Experimental High-Dimensional Greenberger-Horne-Zeilinger Entanglement with Superconducting Transmon Qutrits

Alba Cervera-Lierta, Mario Krenn, Alán Aspuru-Guzik, and Alexey Galda

Phys. Rev. Applied 17, 024062 – Published 23 February 2022

Abstract

Multipartite entanglement is one of the core concepts in quantum information science with broad applications that span from condensed matter physics to quantum physics foundation tests. Although its most studied and tested forms encompass two-dimensional systems, current quantum platforms technically allow the manipulation of additional quantum levels. We report the experimental demonstration and certification of a high-dimensional multipartite entangled state in a superconducting quantum processor. We generate the three-qutrit Greenberger-Horne-Zeilinger state by designing the necessary pulses to perform high-dimensional quantum operations. We obtain the fidelity of $76 % \pm 1 %$ , proving the generation of a genuine three-partite and three-dimensional entangled state. To this date, only photonic devices have been able to create and certify the entanglement of these high-dimensional states. Our work demonstrates that another platform, superconducting systems, is ready to exploit genuine high-dimensional entanglement and that a programmable quantum device accessed on the cloud can be used to design and execute experiments beyond binary quantum computation.

Received 12 July 2021
Revised 1 September 2021
Accepted 5 January 2022

DOI:https://doi.org/10.1103/PhysRevApplied.17.024062

Physics Subject Headings (PhySH)

Circuit quantum electrodynamics Quantum control Quantum entanglement Quantum gates Quantum information with solid state qubits Quantum tomography

Superconducting qubits

Microwave techniques

Quantum Information, Science & Technology

Authors & Affiliations

Alba Cervera-Lierta ^1,2,8,*, Mario Krenn^1,2,3,9,†, Alán Aspuru-Guzik^1,2,3,4,‡, and Alexey Galda^5,6,7,§

¹Department of Chemistry, Chemical Physics Theory Group, University of Toronto, Ontario M5S 3H6, Canada
²Department of Computer Science, University of Toronto, Ontario M5T 3A1, Canada
³Vector Institute for Artificial Intelligence, Toronto, Ontario M5G 1M1, Canada
⁴Canadian Institute for Advanced Research (CIFAR) Lebovic Fellow, Toronto, Ontario M5G 1M1, Canada
⁵Menten AI, Inc., San Francisco, California 94111, USA
⁶James Franck Institute, University of Chicago, Chicago, Illinois 60637, USA
⁷Computational Science Division, Argonne National Laboratory, Lemont, Illinois 60439, USA
⁸Barcelona Supercomputing Center, Barcelona 08034, Spain
⁹Max Planck Institute for the Science of Light (MPL), Erlangen 91058, Germany

^*a.cervera.lierta@gmail.com
^†mario.krenn@mpl.mpg.de
^‡alan@aspuru.com
^§agalda@uchicago.edu

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 17, Iss. 2 — February 2022

Subject Areas

Reuse & Permissions

Access Options

Article Available via CHORUS

Download Accepted Manuscript

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
Map of Entangland. It represents the experimental state-of-the-art generation of high-dimensional multipartite states. Bipartite entangled states have been generated for up to 100 dimensions with photons [37, 38], and recently a qutrit Bell state was generated with superconducting circuits [18] (Bell state region). On the other hand, multipartite qubit states have also been produced in different platforms [39, 40, 41, 42, 43, 44, 45] (Qubitland). The present work demonstrates the experimental generation and certification of a three-dimensional GHZ state with superconducting quantum devices. There are only two photonic experiments that have been able to explore Entangland beyond the Alice and Bob range, i.e., the divide between bipartite and multipartite entanglement, and to cross the Flatland river [36, 46, 47], i.e., the divide between qubit states and $d > 2$ level subsystems. Besides these photonic works, the experiment from Ref. [18] requires high-dimensional entanglement, although the entanglement of the state generated is not certified experimentally.
Reuse & Permissions
Figure 2
Quantum circuit to generate the qutrit GHZ state, its diagrammatic representation through the GHZ clock, and the pulse schedule. The single-qutrit gates act in the (01) and (12) subspaces, and the cnot gates act as described in Table 2. (a) First, a superposition in the (01) subspace is generated in the second qutrit using a $R_{y}^{(01)}$ gate with angle $θ = 2 \arctan (1 / \sqrt{2})$ . (b) Next, two cnot gates are applied to create two-dimensional entanglement across all three transmons. (c) The superposition state is shifted from the (01) to the (12) subspace by using $R_{x}^{(12)}$ and $R_{y}^{(12)}$ gates. (d) The third basis element is created by generating another (01) superposition, followed by repeating the application of cnot gates controlled on the central qutrit to finish the qutrit GHZ state. The bottom panel shows the pulse schedule that executes the circuit. Drive channels $D 1$ , $D 2$ , and $D 3$ are used to implement calibrated microwave pulses of qutrits $Q 2$ , $Q 3$ , and $Q 4$ from the ibmq_rome device, respectively. The cross resonance gates utilized to implement the cnot operations are executed on channels $U 5$ and $U 6$ by transmitting two-transmon control drives.
Reuse & Permissions
Figure 3
Top: experimental results of the density matrix elements needed to compute the fidelity with respect to the GHZ state. Only six elements are required to reconstruct the fidelity: the three diagonal terms $⟨ 000 | ρ | 000 ⟩$ , $⟨ 111 | ρ | 111 ⟩$ , and $⟨ 222 | ρ | 222 ⟩$ , and the off-diagonal terms $⟨ 000 | ρ | 111 ⟩$ , $⟨ 000 | ρ | 222 ⟩$ , and $⟨ 111 | ρ | 222 ⟩$ . The plot shows the absolute value of these elements (in gray; all other elements not measured). The corresponding fidelity obtained is $76 % \pm 1 %$ , exceeding the 2/3 threshold required to certify genuine three-partite entanglement. The real parts of these elements are presented in Appendix pp5. Bottom: probability amplitudes of each computational basis element as a function of the phase introduced with a $R_{z}^{(01)}$ gate between the GHZ generation and the gates used for the tomography. As expected, these amplitudes oscillate due to the reference frame imposed in the (01) subspace.
Reuse & Permissions
Figure 4
Oscillations of the expectation value of each basis element used to compute the tomography of a GHZ experiment as a function of the delay introduced between the two $R^{(12)}$ gates used in the projective measurements. This experiment is carried out using the ibmq_rome chip on transmons $Q 2$ - $Q 3$ - $Q 4$ on a different day than the results shown in the main paper. The fidelities obtained are $F_{expt.} = 0.73 \pm 0.01$ and $F_{raw} = 0.67 \pm 0.01$ .
Reuse & Permissions
Figure 5
Rabi $1 \to 2$ experiments performed to calibrate the amplitude of the $π_{1 \to 2}$ pulse. The amplitudes obtained are 0.278, 0.225, 0.267 for transmons $Q 2$ , $Q 3$ , and $Q 4$ , respectively.
Reuse & Permissions
Figure 6
Calibration matrix for $| 0 ⟩ / | 1 ⟩$ readout on the [ $Q 4$ , $Q 3$ , $Q 2$ ] subset of qutrits of the ibmq_rome chip.
Reuse & Permissions

Physical Review Applied