Skip to main content
SpringerLink
Log in

Robust Quantum Arithmetic Operations with Intermediate Qutrits in the NISQ-era

  • Published:
International Journal of Theoretical Physics Aims and scope Submit manuscript

Abstract

Numerous scientific developments in this NISQ-era (Noisy Intermediate Scale Quantum) have raised the importance for quantum algorithms relative to their conventional counterparts due to its asymptotic advantage. For resource estimates in several quantum algorithms, arithmetic operations are crucial. With resources reported as a number of Toffoli gates or T gates with/without ancilla, several efficient implementations of arithmetic operations, such as addition/subtraction, multiplication/division, square root, etc., have been accomplished in binary quantum systems. More recently, it has been shown that intermediate qutrits may be employed in the ancilla-free frontier zone, enabling us to function effectively there. In order to achieve efficient implementation of all the above-mentioned quantum arithmetic operations with regard to gate count and circuit-depth without T gate and ancilla, we have included an intermediate qutrit method in this paper. Future research aiming at reducing costs while taking into account arithmetic operations for computing tasks might be guided by our resource estimations using intermediate qutrits. Thus the enhancements are examined in relation to the fundamental arithmetic circuits. The intermediate qutrit approach necessitates access to higher energy levels, making the design susceptible to errors. We nevertheless find that our proposed technique produces much fewer errors than the state-of-the-art work, more specifically, there is an approximately 20% decrease in error probability for adder with 20 number of qubits. Similarly for multiplier and square root, the in error probability is nearly 20% for 5 number of qubits and 10 number of qubits respectively. We thus demonstrate that the percentage decrease in the probability of error is significant due to the fact that we achieve circuit efficiency by reducing circuit-depth in comparison to qubit-only works.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Similar content being viewed by others

References

  1. Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information: 10th Anniversary Edition. 10th edn. Cambridge University Press (2011)

  2. Preskill, J.: Quantum computing in the nisq era and beyond. Quantum 2, 79 (2018). https://doi.org/10.22331/q-2018-08-06-79

  3. Gokhale, P., Baker, J.M., Duckering, C., Chong, F.T., Brown, N.C., Brown, K.R.: Extending the frontier of quantum computers with qutrits. IEEE Micro 40(3), 64–72 (2020)

    Article  Google Scholar 

  4. Häner, T., Roetteler, M., Svore, K.M.: Optimizing Quantum Circuits for Arithmetic. (2018). arXiv:1805.12445

  5. Chakrabarti, S., Krishnakumar, R., Mazzola, G., Stamatopoulos, N., Woerner, S., Zeng, W.J.: A threshold for quantum advantage in derivative pricing. Quantum 5, 463 (2021). https://doi.org/10.22331/q-2021-06-01-463

  6. Saha, A., Chatterjee, T., Chattopadhyay, A., Chakrabarti, A.: Intermediate Qutrit-based Improved Quantum Arithmetic Operations with Application on Financial Derivative Pricing. (2022). arXiv:2205.15822

  7. Selinger, P.: Quantum circuits of \(t\)-depth one. Phys. Rev. A 87, 042302 (2013). https://doi.org/10.1103/PhysRevA.87.042302

    Article  ADS  Google Scholar 

  8. Amy, M., Maslov, D., Mosca, M., Roetteler, M.: A meet-in-the-middle algorithm for fast synthesis of depth-optimal quantum circuits. IEEE Trans. Comput. Aided Design Integ. Circ. Syst. 32( 6), 818–830 (2013). https://doi.org/10.1109/TCAD.2013.2244643

  9. Gokhale, P., Baker, J.M., Duckering, C., Brown, N.C., Brown, K.R., Chong, F.T.: Asymptotic improvements to quantum circuits via qutrits. In: Proceedings of the 46th International Symposium on Computer Architecture. ISCA ’19, pp. 554–566. Association for Computing Machinery (2019). https://doi.org/10.1145/3307650.3322253

  10. Baker, J.M., Duckering, C., Chong, F.T.: Efficient Quantum Circuit Decompositions via Intermediate Qudits. (2020). arXiv:2002.10592

  11. Wang, Y., Hu, Z., Sanders, B.C., Kais, S.: Qudits and high-dimensional quantum computing. Front. Phys. 8 (2020). https://doi.org/10.3389/fphy.2020.589504

  12. Koch, J., Yu, T.M., Gambetta, J., Houck, A.A., Schuster, D.I., Majer, J., Blais, A., Devoret, M.H., Girvin, S.M., Schoelkopf, R.J.: Charge-insensitive qubit design derived from the cooper pair box. Phys. Rev. A 76, 042319 (2007). https://doi.org/10.1103/PhysRevA.76.042319

    Article  ADS  Google Scholar 

  13. Klimov, A.B., Guzmán, R., Retamal, J.C., Saavedra, C.: Qutrit quantum computer with trapped ions. Phys. Rev. A 67, 062313 (2003). https://doi.org/10.1103/PhysRevA.67.062313

    Article  ADS  Google Scholar 

  14. Gao, X., Erhard, M., Zeilinger, A., Krenn, M.: Computer-inspired concept for high-dimensional multipartite quantum gates. Phys. Rev. Lett. 125(5) (2020). https://doi.org/10.1103/physrevlett.125.050501

  15. Barenco, A., Bennett, C.H., Cleve, R., DiVincenzo, D.P., Margolus, N., Shor, P., Sleator, T., Smolin, J.A., Weinfurter, H.: Elementary gates for quantum computation. Phys. Rev. A 52, 3457–3467 (1995). https://doi.org/10.1103/PhysRevA.52.3457

    Article  ADS  Google Scholar 

  16. He, Y., Luo, M.-X., Zhang, E., Wang, H.-K., Wang, X.-F.: Decompositions of n-qubit toffoli gates with linear circuit complexity. Int. J. Theoretical Phys. 56(7), 2350–2361 (2017). https://doi.org/10.1007/s10773-017-3389-4

    Article  ADS  MathSciNet  MATH  Google Scholar 

  17. Gidney, C.: Constructing large controlled nots (2015)

  18. Draper, T.G., Kutin, S.A., Rains, E.M., Svore, K.M.: A logarithmic-depth quantum carry-lookahead adder (2004). arXiv:0406142

  19. Li, A., Stein, S., Krishnamoorthy, S., Ang, J.: QASMBench: A Low-level QASM Benchmark Suite for NISQ Evaluation and Simulation. (2020). arXiv:2005.13018

  20. Takahashi, Y., Tani, S., Kunihiro, N.: Quantum addition circuits and unbounded fan-out (2009). arXiv:0910.2530

  21. Baker, J.M., Duckering, C., Gokhale, P., Brown, N.C., Brown, K.R., Chong, F.T.: Improved quantum circuits via intermediate qutrits. ACM Trans. Quant. Comput. 1(1) (2020). https://doi.org/10.1145/3406309

  22. Fowler, A.G., Mariantoni, M., Martinis, J.M., Cleland, A.N.: Surface codes: Towards practical large-scale quantum computation. Phys. Rev. A 86(3), 032324 (2012)

    Article  ADS  Google Scholar 

  23. IBM Quantum. https://quantum-computing.ibm.com/ (2021)

  24. Fischer, L.E., Miller, D., Tacchino, F., Barkoutsos, P.K., Egger, D.J., Tavernelli, I.: Ancilla-free implementation of generalized measurements for qubits embedded in a qudit space. (2022). arXiv:2203.07369

  25. Chatterjee, T., Das, A., Bala, S.K., Saha, A., Chattopadhyay, A., Chakrabarti, A.: QuDiet: A Classical Simulation Platform for Qubit-Qudit Hybrid Quantum Systems. (2022). arXiv:2211.07918

  26. Majumdar, R., Madan, D., Bhoumik, D., Vinayagamurthy, D., Raghunathan, S., Sur-Kolay, S.: Optimizing ansatz design in qaoa for max-cut. arXiv:2106.02812 (2021)

  27. Saha, A., Majumdar, R., Saha, D., Chakrabarti, A., Sur-Kolay, S.: Asymptotically improved circuit for a \(d\)-ary grover’s algorithm with advanced decomposition of the \(n\)-qudit toffoli gate. Phys. Rev. A 105, 062453 (2022). https://doi.org/10.1103/PhysRevA.105.062453

    Article  ADS  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. A. K. Choudhury School of Information Technology, University of Calcutta, 700106, Kolkata, West Bengal, India

    Amit Saha & Amlan Chakrabarti

  2. Atos, 411057, Pune, Maharashtra, India

    Amit Saha

  3. School of Computer Science and Engineering, Nanyang Technological University, Singapore, Singapore

    Anupam Chattopadhyay

Authors
  1. Amit Saha
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Anupam Chattopadhyay
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Amlan Chakrabarti
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Amit Saha.

Ethics declarations

Conflicts of interest

The authors declare that no funds, grants, or other support were received during the preparation of this manuscript. The authors have no relevant financial or non-financial interests to disclose. All authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by Amit Saha, Anupam Chattopadhyay and Amlan Chakrabarti. The first draft of the manuscript was written by Amit Saha and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript. Our manuscript has no associated data.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Saha, A., Chattopadhyay, A. & Chakrabarti, A. Robust Quantum Arithmetic Operations with Intermediate Qutrits in the NISQ-era. Int J Theor Phys 62, 92 (2023). https://doi.org/10.1007/s10773-023-05339-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10773-023-05339-3

Keywords

Search

Navigation