Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

# Deterministic multi-mode gates on a scalable photonic quantum computing platform

## Abstract

Quantum computing can be realized with numerous different hardware platforms and computational protocols. A highly promising, and potentially scalable, idea is to combine a photonic platform with measurement-induced quantum information processing. In this approach, gate operations can be implemented through optical measurements on a multipartite entangled quantum state—a so-called cluster state. Previously, a few quantum gates on non-universal or non-scalable cluster states have been performed, but a full set of gates for universal scalable quantum computing has not been realized. Here we propose and demonstrate the deterministic implementation of a multi-mode set of measurement-induced quantum gates in a large two-dimensional optical cluster state using phase-controlled continuous-variable quadrature measurements. Each gate is programmed into the phases of high-efficiency quadrature measurements, which execute the transformations by teleportation through the cluster state. We further execute a small quantum circuit consisting of 10 single-mode gates and 2 two-mode gates on a three-mode input state. Fault-tolerant universal quantum computing is possible with this platform if the cluster-state entanglement is improved and a supply of states with Gottesman–Kitaev–Preskill encoding is available.

This is a preview of subscription content, access via your institution

## Relevant articles

• ### Taming numerical errors in simulations of continuous variable non-Gaussian state preparation

Scientific Reports Open Access 04 October 2022

• ### Slowing quantum decoherence of oscillators by hybrid processing

npj Quantum Information Open Access 15 June 2022

• ### Quantum computational advantage with a programmable photonic processor

Nature Open Access 01 June 2022

## Access options

\$32.00

All prices are NET prices.

## Data availability

Raw data and corresponding data analysis code that support the findings of this study (Figs. 24) are available at figshare with the identifier https://doi.org/10.11583/DTU.1467357053.

## References

1. Egan, L. et al. Fault-tolerant operation of a quantum error-correction code. Preprint at https://arxiv.org/pdf/2009.11482.pdf (2020).

2. Arute, F. et al. Quantum supremacy using a programmable superconducting processor. Nature 574, 505–510 (2019).

3. Raussendorf, R. & Briegel, H. J. A one-way quantum computer. Phys. Rev. Lett. 86, 5188–5191 (2001).

4. Larsen, M. V., Guo, X., Breum, C. R., Neergaard-Nielsen, J. S. & Andersen, U. L. Deterministic generation of a two-dimensional cluster state. Science 366, 369–372 (2019).

5. Asavanant, W. et al. Generation of time-domain-multiplexed two-dimensional cluster state. Science 366, 373–376 (2019).

6. Asavanant, W. et al. One-hundred step measurement-based quantum computation multiplexed in the time domain with 25-MHz clock frequency. Preprint at https://arxiv.org/pdf/2006.11537.pdf (2020).

7. Lloyd, S. & Braunstein, S. L. Quantum computation over continuous variables. Phys. Rev. Lett. 82, 1784–1787 (1999).

8. Menicucci, N. C. et al. Universal quantum computation with continuous-variable cluster states. Phys. Rev. Lett. 97, 110501 (2006).

9. Lund, A. P., Ralph, T. C. & Haselgrove, H. L. Fault-tolerant linear optical quantum computing with small-amplitude coherent states. Phys. Rev. Lett. 100, 030503 (2008).

10. Gottesman, D., Kitaev, A. & Preskill, J. Encoding a qubit in an oscillator. Phys. Rev. A 64, 012310 (2001).

11. Weedbrook, C. et al. Gaussian quantum information. Rev. Mod. Phys. 84, 621–669 (2012).

12. Andersen, U. L., Neergaard-Nielsen, J. S., van Loock, P. & Furusawa, A. Hybrid discrete- and continuous-variable quantum information. Nat. Phys. 11, 713–719 (2015).

13. Menicucci, N. C. Fault-tolerant measurement-based quantum computing with continuous-variable cluster states. Phys. Rev. Lett. 112, 120504 (2014).

14. Baragiola, B. Q., Pantaleoni, G., Alexander, R. N., Karanjai, A. & Menicucci, N. C. All-Gaussian universality and fault tolerance with the Gottesman–Kitaev–Preskill code. Phys. Rev. Lett. 123, 200502 (2019).

15. Yamasaki, H., Matsuura, T. & Koashi, M. Cost-reduced all-Gaussian universality with the Gottesman–Kitaev–Preskill code: resource-theoretic approach to cost analysis. Phys. Rev. Res. 2, 023270 (2020).

16. Fukui, K., Tomita, A., Okamoto, A. & Fujii, K. High-threshold fault-tolerant quantum computation with analog quantum error correction. Phys. Rev. X 8, 021054 (2018).

17. Noh, K. & Chamberland, C. Fault-tolerant bosonic quantum error correction with the surface Gottesman–Kitaev–Preskil code. Phys. Rev. A 101, 012316 (2020).

18. Ukai, R. et al. Demonstration of unconditional one-way quantum computations for continuous variables. Phys. Rev. Lett. 106, 240504 (2011).

19. Ukai, R., Yokoyama, S., Yoshikawa, J.-i, van Loock, P. & Furusawa, A. Demonstration of a controlled-phase gate for continuous-variable one-way quantum computation. Phys. Rev. Lett. 107, 250501 (2011).

20. Su, X. et al. Gate sequence for continuous variable one-way quantum computation. Nat. Commun. 4, 2828 (2013).

21. Reimer, C. et al. High-dimensional one-way quantum processing implemented on d-level cluster states. Nat. Phys. 15, 148–154 (2019).

22. Gao, W. et al. Experimental measurement-based quantum computing beyond the cluster-state model. Nat. Photon. 5, 117–123 (2011).

23. Walther, W. et al. Experimental one-way quantum computing. Nature 434, 169–176 (2005).

24. Lanyon, B. P. et al. Measurement-based quantum computation with trapped ions. Phys. Rev. Lett. 111, 210501 (2013).

25. Hastrup, J., Larsen, M. V., Neergaard-Nielsen, J. S., Menicucci, N. C. & Andersen, U. L. Cubic phase gates are not suitable for non-Clifford operations on GKP states. Phys. Rev. A 103, 032409 (2021).

26. Walshe, B. W., Baragiola, B. Q., Alexander, R. N. & Menicucci, N. C. Continuous-variable gate teleportation and bosonic-code error correction. Phys. Rev. A 102, 062411 (2020).

27. Menicucci, N. C. Temporal-mode continuous-variable cluster states using linear optics. Phys. Rev. A 83, 062314 (2011).

28. Yokoyama, S. et al. Ultra-large-scale continuous-variable cluster states multiplexed in the time domain. Nat. Photon. 7, 982–986 (2013).

29. Menicucci, N. C., Flammia, S. T. & van Loock, P. Graphical calculus for Gaussian pure states. Phys. Rev. A 83, 042335 (2011).

30. Alexander, R. N., Armstrong, S. C., Ukai, R. & Menicucci, N. C. Noise analysis of single-mode Gaussian operations using continuous-variable cluster states. Phys. Rev. A 90, 062324 (2014).

31. Larsen, M. V., Neergaard-Nielsen, J. S. & Andersen, U. L. Architecture and noise analysis of continuous-variable quantum gates using two-dimensional cluster states. Phys. Rev. A 102, 042608 (2020).

32. Alexander, R. N. et al. One-way quantum computing with arbitrarily large time-frequency continuous-variable cluster states from a single optical parametric oscillator. Phys. Rev. A 94, 032327 (2016).

33. Ukai, R., Yoshikawa, J.-i, Iwata, N., van Loock, P. & Furusawa, A. Universal linear Bogoliubov transformations through one-way quantum computation. Phys. Rev. A 81, 032315 (2010).

34. Gu, M., Weedbrook, C., Menicucci, N. C., Ralph, T. C. & van Loock, P. Quantum computing with continuous-variable clusters. Phys. Rev. A 79, 062318 (2009).

35. Walshe, B. W., Mensen, L. J., Baragiola, B. Q. & Menicucci, N. C. Robust fault tolerance for continuous-variable cluster states with excess antisqueezing. Phys. Rev. A 100, 010301(R) (2019).

36. Tasker, J. F. et al. Silicon photonics interfaced with integrated electronics for 9 GHz measurement of squeezed light. Nat. Photon. 15, 11–15 (2021).

37. Kashiwazaki, T. et al. Continuous-wave 6-dB-squeezed light with 2.5-THz-bandwidth from single-mode PPLN waveguide. APL Photon. 5, 036104 (2020).

38. Preskill, J. Quantum computing in the NISQ era and beyond. Quantum 2, 79 (2018).

39. Su, D. et al. Implementing quantum algorithms on temporal photonic cluster states. Phys. Rev. A 98, 032316 (2018).

40. Cai, Y. et al. Multimode entanglement in reconfigurable graph states using optical frequency combs. Nat. Commun. 8, 15645 (2017).

41. Chen, M., Menicucci, N. C. & Pfister, O. Experimental realization of multipartite entanglement of 60 modes of a quantum optical frequency comb. Phys. Rev. Lett. 112, 120505 (2014).

42. Armstrong, S. et al. Programmable multimode quantum networks. Nat. Commun. 3, 1026 (2012).

43. Larsen, M. V., Chamberland, C., Noh, K., Neergaard-Nielsen, J. S. & Andersen, U. L. A fault-tolerant continuous-variable measurement-based quantum computation architecture. Preprint at https://arxiv.org/pdf/2101.03014.pdf (2021).

44. Vahlbruch, H., Mehmet, M., Danzmann, K. & Schnabel, R. Detection of 15-dB squeezed states of light and their application for the absolute calibration of photoelectric quantum efficiency. Phys. Rev. Lett. 117, 110801 (2016).

45. Flühmann, C. et al. Encoding a qubit in a trapped-ion mechanical oscillator. Nature 566, 513–517 (2019).

46. Campagne-Ibarcq, P. et al. Quantum error correction of a qubit encoded in grid states of an oscillator. Nature 584, 368–372 (2020).

47. Pirandola, S., Mancini, S., Vitali, D. & Tombesi, P. Generating continuous variable quantum codewords in the near-field atomic lithography. J. Phys. B 39, 997 (2006).

48. Motes, K. R., Baragiola, B. Q., Gilchrist, A. & Menicucci, N. C. Encoding qubits into oscillators with atomic ensembles and squeezed light. Phys. Rev. A 95, 053819 (2017).

49. Weigand, D. J. & Terhal, B. M. Generating grid states from Schrödinger-cat states without postselection. Phys. Rev. A 97, 022341 (2018).

50. Eaton, M., Nehra, R. & Pfister, O. Non-Gaussian and Gottesman–Kitaev–Preskill state preparation by photon catalysis. New J. Phys. 21, 113034 (2019).

51. Su, D., Myers, C. R. & Sabapathy, K. K. Conversion of Gaussian states to non-Gaussian states using photon-number-resolving detectors. Phys. Rev. A 100, 052301 (2019).

52. Tzitrin, I., Bourassa, J. E., Menicucci, N. C. & Sabapathy, K. K. Progress towards practical qubit computation using approximate Gottesman–Kitaev–Preskill codes. Phys. Rev. A 101, 032315 (2020).

53. Larsen, M. V., Guo, X., Breum, C. R., Neergaard-Nielsen, J. S. & Andersen, U. L. Deterministic multi-mode gates on a scalable photonic quantum computing platform. figshare https://doi.org/10.11583/DTU.14673570 (2021).

## Acknowledgements

We acknowledge useful discussions with R. N. Alexander and J. Hastrup. The work was supported by the Danish National Research Foundation through the Center for Macroscopic Quantum States (bigQ, DNRF0142).

## Author information

Authors

### Contributions

M.V.L. and U.L.A. conceived the project. J.S.N.-N., X.G., C.R.B. and M.V.L. built the squeezing sources. M.V.L. developed the theoretical background, designed the experiment and built the set-up. M.V.L. performed the experiment and data analysis. The project was supervised by U.L.A. and J.S.N.-N. The manuscript was written by U.L.A., M.V.L. and J.S.N.-N., with feedback from all authors.

### Corresponding authors

Correspondence to Mikkel V. Larsen or Ulrik L. Andersen.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

Peer review informationNature Physics thanks the anonymous reviewers for their contribution to the peer review of this work.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Supplementary information

### Supplementary Information

Supplementary Sections 1–4, Figs. 1–11 and refs. 1–22.

## Rights and permissions

Reprints and Permissions

Larsen, M.V., Guo, X., Breum, C.R. et al. Deterministic multi-mode gates on a scalable photonic quantum computing platform. Nat. Phys. 17, 1018–1023 (2021). https://doi.org/10.1038/s41567-021-01296-y

• Accepted:

• Published:

• Issue Date:

• DOI: https://doi.org/10.1038/s41567-021-01296-y

• ### Slowing quantum decoherence of oscillators by hybrid processing

• Kimin Park
• Jacob Hastrup
• Ulrik L. Andersen

npj Quantum Information (2022)

• ### Quantum computational advantage with a programmable photonic processor

• Fabian Laudenbach
• Jonathan Lavoie

Nature (2022)

• ### Taming numerical errors in simulations of continuous variable non-Gaussian state preparation

• Jan Provazník