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Algebraic and Logical Emulations of Quantum Circuits

  • Kenneth Regan
  • Amlan Chakrabarti
  • Chaowen Guan
Chapter
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10730)

Abstract

Quantum circuits exhibit several features of large-scale distributed systems. They have a concise design formalism but behavior that is challenging to represent let alone predict. Issues of scalability—both in the yet-to-be-engineered quantum hardware and in classical simulators—are paramount. They require sparse representations for efficient modeling. Whereas simulators represent both the system’s current state and its operations directly, emulators manipulate the images of system states under a mapping to a different formalism. We describe three such formalisms for quantum circuits. The first two extend the polynomial construction of Dawson et al. [1] to (i) work for any set of quantum gates obeying a certain “balance” condition and (ii) produce a single polynomial over any sufficiently structured field or ring. The third appears novel and employs only simple Boolean formulas, optionally limited to a form we call “parity-of-AND” equations. Especially the third can combine with off-the-shelf state-of-the-art third-party software, namely model counters and \(\mathrm {\#SAT}\) solvers, that we show capable of vast improvements in the emulation time in natural instances. We have programmed all three constructions to proof-of-concept level and report some preliminary tests and applications. These include algebraic analysis of special quantum circuits and the possibility of a new classical attack on the factoring problem. Preliminary comparisons are made with the libquantum simulator [2, 3, 4].

Notes

Acknowledgments

Most of the initial work on this paper was done while the first author was a sabbatical visitor to the Universitié de Montreal, partly supported by the UdeM Département d’informatique et de recherche opérationnelle, and by the University at Buffalo Computer Science Department. We thank especially Professors Pierre McKenzie, Alain Tapp, and Jin-Yi Cai for insightful discussions, and Igor Markov for further pointers to the literature and a press-time tip that libquantum could be modified to output the entire quantum circuits of thousands of gates for Shor’s algorithm in a format readable by our emulator. We thank the referees and also Michael Nielsen, John Sidles, Wim van Dam, Alex Russell, and Ronald de Wolf for helpful comments.

References

  1. 1.
    Dawson, C., Haselgrove, H., Hines, A., Mortimer, D., Nielsen, M., Osborne, T.: Quantum computing and polynomial equations over the finite field \(Z_2\). Quantum Inf. Comput. 5, 102–112 (2004)MathSciNetMATHGoogle Scholar
  2. 2.
    Butscher, B., Weimer, H.: Simulation eines Quantencomputers (2003). http://www.libquantum.de/files/libquantum.pdf
  3. 3.
    Weimer, H., Müller, M., Lesanovsky, I., Zoller, P., Büchler, H.: A Rydberg quantum simulator. Nature Phys. 6, 382–388 (2010)CrossRefGoogle Scholar
  4. 4.
    Weimer, H., Butscher, B.: libquantum 1.1.1: the C library for quantum computing and quantum simulation (2003–2013 (v. 1.1.1)). http://www.libquantum.de/
  5. 5.
    Wybiral, D., Hwang, J.: Quantum circuit simulator (2012). http://www.davyw.com/quantum/
  6. 6.
    Shor, P.W.: Algorithms for quantum computation: discrete logarithms and factoring. In: Proceedings of the 35th Annual IEEE Symposium on the Foundations of Computer Science, pp. 124–134 (1994)Google Scholar
  7. 7.
    Boneh, D.: Twenty years of attacks on the RSA cryptosystem. Notices Am. Math. Soc. 46, 203–213 (1999)MathSciNetMATHGoogle Scholar
  8. 8.
    Häner, T., Steiger, D., Smelyanskiy, M., Troyer, M.: High performance emulation of quantum circuits. In: Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis, Salt Lake City, Utah. IEEE Press, November 2016. Article 74 in e-volumeGoogle Scholar
  9. 9.
    Greuel, G.M., Pfister, G., Schönemann, H.: Singular version 1.2 user manual. In: Reports on Computer Algebra, vol. 21. Centre for Computer Algebra, University of Kaiserslautern (1998). http://www.singular.uni-kl.de/
  10. 10.
    Greuel, G.M., Pfister, G., Schönemann, H.: Singular 3.0. A Computer Algebra System for Polynomial Computations, Centre for Computer Algebra, University of Kaiserslautern (2005). http://www.singular.uni-kl.de
  11. 11.
    Thurley, M.: sharpSAT – counting models with advanced component caching and implicit BCP. In: Biere, A., Gomes, C.P. (eds.) SAT 2006. LNCS, vol. 4121, pp. 424–429. Springer, Heidelberg (2006).  https://doi.org/10.1007/11814948_38 CrossRefGoogle Scholar
  12. 12.
    Sang, T., Bacchus, F., Beame, P., Kautz, H., Pitassi, T.: Combining component caching and clause learning for effective model counting. In: Seventh International Conference on Theory and Applications of Satisfiability Testing, Vancouver (2004)Google Scholar
  13. 13.
    Sang, T., Beame, P., Kautz, H.: Heuristics for fast exact model counting. In: Eighth International Conference on Theory and Applications of Satisfiability Testing, Edinburgh, Scotland (2005)Google Scholar
  14. 14.
    Sang, T., Beame, P., Kautz, H.: Performing Bayesian inference by weighted model counting. In: Proceedings of the Twentieth National Conference on Artificial Intelligence (AAAI 2005), Pittsburgh, PA (2005)Google Scholar
  15. 15.
    Gerdt, V., Severyanov, V.: A software package to construct polynomial sets over \(Z_2\) for determining the output of quantum computations. Nucl. Instrum. Methods Phys. Res. A 59, 260–264 (2006)CrossRefGoogle Scholar
  16. 16.
    Bacon, D., van Dam, W., Russell, A.: Analyzing algebraic quantum circuits using exponential sums (2008). http://www.cs.ucsb.edu/vandam/LeastAction.pdf
  17. 17.
    Adleman, L., DeMarrais, J., Huang, M.: Quantum computability. SIAM J. Comput. 26, 1524–1540 (1997)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Fortnow, L., Rogers, J.: Complexity limitations on quantum computation. In: Proceedings of the 13th Annual IEEE Conference on Computational Complexity, pp. 202–206 (1998)Google Scholar
  19. 19.
    Barenco, A., Deutsch, D., Ekert, A., Jozsa, R.: Conditional quantum dynamics and logic gates. Phys. Rev. Lett. 74(20), 4083–4086 (1995)CrossRefGoogle Scholar
  20. 20.
    Barenco, A., Bennett, C., Cleve, R., DiVincenzo, D., Margolus, N., Shor, P., Sleator, T., Smolin, J., Weinfurter, H.: Elementary gates for quantum computation. Phys. Rev. A 52(5), 3457–3467 (1995)CrossRefGoogle Scholar
  21. 21.
    Boixo, S., Isakov, S.V., Smelyanskiy, V.N., Babbush, R., Ding, N., Jiang, Z., Bremner, M.J., Martinis, J.M., Neven, H.: Characterizing quantum supremacy in near-term devices (2016). https://arxiv.org/pdf/1608.00263.pdf
  22. 22.
    Häner, T., Steiger, D.: 0.5 petabyte simulation of a 45-qubit quantum circuit (2017). arXiv:1704.01127v1
  23. 23.
    Feynmann, R.: Simulating physics with computers. Int. J. Theor. Phys. 21, 467–488 (1982)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Feynmann, R.: Quantum mechanical computers. Found. Phys. 16, 507–531 (1986)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Deutsch, D.: Quantum theory, the Church-Turing principle, and the universal quantum computer. Proc. Royal Soc. A 400, 97–117 (1985)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Deutsch, D.: Quantum computational networks. Proc. R. Soc. Lond. A 425(1868), 73–90 (1989)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Yamashita, S., Markov, I.: Fast equivalence-checking for quantum circuits. In: Proceedings of the 2010 IEEE/ACM Symposium on Nanoscale Architectures, Anaheim, CA, USA (2010). May 2013 update at https://arxiv.org/pdf/0909.4119.pdf
  28. 28.
    Eggersglüß, S., Wille, R., Drechsler, R.: Improved SAT-based ATPG: more constraints, better compaction. In: Proceedings of the 2013 International Conference on Computer-Aided Design, San José, CA, USA, pp. 85–90 (2013)Google Scholar
  29. 29.
    Schönhage, A., Strassen, V.: Schnelle Multiplikation grosser Zahlen. Comput. Arch. Elektron. Rechnen 7, 281–292 (1971)MathSciNetMATHGoogle Scholar
  30. 30.
    van Meter, R., Itoh, K.: Fast quantum modular exponentiation. Phys. Rev. A 71, 052320 (2005)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Markov, I., Saeedi, M.: Constant-optimized quantum circuits for modular multiplication and exponentiation. Quantum Inf. Comput. 12, 361–394 (2012)MathSciNetMATHGoogle Scholar
  32. 32.
    Pavlidis, A., Gizopoulos, D.: Fast quantum modular exponentiation architecture for shor’s factoring algorithm. Quantum Inf. Comput. 14, 649–682 (2014)MathSciNetGoogle Scholar
  33. 33.
    Cao, Z., Cao, Z., Liu, L.: Remarks on quantum modular exponentiation and some experimental demonstrations of Shor’s algorithm (2014). https://arxiv.org/abs/1408.6252
  34. 34.
    Gottesman, D.: The Heisenberg representation of quantum computers (1998). http://arxiv.org/abs/quant-ph/9807006
  35. 35.
    Aaronson, S., Gottesman, D.: Improved simulation of stabilizer circuits. Phys. Rev. A 70 (2004)Google Scholar
  36. 36.
    Cai, J.-Y., Chen, X., Lipton, R., Lu, P.: On tractable exponential sums. In: Lee, D.-T., Chen, D.Z., Ying, S. (eds.) FAW 2010. LNCS, vol. 6213, pp. 148–159. Springer, Heidelberg (2010).  https://doi.org/10.1007/978-3-642-14553-7_16 CrossRefGoogle Scholar
  37. 37.
    Cai, J.-Y., Chen, X., Lu, P.: Graph homomorphisms with complex values: a dichotomy theorem. SIAM J. Comput. 42, 924–1029 (2013)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Cai, J.Y., Lu, P., Xia, M.: The complexity of complex weighted Boolean #CSP. J. Comp. Syst. Sci. 80, 217–236 (2014)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Jozsa, R.: Invited Talk: embedding classical into quantum computation. In: Calmet, J., Geiselmann, W., Müller-Quade, J. (eds.) Mathematical Methods in Computer Science. LNCS, vol. 5393, pp. 43–49. Springer, Heidelberg (2008).  https://doi.org/10.1007/978-3-540-89994-5_5. arXiv:0812.4511 [quant-ph]CrossRefGoogle Scholar
  40. 40.
    Spec.org, Butscher, B., Weimer, H.: 462.libquantum SPEC CPU2006 benchmark description (2006). https://www.spec.org/cpu2006/Docs/462.libquantum.html
  41. 41.
    Beckman, D., Chari, A., Devabhaktuni, S., Preskill, J.: Efficient networks for quantum factoring. Phys. Rev. A 54, 1034–1063 (1996)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Markov, I., Saeedi, M.: Faster quantum number factoring via circuit synthesis. Phys. Rev. A 87(012310), 1–5 (2013)Google Scholar
  43. 43.
    Beauregard, S.: Circuit for shor’s algorithm using 2n + 3 qubits. Quantum Inf. Comput. 3, 175 (2003)MathSciNetMATHGoogle Scholar
  44. 44.
    Häner, T., Roetteler, M., Svore, K.: Factoring using 2n + 2 qubits with toffoli based modular multiplication. Quantum Inf. Comput. 17, 673–684 (2017)MathSciNetGoogle Scholar
  45. 45.
    Viamontes, G., Rajagopalan, M., Markov, I., Hayes, J.: Gate-level simulation of quantum circuits. In: Proceedings of the ACM/ IEEE Asia and South-Pacific Design Automation Conference (ASPDAC), Kitakyushu, Japan, pp. 295–301, January 2003Google Scholar
  46. 46.
    Viamontes, G., Markov, I., Hayes, J.: Improving gate-level simulation of quantum circuits. Quantum Inf. Process. 2, 347–380 (2003)MathSciNetCrossRefMATHGoogle Scholar
  47. 47.
    Greve, D.: QDD: a quantum computer emulation library (1999–2007). http://thegreves.com/david/QDD/qdd.html
  48. 48.
    Patrzyk, J., Patrzyk, B., Rycerz, K., Bubak, M.: Towards a novel environment for simulation of quantum computing. Comput. Sci. 16, 103–129 (2015)CrossRefGoogle Scholar
  49. 49.
    Lee, Y., Khalil-Hani, M., Marsono, M.: An FPGA-based quantum computing emulation framework based on serial-parallel architecture. J. Reconfigurable Comput. 2016, 18 pages (2016)Google Scholar
  50. 50.
    Barenco, A., Ekert, A., Suominen, K.A., Törmä, P.: Approximate quantum Fourier transform and decoherence. Phys. Rev. A 54, 139–146 (1996)MathSciNetCrossRefGoogle Scholar
  51. 51.
    Zilic, Z., Radecka, K.: Scaling and better approximating quantum fourier transform by higher radices. IEEE Trans. Comp. 56, 202–207 (2007)MathSciNetCrossRefGoogle Scholar
  52. 52.
    Rötteler, M., Beth, T.: Representation-theoretical properties of the approximate quantum Fourier transform. Appl. Algebra Eng. Commun. Comput. 19, 117–193 (2008)MathSciNetCrossRefMATHGoogle Scholar
  53. 53.
    Prokopenya, A.N.: Approximate quantum fourier transform and quantum algorithm for phase estimation. In: Gerdt, V.P., Koepf, W., Seiler, W.M., Vorozhtsov, E.V. (eds.) CASC 2015. LNCS, vol. 9301, pp. 391–405. Springer, Cham (2015).  https://doi.org/10.1007/978-3-319-24021-3_29 CrossRefGoogle Scholar
  54. 54.
    Aaronson, S., Chen, L.: Complexity-theoretic foundations of quantum supremacy experiments (2016). https://arxiv.org/abs/1612.05903

Copyright information

© Springer-Verlag GmbH Germany 2018

Authors and Affiliations

  1. 1.University at Buffalo (SUNY)BuffaloUSA
  2. 2.University of CalcuttaKolkataIndia

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