Improved quantum ripple-carry addition circuit


A serious obstacle to large-scale quantum algorithms is the large number of elementary gates, such as the controlled-NOT gate or Toffoli gate. Herein, we present an improved linear-depth ripple-carry quantum addition circuit, which is an elementary circuit used for quantum computations. Compared with previous addition circuits costing at least two Toffoli gates for each bit of output, the proposed adder uses only a single Toffoli gate. Moreover, our circuit may be used to construct reversible circuits for modular multiplication, C x mod M with x < M, arising as components of Shor’s algorithm. Our modular-multiplication circuits are simpler than previous constructions, and may be used as primitive circuits for quantum computations.

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  1. 1

    Nielsen M, Chuang I. Quantum Computation and Quantum Information. Cambridge: Cambridge University Press, 2000. 150–280

    Google Scholar 

  2. 2

    Zhou C, Bao W S, Fu X Q. Decoy-state quantum key distribution for the heralded pair coherent state photon source with intensity fluctuations. Sci China Inf Sci, 2011, 41: 1136–1145

    MathSciNet  MATH  Google Scholar 

  3. 3

    Wu H, Wang X B, Pan J W. Quantum communication, status and prospects (in Chinese). Sci China Inf Sci, 2014, 44: 296–311

    Google Scholar 

  4. 4

    Luo M X, Ma S Y, Chen X B, et al. Hybrid quantum states joining and splitting assisted by quantum dots in one-side optical microcavities. Phys Rev A, 2015, 91: 042326

    Article  Google Scholar 

  5. 5

    Shor P W. Algorithms for quantum computation: discrete logarithms and factoring. In: Proceedings of the 35th Annual IEEE Symposium on Foundations of Computer Science (FOCS), Santa Fe, 1994. 124–134

    Google Scholar 

  6. 6

    Shor P W. Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM J Comput, 1997, 26: 1484–1509

    MathSciNet  Article  MATH  Google Scholar 

  7. 7

    Rivest R L, Shamir A, Adleman L. A method for obtaining digital signatures and public key cryptography. Comm ACM, 1978, 21: 120–126

    MathSciNet  Article  MATH  Google Scholar 

  8. 8

    Guo P, Wang J, Geng X H, et al. A variable threshold-value authentication architecture for wireless mesh networks. J Internet Tech, 2014, 15: 929–936

    Google Scholar 

  9. 9

    Fu Z, Sun X, Liu Q, et al. Achieving efficient cloud search services: multi-keyword ranked search over encrypted cloud data supporting parallel computing. IEICE Trans Commun, 2015, 98: 190–200

    Article  Google Scholar 

  10. 10

    Ren Y, Shen J, Wang J, et al. Mutual verifiable provable data auditing in public cloud storage. J Internet Tech, 2015, 16: 317–324

    Google Scholar 

  11. 11

    Xia Z, Wang X, Sun X, et al. A secure and dynamic multi-keyword ranked search scheme over encrypted cloud data. IEEE Trans Parall Distr Syst, in press. doi:10.1109/TPDS.2015.2401003

  12. 12

    Li J, Li X, Yang B, et al. Segmentation-based image copy-move forgery detection scheme. IEEE Trans Inf Foren Secur, 2015, 10: 507–518

    Article  Google Scholar 

  13. 13

    Feynman R P. Simulating physics computers. Inter J Theor Phys, 1982, 21: 476–487

    MathSciNet  Article  Google Scholar 

  14. 14

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

    MathSciNet  Article  MATH  Google Scholar 

  15. 15

    van Dam W, Hallgren S I L. Quantum algorithms for some hidden shift problems. SIAM J Comput, 2006, 36: 763–778

    MathSciNet  Article  MATH  Google Scholar 

  16. 16

    Luo M X, Deng Y. The independence of reduced subgroup-state. Inter J Theor Phys, 2014, 53: 3124–3134

    Article  MATH  Google Scholar 

  17. 17

    Luo M X, Chen X B, Yang Y X, et al. Geometry of quantum computation with qudits. Sci Rep, 2014, 4: 4044

    Google Scholar 

  18. 18

    Luo M X, Wang X. Parallel photonic quantum computation assisted by quantum dots in one-side optical microcavities. Sci Rep, 2014, 4: 4732

    Google Scholar 

  19. 19

    Martin-Lopez E, Laing A, Lawson T, et al. Experimental realization of Shor’s quantum factoring algorithm using qubit recycling. Nature Phot, 2012, 6: 773–776

    Article  Google Scholar 

  20. 20

    Lucero E, Barends R, Chen Y, et al. Computing prime factors with a Josephson phase qubit quantum processor. Nature Phys, 2012, 8: 719–723

    Article  Google Scholar 

  21. 21

    Hao L, Long G L. Experimental implementation of a fixed-point duality quantum search algorithm in the nuclear magnetic resonance quantum system. Sci China Phys Mech Astronomy, 2011, 54: 936–941

    Article  Google Scholar 

  22. 22

    Shende V, Bullock S S, Markov I L. Synthesis of quantum-logic circuits. IEEE Tran Comput AID Design, 2006, 26: 1000–1010

    Article  Google Scholar 

  23. 23

    Beauregard S. Circuit for Shor’s algorithm using 2n + 3 qubits. Quantum Inform Comput, 2003, 3: 175–185

    MathSciNet  MATH  Google Scholar 

  24. 24

    Proos J, Zalka C. Shor’s discrete logarithm quantum algorithm for elliptic curves. Quantum Inform Comput, 2003, 3: 317–344

    MathSciNet  MATH  Google Scholar 

  25. 25

    Fowler A G, Devitt S J, Hollenberg L C L. Implementation of Shor’s algorithm on a linear nearest neighbour qubit array. Quantum Inform Comput, 2004, 4: 237–251

    MathSciNet  MATH  Google Scholar 

  26. 26

    Martí-López E, Laing A, Lawson T, et al. Experimental realization of Shor’s quantum factoring algorithm using qubit recycling. Nature Photon, 2012, 6: 773–776

    Article  Google Scholar 

  27. 27

    Takahashi Y, Kunihiro N. A linear-size quantum circuit for addition with no ancillary qubits. Quantum Inform Comput, 2005, 5: 440–448

    MathSciNet  MATH  Google Scholar 

  28. 28

    Takahashi Y, Kunihiro N. A fast quantum circuit for addition with few qubits. Quantum Inform Comput, 2008, 8: 636–649

    MathSciNet  MATH  Google Scholar 

  29. 29

    Draper T G, Kutin S A, Rains E M, et al. A logarithmic-depth quantum carry-lookahead adder. Quantum Inform Comput, 2006, 6: 351–369

    MathSciNet  MATH  Google Scholar 

  30. 30

    Takahashi Y, Tani S, Kunihiro N. Quantum addition circuits and unbounded fan-out. Quantum Inform Comput, 2010, 10: 0872–0890

    MathSciNet  MATH  Google Scholar 

  31. 31

    Cuccaro S A, Draper T G, Kutin S A, et al. A new quantum ripple-carry addition circuit, In: 8th Workshop on Quantum Information Processing, Cambridge, 2005. 1–9

    Google Scholar 

  32. 32

    Thomsen M K, Axelsen H B. Optimization of a reversible (Quantum) ripple-carry adder. LNCS, 2008, 5204: 228–241

    MathSciNet  MATH  Google Scholar 

  33. 33

    Markov I L, Saeedi M. Constant-optimized quantum circuits for modular multiplication and exponentiation. Quantum Infor Comput, 2012, 12: 0361–0394

    MathSciNet  MATH  Google Scholar 

  34. 34

    Yu N K, Duan R Y, Ying M S. Five two-qubit gates are necessary for implementing the Toffoli gate. Phys Rev A, 2013, 88: 010304(R)

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Correspondence to Zhiguo Qu.

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Wang, F., Luo, M., Li, H. et al. Improved quantum ripple-carry addition circuit. Sci. China Inf. Sci. 59, 042406 (2016).

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  • quantum addition circuit
  • modular multiplication
  • CNOT gate
  • Toffoli gate
  • circuit complexity