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Algebraic and Logical Emulations of Quantum Circuits

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Book cover Transactions on Computational Science XXXI

Part of the book series: Lecture Notes in Computer Science ((TCOMPUTATSCIE,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].

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References

  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)

    MathSciNet  MATH  Google Scholar 

  2. Butscher, B., Weimer, H.: Simulation eines Quantencomputers (2003). http://www.libquantum.de/files/libquantum.pdf

  3. Weimer, H., Müller, M., Lesanovsky, I., Zoller, P., Büchler, H.: A Rydberg quantum simulator. Nature Phys. 6, 382–388 (2010)

    Article  Google Scholar 

  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. Wybiral, D., Hwang, J.: Quantum circuit simulator (2012). http://www.davyw.com/quantum/

  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. Boneh, D.: Twenty years of attacks on the RSA cryptosystem. Notices Am. Math. Soc. 46, 203–213 (1999)

    MathSciNet  MATH  Google Scholar 

  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-volume

    Google Scholar 

  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. 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. 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

    Chapter  Google Scholar 

  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. 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. 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. 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)

    Article  Google Scholar 

  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. Adleman, L., DeMarrais, J., Huang, M.: Quantum computability. SIAM J. Comput. 26, 1524–1540 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  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. Barenco, A., Deutsch, D., Ekert, A., Jozsa, R.: Conditional quantum dynamics and logic gates. Phys. Rev. Lett. 74(20), 4083–4086 (1995)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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. Häner, T., Steiger, D.: 0.5 petabyte simulation of a 45-qubit quantum circuit (2017). arXiv:1704.01127v1

  23. Feynmann, R.: Simulating physics with computers. Int. J. Theor. Phys. 21, 467–488 (1982)

    Article  MathSciNet  Google Scholar 

  24. Feynmann, R.: Quantum mechanical computers. Found. Phys. 16, 507–531 (1986)

    Article  MathSciNet  Google Scholar 

  25. Deutsch, D.: Quantum theory, the Church-Turing principle, and the universal quantum computer. Proc. Royal Soc. A 400, 97–117 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  26. Deutsch, D.: Quantum computational networks. Proc. R. Soc. Lond. A 425(1868), 73–90 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  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. 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. Schönhage, A., Strassen, V.: Schnelle Multiplikation grosser Zahlen. Comput. Arch. Elektron. Rechnen 7, 281–292 (1971)

    MathSciNet  MATH  Google Scholar 

  30. van Meter, R., Itoh, K.: Fast quantum modular exponentiation. Phys. Rev. A 71, 052320 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  31. Markov, I., Saeedi, M.: Constant-optimized quantum circuits for modular multiplication and exponentiation. Quantum Inf. Comput. 12, 361–394 (2012)

    MathSciNet  MATH  Google Scholar 

  32. Pavlidis, A., Gizopoulos, D.: Fast quantum modular exponentiation architecture for shor’s factoring algorithm. Quantum Inf. Comput. 14, 649–682 (2014)

    MathSciNet  Google Scholar 

  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. Gottesman, D.: The Heisenberg representation of quantum computers (1998). http://arxiv.org/abs/quant-ph/9807006

  35. Aaronson, S., Gottesman, D.: Improved simulation of stabilizer circuits. Phys. Rev. A 70 (2004)

    Google Scholar 

  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

    Chapter  Google Scholar 

  37. Cai, J.-Y., Chen, X., Lu, P.: Graph homomorphisms with complex values: a dichotomy theorem. SIAM J. Comput. 42, 924–1029 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  38. Cai, J.Y., Lu, P., Xia, M.: The complexity of complex weighted Boolean #CSP. J. Comp. Syst. Sci. 80, 217–236 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  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]

    Chapter  Google Scholar 

  40. Spec.org, Butscher, B., Weimer, H.: 462.libquantum SPEC CPU2006 benchmark description (2006). https://www.spec.org/cpu2006/Docs/462.libquantum.html

  41. Beckman, D., Chari, A., Devabhaktuni, S., Preskill, J.: Efficient networks for quantum factoring. Phys. Rev. A 54, 1034–1063 (1996)

    Article  MathSciNet  Google Scholar 

  42. Markov, I., Saeedi, M.: Faster quantum number factoring via circuit synthesis. Phys. Rev. A 87(012310), 1–5 (2013)

    Google Scholar 

  43. Beauregard, S.: Circuit for shor’s algorithm using 2n + 3 qubits. Quantum Inf. Comput. 3, 175 (2003)

    MathSciNet  MATH  Google Scholar 

  44. Häner, T., Roetteler, M., Svore, K.: Factoring using 2n + 2 qubits with toffoli based modular multiplication. Quantum Inf. Comput. 17, 673–684 (2017)

    MathSciNet  Google Scholar 

  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 2003

    Google Scholar 

  46. Viamontes, G., Markov, I., Hayes, J.: Improving gate-level simulation of quantum circuits. Quantum Inf. Process. 2, 347–380 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  47. Greve, D.: QDD: a quantum computer emulation library (1999–2007). http://thegreves.com/david/QDD/qdd.html

  48. Patrzyk, J., Patrzyk, B., Rycerz, K., Bubak, M.: Towards a novel environment for simulation of quantum computing. Comput. Sci. 16, 103–129 (2015)

    Article  Google Scholar 

  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. Barenco, A., Ekert, A., Suominen, K.A., Törmä, P.: Approximate quantum Fourier transform and decoherence. Phys. Rev. A 54, 139–146 (1996)

    Article  MathSciNet  Google Scholar 

  51. Zilic, Z., Radecka, K.: Scaling and better approximating quantum fourier transform by higher radices. IEEE Trans. Comp. 56, 202–207 (2007)

    Article  MathSciNet  Google Scholar 

  52. Rötteler, M., Beth, T.: Representation-theoretical properties of the approximate quantum Fourier transform. Appl. Algebra Eng. Commun. Comput. 19, 117–193 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  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

    Chapter  Google Scholar 

  54. Aaronson, S., Chen, L.: Complexity-theoretic foundations of quantum supremacy experiments (2016). https://arxiv.org/abs/1612.05903

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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.

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Authors and Affiliations

  1. University at Buffalo (SUNY), Buffalo, USA

    Kenneth Regan & Chaowen Guan

  2. University of Calcutta, Kolkata, India

    Amlan Chakrabarti

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  1. Kenneth Regan
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Correspondence to Kenneth Regan .

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Editors and Affiliations

  1. University of Calgary, Calgary, Alberta, Canada

    Marina L. Gavrilova

  2. Sardina Systems OÜ, Tallinn, Estonia

    C.J. Kenneth Tan

  3. University of Calcutta, Calcutta, India

    Nabendu Chaki

  4. Bialystok University of Technology, Bialystok, Poland

    Khalid Saeed

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Regan, K., Chakrabarti, A., Guan, C. (2018). Algebraic and Logical Emulations of Quantum Circuits. In: Gavrilova, M., Tan, C., Chaki, N., Saeed, K. (eds) Transactions on Computational Science XXXI. Lecture Notes in Computer Science(), vol 10730. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-56499-8_4

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  • DOI: https://doi.org/10.1007/978-3-662-56499-8_4

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