Empowering a qudit-based quantum processor by traversing the dual bosonic ladder

doi:10.1038/s41467-024-51434-2

. 2024 Aug 19;15(1):7117.

doi: 10.1038/s41467-024-51434-2.

Empowering a qudit-based quantum processor by traversing the dual bosonic ladder

Long B Nguyen^#^{1

2}, Noah Goss^#^{3

4}, Karthik Siva^{5

6}, Yosep Kim⁷, Ed Younis⁸, Bingcheng Qing⁵, Akel Hashim^{5

8}, David I Santiago^{5

8}, Irfan Siddiqi^{5

8}

Affiliations

¹ Department of Physics, University of California, Berkeley, CA, USA. longbnguyen@berkeley.edu.
² Computational Research Division, Lawrence Berkeley National Laboratory, Berkeley, CA, USA. longbnguyen@berkeley.edu.
³ Department of Physics, University of California, Berkeley, CA, USA. noahgoss@berkeley.edu.
⁴ Computational Research Division, Lawrence Berkeley National Laboratory, Berkeley, CA, USA. noahgoss@berkeley.edu.
⁵ Department of Physics, University of California, Berkeley, CA, USA.
⁶ IBM Quantum, Yorktown Heights, NY, USA.
⁷ Department of Physics, Korea University, Seoul, Korea.
⁸ Computational Research Division, Lawrence Berkeley National Laboratory, Berkeley, CA, USA.

^# Contributed equally.

PMID: 39160166
PMCID: PMC11333499
DOI: 10.1038/s41467-024-51434-2

Empowering a qudit-based quantum processor by traversing the dual bosonic ladder

Long B Nguyen et al. Nat Commun. 2024.

. 2024 Aug 19;15(1):7117.

doi: 10.1038/s41467-024-51434-2.

Authors

Long B Nguyen^#^{1

2}, Noah Goss^#^{3

4}, Karthik Siva^{5

6}, Yosep Kim⁷, Ed Younis⁸, Bingcheng Qing⁵, Akel Hashim^{5

8}, David I Santiago^{5

8}, Irfan Siddiqi^{5

8}

Affiliations

¹ Department of Physics, University of California, Berkeley, CA, USA. longbnguyen@berkeley.edu.
² Computational Research Division, Lawrence Berkeley National Laboratory, Berkeley, CA, USA. longbnguyen@berkeley.edu.
³ Department of Physics, University of California, Berkeley, CA, USA. noahgoss@berkeley.edu.
⁴ Computational Research Division, Lawrence Berkeley National Laboratory, Berkeley, CA, USA. noahgoss@berkeley.edu.
⁵ Department of Physics, University of California, Berkeley, CA, USA.
⁶ IBM Quantum, Yorktown Heights, NY, USA.
⁷ Department of Physics, Korea University, Seoul, Korea.
⁸ Computational Research Division, Lawrence Berkeley National Laboratory, Berkeley, CA, USA.

^# Contributed equally.

PMID: 39160166
PMCID: PMC11333499
DOI: 10.1038/s41467-024-51434-2

Abstract

High-dimensional quantum information processing has emerged as a promising avenue to transcend hardware limitations and advance the frontiers of quantum technologies. Harnessing the untapped potential of the so-called qudits necessitates the development of quantum protocols beyond the established qubit methodologies. Here, we present a robust, hardware-efficient, and scalable approach for operating multidimensional solid-state systems using Raman-assisted two-photon interactions. We then utilize them to construct extensible multi-qubit operations, realize highly entangled multidimensional states including atomic squeezed states and Schrödinger cat states, and implement programmable entanglement distribution along a qudit array. Our work illuminates the quantum electrodynamics of strongly driven multi-qudit systems and provides the experimental foundation for the future development of high-dimensional quantum applications such as quantum sensing and fault-tolerant quantum computing.

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Conflict of interest statement

The authors declare no competing interests.

Figures

**Fig. 1. A qudit-based quantum processor.**
a Schematic of the high-dimensional system which is constructed by linking individual qudits into an array. Each qudit in the chain is a nonlinear harmonic oscillator comprising of a Josephson junction and a capacitor in parallel, and its eigenstates form a bosonic ladder of d levels within the cosine potential. They are depicted as non-cyclical gears that can be rotated either individually using local controls or simultaneously using two-photon drives. b Local control of a qudit. (Top) Rabi oscillations of the populations under resonant microwave drives. The solid lines represent cosine fits. (Bottom) Randomized benchmarking of single-qudit gates. The solid lines represent exponential fits. c Readout of a qudit. (Top) A resonator is dispersively coupled to a qudit to measure its state. The probe signal from the resonator distinctly separates into individual blobs on the IQ plane, corresponding to the qudit being in the states $∣0⟩, ∣1⟩, ∣2⟩$ , and $∣3⟩$ . (Bottom) Preparation and measurement confusion matrix of qudit states, which reflects the readout fidelities (Supplementary Note 1). d Simplified depiction of the Raman-assisted two-photon-driven dynamics. The coupled-qudits pair form a set of eigenstates, visualized as a dual bosonic ladder. The entangling dynamics takes place in a four-level manifold within this structure. The effect of microwave drives applied to both qudits at a frequency close to the average of the single-excitation frequencies in this subspace is twofold: they first map the relevant qudit states to the dressed frame, then induce a Raman-assisted transition between the bipartite states $∣k, l⟩$ and $∣k + 1, l + 1⟩$ (inset), resulting in an interaction at a rate Ω_2p.

**Fig. 2. Traversing the dual bosonic ladder.**
a Coherent $∣00⟩ \leftrightarrow ∣11⟩$ population exchange between two coupled qudits. The interaction is induced by monochromatic microwave drives. Here, the drive frequency is shown with respect to the optimal point where the oscillation is the most coherent. The chevron is also quasi-symmetric about this frequency point. b Coherent $∣11⟩ \leftrightarrow ∣22⟩$ and $∣11⟩ \leftrightarrow ∣02⟩$ population exchange between two coupled qudits. The former interaction is induced by the two-photon Raman process, while the latter originates from the resonant condition between $∣11⟩$ and $∣02⟩$ in the driven frame. c Coherent flip-flop oscillation between $∣11⟩$ and $∣22⟩$ states obtained at the optimal drive frequency, indicated by the red arrows in panel b. The solid lines represent cosine fits. d Optimal driving frequencies at various drive amplitudes. The data are shown with respect to ω_avg = (ω_k,l+1 + ω_k+1,l)/2 in the y-axis. The x-axis is expressed in terms of the single-photon Rabi rate in the 0-1 subspace of the first qudit. The solid lines are analytical results. e Interaction rates of the dual bosonic transitions up to d = 4 at various drive amplitudes. All data points are measured at the optimal driving frequencies. Analytical results are shown as solid lines.

**Fig. 3. Implementation and verification of multi-qubit gates.**
a Gate sequence to implement a CCZ unitary. The qutrit swap gates (pink) are used to shelve and then retrieve the control state ${∣11⟩}_{c}$ . They sandwich a cross-Kerr gate (green) that induces a Z gate on Q₁ if and only if Q₂ is in $∣2⟩$ . The final stage (blue) is used to correct the residual ZZ phases between the qubits. b The three-body operation manifests as the phase shift of the target qubit (Q₁) when the control qubits (Q₂ and Q₃) are in ${∣11⟩}_{c}$ . The solid lines represent cosine fits. c Pauli fidelities of the dressed cycle and the reference cycle from cycle benchmarking. The resulting gate fidelity is $F_{CCZ} = 96.0 (3) %$ . d Gate sequence to implement a CCCZ unitary. A cascade of qutrit swap interactions (pink) is used to shelve and retrieve the respective ${∣11⟩}_{c}$ states. In the middle of the sequence is a cross-Kerr gate (green) that induces a Z gate on Q₄ if and only if Q₃ is in $∣2⟩$ . All the residual correlated phases are corrected in the final stage (blue). e The four-body operation is effectively revealed through a π-phase-shift of the target qubit (Q₄) for the control state ${∣110⟩}_{c}$ . The solid lines represent cosine fits. f Truth table of the four-qubit Toffoli gate. The target state is shown to be flipped when the control state is $∣Q_{3} Q_{2} Q_{1}⟩ = {∣110⟩}_{c}$ ( $∣0110⟩ \leftrightarrow ∣1110⟩$ ). The corresponding truth table fidelity is $F_{CCCZ} = 92 (1) %$ .

**Fig. 4. High-dimensional entanglement.**
a Density matrix of the qubit (d = 2) Bell state with a raw (purified) fidelity of 99.2% (99.9%). b Density matrix of the qutrit (d = 3) Bell state with $F = 97.7 % (99.6 %)$ . c Density matrix (left) and Husimi-Q distribution (right) of the ququart (d = 4) Bell state with $F = 94.3 % (99.3 %)$ . d Husimi-Q distribution (left) and density matrix (right) of the ququart (d = 4) NOON state $∣30⟩ + e^{i α} ∣03⟩ () / \sqrt{2}$ with $F = 94.6 % (97.3 %)$ . e Wigner functions of the high-dimensional atomic cat states, $(∣00⟩ + e^{i α} ∣33⟩) / \sqrt{2}$ with $F = 98.6 % (99.3 %)$ (left) and $(∣000⟩ + e^{i α} ∣333⟩) / \sqrt{2}$ with $F = 80.1 % (90.9 %)$ (right).

**Fig. 5. Distribution of qudit entanglement.**
Measured populations of (a) Q₁-Q₂ Bell₃ state, (b) Q₁-Q₃ Bell₃ state, (c) Q₁-Q₄ Bell₃ state, (d) Q₁-Q₂-Q₃ GHZ₃ state, and (e) Q₁-Q₂-Q₃-Q₄ GHZ₃ state. (f) Density matrix of the four-qudit GHZ₃ state. The raw and purified fidelities are 70.6% and 82.7%, respectively.

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References

1. Wang, Y., Hu, Z., Sanders, B. C. & Kais, S. Qudits and high-dimensional quantum computing. Front. Phys.8, 589504 (2020).10.3389/fphy.2020.589504 - DOI
1. Gokhale, P. et al. Asymptotic improvements to quantum circuits via qutrits. In Proceedings of the 46th International Symposium on Computer Architecture, ISCA ’19, p. 554–566 (Association for Computing Machinery, 2019).
1. Gustafson, E. Noise improvements in quantum simulations of sQED using qutrits. arXiv10.48550/arXiv.2201.04546 (2022).
1. Cozzolino, D., Da Lio, B., Bacco, D. & Oxenløwe, LeifKatsuo High-dimensional quantum communication: benefits, progress, and future challenges. Adv. Quant. Technol.2, 1900038 (2019).10.1002/qute.201900038 - DOI
1. Campbell, E. T. Enhanced fault-tolerant quantum computing in d-level systems. Phys. Rev. Lett.113, 230501 (2014). 10.1103/PhysRevLett.113.230501 - DOI - PubMed

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[1] Wang, Y., Hu, Z., Sanders, B. C. & Kais, S. Qudits and high-dimensional quantum computing. Front. Phys.8, 589504 (2020).10.3389/fphy.2020.589504 - DOI

[2] Wang, Y., Hu, Z., Sanders, B. C. & Kais, S. Qudits and high-dimensional quantum computing. Front. Phys.8, 589504 (2020).10.3389/fphy.2020.589504 - DOI

[3] Gokhale, P. et al. Asymptotic improvements to quantum circuits via qutrits. In Proceedings of the 46th International Symposium on Computer Architecture, ISCA ’19, p. 554–566 (Association for Computing Machinery, 2019).

[4] Gokhale, P. et al. Asymptotic improvements to quantum circuits via qutrits. In Proceedings of the 46th International Symposium on Computer Architecture, ISCA ’19, p. 554–566 (Association for Computing Machinery, 2019).

[5] Gustafson, E. Noise improvements in quantum simulations of sQED using qutrits. arXiv10.48550/arXiv.2201.04546 (2022).

[6] Gustafson, E. Noise improvements in quantum simulations of sQED using qutrits. arXiv10.48550/arXiv.2201.04546 (2022).

[7] Cozzolino, D., Da Lio, B., Bacco, D. & Oxenløwe, LeifKatsuo High-dimensional quantum communication: benefits, progress, and future challenges. Adv. Quant. Technol.2, 1900038 (2019).10.1002/qute.201900038 - DOI

[8] Cozzolino, D., Da Lio, B., Bacco, D. & Oxenløwe, LeifKatsuo High-dimensional quantum communication: benefits, progress, and future challenges. Adv. Quant. Technol.2, 1900038 (2019).10.1002/qute.201900038 - DOI

[9] Campbell, E. T. Enhanced fault-tolerant quantum computing in d-level systems. Phys. Rev. Lett.113, 230501 (2014). 10.1103/PhysRevLett.113.230501 - DOI - PubMed

[10] Campbell, E. T. Enhanced fault-tolerant quantum computing in d-level systems. Phys. Rev. Lett.113, 230501 (2014). 10.1103/PhysRevLett.113.230501 - DOI - PubMed

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Empowering a qudit-based quantum processor by traversing the dual bosonic ladder

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Empowering a qudit-based quantum processor by traversing the dual bosonic ladder

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