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Selective and efficient quantum state tomography for multiqubit systems

Aniket Patel, Akshay Gaikwad, Tangyou Huang, Anton Frisk Kockum, and Tahereh Abad*

  • *Contact author: tahereh.abad@chalmers.se
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Phys. Rev. Research 8, 013339 – Published 30 March, 2026

DOI: https://doi.org/10.1103/hynl-kxl2

Abstract

Quantum state tomography (QST) is a crucial tool for characterizing quantum states. However, performing full QST becomes impractical for reconstructing multiqubit density matrices since datasets and computational costs grow exponentially with qubit number. To bypass full QST, we introduce selective and efficient QST (SEEQST)—a method that enables efficient estimation of multiple selected elements of an arbitrary multiqubit density matrix. We show that any N-qubit density matrix can be partitioned into 2N disjoint subsets, each containing 2N elements. With SEEQST, any such subset can be accurately estimated from just two experiments with only single-qubit measurements. The complexity for estimating any subset remains constant regardless of Hilbert-space dimension, so, if desired, SEEQST can reconstruct the full density matrix, using 2N+1−1 experiments, where standard methods would use 3N experiments. We provide a circuit decomposition for the SEEQST experiments, demonstrating that their maximum circuit depth scales logarithmically with N assuming all-to-all connectivity.

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References (88)

  1. D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, Measurement of qubits, Phys. Rev. A 64, 052312 (2001).
  2. M. Paris and J. Řeháček, Quantum State Estimation, Lecture Notes in Physics (Springer, Berlin, 2004), Vol. 649.
  3. Y.-X. Liu, L. F. Wei, and F. Nori, Tomographic measurements on superconducting qubit states, Phys. Rev. B 72, 014547 (2005).
  4. A. I. Lvovsky and M. G. Raymer, Continuous-variable optical quantum-state tomography, Rev. Mod. Phys. 81, 299 (2009).
  5. T. Baumgratz, D. Gross, M. Cramer, and M. B. Plenio, Scalable reconstruction of density matrices, Phys. Rev. Lett. 111, 020401 (2013).
  6. V. Gebhart, R. Santagati, A. A. Gentile, E. M. Gauger, D. Craig, N. Ares, L. Banchi, F. Marquardt, L. Pezzè, and C. Bonato, Learning quantum systems, Nat. Rev. Phys. 5, 141 (2023).
  7. A. Hashim, L. B. Nguyen, N. Goss, B. Marinelli, R. K. Naik, T. Chistolini, J. Hines, J. P. Marceaux, Y. Kim, P. Gokhale, T. Tomesh, S. Chen, L. Jiang, S. Ferracin, K. Rudinger, T. Proctor, K. C. Young, R. Blume-Kohout, and I. Siddiqi, A practical introduction to benchmarking and characterization of quantum computers, PRX Quantum 6, 030202 (2025).
  8. R. Blume-Kohout, T. Proctor, and K. Young, Quantum characterization, verification, and validation, arXiv:2503.16383.
  9. C. A. Riofrío, D. Gross, S. T. Flammia, T. Monz, D. Nigg, R. Blatt, and J. Eisert, Experimental quantum compressed sensing for a seven-qubit system, Nat. Commun. 8, 15305 (2017).
  10. J. Li, S. Huang, Z. Luo, K. Li, D. Lu, and B. Zeng, Optimal design of measurement settings for quantum-state-tomography experiments, Phys. Rev. A 96, 032307 (2017).
  11. J. Cotler and F. Wilczek, Quantum overlapping tomography, Phys. Rev. Lett. 124, 100401 (2020).
  12. Z. Hradil, Quantum-state estimation, Phys. Rev. A 55, R1561 (1997).
  13. K. Banaszek, G. M. D’Ariano, M. G. A. Paris, and M. F. Sacchi, Maximum-likelihood estimation of the density matrix, Phys. Rev. A 61, 010304(R) (1999).
  14. A. I. Lvovsky, Iterative maximum-likelihood reconstruction in quantum homodyne tomography, J. Opt. B: Quantum Semiclassical Opt. 6, S556 (2004).
  15. J. Řeháček, Z. Hradil, E. Knill, and A. I. Lvovsky, Diluted maximum-likelihood algorithm for quantum tomography, Phys. Rev. A 75, 042108 (2007).
  16. J. A. Smolin, J. M. Gambetta, and G. Smith, Efficient method for computing the maximum-likelihood quantum state from measurements with additive gaussian noise, Phys. Rev. Lett. 108, 070502 (2012).
  17. J. Shang, Z. Zhang, and H. K. Ng, Superfast maximum-likelihood reconstruction for quantum tomography, Phys. Rev. A 95, 062336 (2017).
  18. C. Ferrie, Self-guided quantum tomography, Phys. Rev. Lett. 113, 190404 (2014).
  19. E. Bolduc, G. C. Knee, E. M. Gauger, and J. Leach, Projected gradient descent algorithms for quantum state tomography, npj Quantum Inf. 3, 44 (2017).
  20. M.-C. Hsu, E.-J. Kuo, W.-H. Yu, J.-F. Cai, and M.-H. Hsieh, Quantum state tomography via nonconvex riemannian gradient descent, Phys. Rev. Lett. 132, 240804 (2024).
  21. Y. Wang, L. Liu, S. Cheng, L. Li, and J. Chen, Efficient factored gradient descent algorithm for quantum state tomography, Phys. Rev. Res. 6, 033034 (2024).
  22. A. Gaikwad, M. S. Torres, S. Ahmed, and A. F. Kockum, Gradient-descent methods for fast quantum state tomography, Quantum Sci. Technol. 10, 045055 (2025).
  23. D. Hoshi, T. Nagase, S. Kwon, D. Iyama, T. Kamiya, S. Fujii, H. Mukai, S. Ahmed, A. F. Kockum, S. Watabe, F. Yoshihara, and J.-S. Tsai, Entangling Schrödinger's cat states by bridging discrete- and continuous-variable encoding, Nat. Commun. 16, 1309 (2025).
  24. Y. Quek, S. Fort, and H. K. Ng, Adaptive quantum state tomography with neural networks, npj Quantum Inf. 7, 105 (2021).
  25. S. Lohani, B. T. Kirby, M. Brodsky, O. Danaci, and R. T. Glasser, Machine learning assisted quantum state estimation, Mach. Learn.: Sci. Technol. 1, 035007 (2020).
  26. A. Gaikwad, O. Bihani, Arvind, and K. Dorai, Neural-network-assisted quantum state and process tomography using limited data sets, Phys. Rev. A 109, 012402 (2024).
  27. T. Schmale, M. Reh, and M. Gärttner, Efficient quantum state tomography with convolutional neural networks, npj Quantum Inf. 8, 115 (2022).
  28. S. Ahmed, C. Sánchez Muñoz, F. Nori, and A. F. Kockum, Quantum state tomography with conditional generative adversarial networks, Phys. Rev. Lett. 127, 140502 (2021).
  29. S. Ahmed, C. Sánchez Muñoz, F. Nori, and A. F. Kockum, Classification and reconstruction of optical quantum states with deep neural networks, Phys. Rev. Res. 3, 033278 (2021).
  30. S. Yu, F. Albarrán-Arriagada, J. C. Retamal, Y.-T. Wang, W. Liu, Z.-J. Ke, Y. Meng, Z.-P. Li, J.-S. Tang, E. Solano, L. Lamata, C.-F. Li, and G.-C. Guo, Reconstruction of a photonic qubit state with reinforcement learning, Adv. Quantum Technol. 2, 1800074 (2019).
  31. N. Innan, O. I. Siddiqui, S. Arora, T. Ghosh, Y. P. Koçak, D. Paragas, A. A. O. Galib, M. A.-Z. Khan, and M. Bennai, Quantum state tomography using quantum machine learning, Quantum Mach. Intell. 6, 28 (2024).
  32. 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, 024013 (2020).
  33. Y. Liu, D. Wang, S. Xue, A. Huang, X. Fu, X. Qiang, P. Xu, H.-L. Huang, M. Deng, C. Guo, X. Yang, and J. Wu, Variational quantum circuits for quantum state tomography, Phys. Rev. A 101, 052316 (2020).
  34. D. Gross, Y.-K. Liu, S. T. Flammia, S. Becker, and J. Eisert, Quantum state tomography via compressed sensing, Phys. Rev. Lett. 105, 150401 (2010).
  35. A. Steffens, C. A. Riofrao, W. McCutcheon, I. Roth, B. A. Bell, A. McMillan, M. S. Tame, J. G. Rarity, and J. Eisert, Experimentally exploring compressed sensing quantum tomography, Quantum Sci. Technol. 2, 025005 (2017).
  36. I. Strandberg, Simple, reliable, and noise-resilient continuous-variable quantum state tomography with convex optimization, Phys. Rev. Appl. 18, 044041 (2022).
  37. A. Gaikwad, Arvind, and K. Dorai, True experimental reconstruction of quantum states and processes via convex optimization, Quantum Inf. Proc. 20, 19 (2021).
  38. R. Nehra, M. Eaton, C. González-Arciniegas, M. S. Kim, T. Gerrits, A. Lita, S. W. Nam, and O. Pfister, Generalized overlap quantum state tomography, Phys. Rev. Res. 2, 042002(R) (2020).
  39. M. Cramer, M. B. Plenio, S. T. Flammia, R. Somma, D. Gross, S. D. Bartlett, O. Landon-Cardinal, D. Poulin, and Y.-K. Liu, Efficient quantum state tomography, Nat. Commun. 1, 149 (2010).
  40. B. P. Lanyon, C. Maier, M. Holzäpfel, T. Baumgratz, C. Hempel, P. Jurcevic, I. Dhand, A. S. Buyskikh, A. J. Daley, M. Cramer, M. B. Plenio, R. Blatt, and C. F. Roos, Efficient tomography of a quantum many-body system, Nat. Phys. 13, 1158 (2017).
  41. G. Tóth, W. Wieczorek, D. Gross, R. Krischek, C. Schwemmer, and H. Weinfurter, Permutationally invariant quantum tomography, Phys. Rev. Lett. 105, 250403 (2010).
  42. A. Kyrillidis, A. Kalev, D. Park, S. Bhojanapalli, C. Caramanis, and S. Sanghavi, Provable compressed sensing quantum state tomography via non-convex methods, npj Quantum Inf. 4, 36 (2018).
  43. S. T. Flammia and Y.-K. Liu, Direct fidelity estimation from few pauli measurements, Phys. Rev. Lett. 106, 230501 (2011).
  44. H.-Y. Huang, R. Kueng, and J. Preskill, Predicting many properties of a quantum system from very few measurements, Nat. Phys. 16, 1050 (2020).
  45. S. N. Hearth, M. O. Flynn, A. Chandran, and C. R. Laumann, Efficient local classical shadow tomography with number conservation, Phys. Rev. Lett. 133, 060802 (2024).
  46. X. Bonet-Monroig, R. Babbush, and T. E. O’Brien, Nearly optimal measurement scheduling for partial tomography of quantum states, Phys. Rev. X 10, 031064 (2020).
  47. J. S. Lundeen and C. Bamber, Procedure for direct measurement of general quantum states using weak measurement, Phys. Rev. Lett. 108, 070402 (2012).
  48. S. Wu, State tomography via weak measurements, Sci. Rep. 3, 1193 (2013).
  49. A. Gaikwad, G. Singh, K. Dorai, and Arvind, Direct tomography of quantum states and processes via weak measurements of Pauli spin operators on an NMR quantum processor, Eur. Phys. J. D 77, 209 (2023).
  50. Y. Kim, Y.-S. Kim, S.-Y. Lee, S.-W. Han, S. Moon, Y.-H. Kim, and Y.-W. Cho, Direct quantum process tomography via measuring sequential weak values of incompatible observables, Nat. Commun. 9, 192 (2018).
  51. L. Calderaro, G. Foletto, D. Dequal, P. Villoresi, and G. Vallone, Direct reconstruction of the quantum density matrix by strong measurements, Phys. Rev. Lett. 121, 230501 (2018).
  52. G. S. Thekkadath, L. Giner, Y. Chalich, M. J. Horton, J. Banker, and J. S. Lundeen, Direct measurement of the density matrix of a quantum system, Phys. Rev. Lett. 117, 120401 (2016).
  53. S. Zhang, Y. Zhou, Y. Mei, K. Liao, Y.-L. Wen, J. Li, X.-D. Zhang, S. Du, H. Yan, and S.-L. Zhu, δ-quench measurement of a pure quantum-state wave function, Phys. Rev. Lett. 123, 190402 (2019).
  54. T. Feng, C. Ren, and X. Zhou, Direct measurement of density-matrix elements using a phase-shifting technique, Phys. Rev. A 104, 042403 (2021).
  55. C. Li, Y. Wang, T. Feng, Z. Li, C. Ren, and X. Zhou, Direct measurement of density-matrix elements with a phase-shifting technique on a quantum photonic chip, Phys. Rev. A 105, 062414 (2022).
  56. A. K. Ekert, C. M. Alves, D. K. L. Oi, M. Horodecki, P. Horodecki, and L. C. Kwek, Direct estimations of linear and nonlinear functionals of a quantum state, Phys. Rev. Lett. 88, 217901 (2002).
  57. A. Bendersky and J. P. Paz, Selective and efficient quantum state tomography and its application to quantum process tomography, Phys. Rev. A 87, 012122 (2013).
  58. E. Bolduc, G. Gariepy, and J. Leach, Direct measurement of large-scale quantum states via expectation values of non-Hermitian matrices, Nat. Commun. 7, 10439 (2016).
  59. V. Feldman and A. Bendersky, Selective quantum state tomography for continuous-variable systems, Phys. Rev. A 105, 032453 (2022).
  60. J. Morris and B. Dakić, Selective quantum state tomography, arXiv:1909.05880.
  61. V. Feldman and A. Bendersky, Selective continuous-variable quantum process tomography, Phys. Rev. A 111, 042409 (2025).
  62. H. Laurell, S. Luo, R. Weissenbilder, M. Ammitzböll, S. Ahmed, H. Söderberg, C. Leon. M. Petersson, V. Poulain, C. Guo, C. Dittel, D. Finkelstein-Shapiro, R. J. Squibb, R. Feifel, M. Gisselbrecht, C. L. Arnold, A. Buchleitner, E. Lindroth, A. Frisk Kockum, A. L’Huillier, and D. Busto, Measuring the quantum state of photoelectrons, Nat. Photon. 19, 352 (2025).
  63. K. He, M. Yuan, Y. Wong, S. Chakram, A. Seif, L. Jiang, and D. I. Schuster, Efficient multimode Wigner tomography, Nat. Commun. 15, 4138 (2024).
  64. Z. Kis and S. Stenholm, Measuring the density matrix by local addressing, Phys. Rev. A 64, 065401 (2001).
  65. A. Sengupta, A. Chatterjee, G. J. Sreejith, and T. S. Mahesh, Partial quantum shadow tomography for structured operators and its experimental demonstration using NMR, Phys. Rev. A 113, 032419 (2026).
  66. D. Goyeneche, G. Cañas, S. Etcheverry, E. S. Gómez, G. B. Xavier, G. Lima, and A. Delgado, Five measurement bases determine pure quantum states on any dimension, Phys. Rev. Lett. 115, 090401 (2015).
  67. T. Feng, T. Xiao, Y. Wang, S. Pang, F. Hanif, X. Zhou, Q. Zhao, M. S. Kim, and J. Sun, Two measurement bases are asymptotically informationally complete for any pure state tomography, arXiv:2501.17061v1.
  68. B. Reggio, N. Butt, A. Lytle, and P. Draper, Fast partitioning of Pauli strings into commuting families for optimal expectation value measurements of dense operators, Phys. Rev. A 110, 022606 (2024).
  69. E. van den Berg and K. Temme, Circuit optimization of Hamiltonian simulation by simultaneous diagonalization of Pauli clusters, Quantum 4, 322 (2020).
  70. Python code for SEEQST, https://github.com/aniket-ae/SEEQST.
  71. A. Gaikwad, D. Rehal, A. Singh, Arvind, and K. Dorai, Experimental demonstration of selective quantum process tomography on an NMR quantum information processor, Phys. Rev. A 97, 022311 (2018).
  72. A. Gaikwad, K. Shende, Arvind, and K. Dorai, Implementing efficient selective quantum process tomography of superconducting quantum gates on IBM quantum experience, Sci. Rep. 12, 3688 (2022).
  73. D. M. Greenberger, M. A. Horne, and A. Zeilinger, Going beyond Bell's theorem, in Bell's Theorem, Quantum Theory and Conceptions of the Universe, edited by M. Kafatos (Springer, Netherlands, 1989), p. 69.
  74. W. Dür, G. Vidal, and J. I. Cirac, Three qubits can be entangled in two inequivalent ways, Phys. Rev. A 62, 062314 (2000).
  75. M. Ippoliti, Classical shadows based on locally-entangled measurements, Quantum 8, 1293 (2024).
  76. J. F. Kam, H. Kang, C. D. Hill, G. J. Mooney, and L. C. L. Hollenberg, Characterization of entanglement on superconducting quantum computers of up to 414 qubits, Phys. Rev. Res. 6, 033155 (2024).
  77. G. J. Mooney, G. A. L. White, C. D. Hill, and L. C. L. Hollenberg, Generation and verification of 27-qubit Greenberger-Horne-Zeilinger states in a superconducting quantum computer, J. Phys. Commun. 5, 095004 (2021).
  78. W. Feng, G.-Q. Zhang, Q.-P. Su, J.-X. Zhang, and C.-P. Yang, Generation of Greenberger-Horne-Zeilinger states on two-dimensional superconducting-qubit lattices via parallel multiqubit-gate operations, Phys. Rev. Appl. 18, 064036 (2022).
  79. H. Liao, G. S. Hartnett, A. Kakkar, A. Tan, M. Hush, P. S. Mundada, M. J. Biercuk, and Y. Baum, Achieving computational gains with quantum error-correction primitives: Generation of long-range entanglement enhanced by error detection, PRX Quantum 6, 020331 (2025).
  80. ibm_marrakesh processor specifications, https://quantum.ibm.com/services/resources?type=Heron&system=ibm_marrakesh.
  81. R. Jozsa, Fidelity for mixed quantum states, J. Mod. Opt. 41, 2315 (1994).
  82. D. Binosi, G. Garberoglio, D. Maragnano, M. Dapor, and M. Liscidini, A tailor-made quantum state tomography approach, APL Quantum 1, 036112 (2024).
  83. M.-D. Choi, Completely positive linear maps on complex matrices, Linear Algebra Appl. 10, 285 (1975).
  84. D. W. Leung, Choi's proof as a recipe for quantum process tomography, J. Math. Phys. 44, 528 (2003).
  85. M. Jiang, S. Luo, and S. Fu, Channel-state duality, Phys. Rev. A 87, 022310 (2013).
  86. M. Cattaneo, M. A. C. Rossi, K. Korhonen, E.-M. Borrelli, G. García-Pérez, Z. Zimborás, and D. Cavalcanti, Self-consistent quantum measurement tomography based on semidefinite programming, Phys. Rev. Res. 5, 033154 (2023).
  87. J. Barberà-Rodríguez, L. Zambrano, A. Acín, and D. Farina, Boosting projective methods for quantum process and detector tomography, Phys. Rev. Res. 7, 013208 (2025).
  88. L. Zambrano, S. Ramos-Calderer, and R. Kueng, Fast quantum measurement tomography with dimension-optimal error bounds, arXiv:2507.04500.

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Phys. Rev. Research 8, 013339– Published 30 March, 2026
  • Received 8 July 2025
  • Revised 15 December 2025
  • Accepted 13 February 2026
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