Experimental benchmarking of quantum state overlap estimation strategies with photonic systems

Hao Zhan; Ben Wang; Minghao Mi; Jie Xie; Liang Xu; Aonan Zhang; Lijian Zhang

doi:10.1038/s41377-025-01755-8

Go to Nature Website

Your Location：

Home >

Browse articles >

Experimental benchmarking of quantum state overlap estimation strategies with photonic systems

Original Articles

- Experimental benchmarking of quantum state overlap estimation strategies with photonic systems
- Light: Science & Applications Vol. 14, Issue 3, Pages: 849-860(2025)
- Affiliations：
  
  1.National Laboratory of Solid State Microstructures, Key Laboratory of Intelligent Optical Sensing and Manipulation, College of Engineering and Applied Sciences, Jiangsu Physical Science Research Center, and Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093 Jiangsu, China
  2.Department of Physics, Imperial College London, Prince Consort Road, London SW7 2AZ, UK
- Author bio：
  
  Aonan Zhang (a.zhang@imperial.ac.uk)
  Lijian Zhang (lijian.zhang@nju.edu.cn)
- Funds：
- DOI：10.1038/s41377-025-01755-8
  CLC：
- Received：16 April 2024，
  
  Revised：09 January 2025，
  
  Accepted：2025-01-10，
  
  Published Online：12 February 2025，
  
  Published：31 March 2025
- Accepted：
Scan QR Code
Zhan, H. et al. Experimental benchmarking of quantum state overlap estimation strategies with photonic systems. Light: Science & Applications, 14, 849-860 (2025).
DOI：

Zhan, H. et al. Experimental benchmarking of quantum state overlap estimation strategies with photonic systems. Light: Science & Applications, 14, 849-860 (2025). DOI： 10.1038/s41377-025-01755-8.

摘要

Abstract

Accurately estimating the overlap between quantum states is a fundamental task in quantum information processing. While various strategies using distinct quantum measurements have been proposed for overlap estimation

the lack of experimental benchmarks on estimation precision limits strategy selection in different situations. Here we compare the performance of four practical strategies for overlap estimation

including tomography-tomography

tomography-projection

Schur collective measurement and optical swap test using photonic quantum systems. We encode the quantum states on the polarization and path degrees of freedom of single photons. The corresponding measurements are performed by photon detection on certain modes following single-photon mode transformation or two-photon interference. We further propose an adaptive strategy with optimized precision in full-range overlap estimation. Our results shed new light on extracting the parameter of interest from quantum systems

prompting the design of efficient quantum protocols.

关键词

Keywords

references

Bartlett, S. D., Rudolph, T. & Spekkens, R. W. Optimal measurements for relative quantum information. Phys. Rev. A 70 , 032321, https://doi.org/10.1103/PhysRevA.70.032321 (2004)..

Bagan, E., Iblisdir, S. & Muñoz-Tapia, R. Relative states, quantum axes, and quantum references. Phys. Rev. A 73 , 022341, https://doi.org/10.1103/PhysRevA.73.022341 (2006)..

Bartlett, S. D., Rudolph, T. & Spekkens, R. W. Reference frames, superselection rules, and quantum information. Rev. Mod. Phys. 79 , 555–609, https://doi.org/10.1103/RevModPhys.79.555 (2007)..

Tsang, M., Albarelli, F. & Datta, A. Quantum semiparametric estimation. Phys. Rev. X 10 , 031023, https://doi.org/10.1103/PhysRevX.10.031023 (2020)..

Giordani, T. et al. Experimental certification of contextuality, coherence, and dimension in a programmable universal photonic processor. Sci. Adv. 9 , eadj4249 , https://doi.org/10.1126/sciadv.adj4249 (2023)..

Ekert, A. K. et al. Direct estimations of linear and nonlinear functionals of a quantum state. Phys. Rev. Lett. 88 , 217901, https://doi.org/10.1103/PhysRevLett.88.217901 (2002)..

Mintert, F., Kuś, M. & Buchleitner, A. Concurrence of mixed multipartite quantum states. Phys. Rev. Lett. 95 , 260502, https://doi.org/10.1103/PhysRevLett.95.260502 (2005)..

Walborn, S. P. et al. Experimental determination of entanglement by a projective measurement. Phys. Rev. A 75 , 032338, https://doi.org/10.1103/PhysRevA.75.032338 (2007)..

Consiglio, M., Apollaro, T. J. G. & Wieśniak, M. Variational approach to the quantum separability problem. Phys. Rev. A 106 , 062413, https://doi.org/10.1103/PhysRevA.106.062413 (2022)..

Elben, A. et al. Cross-platform verification of intermediate scale quantum devices. Phys. Rev. Lett. 124 , 010504, https://doi.org/10.1103/PhysRevLett.124.010504 (2020)..

Jayakumar, A. et al. Quantum algorithm implementations for beginners. ACM Trans. Quantum Comput. 3 , 18, https://doi.org/10.1145/3517340 (2022)..

Cerezo, M. et al. Variational quantum algorithms. Nat. Rev. Phys. 3 , 625–644, https://doi.org/10.1038/s42254-021-00348-9 (2021)..

Biamonte, J. et al. Quantum machine learning. Nature 549 , 195–202, https://doi.org/10.1038/nature23474 (2017)..

Zeguendry, A., Jarir, Z. & Quafafou, M. Quantum machine learning: a review and case studies. Entropy 25 , 287, https://doi.org/10.3390/e25020287 (2023)..

Wan, K. H. et al. Quantum generalisation of feedforward neural networks. Npj Quantum Inf. 3 , 36, https://doi.org/10.1038/s41534-017-0032-4 (2017)..

Abbas, A. et al. The power of quantum neural networks. Nat. Comput. Sci. 1 , 403–409, https://doi.org/10.1038/s43588-021-00084-1 (2021)..

Pan, X. X. et al. Deep quantum neural networks on a superconducting processor. Nat. Commun. 14 , 4006, https://doi.org/10.1038/s41467-023-39785-8 (2023)..

Beer, K. et al. Training deep quantum neural networks. Nat. Commun. 11 , 808, https://doi.org/10.1038/s41467-020-14454-2 (2020)..

Hubregtsen, T. et al. Training quantum embedding kernels on near-term quantum computers. Phys. Rev. A 106 , 042431, https://doi.org/10.1103/PhysRevA.106.042431 (2022)..

Rebentrost, P., Mohseni, M. & Lloyd, S. Quantum support vector machine for big data classification. Phys. Rev. Lett. 113 , 130503, https://doi.org/10.1103/PhysRevLett.113.130503 (2014)..

Havlíček, V. et al. Supervised learning with quantum-enhanced feature spaces. Nature 567 , 209–212, https://doi.org/10.1038/s41586-019-0980-2 (2019)..

Schuld, M. & Killoran, N. Quantum machine learning in feature Hilbert spaces. Phys. Rev. Lett. 122 , 040504, https://doi.org/10.1103/PhysRevLett.122.040504 (2019)..

Liu, Y. C., Arunachalam, S. & Temme, K. A rigorous and robust quantum speed-up in supervised machine learning. Nat. Phys. 17 , 1013–1017, https://doi.org/10.1038/s41567-021-01287-z (2021)..

Sentís, G. et al. Unsupervised classification of quantum data. Phys. Rev. X 9 , 041029, https://doi.org/10.1103/PhysRevX.9.041029 (2019)..

Tancara, D. et al. Kernel-based quantum regressor models learning non-Markovianity. Phys. Rev. A 107 , 022402, https://doi.org/10.1103/PhysRevA.107.022402 (2023)..

Hu, L. et al. Quantum generative adversarial learning in a superconducting quantum circuit. Sci. Adv. 5 , eaav2761, https://doi.org/10.1126/sciadv.aav2761 (2019)..

Carolan, J. et al. Variational quantum unsampling on a quantum photonic processor. Nat. Phys. 16 , 322–327, https://doi.org/10.1038/s41567-019-0747-6 (2020)..

Liang, J. M. et al. Variational quantum algorithms for dimensionality reduction and classification. Phys. Rev. A 101 , 032323, https://doi.org/10.1103/PhysRevA.101 .032323 (2020)..

Buhrman, H. et al. Quantum fingerprinting. Phys. Rev. Lett. 87 , 167902, https://doi.org/10.1103/PhysRevLett.87.167902 (2001)..

Ripper, P., Amaral, G. & Temporão, G. Swap test-based characterization of decoherence in universal quantum computers. Quantum Inf. Process. 22 , 220, https://doi.org/10.1007/s11128-023-03961-y (2023)..

Agliardi, G. et al. Conditions for a quadratic quantum speedup in nonlinear transforms with applications to energy contract pricing. Quantum Sci. Technol. 10 , 025005, https://doi.org/10.1088/2058-9565/ada08c (2025)..

White, S. J. U. et al. A Robust Approach for Time-bin Encoded Photonic Quantum Information Protocols . Preprint at https://doi.org/10.48550/arXiv.2404.16106 https://doi.org/10.48550/arXiv.2404.16106 (2024)..

Nguyen, C. H. et al. Experimental SWAP Test of Infinite Dimensional Quantum States . Preprint at https://doi.org/10.48550/arXiv.2103.10219 https://doi.org/10.48550/arXiv.2103.10219 (2021)..

Li, Y. D. et al. Swap test with quantum dot charge qubits. Phys. Rev. Appl. 18 , 014047, https://doi.org/10.1103/PhysRevApplied.18.014047 (2022)..

Hong, C. K., Ou, Z. Y. & Mandel, L. Measurement of subpicosecond time intervals between two photons by interference. Phys. Rev. Lett. 59 , 2044–2046, https://doi.org/10.1103/PhysRevLett.59.2044 (1987)..

Garcia-Escartin, J. C. & Chamorro-Posada, P. SWAP test and Hong-Ou-Mandel effect are equivalent. Phys. Rev. A 87 , 052330, https://doi.org/10.1103/PhysRevA.87.052330 (2013)..

Zhang, A. et al. Quantum verification of NP problems with single photons and linear optics. Light Sci. Appl. 10 , 169, https://doi.org/10.1038/s41377-021-00608-4 (2021)..

Cincio, L. et al. Learning the quantum algorithm for state overlap. N. J. Phys. 20 , 113022, https://doi.org/10.1088/1367-2630/aae94a (2018)..

Huggins, W. J. et al. A non-orthogonal variational quantum eigensolver. N. J. Phys. 22 , 073009, https://doi.org/10.1088/1367-2630/ab867b (2020)..

Fanizza, M. et al. Beyond the swap test: optimal estimation of quantum state overlap. Phys. Rev. Lett. 124 , 060503, https://doi.org/10.1103/PhysRevLett.124.060503 (2020)..

Steinbrecher, G. R. et al. Quantum optical neural networks. Npj Quantum Inf. 5 , 60, https://doi.org/10.1038/s41534-019-0174-7 (2019)..

Ewaniuk, J. et al. Imperfect quantum photonic neural networks. Adv. Quantum Technol. 6 , 2200125, https://doi.org/10.1002/qute.202200125 (2023)..

Wetzstein, G. et al. Inference in artificial intelligence with deep optics and photonics. Nature 588 , 39–47, https://doi.org/10.1038/s41586-020-2973-6 (2020)..

Zuo, Y. et al. Optical neural network quantum state tomography. Adv. Photonics 4 , 026004, https://doi.org/10.1117/1.AP.4.2.026004 (2022)..

Bernstein, L. et al. Single-shot optical neural network. Sci. Adv. 9 , eadg7904, https://doi.org/10.1126/sciadv.adg7904 (2023)..

Chabaud, U., Markham, D. & Sohbi, A. Quantum machine learning with adaptive linear optics. Quantum 5 , 496, https://doi.org/10.22331/q-2021-07-05-496 (2021)..

Durt, T. et al. On mutually unbiased bases. Int. J. Quantum Inf. 8 , 535–640, https://doi.org/10.1142/S0219749910006502 (2010)..

Hou, Z. B. et al. Deterministic realization of collective measurements via photonic quantum walks. Nat. Commun. 9 , 1414, https://doi.org/10.1038/s41467-018-03849-x (2018)..

Bacon, D., Chuang, I. L. & Harrow, A. W. Efficient quantum circuits for schur and clebsch-gordan transforms. Phys. Rev. Lett. 97 , 170502, https://doi.org/10.1103/PhysRevLett.97.170502 (2006)..

Grice, W. P. & Walmsley, I. A. Spectral information and distinguishability in type-Ⅱ down-conversion with a broadbandpump. Phys. Rev. A 56 , 1627–1634, https://doi.org/10.1103/PhysRevA.56.1627 (1997)..

Cramér, H. Mathematical Methods of Statistics https://doi.org/10.1515/9781400883868 https://doi.org/10.1515/9781400883868 (Princeton: Princeton University Press, 1946)..

Braunstein, S. L. & Caves, C. M. Statistical distance and the geometry of quantum states. Phys. Rev. Lett. 72 , 3439–3443, https://doi.org/10.1103/PhysRevLett.72.3439 (1994)..

Hayashi, A., Hashimoto, T. & Horibe, M. Reexamination of optimal quantum state estimation of pure states. Phys. Rev. A 72 , 032325, https://doi.org/10.1103/PhysRevA.72.032325 (2005)..

Kueng, R., Rauhut, H. & Terstiege, U. Low rank matrix recovery from rank one measurements. Appl. Comput. Harmonic Anal. 42 , 88–116, https://doi.org/10.1016/j.acha.2015.07.007 (2017)..

Haah, J. et al. Sample-optimal tomography of quantum states. In Proceedings of the Forty-Eighth Annual ACM Symposium on Theory of Computing 913–925 https://doi.org/10.1145/2897518.2897585 https://doi.org/10.1145/2897518.2897585 (Cambridge, MA, USA: ACM, 2016)..

Flammia, S. T. et al. Quantum tomography via compressed sensing: error bounds, sample complexity and efficient estimators. N. J. Phys. 14 , 095022, https://doi.org/10.1088/1367-2630/14/9/095022 (2012)..

Bagan, E., Baig, M. & Muñoz-Tapia, R. Optimal scheme for estimating a pure qubit state via local measurements. Phys. Rev. Lett. 89 , 277904, https://doi.org/10.1103/PhysRevLett.89.277904 (2002)..

Mahler, D. H. et al. Adaptive quantum state tomography improves accuracy quadratically. Phys. Rev. Lett. 111 , 183601, https://doi.org/10.1103/PhysRevLett.111.183601 (2013)..

Qi, B. et al. Adaptive quantum state tomography via linear regression estimation: theory and two-qubit experiment. Npj Quantum Inf. 3 , 19, https://doi.org/10.1038/s41534-017-0016-4 (2017)..

Wang, X. W. et al. Generalized quantum measurements on a higher-dimensional system via quantum walks. Phys. Rev. Lett. 131 , 150803, https://doi.org/10.1103/PhysRevLett.131.150803 (2023)..

Zhang, C. N. et al. Hong-ou-mandel interferenc e linking independent room-temperature quantum memories. Photonics Res. 10 , 2388–2393, , https://doi.org/10.1364/PRJ.463404 (2022)..

Ichihara, M. et al. Frequency-multiplexed hong-ou-mandel interference. Phys. Rev. A 107 , 032608, https://doi.org/10.1103/PhysRevA.107.032608 (2023)..

Schuld, M. Supervised Quantum Machine Learning Models are Kernel Methods . Preprint at https://doi.org/10.48550/arXiv.2101.11020 https://doi.org/10.48550/arXiv.2101.11020 (2021)..

Bowie, C., Shrapnel, S. & Kewming, M. J. Quantum kernel evaluation via Hong–Ou–Mandel interference. Quantum Sci. Technol. 9 , 015001, https://doi.org/10.1088/2058-9565/acfba9 (2024)..

Paine, A. E., Elfving, V. E. & Kyriienko, O. Quantum kernel methods for solving regression problems and differential equations. Phys. Rev. A 107 , 032428, https://doi.org/10.1103/PhysRevA.107.032428 (2023)..

Liu, J. G. & Wang, L. Differentiable learning of quantum circuit born machines. Phys. Rev. A 98 , 062324, https://doi.org/10.1103/PhysRev A.98.062324 (2018)..

Sancho-Lorente, T., Román-Roche, J. & Zueco, D. Quantum kernels to learn the phases of quantum matter. Phys. Rev. A 105 , 042432, https://doi.org/10.1103/PhysRevA.105.042432 (2022)..

Roncallo, S. et al. Quantum Optical Classifier with Superexponential Speedup . Preprint at https://doi.org/10.48550/arXiv.2404.15266 https://doi.org/10.48550/arXiv.2404.15266 (2024)..

Views

Downloads

CSCD

Alert me when the article has been cited

Submit

Tools

Publicity Resources

No data

Related Author

No data

Related Institution

No data

⁰