$, where $F_\mathrm{Z\_ED}=0.957$ is the measured fidelity including the encoding and decoding processes. The measured and the ideal Pauli transfer $R$ matrices of the S gate and T gate are shown in Fig. \[fig:fig2\](c), where $\mathrm{S} = {\ensuremath{\left|0\right\rangle}}_L {\ensuremath{\left\langle0\right|}} + i {\ensuremath{\left|1\right\rangle}}_L {\ensuremath{\left\langle1\right|}}$ and $\mathrm{T} =  {\ensuremath{\left|0\right\rangle}}_L{\ensuremath{\left\langle0\right|}} + \exp{(i\pi/4)} {\ensuremath{\left|1\right\rangle}}_L {\ensuremath{\left\langle1\right|}}$.

Our method can be directly generalized to implementation of controlled-phase gates between two photonic qubits encoded in two cavities that are dispersively coupled to one common superconducting qubit [@Wang2016Schrodinger; @Gao2018]. The photon-number-dependent qubit frequency shift allows for the qubit $2\pi$ rotation conditional on both cavities being in the vacuum state ${\ensuremath{\left|00\right\rangle}}$ with a drive at the corresponding qubit frequency. With the encoding ${\ensuremath{\left|0\right\rangle}}_L = {\ensuremath{\left|0\right\rangle}}$, this rotation produces a $\pi$-phase shift if and only if both photonic qubits are in ${\ensuremath{\left|0\right\rangle}}_L$; the other logic basis state of each qubit can be any nonzero photon-number state or a coherent state with a sufficiently large amplitude. If the two photonic basis states in each cavity correspond to two coherent states, the controlled-Z (CZ) gate can be achieved by sandwiching this conditional qubit rotation between two pairs of displacement operations: the first pair of displacements transform the coherent state of each cavity used as the logic basis state ${\ensuremath{\left|0\right\rangle}}_L $ to the vacuum state, and the second pair restore the original coherent states.

Figure \[fig:fig3\] shows the two-cavity geometric phase gates based on Device B, whose schematic is shown in Fig. \[fig:fig3\](a). Besides the transmon qubit simultaneously coupled to both cavities, each cavity also dispersively couples to another ancillary transmon qubit for encoding/decoding and measurement purposes. Figure \[fig:fig3\](b) shows the experimental sequence for the process tomography to benchmark the gate performance. In our experiment, the CZ gate is implemented with two different encoding schemes: coherent state encoding {${\ensuremath{\left|0\right\rangle}}$, ${\ensuremath{\left|2\alpha\right\rangle}}$} and Fock state encoding {${\ensuremath{\left|0\right\rangle}}$, ${\ensuremath{\left|1\right\rangle}}$} for both cavities. The measured Pauli transfer $R$ matrices of the CZ gates, together with the $R$ matrix for the ideal CZ gate, are shown in Fig. \[fig:fig3\](c). The process $R$ matrix fidelities, $F_{\mathrm{CZ\_ED}}$ ($F_\mathrm{ED}$), for these two encodings are 0.727 (0.869) and 0.862 (0.905), respectively. We note that for the Fock state encoding the infidelity dominantly comes from the encoding-decoding process; for the coherent state encoding, besides the encoding-decoding error the fidelity loss mainly comes from the Kerr effects that deform the coherent states during the gate operation.

Our approach can also be used to realize a CNOT gate between two cavities. As a particular example, the control cavity $S_1$ is encoded with Fock states {${\ensuremath{\left|0\right\rangle}}$, ${\ensuremath{\left|1\right\rangle}}$} and the target cavity $S_2$ is encoded with {${\ensuremath{\left|0\right\rangle}}_L = \left({\ensuremath{\left|0\right\rangle}}+{\ensuremath{\left|2\right\rangle}}\right)/\sqrt{2}$, ${\ensuremath{\left|1\right\rangle}}_L = \left({\ensuremath{\left|0\right\rangle}}-{\ensuremath{\left|2\right\rangle}}\right)/\sqrt{2}$}. We note that this gate is equivalent to the CZ gate with the logic basis states of the target qubit encoded in the zero- and two-photon states. The measured Pauli transfer $R$ matrix of the two-cavity CNOT gate is shown in Fig. \[fig:fig3\](d), corresponding to a process $R$ matrix fidelity, $F_{\mathrm{CNOT\_ED}}$ ($F_\mathrm{ED}$), of 0.829 (0.857).

![Two-cavity SNAP gate to create single-photon Bell states ${\ensuremath{\left|\Phi_{\pm}\right\rangle}} = \left({\ensuremath{\left|01\right\rangle}}\pm{\ensuremath{\left|10\right\rangle}}\right)/\sqrt{2}$. (a) Experimental sequence, which consists of a conditional $2\pi$ rotation on the qubit $Q_3$ and four displacements of the cavity states, followed by joint Wigner tomography measurements. (b) and (c) The measured joint Wigner function $W_{12}$ of the Bell states ${\ensuremath{\left|\Phi_{+}\right\rangle}}$ and ${\ensuremath{\left|\Phi_{-}\right\rangle}}$ on the Re-Re and Im-Re planes, respectively. (d) and (e) Real parts of the density matrices of the states ${\ensuremath{\left|\Phi_{+}\right\rangle}}$ and ${\ensuremath{\left|\Phi_{-}\right\rangle}}$ measured with the decoding and state tomography sequence as shown in Fig. \[fig:fig3\](b), respectively. Solid black outlines are for the ideal density matrices. Measured imaginary parts for both states are smaller than 0.04 and not shown. The fidelities for ${\ensuremath{\left|\Phi_{\pm}\right\rangle}}$ are 0.957 and 0.930, respectively.[]{data-label="fig:fig4"}](Figure4_final.pdf)

We finally show that our conditional dynamics can be used to deterministically create high-fidelity single-photon Bell states ${\ensuremath{\left|\Phi_{\pm}\right\rangle}} = \left({\ensuremath{\left|01\right\rangle}}\pm{\ensuremath{\left|10\right\rangle}}\right)/\sqrt{2}$. The approach is an extension of the previously reported SNAP operation for universal control of one cavity [@Krastanov2015; @Heeres2015] to two cavities. When combined with the single-cavity SNAP gates, our method can be used to realize arbitrary universal multi-cavity control. The experimental sequence is shown in Fig. \[fig:fig4\](a), where a conditional $2\pi$ rotation on qubit $Q_3$ is sandwiched in between two pairs of phase-space displacements of the cavities. With help of the two ancillary qubits, joint Wigner tomography of the two cavities is performed and two slice cuts of the measured two-mode Wigner function are shown in Figs. \[fig:fig4\](b-c). The density matrices of ${\ensuremath{\left|\Phi_{+}\right\rangle}}$ and ${\ensuremath{\left|\Phi_{-}\right\rangle}}$, reconstructed by mapping the state of the two cavities to qubits $Q_1$ and $Q_2$ and then jointly measuring the state of these qubits \[as in the tomography measurement of Fig. \[fig:fig3\](b)\], are displayed in Figs. \[fig:fig4\](d-e), with state fidelities of 0.957 and 0.930, respectively.

One distinct feature of our gate dynamics is that it is compatible with error correction schemes. For logic qubits whose basis states are encoded in even cat states, the photon-number parity can be used as an error syndrome of the single-photon loss [@LeghtasPRL2013; @Mirrahimi2014; @Ofek2016]. With this encoding, each of the two-qubit logic basis states is composed of four two-mode coherent state components, and a CZ gate can be realized by subsequently performing four conditional phase operations. Each operation inverts the phase of one of the four components forming a specific logic basis state, and can be realized by sandwiching a qubit $2\pi$ rotation conditional on the cavities’ vacuum state between two pairs of phase-space displacements of the cavities. For another kind of error-correctable logic qubits binomially encoded as $\{{\ensuremath{\left|0\right\rangle}}_L=({\ensuremath{\left|0\right\rangle}}+{\ensuremath{\left|4\right\rangle}})/\sqrt{2}, {\ensuremath{\left|1\right\rangle}}_L=~{\ensuremath{\left|2\right\rangle}}\}$ [@Michael2016; @Hu2018], the CZ gate can be realized by a qubit $2\pi$ rotation conditional on the joint cavities’ state ${\ensuremath{\left|2\right\rangle}}{\ensuremath{\left|2\right\rangle}}$, enabled with a drive at the qubit’s frequency associated with this cavities’ state. For our device, the error due to Kerr effects is larger than that caused by single-photon loss. With the improvement of the device design, the Kerr strengths can be significantly mitigated [@Rosenblum2018]. We plan to design and fabricate a device with improved performance, and demonstrate gates with error-correctable logic qubits.

For a setup with three or more cavities coupled to one common qubit, the qubit $2\pi $ rotation conditional on all cavities being in the vacuum state directly leads to a phase gate acting on these cavities if the vacuum state in each cavity acts as the logic basis state ${\ensuremath{\left|0\right\rangle}}_L$. This kind of gates is useful for quantum error correction [@Reed2012Realization] and serves as a central element for implementation of the quantum search algorithm [@Nielsen]. In addition to implementation of quantum gates, our method can be used for stabilizing the parity of a cat state: When an environmentally-induced parity jump occurs, one can correct for it through the combination of a conditional qubit $2\pi$ rotation and two displacement operations.

We thank valuable discussions with Chen Wang. This work was supported by National Key Research and Development Program of China No.2017YFA0304303 and the National Natural Science Foundation of China under Grants No.11474177, No. 11874114, No. 11674060, and No. 11875108.

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