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Dimension-adaptive machine learning-based quantum state reconstruction

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Abstract

We introduce an approach for performing quantum state reconstruction on systems of n qubits using a machine learning-based reconstruction system trained exclusively on m qubits, where m ≥ n. This approach removes the necessity of exactly matching the dimensionality of a system under consideration with the dimension of a model used for training. We demonstrate our technique by performing quantum state reconstruction on randomly sampled systems of one, two, and three qubits using machine learning-based methods trained exclusively on systems containing at least one additional qubit. The reconstruction time required for machine learning-based methods scales significantly more favorably than the training time; hence this technique can offer an overall saving of resources by leveraging a single neural network for dimension-variable state reconstruction, obviating the need to train dedicated machine learning systems for each Hilbert space.

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References

  • Ahmed S, Muñoz CS, Nori F, Kockum AF (2021) Quantum state tomography with conditional generative adversarial networks. Phys Rev Lett 127:140502

    Article  Google Scholar 

  • Ahmed S, Muñoz CS, Nori F, Kockum AF (2021) Classification and reconstruction of optical quantum states with deep neural networks. Phys Rev Research 3:033278

    Article  Google Scholar 

  • Al Osipov V, Sommersand H-J, Zyczkowski K (2010) Random Bures mixed states and the distribution of their purity. J Phys A: Math Theor 43:055302

    Article  MathSciNet  MATH  Google Scholar 

  • Altepeter JB, Jeffrey ER, Kwiat PG (2005) Photonic state tomography. Adv At Mol Opt Phys 52:105

    Article  Google Scholar 

  • Banaszek K, D’Ariano GM, Paris MGA, Sacchi MF (1999) Maximum-likelihood estimation of the density matrix. Phys Rev A 61:010304

    Article  Google Scholar 

  • Blume-Kohout R (2010) Optimal, reliable estimation of quantum states. New J Phys 12:043034

    Article  MATH  Google Scholar 

  • Borah S, Sarma B, Kewming M, Milburn GJ, Twamley J (2021) Measurement-Based Feedback Quantum Control with Deep Reinforcement Learning for a Double-Well Nonlinear Potential. Phys Rev Lett 127:190403

    Article  Google Scholar 

  • Carrasquilla J, Torlai G, Melko RG, Aolita L (2019) Reconstructing quantum states with generative models. Nat Mach Intell 1:155

    Article  Google Scholar 

  • Chapman JC, Lukens JM, Qi B, Pooser RC, Peters NA (2022) Bayesian homodyne and heterodyne tomography. Opt Express 30:15184

    Article  Google Scholar 

  • Danaci O, Lohani S, Kirby BT, Glasser RT (2021) Machine learning pipeline for quantum state estimation with incomplete measurements. Mach Learn Sci Technol 2:035014

    Article  Google Scholar 

  • Eisert J, Hangleiter D, Walk N, Roth I, Markham D, Parekh R, Chabaud U, Kashefi E (2020) Quantum certification and benchmarking. Nat Rev Phys 2:382

    Article  Google Scholar 

  • Ekert AK, Alves CM, Oi DK, Horodecki M, Horodecki P, Kwek LC (2002) Direct estimations of linear and nonlinear functionals of a quantum state. Phys Rev Lett 88:217901

    Article  Google Scholar 

  • Flammia ST , Liu Y-K (2011) Direct Fidelity Estimation from Few Pauli Measurements. Phys Rev Lett 106:230501

    Article  Google Scholar 

  • Genois É, Gross JA, Di Paolo A, Stevenson NJ, Koolstra G, Hashim A, Siddiqi I, Blais A (2021) Quantum-tailored machine-learning characterization of a superconducting qubit. PRX Quantum 2:040355

    Article  Google Scholar 

  • Giordani T, Suprano A, Polino E, Acanfora F, Innocenti L, Ferraro A, Paternostro M, Spagnolo N, Sciarrino F (2020) Machine learning-based classification of vector vortex beams. Phys Rev Lett 124:160401

    Article  Google Scholar 

  • Granade C, Combes J, Cory DG (2016) Practical Bayesian tomography. New J Phys 18:033024

    Article  MATH  Google Scholar 

  • Gross D, Liu Y-K, Flammia ST, Becker S, Eisert J (2010) Quantum state tomography via compressed sensing. Phys Rev Lett 105:150401

    Article  Google Scholar 

  • Gupta R, Xia R, Levine RD, Kais S (2021) Maximal entropy approach for quantum state tomography. PRX Quantum 2:010318

    Article  Google Scholar 

  • Häffner H, Hänsel W, Roos CF, Benhelm J, Chek-al-kar D, Chwalla M, Körber T, Rapol UD, Riebe M, Schmidt PO, Becher C, Gühne O, Dür W, Blatt R (2005) Scalable multiparticle entanglement of trapped ions. Nature 438:643

    Article  Google Scholar 

  • Hou Z, Zhong H-S, Tian Y, Dong D, Qi B, Li L, Wang Y, Nori F, Xiang G-Y, Li C-F, Guo G-C (2016) Full reconstruction of a 14-qubit state within four hours. New J Phys 18:083036

    Article  Google Scholar 

  • Hradil Z (1997) Quantum-state estimation. Phys Rev A 55:R1561

    Article  MathSciNet  Google Scholar 

  • Hsieh H-Y, Ning J, Chen Y-R, Wu H-C, Chen HL, Wu C-M, Lee R-K (2022) Direct Parameter Estimations from Machine Learning-Enhanced Quantum State Tomography. Symmetry 14: 874

    Article  Google Scholar 

  • Huang H-Y, kuengand R, Preskill J (2020) Predicting many properties of a quantum system from very few measurements. Nat Phys 16:1050

    Article  Google Scholar 

  • Huszár F , Houlsby NMT (2012) Adaptive Bayesian quantum tomography. Phys Rev A 85:052120

    Article  Google Scholar 

  • James DFV, Kwiat PG, Munro WJ, White AG (2001) Measurement of qubits. Phys Rev A 64:052312

    Article  Google Scholar 

  • James DF, Kwiat PG, Munro WJ, White AG (2005) On the measurement of qubits. In: Asymptotic theory of quantum statistical inference: selected papers. World Scientific, pp 509–538

  • Koutnỳ D, Motka L, Hradil Z, Ř,eháček J, Sánchez-Soto LL (2022) Neural-network quantum state tomography. Phys Rev A 106:012409

    Article  MathSciNet  Google Scholar 

  • Kravtsov KS, Straupe SS, Radchenko IV, Houlsby NMT, Huszár F, Kulik SP (2013) Experimental adaptive Bayesian tomography. Phys Rev A 87:062122

    Article  Google Scholar 

  • Liu W-T, Zhang T, Liu J-Y, Chen P-X, Yuan J-M (2012) Experimental quantum state tomography via compressed sampling. Phys Rev Lett 108:170403

    Article  Google Scholar 

  • Lohani S, Kirby BT, Brodsky M, Danaci O, Glasser RT (2020) Machine learning assisted quantum state estimation. Mach Learn Sci Technol 1:035007

    Article  Google Scholar 

  • Lohani S, Knutson EM, Glasser RT (2020) Generative machine learning for robust free-space communication. Commun Phys 3:177

    Article  Google Scholar 

  • Lohani S, Lukens JM, Glasser RT, Searles TA, Kirby BT (2022) Data-centric machine learning in quantum information science. Mach Learn Sci Technol 3:04LT01

    Article  Google Scholar 

  • Lohani S, Lukens JM, Jones DE, Searles TA, Glasser RT, Kirby BT (2021) Improving application performance with biased distributions of quantum states. Phys Rev Research 3: 043145

    Article  Google Scholar 

  • Lohani S, Searles TA, Kirby BT, Glasser R (2021) On the experimental feasibility of quantum state reconstruction via machine learning. IEEE Trans Quantum Eng 2:2103410

    Article  Google Scholar 

  • Lu S, Huang S, Li K, Li J, Chen J, Lu D, Ji Z, shen Y, Zhou D, Zeng B (2018) Separability-entanglement classifier via machine learning. Phys Rev A 98:012315

    Article  Google Scholar 

  • Lu H-H, Myilswamy KV, Bennink RS, Seshadri S, Alshaykh MS, Liu J, Kippenberg TJ, Leaird DE, Weiner AM, Lukens JM (2022) Bayesian tomography of high-dimensional on-chip biphoton frequency combs with randomized measurements. Nat Commun 13:4338

    Article  Google Scholar 

  • Lukens JM, Law KJ, Bennink RS (2021) A Bayesian analysis of classical shadows. npj Quantum Inf 7:113

    Article  Google Scholar 

  • Lukens JM, Law KJH, Jasra A, Lougovski P (2020) A practical and efficient approach for Bayesian quantum state estimation. New J Phys 22:063038

    Article  MathSciNet  Google Scholar 

  • Lvovsky AI (2004) Iterative maximum-likelihood reconstruction in quantum homodyne tomography. J Opt B: Quantum Semiclass Opt 6:S556

    Article  Google Scholar 

  • Mai TT, Alquier P (2017) Pseudo-Bayesian quantum tomography with rank-adaptation. J Stat Plan Inference 184:62. ISSN 0378-3758

    Article  MathSciNet  MATH  Google Scholar 

  • Melkani A, Gneiting C, Nori F (2020) Eigenstate extraction with neural-network tomography. Phys Rev A 102:022412

    Article  MathSciNet  Google Scholar 

  • Merkel ST, Riofrio CA, Flammia ST, Deutsch IH (2010) Random unitary maps for quantum state reconstruction. Phys Rev A 81:032126

    Article  Google Scholar 

  • Neugebauer M, Fischer L, Jäger A, Czischek S, Jochim S, Weidemüller M, Gärttner M (2020) Neural-network quantum state tomography in a two-qubit experiment. Phys Rev A 102:042604

    Article  Google Scholar 

  • Nielsen MA (1996) The entanglement fidelity and quantum error correction. arXiv:quant-ph/9606012

  • Olivares S, Paris MG (2007) Quantum estimation via the minimum Kullback entropy principle. Phys Rev A 76:042120

    Article  Google Scholar 

  • Palmieri AM, Kovlakov E, Bianchi F, Yudin D, Straupe S, Biamonte JD, Kulik S (2020) Experimental neural network enhanced quantum tomography. npj Quantum Inf 6:20

    Article  Google Scholar 

  • Qi B, Hou Z, Li L, Dong D, Xiang G, Guo G (2013) Quantum state tomography via linear regression estimation. Sci Rep 3:3496

    Article  Google Scholar 

  • Quek Y, Fort S, Ng HK (2021) Adaptive quantum state tomography with neural networks. npj Quantum Inf 7:105

    Article  Google Scholar 

  • Řeháček J, Englert B-G , Kaszlikowski D (2004) Minimal qubit tomography. Phys Rev A 70:052321

    Article  Google Scholar 

  • Rosset D, Ferretti-Schöbitz R, Bancal J-D, Gisin N, Liang Y-C (2012) Imperfect measurement settings: implications for quantum state tomography and entanglement witnesses. Phys Rev A 86:062325

    Article  Google Scholar 

  • Seah Y-L, Shang J, Ng HK, Nott DJ, Englert B-G (2015) Monte Carlo sampling from the quantum state space, II. New J Phys 17:043018

    Article  MATH  Google Scholar 

  • Simmerman EM, Lu H-H, Weiner AM, Lukens JM (2020) Efficient compressive and Bayesian characterization of biphoton frequency spectra. Opt Lett 45:2886

    Article  Google Scholar 

  • Smolin JA, Gambetta JM, Smith G (2012) Efficient method for computing the maximum-likelihood quantum state from measurements with additive Gaussian noise. Phys Rev Lett 108:070502

    Article  Google Scholar 

  • Sommers H-J, Zyczkowski K (2003) Bures volume of the set of mixed quantum states. J Phys A: Math Gen 36:10083

    Article  MathSciNet  MATH  Google Scholar 

  • Spengler C, Huber M, Brierley S, Adaktylos T, Hiesmayr BC (2012) Entanglement detection via mutually unbiased bases. Phys Rev A 86:022311

    Article  Google Scholar 

  • Teo YS, Shin S, Jeong H, Kim Y, Kim Y-H, Struchalin GI, Kovlakov EV, Straupe SS, Kulik SP, Leuchs G, Sánchez-Soto LL (2021) Benchmarking quantum tomography completeness and fidelity with machine learning. New J Phys 23:103021

    Article  Google Scholar 

  • Thew RT, Nemoto K, White AG, Munro WJ (2002) Qudit quantum-state tomography. Phys Rev A 66:012303

    Article  MathSciNet  Google Scholar 

  • Tiunov ES, Tiunova V, Ulanov AE, Lvovsky A (2020) Experimental quantum homodyne tomography via machine learning. A Fedorov, Optica 7:448

    Google Scholar 

  • Torlai G, Mazzola G, Carrasquilla J, Troyer M, Melkoand R, Carleo G (2018) Neural-network quantum state tomography. Nat Phys 14:447

    Article  Google Scholar 

  • Torlai G, Timar B, van Nieuwenburg EPL, Levine H, Omran A, Keesling A, Bernien Greiner H, Vuletić V, Lukin MD, Melko RG, Endres M (2019) Integrating Neural Networks with a Quantum Simulator for State Reconstruction. Phys Rev Lett 123: 230504

    Article  Google Scholar 

  • Wang R, Hernani-Morales C, Martín-guerrero JD, Solano E, Albarrán-arriagada F (2021) Quantum pattern recognition in photonic circuits. Quantum Sci Technol 7:015010

    Article  Google Scholar 

  • Wiatowski T, Bölcskei H (2017) A mathematical theory of deep convolutional neural networks for feature extraction. IEEE Trans Inf Theory 64:1845

    Article  MathSciNet  MATH  Google Scholar 

  • Wilde MM (2011) From classical to quantum Shannon theory. arXiv:1106.1445

  • Williams BP, Lougovski P (2017) Quantum state estimation when qubits are lost: a no-data-left-behind approach. New J Phys 19:043003

    Article  MATH  Google Scholar 

  • Wootters WK, Fields BD (1989) Optimal state-determination by mutually unbiased measurements. Ann Phys 191:363

    Article  MathSciNet  Google Scholar 

  • Zia D, Checchinato R, Suprano A, Giordani T, Polino E, Innocenti L, Ferraro A, Paternostro M, Spagnoloand N, Sciarrino F (2022) Regression of high dimensional angular momentum states of light. arXiv:2206.09873

  • Zyczkowski K, Sommers H-J (2003) Hilbert–Schmidt volume of the set of mixed quantum states. J Phys A: Math Gen 36:10115

    Article  MathSciNet  MATH  Google Scholar 

  • Zyczkowski K, Sommers H-J (2005) Average fidelity between random quantum states. Phys Rev A 71:032313

    Article  MathSciNet  MATH  Google Scholar 

Download references

Funding

Work by S. Lohani and T. A. Searles was supported in part by the U.S. Department of Energy, Office of Science, National Quantum Information Science Research Centers, Co-design Center for Quantum Advantage (C2QA) under contract number DE-SC0012704. A portion of this work was performed at Oak Ridge National Laboratory, operated by UT-Battelle for the U.S. Department of Energy under contract no. DE-AC05-00OR22725. J. M. Lukens acknowledges funding by the U.S. Department of Energy, Office of Science, Advanced Scientific Computing Research, through the Early Career Research Program (Field Work Proposal ERKJ353). The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Laboratory or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation herein. Additionally, this material is based upon work supported by, or in part by, the Army Research Laboratory and the Army Research Office under contract/grant numbers W911NF-19-2-0087 and W911NF-20-2-0168.

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Authors and Affiliations

  1. Dept. of Electrical & Computer Engineering, University of Illinois Chicago, Chicago, IL, 60607, USA

    Sanjaya Lohani, Sangita Regmi & Thomas A. Searles

  2. Research Technology Office, Arizona State University, Tempe, AZ, 85287, USA

    Joseph M. Lukens

  3. Quantum Information Science Section, Oak Ridge National Laboratory, Oak Ridge, TN, 37831, USA

    Joseph M. Lukens

  4. Tulane University, New Orleans, LA, 70118, USA

    Ryan T. Glasser & Brian T. Kirby

  5. DEVCOM Army Research Laboratory, Adelphi, MD, 20783, USA

    Brian T. Kirby

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  1. Sanjaya Lohani
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Correspondence to Sanjaya Lohani.

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Data availability

The code and a web-app to generate all datasets are, respectively, openly available at the following URLs: https://github.com/slohani-ai/machine-learning-for-physical-sciences, and https://mlphys.streamlit.app/.

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Appendices

Appendix A. Results when training and testing states are sampled according to the Bures distribution

In order to illustrate the concept for other cases, we sample 35,500 random quantum states ρ according to the Bures measure for the given m (Al Osipov et al. 2010). Similarly, we also simulate the associated 6m Pauli measurement outcomes for systems with m ∈{2, 3, 4} qubits directly from expectation values. As described in the main text, we split the sampled data into a training set of size 35,000 and a validation set of size 500 to cross-validate the network performance per epoch. We implement a batch size of 100 in the training of a network. After training, we generate test sets, again, using the Bures measure for the same and fewer-qubit systems that are entirely unknown to the trained network. Finally, the reconstruction fidelity with respect to subsystem size and number of qubits are, respectively, shown in Fig. 4(a) and (b). Although the average reconstruction fidelities for states sampled according to the Bures measure are slightly lower than those drawn from the HS measure (see Fig. 2 in the main text), the same important scaling trends hold.

Fig. 4
figure 4

Test and train with random quantum states sampled from the Bures measure. (a) Reconstruction fidelity with respect to subsystem of predicted quantum states. (b) Reconstruction fidelity versus number of qubits

Full size image

Appendix B. Average fidelity between random quantum states

For completeness we include here the expression for the average fidelity 〈F〉N between two random mixed states of dimension N generated according to the HS measure. We take our results from Zyczkowski and Sommers (2005) where a more general expression applicable to two random mixed states chosen according to an arbitrary induced measure is presented. In simplifying the results of Zyczkowski and Sommers (2005) we find

$$ \begin{aligned} \langle F\rangle_{N}&=\frac{1}{N^{4}}\left[\text{Tr}\left( X_{0}^{-1}X_{1}\right) \right.\\ &+\left.\left( \text{Tr}\left[X_{0}^{-1}X_{1/2}\right] \right)^{2}-\text{Tr}\left( \left[X_{0}^{-1}X_{1/2}\right]^{2}\right) \right] \end{aligned} $$
(B1)

where Xn defines a matrix with entries

$$ (X_{n})_{k,l}={\Gamma}\left( n+k+l-1\right){\Gamma}(n+1), $$
(B2)

for k,l ∈{1, 2,...,N}, and Γ(⋅) is the usual gamma function. Using these expressions we find for one, two, and three qubit states, respectively, that 〈F〉2 = 0.67, 〈F〉4 = 0.59, and 〈F〉8 = 0.57.

In addition to the average fidelity between two random density matrices chosen according to the HS measure, we also show three other average fidelities in Figs. 3 and 4(b). Figure 4(b) includes the average fidelity between two random quantum states chosen according to the Bures measure. In Zyczkowski and Sommers (2005), an analytical result is given for this situation in the case of single-qubit states: 〈F〉2 = 0.590, which we supplement with numerical results for two and three qubits to create the relevant curve in Fig. 4b. Finally, in Figs. 3 and 4(b), we also show the average fidelity between random states chosen according to either the HS or Bures measures against the maximally mixed state. We obtain these values numerically but note that asymptotic results for these situations are available (Zyczkowski and Sommers 2005).

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Lohani, S., Regmi, S., Lukens, J.M. et al. Dimension-adaptive machine learning-based quantum state reconstruction. Quantum Mach. Intell. 5, 1 (2023). https://doi.org/10.1007/s42484-022-00088-8

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