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  • Open Access

Space-time trade-off in networked virtual distillation

Tenzan Araki1,2,*, Joseph F. Goodwin1, and Bálint Koczor2

  • 1Department of Physics, Clarendon Laboratory, University of Oxford, Parks Road, Oxford OX1 3PU, United Kingdom
  • 2Mathematical Institute, University of Oxford, Woodstock Road, Oxford OX2 6GG, United Kingdom

  • *Contact author: tenzan.araki@physics.ox.ac.uk
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Phys. Rev. A 112, 042619 – Published 27 October, 2025

DOI: https://doi.org/10.1103/t828-wvgl

Abstract

In contrast to monolithic devices, modular, networked quantum architectures are based on interconnecting smaller quantum hardware nodes using quantum communication links and offer a promising approach to scalability. Virtual distillation (VD) is a technique that can, under ideal conditions, suppress errors exponentially as the number of quantum state copies increases. However, additional gate operations required for VD introduce further errors, which may limit its practical effectiveness. In this work, we analyze three practical implementations of VD that correspond to edge cases that maximize space-time trade-offs. Specifically, we consider an implementation that minimizes the number of qubits but introduces significantly deeper quantum circuits and contrast it with implementations that parallelize the preparation of copies using additional qubits, including a constant-depth implementation. We rigorously characterize their circuit depth and gate count requirements and develop explicit architectures for implementing them in networked quantum systems—while also detailing implementations in early fault-tolerant quantum architectures. We numerically compare the performance of the three implementations under realistic noise characteristics of networked ion trap systems and conclude the following. First, VD effectively suppresses errors even for very noisy states. Second, the constant-depth implementation consistently outperforms the implementation that minimizes the number of qubits. Finally, the approach is highly robust to errors in remote entangling operations, with noise in local gates being the main limiting factor to its performance.

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References (74)

  1. M. Cerezo, A. Arrasmith, R. Babbush, S. C. Benjamin, S. Endo, K. Fujii, J. R. McClean, K. Mitarai, X. Yuan, L. Cincio, and P. J. Coles, Variational quantum algorithms, Nat. Rev. Phys. 3, 625 (2021).
  2. K. Bharti, A. Cervera-Lierta, T. H. Kyaw, T. Haug, S. Alperin-Lea, A. Anand, M. Degroote, H. Heimonen, J. S. Kottmann, T. Menke, W.-K. Mok, S. Sim, L.-C. Kwek, and A. Aspuru-Guzik, Noisy intermediate-scale quantum algorithms, Rev. Mod. Phys. 94, 015004 (2022).
  3. Z. Zimborás, B. Koczor, Z. Holmes, E.-M. Borrelli, A. Gilyén, H.-Y. Huang, Z. Cai, A. Acín, L. Aolita, L. Banchi, F. G. S. L. Brandão, D. Cavalcanti, T. Cubitt, S. N. Filippov, G. García-Pérez, J. Goold, O. Kálmán, E. Kyoseva, M. A. C. Rossi, B. Sokolov et al., Myths around quantum computation before full fault tolerance: What no-go theorems rule out and what they don't, arXiv:2501.05694.
  4. C. Piveteau, D. Sutter, S. Bravyi, J. M. Gambetta, and K. Temme, Error mitigation for universal gates on encoded qubits, Phys. Rev. Lett. 127, 200505 (2021).
  5. Y. Suzuki, S. Endo, K. Fujii, and Y. Tokunaga, Quantum error mitigation as a universal error reduction technique: Applications from the NISQ to the fault-tolerant quantum computing eras, PRX Quantum 3, 010345 (2022).
  6. Y. Akahoshi, K. Maruyama, H. Oshima, S. Sato, and K. Fujii, Partially fault-tolerant quantum computing architecture with error-corrected Clifford gates and space-time efficient analog rotations, PRX Quantum 5, 010337 (2024).
  7. Z. Cai, R. Babbush, S. C. Benjamin, S. Endo, W. J. Huggins, Y. Li, J. R. McClean, and T. E. O'Brien, Quantum error mitigation, Rev. Mod. Phys. 95, 045005 (2023).
  8. W. J. Huggins, S. McArdle, T. E. O'Brien, J. Lee, N. C. Rubin, S. Boixo, K. B. Whaley, R. Babbush, and J. R. McClean, Virtual distillation for quantum error mitigation, Phys. Rev. X 11, 041036 (2021).
  9. B. Koczor, Exponential error suppression for near-term quantum devices, Phys. Rev. X 11, 031057 (2021).
  10. In practice, even if identical gates are applied, the copies may not be identically prepared due to variations in the noise channels. VD remains effective as long as |ψ〉 is the dominant eigenvector across all copies, and its performance is limited by the lowest dominant eigenvalue amongst the copies as shown in Ref. [9].
  11. B. Koczor, The dominant eigenvector of a noisy quantum state, New J. Phys. 23, 123047 (2021).
  12. M. Silva, E. Magesan, D. W. Kribs, and J. Emerson, Scalable protocol for identification of correctable codes, Phys. Rev. A 78, 012347 (2008).
  13. E. Magesan, J. M. Gambetta, and J. Emerson, Characterizing quantum gates via randomized benchmarking, Phys. Rev. A 85, 042311 (2012).
  14. Z. Cai and S. C. Benjamin, Constructing smaller Pauli twirling sets for arbitrary error channels, Sci. Rep. 9, 11281 (2019).
  15. Z. Cai, X. Xu, and S. C. Benjamin, Mitigating coherent noise using Pauli conjugation, npj Quantum Inf. 6, 17 (2020).
  16. N. Yoshioka, H. Hakoshima, Y. Matsuzaki, Y. Tokunaga, Y. Suzuki, and S. Endo, Generalized quantum subspace expansion, Phys. Rev. Lett. 129, 020502 (2022).
  17. B. Yang, N. Yoshioka, H. Harada, S. Hakkaku, Y. Tokunaga, H. Hakoshima, K. Yamamoto, and S. Endo, Resource-efficient generalized quantum subspace expansion, Phys. Rev. Appl. 23, 054021 (2025).
  18. B. Koczor, Sparse probabilistic synthesis of quantum operations, PRX Quantum 5, 040352 (2024).
  19. B. Koczor, J. J. L. Morton, and S. C. Benjamin, Probabilistic interpolation of quantum rotation angles, Phys. Rev. Lett. 132, 130602 (2024).
  20. M. Huo and Y. Li, Dual-state purification for practical quantum error mitigation, Phys. Rev. A 105, 022427 (2022).
  21. Z. Cai, Resource-efficient purification-based quantum error mitigation, arXiv:2107.07279.
  22. P. Czarnik, A. Arrasmith, L. Cincio, and P. J. Coles, Qubit-efficient exponential suppression of errors, arXiv:2102.06056.
  23. D. Gottesman and I. L. Chuang, Demonstrating the viability of universal quantum computation using teleportation and single-qubit operations, Nature (London) 402, 390 (1999).
  24. C. Monroe, R. Raussendorf, A. Ruthven, K. R. Brown, P. Maunz, L.-M. Duan, and J. Kim, Large-scale modular quantum-computer architecture with atomic memory and photonic interconnects, Phys. Rev. A 89, 022317 (2014).
  25. L. Jiang, J. M. Taylor, A. S. Sørensen, and M. D. Lukin, Distributed quantum computation based on small quantum registers, Phys. Rev. A 76, 062323 (2007).
  26. M. Caleffi, M. Amoretti, D. Ferrari, J. Illiano, A. Manzalini, and A. S. Cacciapuoti, Distributed quantum computing: A survey, Comput. Netw. 254, 110672 (2024).
  27. H. Jnane, B. Undseth, Z. Cai, S. C. Benjamin, and B. Koczor, Multicore quantum computing, Phys. Rev. Appl. 18, 044064 (2022).
  28. J. Dénes, The representation of a permutation as the product of a minimal number of transpositions and its connection with the theory of graphs, Publ. Math. Inst. Hung. Acad. Sci. 4, 63 (1959).
  29. J.-M. Liang, Q.-Q. Lv, Z.-X. Wang, and S.-M. Fei, Unified multivariate trace estimation and quantum error mitigation, Phys. Rev. A 107, 012606 (2023).
  30. H. Hakoshima, S. Endo, K. Yamamoto, Y. Matsuzaki, and N. Yoshioka, Localized virtual purification, Phys. Rev. Lett. 133, 080601 (2024).
  31. Y. Quek, E. Kaur, and M. M. Wilde, Multivariate trace estimation in constant quantum depth, Quantum 8, 1220 (2024).
  32. T. Häner, D. S. Steiger, T. Hoefler, and M. Troyer, Distributed quantum computing with QMPI, in Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis (Association for Computing Machinery, New York, 2021).
  33. D. Main, P. Drmota, D. P. Nadlinger, E. M. Ainley, A. Agrawal, B. C. Nichol, R. Srinivas, G. Araneda, and D. M. Lucas, Distributed quantum computing across an optical network link, Nature (London) 638, 383 (2025).
  34. L. J. Stephenson, D. P. Nadlinger, B. C. Nichol, S. An, P. Drmota, T. G. Ballance, K. Thirumalai, J. F. Goodwin, D. M. Lucas, and C. J. Ballance, High-rate, high-fidelity entanglement of qubits across an elementary quantum network, Phys. Rev. Lett. 124, 110501 (2020).
  35. V. Krutyanskiy, M. Galli, V. Krcmarsky, S. Baier, D. A. Fioretto, Y. Pu, A. Mazloom, P. Sekatski, M. Canteri, M. Teller, J. Schupp, J. Bate, M. Meraner, N. Sangouard, B. P. Lanyon, and T. E. Northup, Entanglement of trapped-ion qubits separated by 230 meters, Phys. Rev. Lett. 130, 050803 (2023).
  36. S. Saha, M. Shalaev, J. O'Reilly, I. Goetting, G. Toh, A. Kalakuntla, Y. Yu, and C. Monroe, High-fidelity remote entanglement of trapped atoms mediated by time-bin photons, Nat. Commun. 16, 2533 (2025).
  37. S. de Bone, R. Ouyang, K. Goodenough, and D. Elkouss, Protocols for creating and distilling multipartite GHZ states with Bell pairs, IEEE Trans. Quant. Eng. 1, 1 (2020).
  38. E. M. Ainley, A. Agrawal, D. Main, P. Drmota, D. P. Nadlinger, B. C. Nichol, R. Srinivas, and G. Araneda, Multipartite entanglement for multi-node quantum networks, arXiv:2408.00149.
  39. N. H. Nickerson, Y. Li, and S. C. Benjamin, Topological quantum computing with a very noisy network and local error rates approaching one percent, Nat. Commun. 4, 1756 (2013).
  40. N. H. Nickerson, J. F. Fitzsimons, and S. C. Benjamin, Freely scalable quantum technologies using cells of 5-to-50 qubits with very lossy and noisy photonic links, Phys. Rev. X 4, 041041 (2014).
  41. C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wootters, Purification of noisy entanglement and faithful teleportation via noisy channels, Phys. Rev. Lett. 76, 722 (1996).
  42. C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Wootters, Mixed-state entanglement and quantum error correction, Phys. Rev. A 54, 3824 (1996).
  43. W. Dür and H.-J. Briegel, Entanglement purification for quantum computation, Phys. Rev. Lett. 90, 067901 (2003).
  44. K. Yamamoto, Y. Matsuzaki, Y. Suzuki, Y. Tokunaga, and S. Endo, Virtual entanglement purification via noisy entanglement, arXiv:2411.10024.
  45. E. Campbell, Random compiler for fast Hamiltonian simulation, Phys. Rev. Lett. 123, 070503 (2019).
  46. C. Kiumi and B. Koczor, TE-PAI: Exact time evolution by sampling random circuits, Quantum Sci. Technol. 10, 045071 (2025).
  47. V. Kliuchnikov, K. Lauter, R. Minko, A. Paetznick, and C. Petit, Shorter quantum circuits via single-qubit gate approximation, Quantum 7, 1208 (2023).
  48. P. Selinger, Quantum circuits of T-depth one, Phys. Rev. A 87, 042302 (2013).
  49. M. Amy, D. Maslov, M. Mosca, and M. Roetteler, A meet-in-the-middle algorithm for fast synthesis of depth-optimal quantum circuits, IEEE Trans. Comput.-Aided Des. Integr. Circ. Syst. 32, 818 (2013).
  50. D. J. Luitz, N. Laflorencie, and F. Alet, Many-body localization edge in the random-field Heisenberg chain, Phys. Rev. B 91, 081103(R) (2015).
  51. A. M. Childs, D. Maslov, Y. Nam, N. J. Ross, and Y. Su, Toward the first quantum simulation with quantum speedup, Proc. Natl. Acad. Sci. USA 115, 9456 (2018).
  52. H. F. Trotter, On the product of semi-groups of operators, Proc. Am. Math. Soc. 10, 545 (1959).
  53. M. Suzuki, Generalized trotter's formula and systematic approximants of exponential operators and inner derivations with applications to many-body problems, Commun. Math. Phys. 51, 183 (1976).
  54. Any residual coherent contribution from single-qubit gates can be converted into incoherent errors by the application of twirling techniques.
  55. A. S. Sotirova, J. D. Leppard, A. Vazquez-Brennan, S. M. Decoppet, F. Pokorny, M. Malinowski, and C. J. Ballance, High-fidelity heralded quantum state preparation and measurement, arXiv:2409.05805.
  56. Y. Seis, B. J. Brown, A. S. Sørensen, and J. F. Goodwin, Improving trapped-ion-qubit memories via code-mediated error-channel balancing, Phys. Rev. A 107, 052417 (2023).
  57. T. Jones, A. Brown, I. Bush, and S. C. Benjamin, QuEST and high performance simulation of quantum computers, Sci. Rep. 9, 10736 (2019).
  58. T. Jones and S. Benjamin, QuESTlink—Mathematica embiggened by a hardware-optimised quantum emulator, Quantum Sci. Technol. 5, 034012 (2020).
  59. A. Kay, Tutorial on the Quantikz package, arXiv:1809.03842.
  60. A. Richards, University of Oxford Advanced Research Computing, Zenodo (2015), doi:10.5281/zenodo.22558.
  61. J. C. Garcia-Escartin and P. Chamorro-Posada, Swap test and Hong-Ou-Mandel effect are equivalent, Phys. Rev. A 87, 052330 (2013).
  62. L. Cincio, Y. Subaşı, A. T. Sornborger, and P. J. Coles, Learning the quantum algorithm for state overlap, New J. Phys. 20, 113022 (2018).
  63. C. Langer, R. Ozeri, J. D. Jost, J. Chiaverini, B. DeMarco, A. Ben-Kish, R. B. Blakestad, J. Britton, D. B. Hume, W. M. Itano, D. Leibfried, R. Reichle, T. Rosenband, T. Schaetz, P. O. Schmidt, and D. J. Wineland, Long-lived qubit memory using atomic ions, Phys. Rev. Lett. 95, 060502 (2005).
  64. S. Olmschenk, K. C. Younge, D. L. Moehring, D. N. Matsukevich, P. Maunz, and C. Monroe, Manipulation and detection of a trapped Yb+ hyperfine qubit, Phys. Rev. A 76, 052314 (2007).
  65. M. A. Sepiol, A. C. Hughes, J. E. Tarlton, D. P. Nadlinger, T. G. Ballance, C. J. Ballance, T. P. Harty, A. M. Steane, J. F. Goodwin, and D. M. Lucas, Probing qubit memory errors at the part-per-million level, Phys. Rev. Lett. 123, 110503 (2019).
  66. Y. Wang, M. Um, J. Zhang, S. An, M. Lyu, J.-N. Zhang, L.-M. Duan, D. Yum, and K. Kim, Single-qubit quantum memory exceeding ten-minute coherence time, Nat. Photon. 11, 646 (2017).
  67. Y. Wang, S. Crain, C. Fang, B. Zhang, S. Huang, Q. Liang, P. H. Leung, K. R. Brown, and J. Kim, High-fidelity two-qubit gates using a microelectromechanical-system-based beam steering system for individual qubit addressing, Phys. Rev. Lett. 125, 150505 (2020).
  68. D. Kielpinski, C. Monroe, and D. J. Wineland, Architecture for a large-scale ion-trap quantum computer, Nature (London) 417, 709 (2002).
  69. C. J. Ballance, T. P. Harty, N. M. Linke, M. A. Sepiol, and D. M. Lucas, High-fidelity quantum logic gates using trapped-ion hyperfine qubits, Phys. Rev. Lett. 117, 060504 (2016).
  70. D. C. McKay, C. J. Wood, S. Sheldon, J. M. Chow, and J. M. Gambetta, Efficient Z gates for quantum computing, Phys. Rev. A 96, 022330 (2017).
  71. C. F. Roos, Ion trap quantum gates with amplitude-modulated laser beams, New J. Phys. 10, 013002 (2008).
  72. J. P. Gaebler, T. R. Tan, Y. Lin, Y. Wan, R. Bowler, A. C. Keith, S. Glancy, K. Coakley, E. Knill, D. Leibfried, and D. J. Wineland, High-fidelity universal gate set for Be9+ ion qubits, Phys. Rev. Lett. 117, 060505 (2016).
  73. V. M. Schäfer, C. J. Ballance, K. Thirumalai, L. J. Stephenson, T. G. Ballance, A. M. Steane, and D. M. Lucas, Fast quantum logic gates with trapped-ion qubits, Nature (London) 555, 75 (2018).
  74. J. Schupp, V. Krcmarsky, V. Krutyanskiy, M. Meraner, T. E. Northup, and B. P. Lanyon, Interface between trapped-ion qubits and traveling photons with close-to-optimal efficiency, PRX Quantum 2, 020331 (2021).

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Phys. Rev. A 112, 042619– Published 27 October, 2025
  • Received 20 May 2025
  • Accepted 29 September 2025
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