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Nearly Optimal Circuit Size for Sparse Quantum State Preparation

Authors Lvzhou Li , Jingquan Luo

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ACM Subject Classification
  • Theory of computation → Circuit complexity
  • Theory of computation → Quantum computation theory
Keywords
  • Quantum computing
  • quantum state preparation
  • circuit complexity

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Abstract

Quantum state preparation is a fundamental and significant subroutine in quantum computing. In this paper, we conduct a systematic investigation of the circuit size (the total count of elementary gates in the circuit) for sparse quantum state preparation. A quantum state is said to be d-sparse if it has only d non-zero amplitudes. For the task of preparing an n-qubit d-sparse quantum state, we obtain the following results:  
- Without ancillary qubits: Any n-qubit d-sparse quantum state can be prepared by a quantum circuit of size O(nd/(log n) + n) without using ancillary qubits, which improves the previous best results. It is asymptotically optimal when d = poly(n), and this optimality holds for a broader scope under some reasonable assumptions. 
- With limited ancillary qubits: (i) Based on the first result, we prove for the first time a trade-off between the number of ancillary qubits and the circuit size: any n-qubit d-sparse quantum state can be prepared by a quantum circuit of size O((nd)/(log(n + m)) + n) using m ancillary qubits for any m ∈ O((nd)/(log nd) + n). (ii) We establish a matching lower bound Ω((nd)/(log(n+m))+n) under some reasonable assumptions, and obtain a slightly weaker lower bound Ω((nd)/(log(n+m)+log d) + n) without any assumptions. 
- With unlimited ancillary qubits: Given an arbitrary amount of ancillary qubits available, the circuit size for preparing n-qubit d-sparse quantum states is Θ((nd)/(log nd) + n).

Cite As Get BibTex

Lvzhou Li and Jingquan Luo. Nearly Optimal Circuit Size for Sparse Quantum State Preparation. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 113:1-113:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ICALP.2025.113

Author Details

Lvzhou Li
  • Institute of Quantum Computing and Software, School of Computer Science and Engineering, Sun Yat-sen University, Guangzhou, China
Jingquan Luo
  • Institute of Quantum Computing and Software, School of Computer Science and Engineering, Sun Yat-sen University, Guangzhou, China

Funding

Lvzhou Li and Jingquan Luo were supported by the National Key Research and Development Program of China (Grant No.2024YFB4504004), the National Natural Science Foundation of China (Grants No.92465202 and No.62272492), the Guangdong Provincial Quantum Science Strategic Initiative (Grant No.GDZX2303007), and the Guangzhou Science and Technology Program (Grant No.2024A04J4892).

References

  1. Nabila Abdessaied, Mathias Soeken, Michael Kirkedal Thomsen, and Rolf Drechsler. Upper bounds for reversible circuits based on young subgroups. Information Processing Letters, 114(6):282-286, 2014. URL: https://doi.org/10.1016/j.ipl.2014.01.003.
  2. Israel F Araujo, Carsten Blank, Ismael CS Araújo, and Adenilton J da Silva. Low-rank quantum state preparation. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 2023. URL: https://doi.org/10.1109/TCAD.2023.3297972.
  3. Adriano Barenco, Charles H Bennett, Richard Cleve, David P DiVincenzo, Norman Margolus, Peter Shor, Tycho Sleator, John A Smolin, and Harald Weinfurter. Elementary gates for quantum computation. Physical Review A, 52(5):3457, 1995. URL: https://doi.org/10.1103/PhysRevA.52.3457.
  4. Johannes Bausch. Fast black-box quantum state preparation. Quantum, 6:773, 2022. URL: https://doi.org/10.22331/q-2022-08-04-773.
  5. Ville Bergholm, Juha J Vartiainen, Mikko Möttönen, and Martti M Salomaa. Quantum circuits with uniformly controlled one-qubit gates. Physical Review A, 71(5):052330, 2005. URL: https://doi.org/10.1103/PhysRevA.71.052330.
  6. Dominic W Berry, Andrew M Childs, Richard Cleve, Robin Kothari, and Rolando D Somma. Simulating hamiltonian dynamics with a truncated taylor series. Physical Review Letters, 114(9):090502, 2015. URL: https://doi.org/10.1103/physrevlett.114.090502.
  7. Alex Brodsky. Reversible circuit realizations of Boolean functions. In Exploring New Frontiers of Theoretical Informatics, IFIP 18th World Computer Congress, TC1 3rd International Conference on Theoretical Computer Science, volume 155 of IFIP, pages 67-80. Kluwer/Springer, 2004. URL: https://doi.org/10.1007/1-4020-8141-3_8.
  8. Nai-Hui Chia, András Pal Gilyén, Tongyang Li, Han-Hsuan Lin, Ewin Tang, and Chunhao Wang. Sampling-based sublinear low-rank matrix arithmetic framework for dequantizing quantum machine learning. Journal of the ACM, 69(5):1-72, 2022. URL: https://doi.org/10.1145/3549524.
  9. Nai-Hui Chia, Tongyang Li, Han-Hsuan Lin, and Chunhao Wang. Quantum-inspired sublinear algorithm for solving low-rank semidefinite programming. In Proceedings of the 45th International Symposium on Mathematical Foundations of Computer Science, volume 170 of Leibniz International Proceedings in Informatics (LIPIcs), pages 23:1-23:15, 2020. URL: https://doi.org/10.4230/LIPIcs.MFCS.2020.23.
  10. Andrew M Childs, Dmitri Maslov, Yunseong Nam, Neil J Ross, and Yuan Su. Toward the first quantum simulation with quantum speedup. Proceedings of the National Academy of Sciences, 115(38):9456-9461, 2018. URL: https://doi.org/10.1073/pnas.1801723115.
  11. William Cottrell, Ben Freivogel, Diego M Hofman, and Sagar F Lokhande. How to build the thermofield double state. Journal of High Energy Physics, 2019(2):1-43, 2019. URL: https://doi.org/10.1007/jhep02(2019)058.
  12. Tiago ML de Veras, Leon D da Silva, and Adenilton J da Silva. Double sparse quantum state preparation. Quantum Information Processing, 21(6):204, 2022. URL: https://doi.org/10.1007/s11128-022-03549-y.
  13. Tiago ML de Veras, Ismael CS De Araujo, Daniel K Park, and Adenilton J da Silva. Circuit-based quantum random access memory for classical data with continuous amplitudes. IEEE Transactions on Computers, 70(12):2125-2135, 2020. URL: https://doi.org/10.1109/TC.2020.3037932.
  14. Richard P Feynman. Simulating physics with computers. International Journal of Theoretical Physics, 21(6/7), 1982. Google Scholar
  15. Craig Gidney. Constructing large controlled nots. https://algassert.com/circuits/2015/06/05/Constructing-Large-Controlled-Nots.html, 2015.
  16. András Gilyén, Zhao Song, and Ewin Tang. An improved quantum-inspired algorithm for linear regression. Quantum, 6:754, 2022. URL: https://doi.org/10.22331/q-2022-06-30-754.
  17. Niels Gleinig and Torsten Hoefler. An efficient algorithm for sparse quantum state preparation. In Proceedings of the 58th ACM/IEEE Design Automation Conference, pages 433-438. IEEE, 2021. URL: https://doi.org/10.1109/DAC18074.2021.9586240.
  18. Javier Gonzalez-Conde, Thomas W Watts, Pablo Rodriguez-Grasa, and Mikel Sanz. Efficient quantum amplitude encoding of polynomial functions. Quantum, 8:1297, 2024. URL: https://doi.org/10.22331/q-2024-03-21-1297.
  19. Lov Grover and Terry Rudolph. Creating superpositions that correspond to efficiently integrable probability distributions, 2002. URL: https://arxiv.org/abs/quant-ph/0208112.
  20. Lov K Grover. A fast quantum mechanical algorithm for database search. In Proceedings of the 28th Annual ACM Symposium on Theory of Computing, pages 212-219, 1996. URL: https://doi.org/10.1145/237814.237866.
  21. Marshall Hall. The theory of groups. Courier Dover Publications, 2018. Google Scholar
  22. Aram W Harrow, Avinatan Hassidim, and Seth Lloyd. Quantum algorithm for linear systems of equations. Physical Review Letters, 103(15):150502, 2009. URL: https://doi.org/10.1103/physrevlett.103.150502.
  23. Adam Holmes and Anne Y Matsuura. Efficient quantum circuits for accurate state preparation of smooth, differentiable functions. In Proceedings of IEEE International Conference on Quantum Computing and Engineering, pages 169-179. IEEE, 2020. URL: https://doi.org/10.1109/QCE49297.2020.00030.
  24. Raban Iten, Roger Colbeck, Ivan Kukuljan, Jonathan Home, and Matthias Christandl. Quantum circuits for isometries. Physical Review A, 93(3):032318, 2016. URL: https://doi.org/10.1103/PhysRevA.93.032318.
  25. Jiaqing Jiang, Xiaoming Sun, Shang-Hua Teng, Bujiao Wu, Kewen Wu, and Jialin Zhang. Optimal space-depth trade-off of CNOT circuits in quantum logic synthesis. In Proceedings of the 31st Annual ACM-SIAM Symposium on Discrete Algorithms, pages 213-229. SIAM, 2020. URL: https://doi.org/10.1137/1.9781611975994.13.
  26. Sonika Johri, Shantanu Debnath, Avinash Mocherla, Alexandros Singk, Anupam Prakash, Jungsang Kim, and Iordanis Kerenidis. Nearest centroid classification on a trapped ion quantum computer. npj Quantum Information, 7(1):122, 2021. URL: https://doi.org/10.1038/s41534-021-00456-5.
  27. Iordanis Kerenidis and Jonas Landman. Quantum spectral clustering. Physical Review A, 103(4):042415, 2021. URL: https://doi.org/10.1103/PhysRevA.103.042415.
  28. Iordanis Kerenidis, Jonas Landman, Alessandro Luongo, and Anupam Prakash. q-means: A quantum algorithm for unsupervised machine learning. In Advances in Neural Information Processing Systems, volume 32. Curran Associates, Inc., 2019. URL: https://proceedings.neurips.cc/paper_files/paper/2019/file/16026d60ff9b54410b3435b403afd226-Paper.pdf.
  29. Iordanis Kerenidis and Anupam Prakash. Quantum recommendation systems. In Proceedings of the 8th Innovations in Theoretical Computer Science Conference, volume 67 of Leibniz International Proceedings in Informatics (LIPIcs), pages 49:1-49:21, 2017. URL: https://doi.org/10.4230/LIPIcs.ITCS.2017.49.
  30. Lvzhou Li and Jingquan Luo. Nearly optimal circuit size for sparse quantum state preparation, 2024. URL: https://arxiv.org/abs/2406.16142.
  31. Seth Lloyd, Masoud Mohseni, and Patrick Rebentrost. Quantum principal component analysis. Nature Physics, 10(9):631-633, 2014. URL: https://doi.org/10.1038/nphys3029.
  32. Guang Hao Low and Isaac L Chuang. Optimal hamiltonian simulation by quantum signal processing. Physical Review Letters, 118(1):010501, 2017. URL: https://doi.org/10.1103/physrevlett.118.010501.
  33. Guang Hao Low and Isaac L Chuang. Hamiltonian simulation by qubitization. Quantum, 3:163, 2019. URL: https://doi.org/10.22331/q-2019-07-12-163.
  34. Emanuel Malvetti, Raban Iten, and Roger Colbeck. Quantum circuits for sparse isometries. Quantum, 5:412, 2021. URL: https://doi.org/10.22331/q-2021-03-15-412.
  35. Rui Mao, Guojing Tian, and Xiaoming Sun. Toward optimal circuit size for sparse quantum state preparation. Physical Review A, 110:032439, September 2024. URL: https://doi.org/10.1103/PhysRevA.110.032439.
  36. Gabriel Marin-Sanchez, Javier Gonzalez-Conde, and Mikel Sanz. Quantum algorithms for approximate function loading. Physical Review Research, 5(3):033114, 2023. URL: https://doi.org/10.1103/PhysRevResearch.5.033114.
  37. Ketan Markov, Igor Patel, and John Hayes. Optimal synthesis of linear reversible circuits. Quantum Information and Computation, 8(3&4):0282-0294, 2008. URL: https://doi.org/10.26421/QIC8.3-4-4.
  38. Sam McArdle, András Gilyén, and Mario Berta. Quantum state preparation without coherent arithmetic, 2022. URL: https://arxiv.org/abs/2210.14892.
  39. Cristopher Moore and Martin Nilsson. Parallel quantum computation and quantum codes. SIAM Journal on Computing, 31(3):799-815, 2001. URL: https://doi.org/10.1137/S0097539799355053.
  40. Mudassir Moosa, Thomas W Watts, Yiyou Chen, Abhijat Sarma, and Peter L McMahon. Linear-depth quantum circuits for loading fourier approximations of arbitrary functions. Quantum Science and Technology, 9(1):015002, 2023. URL: https://doi.org/10.1088/2058-9565/acfc62.
  41. Fereshte Mozafari, Giovanni De Micheli, and Yuxiang Yang. Efficient deterministic preparation of quantum states using decision diagrams. Physical Review A, 106(2):022617, 2022. URL: https://doi.org/10.48550/arXiv.2206.08588.
  42. Michael A Nielsen and Isaac L Chuang. Quantum computation and quantum information, volume 2. Cambridge university press Cambridge, 2001. URL: https://doi.org/10.1145/505482.505499.
  43. Daniel K Park, Francesco Petruccione, and June-Koo Kevin Rhee. Circuit-based quantum random access memory for classical data. Scientific Reports, 9(1):3949, 2019. URL: https://doi.org/10.1038/s41598-019-40439-3.
  44. Martin Plesch and Časlav Brukner. Quantum-state preparation with universal gate decompositions. Physical Review A, 83(3):032302, 2011. URL: https://doi.org/10.1103/PhysRevA.83.032302.
  45. Debora Ramacciotti, Andreea I Lefterovici, and Antonio F Rotundo. Simple quantum algorithm to efficiently prepare sparse states. Physical Review A, 110(3):032609, 2024. URL: https://doi.org/10.1103/physreva.110.032609.
  46. Arthur G. Rattew and Bálint Koczor. Preparing arbitrary continuous functions in quantum registers with logarithmic complexity, 2022. URL: https://arxiv.org/abs/2205.00519.
  47. Patrick Rebentrost, Masoud Mohseni, and Seth Lloyd. Quantum support vector machine for big data classification. Physical Review Letters, 113(13):130503, 2014. URL: https://doi.org/10.1103/PhysRevLett.113.130503.
  48. Mehdi Saeedi and Igor L Markov. Synthesis and optimization of reversible circuits—a survey. ACM Computing Surveys, 45(2):1-34, 2013. URL: https://doi.org/10.1145/2431211.2431220.
  49. Mehdi Saeedi, Mehdi Sedighi, and Morteza Saheb Zamani. A library-based synthesis methodology for reversible logic. Microelectronics Journal, 41(4):185-194, 2010. URL: https://doi.org/10.1016/J.MEJO.2010.02.002.
  50. Mehdi Saeedi, Morteza Saheb Zamani, Mehdi Sedighi, and Zahra Sasanian. Reversible circuit synthesis using a cycle-based approach. ACM Journal on Emerging Technologies in Computing Systems, 6(4):1-26, 2010. URL: https://doi.org/10.1145/1877745.1877747.
  51. Yuval R Sanders, Guang Hao Low, Artur Scherer, and Dominic W Berry. Black-box quantum state preparation without arithmetic. Physical Review Letters, 122(2):020502, 2019. URL: https://doi.org/10.1103/PhysRevLett.122.020502.
  52. Vivek V Shende, Stephen S Bullock, and Igor L Markov. Synthesis of quantum logic circuits. In Proceedings of the 2005 Asia and South Pacific Design Automation Conference, pages 272-275, 2005. URL: https://doi.org/10.1145/1120725.1120847.
  53. Vivek V Shende, Igor L Markov, and Stephen S Bullock. Smaller two-qubit circuits for quantum communication and computation. In Proceedings Design, Automation and Test in Europe Conference and Exhibition, volume 2, pages 980-985. IEEE, 2004. URL: https://doi.org/10.1109/DATE.2004.1269020.
  54. Vivek V Shende, Aditya K Prasad, Igor L Markov, and John P Hayes. Synthesis of reversible logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 22(6):710-722, 2003. URL: https://doi.org/10.1109/TCAD.2003.811448.
  55. Peter W Shor. Algorithms for quantum computation: discrete logarithms and factoring. In Proceedings of the 35th Annual Symposium on Foundations of Computer Science, pages 124-134. IEEE, 1994. URL: https://doi.org/10.1109/SFCS.1994.365700.
  56. Xiaoming Sun, Guojing Tian, Shuai Yang, Pei Yuan, and Shengyu Zhang. Asymptotically optimal circuit depth for quantum state preparation and general unitary synthesis. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 42(10):3301-3314, 2023. URL: https://doi.org/10.1109/TCAD.2023.3244885.
  57. Ewin Tang. A quantum-inspired classical algorithm for recommendation systems. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, pages 217-228, 2019. URL: https://doi.org/10.1145/3313276.3316310.
  58. Ewin Tang. Quantum principal component analysis only achieves an exponential speedup because of its state preparation assumptions. Physical Review Letters, 127(6):060503, 2021. URL: https://doi.org/10.1103/PhysRevLett.127.060503.
  59. Carlo A Trugenberger. Probabilistic quantum memories. Physical Review Letters, 87(6):067901, 2001. URL: https://doi.org/10.1103/PhysRevLett.87.067901.
  60. Shengbin Wang, Zhimin Wang, Guolong Cui, Shangshang Shi, Ruimin Shang, Lixin Fan, Wendong Li, Zhiqiang Wei, and Yongjian Gu. Fast black-box quantum state preparation based on linear combination of unitaries. Quantum Information Processing, 20(8):270, 2021. URL: https://doi.org/10.1007/s11128-021-03203-z.
  61. Robert Wille and Rolf Drechsler. Bdd-based synthesis of reversible logic for large functions. In Proceedings of the 46th Annual Design Automation Conference, pages 270-275, 2009. URL: https://doi.org/10.1145/1629911.1629984.
  62. Xian Wu and Lvzhou Li. Asymptotically optimal synthesis of reversible circuits. Information and Computation, 301:105235, 2024. URL: https://doi.org/10.1016/j.ic.2024.105235.
  63. Dmitry V Zakablukov. On asymptotic gate complexity and depth of reversible circuits without additional memory. Journal of Computer and System Sciences, 84:132-143, 2017. URL: https://doi.org/10.1016/j.jcss.2016.09.010.
  64. Xiao-Ming Zhang, Tongyang Li, and Xiao Yuan. Quantum state preparation with optimal circuit depth: Implementations and applications. Physical Review Letters, 129(23):230504, 2022. URL: https://doi.org/10.1103/PhysRevLett.129.230504.
  65. Xiao-Ming Zhang and Xiao Yuan. Circuit complexity of quantum access models for encoding classical data. npj Quantum Information, 10(1):42, 2024. URL: https://doi.org/10.1038/s41534-024-00835-8.
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