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Experimental Realization of Two Qutrits Gate with Tunable Coupling in Superconducting Circuits

Gate-based quantum computation has been extensively investigated using quantum circuits based on qubits. In many cases, such qubits are actually made out of multilevel systems but with only two states being used for computational purpose. While such a strategy has the advantage of being in line with the common binary logic, it in some sense wastes the ready-for-use resources in the large Hilbert space of these intrinsic multi-dimensional systems. Quantum computation beyond qubits (e.g., using qutrits or qudits) has thus been discussed and argued to be more efficient than its qubit counterpart in certain scenarios. However, one of the essential elements for qutrit-based quantum computation, two-qutrit quantum gate, remains a major challenge. In this work, we propose and demonstrate a highly efficient and scalable two-qutrit quantum gate in superconducting quantum circuits. Using a tunable coupler to control the cross-Kerr coupling between two qutrits, our scheme realizes a two-qutrit conditional phase gate with fidelity 89.3% by combining simple pulses applied to the coupler with single-qutrit operations. We further use such a two-qutrit gate to prepare an EPR state of two qutrits with a fidelity of 95.5%. Our scheme takes advantage of a tunable qutrit-qutrit coupling with a large on/off ratio. It therefore offers both high efficiency and low cross talk between qutrits, thus being friendly for scaling up. Our work constitutes an important step towards scalable qutrit-based quantum computation.

Gate-based quantum computation has been extensively investigated using quantum circuits based on qubits. In many cases, such qubits are actually made out of multilevel systems but with only two states being used for computational purpose. While such a strategy has the advantage of being in line with the common binary logic, it in some sense wastes the ready-for-use resources in the large Hilbert space of these intrinsic multi-dimensional systems. Quantum computation beyond qubits (e.g., using qutrits or qudits) has thus been discussed and argued to be more efficient than its qubit counterpart in certain scenarios. However, one of the essential elements for qutrit-based quantum computation, two-qutrit quantum gate, remains a major challenge. In this work, we propose and demonstrate a highly efficient and scalable two-qutrit quantum gate in superconducting quantum circuits. Using a tunable coupler to control the cross-Kerr coupling between two qutrits, our scheme realizes a two-qutrit conditional phase gate with fidelity 89.3% by combining simple pulses applied to the coupler with single-qutrit operations. We further use such a two-qutrit gate to prepare an EPR state of two qutrits with a fidelity of 95.5%. Our scheme takes advantage of a tunable qutritqutrit coupling with a large on/off ratio. It therefore offers both high efficiency and low cross talk between qutrits, thus being friendly for scaling up. Our work constitutes an important step towards scalable qutrit-based quantum computation.
The past two decades have witnessed an extraordinary progress in our capability of precisely controlling various quantum systems for quantum information processing (QIP). As a milestone, for example, the concept of quantum advantage has been demonstrated recently in different scenarios [1][2][3][4]. Most existing research on QIP, both theoretical and experimental, relies heavily on twolevel quantum systems, i.e., qubits, analogue of classical binary bits [5]. However, it has been suggested that QIP using multi-level systems, such as qutrits (three levels) or qudits ( more than three levels), may have significant advantages over qubit-based schemes due to the added access to extra dimensions of the Hilbert space [6][7][8][9][10][11]. Among others are more efficient coding for quantum error correction [12], more flexible simulation of quantum dynamics [13,14], magic state distillation with higher fidelity [15], and more robust quantum cryptography protocols [16][17][18]. These advantages would be particularly welcome in the noisy intermediate-scale quantum era where hardware efficiency is of critical importance [19]. Some initial efforts of using qutrits or qudits for QIP include construction of quantum gates and generation of entanglement in photonic systems [20][21][22][23][24][25] and su-perconducting quantum circuits [13]. These works have already shown encouraging potential applications of multilevel systems in QIP. Further progress requires viable schemes that have good scalability with relatively simple engineering.
Circuit quantum electrodynamics (cQED) is the study of the interaction between superconducting circuits and quantized electromagnetic fields in the microwave frequency region [26,27]. Due to the diversity of superconducting quantum devices and flexibility in circuit design, as well as a constantly improving understanding of decoherence in these systems [28,29], superconducting quantum circuits based on cQED (especially the mainstream architecture with transmon qubits [30]) have nowadays become a leading experimental platform for quantum computation [1,31,32]. While superconducting qubits, including transmons, are intrinsically multilevel systems, most work only utilize two of them to encode quantum information, and the rest are only exploited as ancillary during the logic gate operation or treated as leakage channel [33][34][35][36][37].
principle of entanglement generation [13,38] and quantum gates [39][40][41]. In these works, two-qutrit gate operations involving two transmons are realized based on a small residual cross-Kerr coupling between them, results in slow gate operation and correlation error during idle period. Therefore, scheme for two-qutrit gates featuring a stronger coupling with a large on-off ratio are highly desirable to further improve qutrit-based quantum operation fidelity meanwhile maintain scalability.
In this work, we develop a scalable two-qutrit conditional phase (Cphase) gate using two transmons of fixed frequencies connected to a tunable coupler. The coupler itself is also a transmon but with a tunable energy spacing, which can be used to vary the effective cross-Kerr coupling between the other two transmons connected to it. Such a coupling scheme has been used to construct conditional phase gates of qubits and is highly favored for its scalability [42][43][44][45][46][47][48]. In our study, the cross-Kerr coupling among higher levels of the two qutrits can be tuned in a broad range from about 100 kHz (idling regime for single-qutrit operations) to around 100 MHz (accumulating conditional phase for the two-qutrit gate) by adjusting the frequency of the coupler, achieving a large on/off ratio. Using such a setup, we demonstrate a two-qutrit Cphase gate with a fidelity of 89.3(±1.9)% determined by a quantum process tomography. We note that compared to the qubit case, qutrit quantum gates utilize more and higher energy levels and thus require for more complex control schemes and are also more susceptible to decoherence [13]. With a better understanding and engineering of the transmon-coupler setup [49][50][51], qutrit quantum gates with much improved fidelity can be realized.
Our superconducting quantum processor consists of eight fixed-frequency transmons, arranged in a ring geometry with nearest neighboring transmons interconnected via frequency-tunable couplers. The two transmons and their coupler used in this study is shown in Fig. 1(a) and its simplified quantum circuit is sketched in Fig. 1(b). Each transmon has a dedicated readout resonator for multiplexed dispersive readout. The tunable coupler consists of a superconducting pad capacitively coupled to the ground via a superconducting quantum interference device (SQUID). Diplexed signals for controlling both the qutrit (4-8 GHz) and the coupler (DC-1 GHz) are synthesized at room temperature and transmitted to the device via a shared control line. Such a design substantially reduces wiring complexity and is friendly for scalability. More details about the whole device and the experimental setup can be found in Ref. [52].
Typical parameter settings of a transmon guarantee that it has a good coherence performance while maintaining a sufficiently large anharmonicity, which enables it to be operated as not only a qubit but also a higherdimensional qutrit or qudit [53,54]. In our study, the lowest three energy levels (labeled as |0 , |1 , and |2 ) of a transmon are encoded as logical states of a qutrit 200 µm readout resonator , is compared to the summation (orange) of eigenenergies of the states |100 and |002 . Their difference (shaded area) represents the conditional phase χ102, which increases significantly when the coupler is tuned from its sweet point (about 7.64 GHz) to the frequency range of the qutrits (about 6.6 GHz). Similar behavior (not shown) is observed for χ101, χ201, and χ202.
( Fig. 1(c)). An arbitrary single-qutrit gate operation can be decomposed into operations in the two subspaces of (|0 , |1 ) and (|1 , |2 ) [13,40]. However, in order to realize single-qutrit gates of high fidelity, the AC Stark effect due to a small anharmonicity of transmon should be taken into account [13,55]. For example, a microwave drive resonant to the subspace of (|0 , |1 ) induces an effective Z rotation in the subspace of (|1 , |2 ), and vice versa.
The system shown in Fig. 1 can be effectively described by a model of three coupled nonlinear oscillators. When only the low lying energy levels are considered, its Hamiltonian can be written as [42] (h ≡ 1): Mea.

Q1=
phase of the second Ramsey pulse ()


Coupler frequency  (GHz) Tunable cross-Kerr coupling between qutrits. (a) Pulse sequence of a Ramsey-like experiment on Q1 conditioned by the state of Q2 to extract the cross-Kerr coupling χ102. Two π/2 pulses are applied to Q1. The first one generates a rotation around the X axis in the |0 ,|1 subspace and the second pulse corresponds to a rotation in the |1 ,|2 subspace around an axis with an angle θ off the X axis. Between the two pulses applied to Q1, a square-shaped flux pulse with amplitude V b and duration τ is applied to the coupler, which induces a phase shift to be captured by Q1. Depending on the magnitude of V b , τ is set to different values ranging from 1 to 2000 ns (see SM for details). Prior to the above three pulses, Q2 is prepared into the second or the ground states (with or without applying the pulses enclosed by the red dashed lines). The conditional phase accumulated on Q1 is measured as a function of θ. (b) Experimental data (markers) are fitted by a sinusoidal function (solid lines) to extract the phase difference ∆φ between the cases of Q2 being prepared into the second excited (orange) and the ground (blue) states. The effective cross-Kerr coupling χ102 can be derived with the relation ∆φ = χ102τ at this fixed V b condition. Similar measurements are performed to obtain χ101, χ201, and χ202. (c) Repeated experiments in (b) with different pulse amplitudes V b , and combination the mapping relation pulse amplitudes V b to coupler frequency ωc ( not shown here), we obtain four χi0j(markers) depends on ωc for two qutrits, which is in agreement with numerical results quite well(corresponding same color lines). We choose coupler sweet point ωc ≈ 7.640 GHz as the idling point to keep small residual couplings in this work.
where subscripts 1,2 and C refer to qutrit 1 (Q 1 ), qutrit 2 (Q 2 ), and the tunable coupler (C), respectively. ω 1 /2π = 6.074 GHz (ω 2 /2π = 6.725 GHz) is the transition frequency between the ground and the first exited states of Q 1 (Q 2 ), and is the corresponding anharmonicity. The maximum frequency and anharmonicity of the coupler with a symmetric SQUID are 7.640 and -0.310 GHz, respectively. The three coupling constants determined by the mutual capacitances are designed to be roughly g 1C /2π ∼ g 2C /2π (≈ 100 MHz) g 12 /2π (≈ 5 MHz), which results in a better adiabatic control and a small residual effective cross-Kerr interaction.
For the study of two-qutrit gates in this work, it is convenient to consider the effective form of H in Eq.(1) truncated to the computational subspace: H ef f = ijω i0j |i0j i0j| [43,56], where |i0j andω i0j represent the eigenstates and eigenenergies of H, respectively. The tunable effective cross-Kerr interaction is defined as χ i0j =ω i0j − (ω i00 +ω 00j ) and is the consequence of interactions among the higher levels of the two qutrits. Bothω i0j and χ i0j are dependent on ω c , the frequency of the tunable coupler. A nonzero χ i0j results in an extra phase for the state of |i0j relative to the phases accumulated on |i00 and |00j . Taking |102 as an example, its numerically simulated eigenenergy as a function of ω c is plotted as the blue curve in Fig. 1(d). As a comparison, the sum ofω 002 andω 100 is plotted as the orange curve. When the coupler is far detuned from both qutrits, the blue and orange curves are nearly indistinguishable, indicating a negligible χ 102 (∼ 100 kHz). However, χ 102 increases to well above 100 MHz as the coupler is tuned closely to the qutrits due to the strong compelling interactions of multiple higher eigenstates. A high on-off ratio for χ i0j can thus be achieved by varying the frequency of the coupler.
We first experimentally characterize the four cross-Kerr couplings χ i0j (i, j = 1, 2) using a conditional Ramsey-like experiment. The pulse sequence for measuring χ 102 , for example, is shown in Fig. 2(a). The conditional phase difference ∆φ = χ 102 τ between Q 2 being in the states of |0 and |2 is the cumulative entangled phase of |102 during the Ramsey experiment [43]. To suppress the edge effect, we measure two conditional phases ∆φ 1 and ∆φ 2 corresponding to two different durations of τ 1 and τ 2 with the same pulse amplitude. A more accurate χ 102 can thus be obtained by χ 102 = (∆φ 1 − ∆φ 2 )/(τ 1 − τ 2 ).
By using the correspondence between the pulse amplitude and the coupler frequency, we plot in Fig. 2(c) the four effective cross-Kerr couplings measured in the above conditional Ramsey experiment. The dynamic range of strength of these couplings spans more than two orders of magnitude, from a few hundred kHz to about 100 MHz, enabling fast two-qutrit gate operations as well as a small residual coupling. The results of χ 101 , χ 102 and χ 201 are in good agreement with numerical simulations using our device parameters. The numerical simulation of χ 202 quantitatively deviates from the experimental result at large values of χ 202 . Such a deviation is expected since the numerical simulation is based on the approximate Hamiltonian in Eq. (1), which assumes fixed values for the anharmonicity α i and the coupling g ij . More accurate simulation should take into account the frequency dependence of both quantities [57].
The large tunability of χ i0j over the coupler frequency makes it possible to realize highly efficient conditional phase gates by engineering the pulse applied to the coupler. Here, we experimentally demonstrate a two-qutrit conditional phase gate as an extended version of the two-qubit conditional phase gate [13]: U Cphase = e i 2 3 π (|101 101| + |202 202|) + e −i 2 3 π (|102 102|+|201 201|)+ i or j=0 |i0j i0j|. The previous schemes of two-qutrit U Cphase gate rely on a small and fixed cross-Kerr interaction and are implemented by interspersing the time evolution under the native cross-Kerr Hamiltonian with multiple single-qutrit gates [13,40]. On the contrary, our scheme uses a tunable cross-Kerr interaction with a large on/off ratio so that the efficiency is much improved. The pulse sequence is shown in Fig. 3(a). Two identical flux pulses are applied to the coupler while single-qutrit gates X 12 π are implemented on both qutrits before and after the second flux pulse, which guarantees that φ 101 = φ 202 and φ 102 = φ 201 . For a flux pulse with fixed amplitude and edge length, the four phases accumulated on |101 , |202 , |201 , and |102 are controlled by setting the duration of the flat top of the flux pulse, as shown in Fig. 3(b). When appropriate parameters are chosen so that φ 101 (φ 202 ) mod 2π = 2π/3 and φ 102 (φ 201 ) mod 2π = 4π/3, a two-qutrit U Cphase gate is realized.
Fidelity of the realized gate is characterized by a standard quantum process tomography (QPT) method [59]. The full quantum process matrix χ exp reconstructed experimentally is compared to the theoretically expected process matrix χ 0 as shown in Fig. 3(c). The gate fidelity is determined by F = T r(χ exp χ 0 ) = 89.3(±1.9)%, this result including errors of imperfect single qutrit gates for states preparation and tomography rotation. Numerical simulations using the open-source QuTiP package [60], with or without decoherence, yield a gate fidelity of 91.2% and 99.2%, respectively. These results indicate that decoherence is the main source of error here, whereas other factors such as sates leakage and single qutrit gate infidelity are relatively small. Effective decoherence time of the qutrits can be calculated from the relationship between the lifetimes of qutrits and the frequency of the coupler, detailed discussion is included in ref. [58].
To further benchmark the performance of the above two-qutrit Cphase gate, we use it to prepare a maximally entangled two-qutrit EPR state: Similar to the case of two qubits, the two-qutrit EPR state can also be prepared by combining a two-qutrit Cphase gate with local single-qutrit Hadamard gates, as shown in Fig. 4.(a). A standard quantum state tomography (QST) method is employed to reconstruct the state density matrix ρ. The results are shown in Fig. 4.(b). The EPR state fidelity F = Ψ EP R |ρ|Ψ EP R is calculated to be 95.5(±2.0)%. We note that the fidelity of two-qutrit EPR state is higher than Cphase gate, this is because the special EPR state only has partial part in higher excited states, and thus it is less exposed to decoherence.
In summary, we propose and experimentally demonstrate a new scheme of implementing two-qutrit Cphase gate. This scheme utilizes a tunable coupler to control the cross-Kerr coupling between the two involved qutrits so that a large on/off ratio (about a few hundred) can be achieved. Such a design can largely suppress errors due to the cross-talk among qutrits induced by a non-negligible residual coupling. In addition, the maximum accessible coupling makes fast two-qutrit gates possible. We demonstrate this scheme on a superconducting quantum circuit where two transmons of fixed frequencies serve as qutrits and another transmon with a tunable frequency controls their coupling. The resulted two-qutrit Cphase gate has a fidelity of 89.3(±1.9)%, mainly limited by decoherence. To show potential applications of this Cphase gate in quantum circuits, we use it to prepare a twoqutrit EPR state with a fidelity of 95.5(±2.0)%. We expect the gate performance can be further enhanced with pulse optimization [49] and improved coherence [50,51]. We anticipate that the two qutrits Cphase gate demonstrated at here will inspire further research in the field of ternary quantum information processing [13,[61][62][63]. Note added -We notice a concurrent development of a similar work to implement Qutrit Entangling Gates with stark drives [63].

I. SUPERCONDUCTING QUBITS PROCESSER
There are eight transmon type superconducting qubits on our devices but only two of them and a coupler are used in this work. Processer design is the same as [1]. Qutrits frequency and other key parameters are shown in TABLE.SI.

II. CPHASE GATE PULSE SHAPE
The coupler between two qutrits has two Josephson junction SQUID structure, by which we can adjust the frequency of the coupler through external flux. The residual longitudinal coupling between two qutrits can be adjust by the coupler frequency. In order for the accumulation of phases to increase linearly with time, we use a square shape pulse with adiabatic edge to control the frequency of the coupler. For simplicity, the half-sin shape is chosen for the adiabatic edge in the two qutrits Cphase gate pulse in Fig. 3(a) in the main text. The rising edge and the falling edge have the same length of time τ 1 and the square shape length is τ 2 . A part of the control signal can be written as: In this work, τ 1 = 50ns, τ 2 = 19ns, and the single qutrit gate length is 40ns.

III. DECOHERENCE TIMES
The coupler between two qubit has no readout resonator. In order to estimate the effect of decoherence on the experiment, we measure the relationship between the coupler frequency and Z pulse amplitude (bias) and the effective decoherence times of both qutrits when the coupler at different frequencies, which are shown in Fig.S1 and Fig.S2, respectively. T 2 effectively describes both decoherence processes including energy dissipation time T 1 and pure dephasing time T φ . According to the trajectory of the coupler, the effective decoherence times in 01 subspace of Q 1 (Q 2 ) are T 01 1 = 27.1(16.8)µs and T 01 φ = 23.9(17.8)µs, where T 01 φ is calculated by 1/T 01 2 = 1/(2T 01 1 ) + 1/T 01 φ . Fig.S1. Coupler frequency vs. bias amplitude. A long and strong drive is applied to the qutrit to measure the energy spectrum of the system at the same time as the Z pulse modulating coupler frequency is applied. The blue full line is the coupler frequency fitted by A * cos(B * (bias + C)) + D and the red(green) dotted line is the frequency of Q1 01 (Q2 01 ). Because of the strong interaction between the qutrits and the coupler, the energy spectrum splits when the coupler and the qutrits are close to each other. The minimum value of energy spectrum splitting is equal to two times of coupling strength (coupler-qutrit). Mea.

V. THE LINDBLAD MASTER EQUATION
A commonly used method for calculating the dynamical evolution of a system with decoherence is the Lindblad master equation, which can be written as: where ρ is the system density matrix, H is the system Hamiltonian with drive and C n are collapse operators. In this work, C n ∈ { 1 √ T 01 1,Q1 a 1 , a † 2 a 2 } where a i (a † i ) is the corresponding creation(annihilation) operator of the qutrit i. For simplicity, we use the effective decoherence time T ef f here and omit the collapse operators of the coupler.

VI. TWO QUTRITS READOUT
The multiplexed dispersive readout scheme is used to distinguish two qutrits states. We prepare nine initial states and then measure the population of each state to get the assignment probability matrix M, which is shown in Fig.S4. In QPT experiment, the measured probability P m is multiplied by the inverse of the matrix to mitigate the measurement error, P cor = M −1 · P m .