Reconstructing unknown quantum states using variational layerwise method

Abstract

In order to gain comprehensive knowledge of an arbitrary unknown quantum state, one feasible way is to reconstruct it, which can be realized by finding a series of quantum operations that can refactor the unitary evolution producing the unknown state. We design an adaptive framework that can reconstruct unknown quantum states at high fidelities, which utilizes SWAP test, parameterized quantum circuits (PQCs) and layerwise learning strategy. We conduct benchmarking on the framework using numerical simulations and reproduce states of up to six qubits at more than 96% overlaps with original states on average using PQCs trained by our framework, revealing its high applicability to quantum systems of different scales theoretically. Moreover, we perform experiments on a five-qubit IBM Quantum hardware to reconstruct random unknown single qubit states, illustrating the practical performance of our framework. For a certain reconstructing fidelity, our method can effectively construct a PQC of suitable length, avoiding barren plateaus of shadow circuits and overuse of quantum resources by deep circuits, which is of much significance when the scale of the target state is large and there is no a priori information on it. This advantage indicates that it can learn credible information of unknown states with limited quantum resources, giving a boost to quantum algorithms based on parameterized circuits on near-term quantum processors.

Graphical abstract

Keywords

variational quantum algorithm / layerwise learning / quantum state reconstructing

References

[1] S. Lloyd, Universal quantum simulators, Science 273(5278), 1073 (1996)

[2] X. Qiang, T. Loke, A. Montanaro, K. Aungskunsiri, X. Zhou, J. L. O’Brien, J. B. Wang, and J. C. F. Matthews, Efficient quantum walk on a quantum processor, Nat. Commun. 7(1), 11511 (2016)

[3] N. N. Zhang, M. J. Tao, W. T. He, X. Y. Chen, X. Y. Kong, F. G. Deng, N. Lambert, and Q. Ai, Efficient quantum simulation of open quantum dynamics at various Hamiltonians and spectral densities, Front. Phys. 16(5), 51501 (2021)

[4] P. W. Shor, Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer, SIAM J. Comput. 26(5), 1484 (1997)

[5] L. K. Grover, Quantum computers can search arbitrarily large databases by a single query, Phys. Rev. Lett. 79(23), 4709 (1997)

[6] G. L. Long, General quantum interference principle and duality computer, Commum. Theor. Phys. 45(5), 825 (2006)

[7] G. L. Long, Duality quantum computing and duality quantum information processing, Int. J. Theor. Phys. 50(4), 1305 (2011)

[8] A. W. Harrow, A. Hassidim, and S. Lloyd, Quantum algorithm for linear systems of equations, Phys. Rev. Lett. 103(15), 150502 (2009)

[9] C. H. Bennett and G. Brassard, in: Proceedings of IEEE International Conference on Computers, Systems and Signal Processing, (1984), pp 175–179

[10] A. K. Ekert, Quantum cryptography based on Bell’s theorem, Phys. Rev. Lett. 67(6), 661 (1991)

[11] C. H. Bennett, Quantum cryptography using any two nonorthogonal states, Phys. Rev. Lett. 68(21), 3121 (1992)

[12] G. L. Long and X. S. Liu, Theoretically efficient high-capacity quantum-key-distribution scheme, Phys. Rev. A 65(3), 032302 (2002)

[13] Z. D. Ye, D. Pan, Z. Sun, C. G. Du, L. G. Yin, and G. L. Long, Generic security analysis framework for quantum secure direct communication, Front. Phys. 16(2), 21503 (2021)

[14] Z. X. Cui, W. Zhong, L. Zhou, and Y. B. Sheng, Measurement-device-independent quantum key distribution with hyper-encoding, Sci. China Phys. Mech. Astron. 62(11), 110311 (2019)

[15] Z. Qi, Y. Li, Y. Huang, J. Feng, Y. Zheng, and X. Chen, A 15-user quantum secure direct communication network, Light Sci. Appl. 10(1), 183 (2021)

[16] G. L. Long and H. Zhang, Drastic increase of channel capacity in quantum secure direct communication using masking, Sci. Bull. (Beijing.) 66(13), 1267 (2021)

[17] X. Liu, Z. Li, D. Luo, C. Huang, D. Ma, M. Geng, J. Wang, Z. Zhang, and K. Wei, Practical decoy-state quantum secure direct communication, Sci. China Phys. Mech. Astron. 64(12), 120311 (2021)

[18] Y. B. Sheng, L. Zhou, and G. L. Long, One-step quantum secure direct communication, Sci. Bull. (Beijing.) 67(4), 367 (2022)

[19] G. M. D’Ariano, M. D. Laurentis, M. G. A. Paris, A. Porzio, and S. Solimeno, Quantum tomography as a tool for the characterization of optical devices, J. Opt. B 4(3), S127 (2002)

[20] F. Albarrán-Arriagada, J. C. Retamal, E. Solano, and L. Lamata, Measurement-based adaptation protocol with quantum reinforcement learning, Phys. Rev. A 98(4), 042315 (2018)

[21] X. Xu, J. Sun, S. Endo, Y. Li, S. C. Benjamin, and X. Yuan, Variational algorithms for linear algebra, arXiv: 1909.03898 [quant-ph] (2019)

[22] C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio, and P. J. Coles, Variational quantum linear solver, arXiv: 1909.05820 [quant-ph] (2019)

[23] X. Wang, Z. Song, and Y. Wang, Variational quantum singular value decomposition, arXiv: 2006.02336 [quant49 ph] (2020)

[24] M. Cerezo, K. Sharma, A. Arrasmith, and P. J. Coles, Variational quantum state eigensolver, arXiv: 2004.01372 [quant-ph] (2020)

[25] S. McArdle, S. Endo, A. Aspuru-Guzik, S. Benjamin, and X. Yuan, Quantum computational chemistry, arXiv: 1808.10402 [quant-ph] (2018)

[26] A. Peruzzo, J. McClean, P. Shadbolt, M. H. Yung, X. Q. Zhou, P. J. Love, A. Aspuru-Guzik, and J. L. O’Brien,, A variational eigenvalue solver on a photonic quantum processor, Nat. Commun. 5(1), 4213 (2014)

[27] J. R. McClean, J. Romero, R. Babbush, and A. Aspuru-Guzik, The theory of variational hybrid quantum–classical algorithms, New J. Phys. 18(2), 023023 (2016)

[28] A. Kandala, A. Mezzacapo, K. Temme, M. Takita, M. Brink, J. M. Chow, and J. M. Gambetta, Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets, Nature 549(7671), 242 (2017)

[29] I. G. Ryabinkin, T. C. Yen, S. N. Genin, and A. F. Izmaylov, Qubit coupled cluster method: A systematic approach to quantum chemistry on a quantum computer, J. Chem. Theory Comput. 14(12), 6317 (2018)

[30] I. G. Ryabinkin, R. A. Lang, S. N. Genin, and A. F. Izmaylov, Iterative qubit coupled cluster approach with efficient screening of generators, arXiv: 1906.11192 [quant-ph] (2019)

[31] D. B. Zhang, Z. H. Yuan, and T. Yin, Variational quantum eigensolvers by variance minimization, arXiv: 2006.15781 [quant-ph] (2020)

[32] E. Farhi and H. Neven, Classification with quantum neural networks on near term processors, arXiv: 1802.06002 [quant-ph] (2018)

[33] A. Skolik, J. R. McClean, M. Mohseni, P. van der Smagt, and M. Leib, Layerwise learning for quantum neural networks, arXiv: 2006.14904 [quant-ph] (2020)

[34] K. Mitarai, M. Negoro, M. Kitagawa, and K. Fujii, Quantum circuit learning, Phys. Rev. A 98(3), 032309 (2018)

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

[36] J. Biamonte, P. Wittek, N. Pancotti, P. Rebentrost, N. Wiebe, and S. Lloyd, Quantum machine learning, Nature 549(7671), 195 (2017)

[37] M. Benedetti, E. Lloyd, S. Sack, and M. Fiorentini, Parameterized quantum circuits as machine learning models, Quantum Sci. Technol. 4(4), 043001 (2019)

[38] S. Wei, Y. Chen, Z. Zhou, and G. Long, A quantum convolutional neural network on NISQ devices, arXiv: 2104.06918 [quant-ph] (2021)

[39] F. Hu, B. N. Wang, N. Wang, and C. Wang, Quantum machine learning with D-wave quantum computer, Quantum Engineering 1(2), e12 (2019)

[40] J. Li, X. Yang, X. Peng, and C. P. Sun, Hybrid quantum-classical approach to quantum optimal control, Phys. Rev. Lett. 118(15), 150503 (2017)

[41] M. Schuld, V. Bergholm, C. Gogolin, J. Izaac, and N. Killoran, Evaluating analytic gradients on quantum hardware, Phys. Rev. A 99(3), 032331 (2019)

[42] IBM quantum, 2021

[43] A. Mari, T. R. Bromley, and N. Killoran, Estimating the gradient and higher-order derivatives on quantum hardware, Phys. Rev. A 103(1), 012405 (2021)

[44] T. Xin, X. Nie, X. Kong, J. Wen, D. Lu, and J. Li, Quantum pure state tomography via variational hybrid quantum-classical method, Phys. Rev. Appl. 13(2), 024013 (2020)

[45] J. Xiao, Paulicirq, 2020

[46] Quantum AI Team and Collaborators, Cirq, 2020

[47] M. Broughton, G. Verdon, T. McCourt, A. J. Martinez, J. H. Yoo, S. V. Isakov, P. Massey, M. Y. Niu, R. Halavati, E. Peters, M. Leib, A. Skolik, M. Streif, D. V. Dollen, J. R. McClean, S. Boixo, D. Bacon, A. K. Ho, H. Neven, and M. Mohseni, Tensorflow quantum: A software framework for quantum machine learning, arXiv: 2003.02989 [quant-ph] (2020)

[48] Qiskit: An open-source framework for quantum computing, 2019

[49] J. C. Garcia-Escartin and P. Chamorro-Posada, SWAP test and Hong–Ou–Mandel effect are equivalent, Phys. Rev. A 87(5), 052330 (2013)

[50] F. Arute, K. Arya, R. Babbush, D. Bacon, J. C. Bardin, et al., Quantum supremacy using a programmable superconducting processor, Nature 574(7779), 505 (2019)

[51] X. Z. Luo, J. G. Liu, P. Zhang, and L. Wang, Yao.jl: Extensible, efficient framework for quantum algorithm design, Quantum 4, 341 (2020)

[52] R. E. Plessix, A review of the adjoint-state method for computing the gradient of a functional with geophysical applications, Geophys. J. Int. 167(2), 495 (2006)

[53] T. Dozat, in: ICLR (2016)

[54] D. P. Kingma and J. Ba, Adam: A method for stochastic optimization, arXiv: 1412.6980 [cs.LG] (2014)

RIGHTS & PERMISSIONS

2022 Higher Education Press