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  • Open Access

Computational Complexity of Unitary and State Design Properties

Yoshifumi Nakata1,*, Yuki Takeuchi2,3,†, Martin Kliesch4, and Andrew Darmawan1

  • *Contact author: yoshifumi.nakata@yukawa.kyoto-u.ac.jp
  • †Present address: Information Technology R&D Center, Mitsubishi Electric Corporation.
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PRX Quantum 6, 030345 – Published 9 September, 2025

DOI: https://doi.org/10.1103/21vm-bz3t

Abstract

We investigate unitary and state t-designs from a computational complexity perspective. First, we address the problems of computing frame potentials that characterize (approximate) t-designs. We present a quantum algorithm for computing frame potentials and establish the following: (1) exact computation can be achieved by a single query to a #P oracle and is #P-hard; (2) for state vectors, deciding whether the frame potential is larger than or smaller than certain values is BQP-complete, provided that the promise gap between the two values is inverse polynomial in the number of qubits; and (3) for both state vectors and unitaries, this promise problem is PP-complete if the promise gap is exponentially small. Second, we address the promise problem of deciding whether or not a given set is a good approximation to a design. Given a certain promise gap that could be constant, we show that this problem is PP-hard, highlighting the inherent computational difficulty of determining properties of unitary and state designs. We further identify implications of our results, including variational methods for constructing designs, diagnosing quantum chaos, and exploring emergent designs in Hamiltonian systems.

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Popular Summary

Pseudorandomness is a highly useful resource in information processing. This extends to the quantum realm: quantum pseudorandomness plays a central role in various quantum information tasks. Despite its significance, the computational difficulty of checking quantum pseudorandomness properties has been largely unexplored. In this study, we take a significant step in this direction by initiating a computational-complexity-theoretic analysis of quantum pseudorandomness. Our main result demonstrates that computing key properties of quantum pseudorandomness is fundamentally hard, even quantumly, unveiling its inherently complex structure.

In our analysis, we focus on a standard formulation of quantum pseudorandomness known as unitary and state designs, which are characterized by so-called frame potentials. We begin by examining the complexity of exactly or approximately computing the frame potentials. We develop a quantum algorithm for this purpose and show that the difficulty varies with the required precision: at lower precision, the frame potentials can be computed in quantum polynomial time, but achieving higher precision is unlikely to be efficient. Furthermore, we prove that no efficient quantum algorithm is likely to exist to determine whether a given ensemble is a design. While quantum pseudorandomness is a powerful resource, these findings highlight that checking its properties is computationally challenging, even for quantum computers.

Because of the broad applicability of our approach, numerous applications are expected, such as those for computationally secure quantum cryptography and for understanding complex quantum many-body systems through the framework of quantum pseudorandomness.

Article Text

References (95)

  1. A. Ambainis and A. Smith, in Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. RANDOM APPROX 2004 2004, Lecture Notes in Computer Science, edited by K. Jansen, S. Khanna, J. D. P. Rolim, and D. Ron (Springer, Berlin, Heidelberg, 2004), Vol. 3122.
  2. P. Hayden, D. Leung, P. W. Shor, and A. Winter, Randomizing quantum states: Constructions and applications, Commun. Math. Phys. 250, 371 (2004).
  3. W. Kretschmer, in 16th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2021) (Schloss Dagstuhl—Leibniz-Zentrum für Informatik, Dagstuhl, Germany, 2021), Vol. 197, pp. 2:1–2:20.
  4. T. Morimae and T. Yamakawa, in Advances in Cryptology—CRYPTO 2022 (Springer Nature Switzerland, Cham, 2022), pp. 269–295.
  5. P. Ananth, L. Qian, and H. Yuen, in Advances in Cryptology—CRYPTO 2022 (Springer Nature Switzerland, Cham, 2022), pp. 208–236.
  6. P. Sen, in 21st Annual IEEE Conference on Computational Complexity (CCC’06) (IEEE, Prague, 2006), pp. 14–287.
  7. F. G. S. L. Brandão and M. Horodecki, Exponential quantum speed-ups are generic, Quantum Inf. Comput. 13, 901 (2013).
  8. S. Boixo, S. V. Isakov, V. N. Smelyanskiy, R. Babbush, N. Ding, Z. Jiang, M. J. Bremner, J. M. Martinis, and H. Neven, Characterizing quantum supremacy in near-term devices, Nat. Phys. 14, 595 (2018).
  9. F. Arute et al., Quantum supremacy using a programmable superconducting processor, Nature 574, 505 (2019).
  10. A. Bouland, B. Fefferman, C. Nirkhe, and U. Vazirani, On the complexity and verification of quantum random circuit sampling, Nat. Phys. 15, 159 (2019).
  11. R. Kueng, H. Rauhut, and U. Terstiege, Low rank matrix recovery from rank one measurements, Appl. Comput. Harmon. Anal. 42, 88 (2017).
  12. S. Kimmel and Y.-K. Liu, in 2017 International Conference on Sampling Theory and Applications (SampTA) (IEEE, Tallinn, Estonia, 2017), pp. 345–349.
  13. R. Kueng, H. Zhu, and D. Gross, Distinguishing quantum states using Clifford orbits, arXiv:1609.08595 [quant-ph].
  14. M. Oszmaniec, R. Augusiak, C. Gogolin, J. Kołodyński, A. Acín, and M. Lewenstein, Random bosonic states for robust quantum metrology, Phys. Rev. X 6, 041044 (2016).
  15. I. Devetak, The private classical capacity and quantum capacity of a quantum channel, IEEE Trans. Inf. Theory 51, 44 (2005).
  16. I. Devetak and A. Winter, Relating quantum privacy and quantum coherence: An operational approach, Phys. Rev. Lett. 93, 080501 (2004).
  17. B. Groisman, S. Popescu, and A. Winter, Quantum, classical, and total amount of correlations in a quantum state, Phys. Rev. A 72, 032317 (2005).
  18. A. Abeyesinghe, I. Devetak, P. Hayden, and A. Winter, The mother of all protocols: Restructuring quantum information’s family tree, Proc. R. Soc. A 465, 2537 (2009).
  19. F. Dupuis, M. Berta, J. Wullschleger, and R. Renner, One-shot decoupling, Commun. Math. Phys. 328, 251 (2014).
  20. O. Szehr, F. Dupuis, M. Tomamichel, and R. Renner, Decoupling with unitary approximate two-designs, New J. Phys. 15, 053022 (2013).
  21. M. Horodecki, J. Oppenheim, and A. Winter, Partial quantum information, Nature 436, 673 (2005).
  22. M. Horodecki, J. Oppenheim, and A. Winter, Quantum state merging and negative information, Commun. Math. Phys. 269, 107 (2007).
  23. Y. Nakata, E. Wakakuwa, and H. Yamasaki, One-shot quantum error correction of classical and quantum information, Phys. Rev. A 104, 012408 (2021).
  24. E. Wakakuwa and Y. Nakata, One-shot triple-resource trade-off in quantum channel coding, IEEE Trans. Inf. Theory 69, 2400 (2023).
  25. S. Popescu, A. J. Short, and A. Winter, Entanglement and the foundations of statistical mechanics, Nat. Phys. 2, 754 (2006).
  26. N. Linden, S. Popescu, A. J. Short, and A. Winter, Quantum mechanical evolution towards thermal equilibrium, Phys. Rev. E 79, 061103 (2009).
  27. L. del Rio, A. Hutter, R. Renner, and S. Wehner, Relative thermalization, Phys. Rev. E 94, 022104 (2016).
  28. K. Kaneko, E. Iyoda, and T. Sagawa, Characterizing complexity of many-body quantum dynamics by higher-order eigenstate thermalization, Phys. Rev. A 101, 042126 (2020).
  29. M. Ippoliti and W. W. Ho, Solvable model of deep thermalization with distinct design times, Quantum 6, 886 (2022).
  30. T. Bhore, J.-Y. Desaules, and Z. Papić, Deep thermalization in constrained quantum systems, Phys. Rev. B 108, 104317 (2023).
  31. P. Hayden and J. Preskill, Black holes as mirrors: Quantum information in random subsystems, J. High Energy Phys. 2007, 120 (2007).
  32. Y. Nakata, E. Wakakuwa, and M. Koashi, Black holes as clouded mirrors: The Hayden-Preskill protocol with symmetry, Quantum 7, 928 (2023).
  33. Y. Nakata, T. Matsuura, and M. Koashi, Decoding general error correcting codes and the role of complementarity, npj Quantum Inf. 11, 4 (2025).
  34. Y. Sekino and L. Susskind, Fast scramblers, J. High Energy Phys. 2008, 065 (2008).
  35. N. Lashkari, D. Stanford, M. Hastings, T. Osborne, and P. Hayden, Towards the fast scrambling conjecture, J. High Energy Phys. 2013, 22 (2013).
  36. J. Maldacena, S. H. Shenker, and D. Stanford, A bound on chaos, J. High Energy Phys. 2016, 106 (2016).
  37. D. A. Roberts and B. Yoshida, Chaos and complexity by design, J. High Energ. Phys. 2017, 121 (2017).
  38. J. S. Cotler, D. K. Mark, H.-Y. Huang, F. Hernández, J. Choi, A. L. Shaw, M. Endres, and S. Choi, Emergent quantum state designs from individual many-body wave functions, PRX Quantum 4, 010311 (2023).
  39. D. K. Mark, F. Surace, A. Elben, A. L. Shaw, J. Choi, G. Refael, M. Endres, and S. Choi, Maximum entropy principle in deep thermalization and in Hilbert-space ergodicity, Phys. Rev. X 14, 041051 (2024).
  40. S. Pilatowsky-Cameo, I. Marvian, S. Choi, and W. W. Ho, Hilbert-space ergodicity in driven quantum systems: Obstructions and designs, Phys. Rev. X 14, 041059 (2024).
  41. J. Emerson, R. Alicki, and K. Życzkowski, Scalable noise estimation with random unitary operators, J. Opt. B: Quantum Semiclass. Opt. 7, S347 (2005).
  42. E. Knill et al., Randomized benchmarking of quantum gates, Phys. Rev. A 77, 012307 (2008).
  43. E. Magesan, J. M. Gambetta, and J. Emerson, Scalable and robust randomized benchmarking of quantum processes, Phys. Rev. Lett. 106, 180504 (2011).
  44. E. Magesan, J. M. Gambetta, and J. Emerson, Characterizing quantum gates via randomized benchmarking, Phys. Rev. A 85, 042311 (2012).
  45. J. Kelly et al., Optimal quantum control using randomized benchmarking, Phys. Rev. Lett. 112, 240504 (2014).
  46. S. Sheldon, L. S. Bishop, E. Magesan, S. Filipp, J. M. Chow, and J. M. Gambetta, Characterizing errors on qubit operations via iterative randomized benchmarking, Phys. Rev. A 93, 012301 (2016).
  47. S. Garion, N. Kanazawa, H. Landa, D. C. McKay, S. Sheldon, A. W. Cross, and C. J. Wood, Experimental implementation of non-Clifford interleaved randomized benchmarking with a controlled-S gate, Phys. Rev. Res. 3, 013204 (2021).
  48. E. Onorati, A. H. Werner, and J. Eisert, Randomized benchmarking for individual quantum gates, Phys. Rev. Lett. 123, 060501 (2019).
  49. J. Helsen, I. Roth, E. Onorati, A. Werner, and J. Eisert, General framework for randomized benchmarking, PRX Quantum 3, 020357 (2022).
  50. M. Heinrich, M. Kliesch, and I. Roth, General guarantees for randomized benchmarking with random quantum circuits, arXiv:2212.06181 [quant-ph].
  51. J. Choi, A. L. Shaw, I. S. Madjarov, X. Xie, R. Finkelstein, J. P. Covey, J. S. Cotler, D. K. Mark, H.-Y. Huang, A. Kale, H. Pichler, F. G. S. L. Brandão, S. Choi, and M. Endres, Preparing random states and benchmarking with many-body quantum chaos, Nature 613, 468 (2023).
  52. D. K. Mark, J. Choi, A. L. Shaw, M. Endres, and S. Choi, Benchmarking quantum simulators using ergodic quantum dynamics, Phys. Rev. Lett. 131, 110601 (2023).
  53. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information: 10th Anniversary Edition (Cambridge University Press, New York, 2010), 10th ed.
  54. Z. Ji, Y. Liu, and F. Song, in Advances in Cryptology—CRYPTO 2018 (Springer International Publishing, Cham, 2018), pp. 126–152.
  55. T. Metger, A. Poremba, M. Sinha, and H. Yuen, Pseudorandom unitaries with non-adaptive security, arXiv:2402.14803 [quant-ph].
  56. P. Delsarte, J. Goethals, and J. Seidel, Bounds for systems of lines, and Jacobi polynomials, Philips Res. Rep. 30, 91 (1975).
  57. P. Delsarte, J. M. Goethals, and J. J. Seidel, Spherical codes and designs, Geom. Dedicata 6, 363 (1977).
  58. E. Bannai, Y. Nakata, T. Okuda, and D. Zhao, Explicit construction of exact unitary designs, Adv. Math. 405, 108457 (2022).
  59. Y. Nakata et al., Quantum circuits for exact unitary t-designs and applications to higher-order randomized benchmarking, PRX Quantum 2, 030339 (2021).
  60. A. W. Harrow and R. A. Low, in Proc. RANDOM’09, Lecture Notes in Computer Science (Springer, Berlin, 2009), pp. 548–561.
  61. F. G. S. L. Brandão, A. W. Harrow, and M. Horodecki, Local random quantum circuits are approximate polynomial-designs, Commun. Math. Phys. 346, 397 (2016).
  62. Y. Nakata, C. Hirche, M. Koashi, and A. Winter, Efficient quantum pseudorandomness with nearly time-independent Hamiltonian dynamics, Phys. Rev. X 7, 021006 (2017).
  63. E. Onorati, O. Buerschaper, M. Kliesch, W. Brown, A. H. Werner, and J. Eisert, Mixing properties of stochastic quantum Hamiltonians, Commun. Math. Phys. 355, 905 (2017).
  64. J. Haferkamp, Random quantum circuits are approximate unitary t-designs in depth O(nt5+o(1)), Quantum 6, 795 (2022).
  65. A. W. Harrow and S. Mehraban, Approximate unitary t-designs by short random quantum circuits using nearest-neighbor and long-range gates, Commun. Math. Phys. 401, 1531 (2023).
  66. J. Haferkamp, F. Montealegre-Mora, M. Heinrich, J. Eisert, D. Gross, and I. Roth, Efficient unitary designs with a system-size independent number of non-Clifford gates, Commun. Math. Phys. 397, 995 (2023).
  67. T. Metger, A. Poremba, M. Sinha, and H. Yuen, Simple constructions of linear-depth t-designs and pseudorandom unitaries, arXiv:2404.12647 [quant-ph].
  68. C.-F. Chen, J. Docter, M. Xu, A. Bouland, F. G. S. L. Brandão, and P. Hayden, in 2024 IEEE 65th Annual Symposium on Foundations of Computer Science (FOCS) (Chicago, IL, USA, 2024), pp. 476–484.
  69. N. LaRacuente and F. Leditzky, in 16th Innovations in Theoretical Computer Science Conference (ITCS 2025), Leibniz International Proceedings in Informatics (LIPIcs), Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2025), Vol. 325, pp. 69:1–69:2.
  70. T. Schuster, J. Haferkamp, and H.-Y. Huang, Random unitaries in extremely low depth, Science 389, 92 (2025).
  71. D. Gross, K. Audenaert, and J. Eisert, Evenly distributed unitaries: On the structure of unitary designs, J. Math. Phys. 48, 052104 (2007).
  72. A. Chan and A. D. Luca, Projected state ensemble of a generic model of many-body quantum chaos, J. Phys. A: Math. Theor. 57, 405001 (2024).
  73. W.-K. Mok, T. Haug, A. L. Shaw, M. Endres, and J. Preskill, Phys. Rev. Lett. 134, 180403 (2025).
  74. W. W. Ho and S. Choi, Exact emergent quantum state designs from quantum chaotic dynamics, Phys. Rev. Lett. 128, 060601 (2022).
  75. M. Ippoliti and W. W. Ho, Dynamical purification and the emergence of quantum state designs from the projected ensemble, PRX Quantum 4, 030322 (2023).
  76. S. Pilatowsky-Cameo, C. B. Dag, W. W. Ho, and S. Choi, Complete Hilbert-space ergodicity in quantum dynamics of generalized Fibonacci drives, Phys. Rev. Lett. 131, 250401 (2023).
  77. A. Ambainis and J. Emerson, in Twenty-Second Annual IEEE Conference on Computational Complexity (CCC’07) (IEEE, San Diego, CA, USA, 2007), pp. 129–140.
  78. O. Goldreich, in Theoretical Computer Science: Essays in Memory of Shimon Even, edited by O. Goldreich, A. L. Rosenberg, and A. L. Selman (Springer-Verlag, Berlin, 2006), pp. 254–290.
  79. J. Watrous, Encyclopedia of Complexity and Systems Science (Springer, New York, 2009).
  80. J. T. Gill, in Proceedings of the Sixth Annual ACM Symposium on Theory of Computing, STOC ’74 (Association for Computing Machinery, New York, 1974), pp. 91–95.
  81. J. L. Cox and T. Pay, An overview of some semantic and syntactic complexity classes, arXiv:1806.03501 [cs.CC].
  82. S. Aaronson and D. Gottesman, Improved simulation of stabilizer circuits, Phys. Rev. Appl. 70, 052328 (2004).
  83. A. M. Childs, Y. Su, M. C. Tran, N. Wiebe, and S. Zhu, Theory of Trotter error with commutator scaling, Phys. Rev. X 11, 011020 (2021).
  84. A. Brodsky and S. Hoory, Simple permutations mix even better, Random Struct. Algorithms 32, 274 (2008).
  85. W. Kretschmer, L. Qian, M. Sinha, and A. Tal, in Proceedings of the 55th Annual ACM Symposium on Theory of Computing, STOC 2023 (Association for Computing Machinery, New York, 2023), pp. 1589–1602.
  86. N. Schuch, M. M. Wolf, F. Verstraete, and J. I. Cirac, Computational complexity of projected entangled pair states, Phys. Rev. Lett. 98, 140506 (2007).
  87. Strictly speaking, the state generated by the quantum circuit composed only of the Hadamard and Toffoli gates is exponentially close to the original state. We ignore this approximation error, as it can be made arbitrarily small within the order of Θ(1/exp(n)).
  88. C. M. da Fonseca, M. L. Glasser, and V. Kowalenko, Basic trigonometric power sums with applications, Ramanujan J. 42, 401 (2017).
  89. Y. Nakata, P. S. Turner, and M. Murao, Phase-random states: Ensembles of states with fixed amplitudes and uniformly distributed phases in a fixed basis, Phys. Rev. A 86, 012301 (2012).
  90. Z. Brakerski and O. Shmueli, in Theory of Cryptography, TCC 2019, Lecture Notes in Computer Science, edited by D. Hofheinz and A. Rosen (Springer, Cham, 2019), Vol. 11891.
  91. P. Ananth, A. Gulati, L. Qian, and H. Yuen, in Theory of Cryptography: 20th International Conference, TCC 2022, Chicago, IL, USA, November 7–10, 2022, Proceedings, Part I (Springer-Verlag, Berlin, Heidelberg, 2022), pp. 237–265.
  92. T. Giurgica-Tiron and A. Bouland, Pseudorandomness from subset states, arXiv:2312.09206 [quant-ph].
  93. F. G. Jeronimo, N. Magrafta, and P. Wu, Pseudorandom and pseudoentangled states from subset states, arXiv:2312.15285 [quant-ph].
  94. D. Aharonov and A. Green, A quantum inspired proof of P#P⊆IP, arXiv:1710.09078 [quant-ph].
  95. A. Gilyén and T. Li, in 11th Innovations in Theoretical Computer Science Conference (ITCS 2020), Leibniz International Proceedings in Informatics (LIPIcs), Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020), Vol. 151, pp. 25:1–25:19.

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PRX Quantum 6, 030345– Published 9 September, 2025
  • Received 17 January 2025
  • Revised 27 May 2025
  • Accepted 29 July 2025
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