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

Kai Luo, Wenhui Huang, Ziyu Tao, Libo Zhang, Yuxuan Zhou, Ji Chu, Wuxin Liu, Biying Wang, Jiangyu Cui, Song Liu, Fei Yan, Man-Hong Yung, Yuanzhen Chen, Tongxing Yan, and Dapeng Yu

Phys. Rev. Lett. 130, 030603 – Published 20 January 2023

Abstract

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 multidimensional 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 Letter, 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 crosstalk between qutrits, thus being friendly for scaling up. Our Letter constitutes an important step toward scalable qutrit-based quantum computation.

Received 5 July 2022
Accepted 3 January 2023

DOI:https://doi.org/10.1103/PhysRevLett.130.030603

Physics Subject Headings (PhySH)

Entanglement production Quantum circuits Quantum computation Quantum control Quantum gates

Superconducting qubits

Quantum Information, Science & Technology

Authors & Affiliations

Kai Luo^1,2,3, Wenhui Huang^2,3, Ziyu Tao ^2,3, Libo Zhang^2,3, Yuxuan Zhou^2,3, Ji Chu³, Wuxin Liu ⁴, Biying Wang⁴, Jiangyu Cui⁴, Song Liu^3,5,6,7, Fei Yan^3,5,6,7, Man-Hong Yung^{2,3,4,5,6,7,*}, Yuanzhen Chen^{2,3,5,6,7,†}, Tongxing Yan ^3,5,6,7,‡, and Dapeng Yu^2,3,5,6,7

¹Department of Physics, Harbin Institute of Technology, Harbin 150001, China
²Department of Physics, Southern University of Science and Technology, Shenzhen 518055, China
³Shenzhen Institute for Quantum Science and Engineering, Southern University of Science and Technology, Shenzhen 518055, China
⁴Central Research Institute, 2012 Labs, Huawei Technologies, Shenzhen, 518129, China
⁵Guangdong Provincial Key Laboratory of Quantum Science and Engineering, Southern University of Science and Technology, Shenzhen 518055, China
⁶Shenzhen Key Laboratory of Quantum Science and Engineering, Southern University of Science and Technology, Shenzhen 518055, China
⁷International Quantum Academy, Shenzhen 518048, China

^*yung@sustech.edu.cn
^†chenyz@sustech.edu.cn
^‡yantx@sustech.edu.cn

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Vol. 130, Iss. 3 — 20 January 2023

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Figure 1
Schematics of device and two-qutrit Cphase gate. (a) False-colored micrograph of the superconducting circuit used in this Letter. Two transmon qutrits (pink and orange) with fixed frequencies are connected to another transmon with tunable frequency serving as a coupler (blue). Green lines are for control. The left control line is shared by the left transmon for its single-qutrit operations and the coupler for tuning its frequency. Each qutrit is capacitively coupled to a dedicated $λ / 4$ resonator for readout. (b) Simplified circuit model of the device in (a). (c) Energy diagram of a typical transmon as a multilevel system. The lowest three levels ( $| 0 ⟩, | 1 ⟩, | 2 ⟩$ ) span the computational subspace of a single qutrit. (d) Partial energy diagram of the qutrit-coupler-qutrit system vs coupler frequency. The eigenenergy of state $| 102 ⟩$ (dark blue curve), ${\tilde{ω}}_{102}$ , is compared to the summation (orange) of eigenenergies of the states $| 100 ⟩$ and $| 002 ⟩$ . Their difference (shaded area) represents the conditional phase $χ_{102}$ , 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 $χ_{101}$ , $χ_{201}$ , and $χ_{202}$ .
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Figure 2
Tunable cross-Kerr coupling between qutrits. (a) Pulse sequence of a Ramsey-like experiment on $Q_{1}$ conditioned by the state of $Q_{2}$ to extract the cross-Kerr coupling $χ_{102}$ . Two $π / 2$ pulses are applied to $Q_{1}$ . 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 $Q_{1}$ , 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 $Q_{1}$ . Depending on the magnitude of $V_{b}$ , $τ$ is set to different values ranging from 1 to 2000 ns (see Supplemental Material [57] for details). Prior to the above three pulses, $Q_{2}$ 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 $Q_{1}$ is measured as a function of $θ$ . (b) Experimental data (markers) are fitted by a sinusoidal function (solid lines) to extract the phase difference $Δ ϕ$ between the cases of $Q_{2}$ being prepared into the second excited (orange) and the ground (blue) states. The effective cross-Kerr coupling $χ_{102}$ can be derived with the relation $Δ ϕ = χ_{102} τ$ at this fixed $V_{b}$ condition. Similar measurements are performed to obtain $χ_{101}$ , $χ_{201}$ , and $χ_{202}$ . (c) Repeated experiments in (b) with different pulse amplitudes $V_{b}$ , and combination the mapping relation pulse amplitudes $V_{b}$ to coupler frequency $ω_{c}$ (not shown here), we obtain four $χ_{i 0 j}$ (markers) depends on $ω_{c}$ for two qutrits, which is in agreement with numerical results quite well (corresponding same color lines). We choose coupler sweet point $ω_{c} \approx 7.640 GHz$ as the idling point to keep small residual couplings in this Letter.
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Figure 3
Realization of two-qutrits Cphase gate. (a) The complete gate sequence consists of two identical pulses controlling the coupler frequency and four single-qutrit $π$ rotations in their $| 1 ⟩$ , $| 2 ⟩$ subspaces. The pulses applied to the coupler have flat tops with a duration of $τ$ and two half-sine shaped edges. Each edge has a duration of 50 ns [57], time axis is not to scale. (b) Phases accumulated on different states as a function of $τ$ . Experimental results (markers) are fitted linearly with the error bars indicating one standard deviation. Notice that the two fitting lines are not parallel to each other and they reach $\frac{2}{3} π$ and $\frac{4}{3} π$ at $τ = 19 ns$ , where the desired phase accumulation for the Cphase gate is fulfilled. (c) Full QPT results of the Cphase gate. Left and right panels are the ideal and experimental quantum process matrix $| χ_{m n} |$ . Coordinates on both axes number the orthogonal basis operators $E_{i}$ [57].
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Figure 4
Preparation of two-qutrit EPR state. (a) Quantum circuit for generating a two-qutrit EPR state using the Cphase gate (green box) and single-qutrit Hadamard gates (blue boxes), followed by a QST characterization. (b) Real and imaginary parts of the density matrix of the two-qutrit EPR state. Solid bars and black outlines are experimental and ideal results, respectively.
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