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Constant-overhead fault-tolerant quantum computation with reconfigurable atom arrays

Nature Physics (2024)Cite this article

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Abstract

Quantum low-density parity-check (qLDPC) codes can achieve high encoding rates and good code distance scaling, potentially enabling low-overhead fault-tolerant quantum computing. However, implementing qLDPC codes involves nonlocal operations that require long-range connectivity between qubits. This makes their physical realization challenging in comparison to geometrically local codes, such as the surface code. Here we propose a hardware-efficient scheme for fault-tolerant quantum computation with high-rate qLDPC codes that is compatible with the recently demonstrated capabilities of reconfigurable atom arrays. Our approach utilizes the product structure inherent in many qLDPC codes to implement the nonlocal syndrome extraction circuit through atom rearrangement, resulting in an effectively constant overhead. We prove the fault tolerance of these protocols, and our simulations show that the qLDPC-based architecture starts to outperform the surface code with as few as several hundred physical qubits. We further find that quantum algorithms involving thousands of logical qubits can be performed using less than 105 physical qubits. Our work suggests that low-overhead quantum computing with qLDPC codes is within reach using current experimental technologies.

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Fig. 1: Architecture of a qLDPC-based fault-tolerant quantum computer using reconfigurable atom arrays.
Fig. 2: Efficient implementation of quantum LDPC codes with atom arrays.
Fig. 3: qLDPC memory performance.
Fig. 4: Fault-tolerant teleportation from surface to qLDPC code.

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Data availability

The data collected and analysed in this article are available at https://doi.org/10.5281/zenodo.8278063 (ref. 65). Source data are provided with this paper.

Code availability

All codes used to generate the figures are available upon request.

References

  1. Fowler, A. G., Mariantoni, M., Martinis, J. M. & Cleland, A. N. Surface codes: towards practical large-scale quantum computation. Phys. Rev. A 86, 032324 (2012).

    ADS  Google Scholar 

  2. Litinski, D. A game of surface codes: large-scale quantum computing with lattice surgery. Quantum 3, 128 (2019).

    Google Scholar 

  3. Beverland, M. E. et al. Assessing requirements to scale to practical quantum advantage. Preprint at https://arxiv.org/abs/2211.07629 (2022).

  4. Gidney, C. & Ekerå, M. How to factor 2048 bit RSA integers in 8 hours using 20 million noisy qubits. Quantum 5, 433 (2019).

    Google Scholar 

  5. Kitaev, A. Y. Fault-tolerant quantum computation by anyons. Ann. Phys. 303, 2–30 (2003).

    ADS  MathSciNet  Google Scholar 

  6. Tillich, J. P. & Zemor, G. Quantum LDPC codes with positive rate and minimum distance proportional to the square root of the blocklength. IEEE Trans. Inf. Theory 60, 1193–1202 (2014).

    MathSciNet  Google Scholar 

  7. Panteleev, P. & Kalachev, G. Degenerate quantum LDPC codes with good finite length performance. Quantum 5, 585 (2019).

    Google Scholar 

  8. Panteleev, P. & Kalachev, G. Quantum LDPC codes with almost linear minimum distance. IEEE Trans. Inf. Theory 68, 213–229 (2022).

    MathSciNet  Google Scholar 

  9. Breuckmann, N. P. & Eberhardt, J. N. Balanced product quantum codes. IEEE Trans. Inf. Theory 67, 6653–6674 (2020).

    MathSciNet  Google Scholar 

  10. Panteleev, P. & Kalachev, G. Asymptotically good quantum and locally testable classical LDPC codes. In Proc. Annual ACM Symposium on Theory of Computing 375–388 (ACM, 2022).

  11. Bravyi, S., Poulin, D. & Terhal, B. Tradeoffs for reliable quantum information storage in 2D systems. Phys. Rev. Lett. 104, 050503 (2010).

    ADS  Google Scholar 

  12. Baspin, N. & Krishna, A. Connectivity constrains quantum codes. Quantum 6, 711 (2021).

    Google Scholar 

  13. Baspin, N. & Krishna, A. Quantifying nonlocality: how outperforming local quantum codes is expensive. Phys. Rev. Lett. 129, 050505 (2022).

    ADS  MathSciNet  Google Scholar 

  14. Tremblay, M. A., Delfosse, N. & Beverland, M. E. Constant-overhead quantum error correction with thin planar connectivity. Phys. Rev. Lett. 129, 050504 (2022).

    ADS  MathSciNet  Google Scholar 

  15. Delfosse, N., Beverland, M. E. & Tremblay, M. A. Bounds on stabilizer measurement circuits and obstructions to local implementations of quantum LDPC codes. Preprint at https://arxiv.org/abs/2109.14599v1 (2021).

  16. Strikis, A. & Berent, L. Quantum low-density parity-check codes for modular architectures. PRX Quantum 4, 020321 (2023).

    ADS  Google Scholar 

  17. Gottesman, D. Fault-tolerant quantum computation with constant overhead. Quantum Inf. Comput. 14, 1338–1372 (2013).

    MathSciNet  Google Scholar 

  18. Cohen, L. Z., Kim, I. H., Bartlett, S. D. & Brown, B. J. Low-overhead fault-tolerant quantum computing using long-range connectivity. Sci. Adv. 8, eabn1717 (2022).

    Google Scholar 

  19. Krishna, A. & Poulin, D. Fault-tolerant gates on hypergraph product codes. Phys. Rev. X 11, 011023 (2021).

    Google Scholar 

  20. Breuckmann, N. P. & Burton, S. Fold-transversal Clifford gates for quantum codes. Preprint at https://arxiv.org/abs/2202.06647v2 (2022).

  21. Quintavalle, A. O., Webster, P. & Vasmer, M. Partitioning qubits in hypergraph product codes to implement logical gates. Preprint at https://arxiv.org/abs/2204.10812v2 (2022).

  22. Bluvstein, D. et al. A quantum processor based on coherent transport of entangled atom arrays. Nature 604, 451–456 (2022).

    ADS  Google Scholar 

  23. Bluvstein, D. et al. Logical quantum processor based on reconfigurable atom arrays. Nature 626, 58–65 (2024).

    ADS  Google Scholar 

  24. Breuckmann, N. P. & Eberhardt, J. N. Quantum low-density parity-check codes. PRX Quantum 2, 040101 (2021).

    ADS  Google Scholar 

  25. Jenkins, A., Lis, J. W., Senoo, A., McGrew, W. F. & Kaufman, A. M. Ytterbium nuclear-spin qubits in an optical tweezer array. Phys. Rev. X 12, 021027 (2022).

    Google Scholar 

  26. Ma, S. et al. Universal gate operations on nuclear spin qubits in an optical tweezer array of 171Yb atoms. Phys. Rev. X 12, 021028 (2022).

    Google Scholar 

  27. Barnes, K. et al. Assembly and coherent control of a register of nuclear spin qubits. Nat. Commun. 13, 2779 (2022).

    ADS  Google Scholar 

  28. Evered, S. J. et al. High-fidelity parallel entangling gates on a neutral atom quantum computer. Nature 622, 268–272 (2023).

    ADS  Google Scholar 

  29. Ma, S. et al. High-fidelity gates with mid-circuit erasure conversion in a metastable neutral atom qubit. Preprint at https://arxiv.org/abs/2305.05493v1 (2023).

  30. Scholl, P. et al. Erasure-cooling, control, and hyper-entanglement of motion in optical tweezers. Preprint at https://arxiv.org/abs/2311.15580v1 (2023).

  31. Beugnon, J. et al. Two-dimensional transport and transfer of a single atomic qubit in optical tweezers. Nat. Phys. 3, 696–699 (2007).

    Google Scholar 

  32. Hashizume, T., Bentsen, G. S., Weber, S. & Daley, A. J. Deterministic fast scrambling with neutral atom arrays. Phys. Rev. Lett. 126, 200603 (2021).

    ADS  Google Scholar 

  33. Pattison, C. A., Krishna, A. & Preskill, J. Hierarchical memories: simulating quantum LDPC codes with local gates. Preprint at https://arxiv.org/abs/2303.04798 (2023).

  34. Leverrier, A., Tillich, J. P. & Zemor, G. Quantum expander codes. In Proc. Annual IEEE Symposium on Foundations of Computer Science 810–824 (IEEE, 2015).

  35. Quintavalle, A. O., Vasmer, M., Roffe, J. & Campbell, E. T. Single-shot error correction of three-dimensional homological product codes. PRX Quantum 2, 020340 (2020).

  36. Bombin, H. Single-shot fault-tolerant quantum error correction. Phys. Rev. X 5, 031043 (2015).

    Google Scholar 

  37. Gidney, C. Stim: a fast stabilizer circuit simulator. Quantum 5, 497 (2021).

    Google Scholar 

  38. Roffe, J., White, D. R., Burton, S. & Campbell, E. Decoding across the quantum low-density parity-check code landscape. Phys. Rev. Res. 2, 043423 (2020).

    Google Scholar 

  39. Higgott, O. & Breuckmann, N. P. Improved single-shot decoding of higher-dimensional hypergraph-product codes. PRX Quantum 4, 020332 (2023).

    ADS  Google Scholar 

  40. Kuo, K. Y. & Lai, C. Y. Exploiting degeneracy in belief propagation decoding of quantum codes. npj Quantum Inf. 8, 111 (2022).

    ADS  Google Scholar 

  41. Higgott, O., Bohdanowicz, T. C., Kubica, A., Flammia, S. T. & Campbell, E. T. Improved decoding of circuit noise and fragile boundaries of tailored surface codes. Phys. Rev. X 13, 031007 (2023).

    Google Scholar 

  42. Delfosse, N. & Paetznick, A. Spacetime codes of Clifford circuits. Preprint at https://arxiv.org/abs/2304.05943v2 (2023).

  43. Grospellier, A., Grouès, L., Krishna, A. & Leverrier, A. Combining hard and soft decoders for hypergraph product codes. Quantum 5, 432 (2021).

    Google Scholar 

  44. Ebadi, S. et al. Quantum optimization of maximum independent set using Rydberg atom arrays. Science 376, 1209–1215 (2022).

    ADS  Google Scholar 

  45. Wang, D. S., Fowler, A. G. & Hollenberg, L. C. L. Surface code quantum computing with error rates over 1. Phys. Rev. A 83, 020302 (2011).

    ADS  Google Scholar 

  46. Horsman, C., Fowler, A. G., Devitt, S. & van Meter, R. Surface code quantum computing by lattice surgery. New J. Phys. 14, 123011 (2012).

    ADS  MathSciNet  Google Scholar 

  47. Breuckmann, N. P., Vuillot, C., Campbell, E., Krishna, A. & Terhal, B. M. Hyperbolic and semi-hyperbolic surface codes for quantum storage. Quantum Sci. Technol. 2, 035007 (2017).

    ADS  Google Scholar 

  48. Poulsen Nautrup, H., Friis, N. & Briegel, H. J. Fault-tolerant interface between quantum memories and quantum processors. Nat. Commun. 8, 1321 (2017).

    ADS  Google Scholar 

  49. Bravyi, S. et al. High-threshold and low-overhead fault-tolerant quantum memory. Nature 627, 778–782 (2024).

    Google Scholar 

  50. Hong, Y., Marinelli, M., Kaufman, A. M. & Lucas, A. Long-range-enhanced surface codes. Preprint at https://arxiv.org/abs/2309.11719v1 (2023).

  51. Fahimniya, A. et al. Fault-tolerant hyperbolic Floquet quantum error correcting codes. Preprint at https://arxiv.org/abs/2309.10033v2 (2023).

  52. Higgott, O. & Breuckmann, N. P. Constructions and performance of hyperbolic and semi-hyperbolic Floquet codes. Preprint at https://arxiv.org/abs/2308.03750v1 (2023).

  53. Bombín, H. Gauge color codes: optimal transversal gates and gauge fixing in topological stabilizer codes. New J. Phys. 17 083002 (2015).

  54. Anshu, A., Breuckmann, N. P. & Nirkhe, C. NLTS Hamiltonians from good quantum codes. Preprint at https://arxiv.org/abs/2206.13228v2 (2022).

  55. Leverrier, A. & Zémor, G. Quantum Tanner codes. Preprint at https://arxiv.org/abs/2202.13641 (2022).

  56. Gu, S., Pattison, C. A. & Tang, E. An efficient decoder for a linear distance quantum LDPC code. In Proc. Annual ACM Symposium on Theory of Computing 919–932 (ACM, 2022).

  57. Dinur, I., Hsieh, M.-H. H., Lin, T.-C. C. & Vidick, T. Good quantum LDPC codes with linear time decoders. In Proc. Annual ACM Symposium on Theory of Computing 905–918 (ACM, 2022).

  58. Lin, T.-C. & Hsieh, M.-H. Good quantum LDPC codes with linear time decoder from lossless expanders. Preprint at https://arxiv.org/abs/2203.03581v1 (2022).

  59. Richardson, T. & Urbanke, R. Modern Coding Theory (Cambridge Univ. Press, 2008).

  60. Fawzi, O., Grospellier, A. & Leverrier, A. Efficient decoding of random errors for quantum expander codes. In Proc. Annual ACM Symposium on Theory of Computing 878–889 (ACM, 2017).

  61. Raveendran, N. et al. Finite rate QLDPC-GKP coding scheme that surpasses the CSS Hamming bound. Quantum 6, 767 (2022).

    Google Scholar 

  62. Dordevic, T. et al. Entanglement transport and a nanophotonic interface for atoms in optical tweezers. Science 373, 1511–1514 (2021).

    ADS  MathSciNet  Google Scholar 

  63. Levine, H. et al. Parallel implementation of high-fidelity multiqubit gates with neutral atoms. Phys. Rev. Lett. 123, 170503 (2019).

    ADS  Google Scholar 

  64. Baspin, N., Fawzi, O. & Shayeghi, A. A lower bound on the overhead of quantum error correction in low dimensions. Preprint at https://arxiv.org/abs/2302.04317v1 (2023).

  65. Xu, Q. et al. Constant-overhead fault-tolerant quantum computation with reconfigurable atom arrays. Zenodo https://doi.org/10.5281/zenodo.8278063

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Acknowledgements

We acknowledge helpful discussions with D. Bandyopadhyay, N. Breuckmann, M. Cain, L. Cohen, C. Duckering, S. Ebadi, S. Evered, X. Gao, S. Geim, M. Kalinowski, S. Li, J. Liu, T. Manovitz, B. Matuz, N. Maskara, Q. Nguyen, H. P. Nautrup, D. Orsucci, N. Rengaswamy, M. Vasmer, P. Yu and H. Zheng, among others. We particularly thank A. Krishna for detailed feedback on our results and paper. We are grateful for the support from the University of Chicago Research Computing Center for assistance with numerical simulations. This work was supported by the Army Research Office (ARO; Grant No. W911NF-23-1-0077 to Q.X. and L.J.), ARO Multidisciplinary University Research Initiatives (MURI; Grant No. W911NF-21-1-0325 to Q.X. and L.J. and Grant No. W911NF-20-1-0082 to J.P.B.A., D.B., M.D.L. and H.Z.), Air Force Office of Scientific Research MURI (Grant Nos. FA9550-19-1-0399 and FA9550-21-1-0209 to Q.X. and L.J. and Grant No. FA9550-19-1-0360 to C.A.P.), the National Science Foundation (NSF; Grant Nos. OMA-1936118, ERC-1941583 and OMA-2137642 to Q.X. and L.J. and Grant Nos. CCF-2313084, CIF-1855879, CCF-2100013 and CIF-2106189 to N.R. and B.V.), NTT Research (L.J.), the Packard Foundation (Grant No. 2020-71479 to L.J.), the Quantum Systems Accelerator Center of the US Department of Energy (Grant Nos. 7568717 and DE-SC0021013 to J.P.B.A., D.B., M.D.L. and H.Z.), the Center for Ultracold Atoms (Grant No. NSF PHY-1734011 to J.P.B.A., D.B., M.D.L. and H.Z.), the NSF Institute for Quantum Information and Matter (C.A.P.), the Optimization with Noisy Intermediate-Scale Quantum Devices programme of the Defense Advanced Research Projects Agency (Grant No. W911NF2010021 to D.B., J.W., M.D.L. and H.Z.), the NSF Graduate Research Fellowship Program (Grant No. DGE1745303 to D.B.), the Fannie and John Hertz Foundation (D.B.), and the Ramsay Centre for Western Civilisation (J.P.B.A.).

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Author notes

  1. These authors contributed equally: Qian Xu, J. Pablo Bonilla Ataides.

Authors and Affiliations

  1. Pritzker School of Molecular Engineering, The University of Chicago, Chicago, IL, USA

    Qian Xu & Liang Jiang

  2. Department of Physics, Harvard University, Cambridge, MA, USA

    J. Pablo Bonilla Ataides, Dolev Bluvstein, Mikhail D. Lukin & Hengyun Zhou

  3. Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA, USA

    Christopher A. Pattison

  4. Department of Electrical and Computer Engineering, University of Arizona, Tucson, AZ, USA

    Nithin Raveendran & Bane Vasić

  5. QuEra Computing Inc., Boston, MA, US

    Jonathan Wurtz & Hengyun Zhou

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Contributions

M.D.L. and L.J. conceived the project. Q.X., J.P.B.A. and C.A.P. performed the numerical simulations. Q.X., J.P.B.A., C.A.P. and H.Z. proved the fault tolerance of the schemes. H.Z. identified the correspondence between product codes and product optical tools, H.Z. and Q.X. devised efficient implementations of various codes, and D.B. verified the experimental feasibility of the proposal. J.W. and H.Z. devised the 1D log-depth rearrangement algorithm. N.R. constructed the LP codes used in the simulations. All authors contributed to the design of the methodology and data analysis. All authors contributed to the writing of the paper. All work was supervised by B.V., M.D.L., L.J. and H.Z.

Corresponding authors

Correspondence to Liang Jiang or Hengyun Zhou.

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Competing interests

M.D.L. is a co-founder and shareholder of QuEra Computing. J.W and H.Z. are employees of QuEra Computing. The other authors declare no competing interests.

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Extended data

Extended Data Fig. 1 Product structure of HGP codes and LP codes.

(a) The HGP code is constructed from two classical LDPC codes. The classical codes are illustrated on the left and top, where circles indicate classical bits and squares indicate classical checks. A data qubit is placed at each intersection of two classical bits (filled orange circles) and of two classical checks (filled blue circles). Z stabilizer generators are placed at the intersection of horizontal bits and vertical checks, while X stabilizer generators are placed at the intersection of horizontal checks and vertical bits. Each stabilizer is connected to data qubits along the same row or column, with the same connectivity as the classical codes, as illustrated for the top left Z stabilizer. We have omitted other connections for ease of visualization. (b) The LP code is constructed by taking a lift over the hypergraph product of two classical protographs. The protographs and their hypergraph product are indicated by the dashed nodes and the lift is illustrated by the multiple inner nodes within each dashed node. The inner connectivity between two dashed nodes is given by the circulant-matrix representation of the ring elements in Eq. (4). When flattening the inner nodes vertically (horizontally), the vertical (horizontal) connectivity between the qubits and the checks for each column (row) is the same as the left (top) lifted classical code.

Extended Data Fig. 2 Efficient non-intersecting rearrangement in log-depth.

By using a divide and conquer algorithm, we can perform an arbitrary 1D rearrangement in depth logarithmic in the number of qubits. Repeating this across the array yields an efficient implementation of the desired rearrangements, without requiring intersecting atom trajectories that may lead to additional loss and decoherence. Here, we illustrate the full set of movements required in a small example. Similar to the earlier figures, blue squares indicate classical checks and orange circles indicate classical bits. When a blue square and orange circle are moved to be neighboring at the end of the rearrangement, they execute an entangling gate. The top panel indicates the desired change of configuration, where the ordering of neighboring atoms in the top row needs to be modified to that in the bottom row via parallel rearrangement, as illustrated by the crossing gray lines. The bottom figure illustrates how we decompose the arbitrary rearrangement into a non-crossing rearrangement, where the gray lines no longer intersect.

Extended Data Fig. 3 Illustration of ordering of operations in pipelined syndrome extraction.

(a) Successive steps of entangling gates for the pipelined product coloration circuit described in Alg. 3, with d = 3 rounds of syndrome extraction. Numbers at the corners of the X and Z ancilla qubits denote the round of syndrome extraction they correspond to. (b) Illustration of a local circuit that data qubits and ancilla qubits see, with dashed lines indicating different circuit moments. As the X stabilizer interacts with both qubits before the Z stabilizer, the syndrome extraction order is valid. Similar analysis can be performed for the commutation relations with the next round of ancilla qubits.

Extended Data Fig. 4 Achievable logical failure rates of the HGP codes with different idling error strengths.

We characterize the idling error strengths as the relative ratio between the idling error rate pi(n) at n = 4 and the gate error rate pg. This idling error strength can potentially be reduced by, for example, increasing the coherence time and accelerating the atom shuttling.

Supplementary information

Supplementary Information

Supplementary Discussion and Figs. 1–10.

Supplementary Video 1

Video of one syndrome extraction cycle for the hypergraph product LDPC code. Black dots indicate atoms (qubits), green dots indicate static spatial light modulators traps, dashed lines indicate AODs used for atom moving and red flashes indicate parallel two-qubit gates.

Source data

Source Data for Fig. 1

Data files and Python notebook for plotting.

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Xu, Q., Bonilla Ataides, J.P., Pattison, C.A. et al. Constant-overhead fault-tolerant quantum computation with reconfigurable atom arrays. Nat. Phys. (2024). https://doi.org/10.1038/s41567-024-02479-z

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