A Faster Hafnian Formula for Complex Matrices and Its Benchmarking on a Supercomputer
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We introduce new and simple algorithms for the calculation of the number of perfect matchings of complex weighted, undirected graphs with and without loops. Our compact formulas for the hafnian and loop hafnian of n × n complex matrices run in O ( n 3 2 n /2 ) time, are embarrassingly parallelizable and, to the best of our knowledge, are the fastest exact algorithms to compute these quantities. Despite our highly optimized algorithm, numerical benchmarks on the Titan supercomputer with matrices up to size 56 × 56 indicate that one would require the 288,000 CPUs of this machine for about 6 weeks to compute the hafnian of a 100 × 100 matrix.
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A more general experiment where the mean of the Gaussian states is non-zero can be used to sample from the loop Hafnian.
“A more general experiment instead, where the mean of the Gaussian states is non-zero, can be used to sample from the loop Hafnian (Björklund et al. , 2019).”
From Noisy intermediate-scale quantum algorithms (2022) · cites this paper as Björklund et al., 2019
Cites
Detailed study of Gaussian boson sampling
Kruse, R., Hamilton, C. S., Sansoni, L., Barkhofen, S., Silberhorn, C., & Jex, I. (2019) · Physical Review A · doi:10.1103/physreva.100.032326 · Cited by 180 · see citing works
Gaussian boson sampling using threshold detectors
Quesada, N., Arrazola, J. M., & Killoran, N. (2018) · Physical Review A · doi:10.1103/physreva.98.062322 · Cited by 120 · see citing works
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Noisy intermediate-scale quantum algorithms
Bharti, K., Cervera-Lierta, A., Kyaw, T. H., Haug, T., Alperin-Lea, S., Anand, A., Degroote, M., Heimonen, H., Kottmann, J. S., Menke, T., et al. (2022) · Reviews of Modern Physics · doi:10.1103/revmodphys.94.015004 · Cited by 1,726 · see citing works
Quantum computational advantage with a programmable photonic processor
Madsen, L. S., Laudenbach, F., Askarani, M. F., Rortais, F., Vincent, T., Bulmer, J. F. F., Miatto, F. M., Neuhaus, L., Helt, L. G., Collins, M. J., et al. (2022) · Nature · doi:10.1038/s41586-022-04725-x · Cited by 1,031 · see citing works
Self-citationComputational advantage of quantum random sampling
Hangleiter, D., & Eisert, J. (2023) · Reviews of Modern Physics · doi:10.1103/revmodphys.95.035001 · Cited by 145 · see citing works
Gaussian boson sampling using threshold detectors
Quesada, N., Arrazola, J. M., & Killoran, N. (2018) · Physical Review A · doi:10.1103/physreva.98.062322 · Cited by 120 · see citing works
Self-citationApplications of near-term photonic quantum computers: software and algorithms
Bromley, T. R., Arrazola, J. M., Jahangiri, S., Izaac, J., Quesada, N., Gran, A. D., Schuld, M., Swinarton, J., Zabaneh, Z., & Killoran, N. (2020) · Quantum Science and Technology · doi:10.1088/2058-9565/ab8504 · Cited by 128 · see citing works
Self-citationQuadratic Speed-Up for Simulating Gaussian Boson Sampling
Quesada, N., Chadwick, R. S., Bell, B. A., Arrazola, J. M., Vincent, T., Qi, H., & García−Patrón, R. (2022) · PRX Quantum · doi:10.1103/prxquantum.3.010306 · Cited by 48 · see citing works
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- Authors
- Björklund, A., Gupt, B., & Quesada, N.
- Affiliations
- Lund University (SE)
- Xanadu Quantum Technologies (Canada) (CA)
- Venue
- ACM Journal of Experimental Algorithmics
- Published
- 2019-06-11
- Volume
- 24
- Pages
- 1-17
- ISSN
- 1084-6654 (ISSN-L)
- OpenAlex
- W2806308536
- Cited by
- 74
- References
- 2
- Citations by year
- 2026: 6 · 2025: 8 · 2024: 11 · 2023: 5 · 2022: 15 · 2021: 8 · 2020: 14 · 2019: 6 · 2018: 1
- FWCI
- 7.31
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- ACM Journal of Experimental Algorithmics (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- Lund University Publications (Lund University) (submittedVersion)
- PolyPublie (École Polytechnique de Montréal) (submittedVersion)
- arXiv (Cornell University)
- Funders
- Office of Science of the U.S. Department of Energy (DE-AC05-00OR22725)
- Topics
- Physical Sciences › Computer Science › Computational Theory and Mathematics › Complexity and Algorithms in Graphs
- Physical Sciences › Mathematics › Discrete Mathematics and Combinatorics › Advanced Combinatorial Mathematics
- Physical Sciences › Computer Science › Computer Vision and Pattern Recognition › Graph Theory and Algorithms
- Keywords
- hafnian, perfect matchings, complex matrices, weighted undirected graphs, parallel algorithms, supercomputer benchmarking, computational complexity
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