Skip to main content

Quantum pattern recognition with multi-neuron interactions

Abstract

We present a quantum neural network with multi-neuron interactions for pattern recognition tasks by a combination of extended classic Hopfield network and adiabatic quantum computation. This scheme can be used as an associative memory to retrieve partial patterns with any number of unknown bits. Also, we propose a preprocessing approach to classifying the pattern space S to suppress spurious patterns. The results of pattern clustering show that for pattern association, the number of weights (\(\eta \)) should equal the numbers of unknown bits in the input pattern (d). It is also remarkable that associative memory function depends on the location of unknown bits apart from the d and load parameter \(\alpha \).

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3

Notes

  1. 1.

    This formula is obtained from permutation of pairwise interacted neuron in each model, for \(N>3\) and \(d>2\).

References

  1. 1.

    Yegnanarayana, B.: Artificial Neural Networks, pp. 77–87. PHI Learning Pvt. Ltd, New Delhi (2009)

    Google Scholar 

  2. 2.

    Hopfield, J.J.: Neural networks and physical systems with emergent collective computational abilities. Proc. Natl. Acad. Sci. 79(8), 2554–2558 (1982)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    Hagan, M.T., Demuth, H.B., Beale, M.H., Jesus, O.D.: Neural Network Design, 2nd edn, Martin Hagan, chap. 21, p. 16 (2014)

  4. 4.

    Bruck, J., Roychowdhury, V.P.: On the number of spurious memories in the Hopfield model (neural network). IEEE Trans. Inf. Theory 36(2), 393–397 (1990)

    Article  MATH  Google Scholar 

  5. 5.

    Hopfield, J.J., Feinstein, D.I., Palmer, R.G.: “Unlearning” has a stabilizing effect in collective memories. Nature 304, 158–159 (1983)

    ADS  Article  Google Scholar 

  6. 6.

    Amit, D.J., Gutfreund, H., Sompolinsky, H.: Spin-glass models of neural networks. Phys. Rev. A 32(2), 1007 (1985)

    ADS  MathSciNet  Article  Google Scholar 

  7. 7.

    Amit, D.J., Gutfreund, H., Sompolinsky, H.: Storing infinite numbers of patterns in a spin-glass model of neural networks. Phys. Rev. Lett. 55(14), 1530 (1985)

    ADS  Article  Google Scholar 

  8. 8.

    Amit, D.J., Gutfreund, H., Sompolinsky, H.: Information storage in neural networks with low levels of activity. Phys. Rev. A 35(5), 2293 (1987)

    ADS  MathSciNet  Article  Google Scholar 

  9. 9.

    Morita, M.: Associative memory with nonmonotone dynamics. Neural Netw. 6(1), 115–126 (1993)

    Article  Google Scholar 

  10. 10.

    Nishikawa, T., Lai, Y.C., Hoppensteadt, F.C.: Capacity of oscillatory associative-memory networks with error-free retrieval. Phys. Rev. Lett. 92(10), 108101 (2004)

    ADS  Article  Google Scholar 

  11. 11.

    Amari, S.I., Maginu, K.: Statistical neurodynamics of associative memory. Neural Netw. 1(1), 63–73 (1988)

    Article  Google Scholar 

  12. 12.

    Okada, M.: A hierarchy of macrodynamical equations for associative memory. Neural Netw. 8(6), 833–838 (1995)

    Article  Google Scholar 

  13. 13.

    Coolen, A.C.C., Sherrington, D.: Order-parameter flow in the fully connected Hopfield model near saturation. Phys. Rev. E 49(3), 1921 (1994)

    ADS  Article  Google Scholar 

  14. 14.

    Coolen, A.C.C., Laughton, S.N., Sherrington, D.: Dynamical replica theory for disordered spin systems. Phys. Rev. B 53(13), 8184 (1996)

    ADS  Article  Google Scholar 

  15. 15.

    Ma, Y.Q.: Statics in the random quantum asymmetric Sherrington–Kirkpatrick model. Phys. Rev. B 45(2), 793 (1992)

    ADS  Article  Google Scholar 

  16. 16.

    Xi, Q., Ma, Y.Q.: Quantum Hopfield model with a random transverse field and a random neuronal threshold. Phys. Lett. A 254(6), 355–360 (1999). Vancouver

    ADS  Article  Google Scholar 

  17. 17.

    Nishimori, H., Nonomura, Y.: Quantum effects in neural networks. J. Phys. Soc. Jpn. 65(12), 3780–3796 (1996)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  18. 18.

    Inoue, J.I.: Pattern-recalling processes in quantum Hopfield networks far from saturation. J. Phys. Conf. Ser. 297(1), 012012 (2011)

    Article  Google Scholar 

  19. 19.

    Suzuki, M.: Relationship between d-dimensional quantal spin systems and (\(d+1\))-dimensional Ising systems: equivalence, critical exponents and systematic approximants of the partition function and spin correlations. Progress of theoretical physics 56(5), 1454–1469 (1976)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  20. 20.

    Farhi, E., Goldstone, J., Gutmann, S., Lapan, J., Lundgren, A., Preda, D.: A quantum adiabatic evolution algorithm applied to random instances of an NP-complete problem. Science 292(5516), 472–475 (2001)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  21. 21.

    Santra, S., Quiroz, G., Ver Steeg, G., Lidar, D.A.: MAX 2-SAT with up to 108 qubits. New J. Phys. 16(4), 045006 (2014)

    ADS  Article  Google Scholar 

  22. 22.

    Aharonov, D., Van Dam, W., Kempe, J., Landau, Z., Lloyd, S., Regev, O.: Adiabatic quantum computation is equivalent to standard quantum computation. In: Proceedings of the 45th Annual Symposium on the Foundations of Computer Science, 2004, Rome, Italy IEEE Computer Society Press, New York, p. 4251 (2004)

  23. 23.

    Das, S., Kobes, R., Kunstatter, G.: Adiabatic quantum computation and Deutschs algorithm. Phys. Rev. A 65(6), 062310 (2002)

    ADS  MathSciNet  Article  Google Scholar 

  24. 24.

    Wei, Z., Ying, M.: A modified quantum adiabatic evolution for the Deutsch–Jozsa problem. Phys. Lett. A 354(4), 271–273 (2006)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  25. 25.

    Roland, J., Cerf, N.J.: Quantum search by local adiabatic evolution. Phys. Rev. A 65(4), 042308 (2002)

    ADS  Article  Google Scholar 

  26. 26.

    Hen, I.: Period finding with adiabatic quantum computation. EPL Europhys. Lett. 105(5), 50005 (2014)

    ADS  Article  Google Scholar 

  27. 27.

    Santra, S., Shehab, O., Balu, R.: Exponential capacity of associative memories under quantum annealing recall. arXiv preprint arXiv:1602.08149 (2016)

  28. 28.

    Seddiqi, H., Humble, T.S.: Adiabatic quantum optimization for associative memory recall. arXiv preprint arXiv:1407.1904 (2014)

  29. 29.

    Seki, Y., Nishimori, H.: Quantum annealing with antiferromagnetic transverse interactions for the Hopfield model. J. Phys. A Math. Theor. 48(33), 335301 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  30. 30.

    Neigovzen, R., Neves, J.L., Sollacher, R., Glaser, S.J.: Quantum pattern recognition with liquid-state nuclear magnetic resonance. Phys. Rev. A 79(4), 042321 (2009)

    ADS  Article  Google Scholar 

  31. 31.

    Fard, E.R., Aghayar, K.: Quantum adiabatic evolution for pattern recognition problem. Chin. Phys. Lett. 34(12), 120302 (2017)

    Article  Google Scholar 

  32. 32.

    Wittek, P.: Quantum Machine Learning: What Quantum Computing Means to Data Mining, pp. 116–117. Academic Press, New York (2014)

    MATH  Google Scholar 

  33. 33.

    Gardner, E.: Multiconnected neural network models. J. Phys. A Math. Gen. 20(11), 3453 (1987)

    ADS  MathSciNet  Article  Google Scholar 

  34. 34.

    Hebb, D.O.: The Organization of Behavior: A Neuropsychological Theory. Psychology Press, London (2005)

    Google Scholar 

  35. 35.

    McEliece, R., Posner, E., Rodemich, E., Venkatesh, S.: The capacity of the Hopfield associative memory. IEEE Trans. Inf. Theory 33(4), 461–482 (1987)

    MathSciNet  Article  MATH  Google Scholar 

  36. 36.

    Gardner, E.: Spin glasses with p-spin interactions. Nucl. Phys. B 257, 747–765 (1985)

    ADS  MathSciNet  Article  Google Scholar 

  37. 37.

    Gross, D.J., Mzard, M.: The simplest spin glass. Nucl. Phys. B 240(4), 431–452 (1984)

    ADS  MathSciNet  Article  Google Scholar 

  38. 38.

    Farhi, E., Goldstone, J., Gutmann, S., Sipser, M.: Quantum computation by adiabatic evolution. arXiv preprint arXiv:quant-ph/0001106 (2000)

  39. 39.

    Born, M., Fock, V.: Beweis des adiabatensatzes. Z. Phys. Hadrons Nucl. 51(3), 165–180 (1928)

    MATH  Google Scholar 

  40. 40.

    Zhang, D.J., Yu, X.D., Tong, D.M.: Theorem on the existence of a nonzero energy gap in adiabatic quantum computation. Phys. Rev. A 90(4), 042321 (2014)

    ADS  Article  Google Scholar 

  41. 41.

    Albash, T., Lidar, D.A.: Adiabatic quantum computing. arXiv preprint arXiv:1611.04471 (2016)

  42. 42.

    McMahon, D.: Quantum Computing Explained, John Wiley & Sons, pp. 305–313 (2007)

  43. 43.

    Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information, 1st edn. Cambridge University Press, Cambridge (2002)

  44. 44.

    Choi, V.: Minor-embedding in adiabatic quantum computation: II. Minor-universal graph design. Quantum Inf. Process. 10(3), 343–353 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  45. 45.

    Hen, I., Young, A.P.: Exponential complexity of the quantum adiabatic algorithm for certain satisfiability problems. Phys. Rev. E 84, 061152 (2011)

    ADS  Article  Google Scholar 

  46. 46.

    Childs, A.M., Farhi, E., Preskill, J.: Robustness of adiabatic quantum computation. Phys. Rev. A 65(1), 012322 (2001)

    ADS  Article  Google Scholar 

  47. 47.

    Roland, J., Cerf, N.J.: Noise resistance of adiabatic quantum computation using random matrix theory. Phys. Rev. A 71(3), 032330 (2005)

    ADS  MathSciNet  Article  MATH  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to K. Aghayar.

Adiabatic condition of QNN

Adiabatic condition of QNN

The behavior of the QNN depends on runtime AQC. Here, we calculate numerically runtime of the algorithm with high probability in two broad case. Previously, the runtime of AQC for the two-qubit neural network [30] has been calculated analytically for recalling pattern process and pattern association separately in more details [31].

Runtime of AQC for pattern recalling process

Let us consider Hamiltonian (19) belong to Sect. 4.1 for recalling patterns from the memory of QNN. To do this work, the time-dependent Hamiltonian (8) by defining a dimensionless quantities \(s=\frac{t}{T}\) can be written as

$$\begin{aligned} H(t)=(1-s)\frac{1}{2}\sum _{i=1}^4(I-\sigma _i^x)\,+\,s\left( -J(\sigma ^z\otimes \sigma ^z\otimes \sigma ^z\otimes \sigma ^z)+\Gamma \sum _{i=1}^4 \xi _{i}^{\mathrm{inp}} \sigma _{i}^z\right) . \end{aligned}$$
(23)

As an instance, suppose the patterns into the first cluster (\(S_1\)) with \(J=1\) load in the memory and the input pattern (\(\xi ^{\mathrm{inp}}\)) and \(\Gamma \) are \((1,1,1,1)^t\) and 1 / 2 respectively so as H(t) would be

$$\begin{aligned} H(t)=(1-s)\frac{1}{2}\sum _{i=1}^4(I-\sigma _i^x)+s\left( -(\sigma ^z\otimes \sigma ^z\otimes \sigma ^z\otimes \sigma ^z)+\frac{1}{2}(\sigma _{1}^z+\sigma _{2}^z+\sigma _{3}^z+\sigma _{4}^z)\right) . \end{aligned}$$
(24)
Fig. 4
figure4

Eigenvalues of the time-dependent Hamiltonian H(s) as a function of the reduced time s for \(N=4\) (With the convention \(\hbar =1\), the energy as well as the reduced time are dimensionless quantities)

The eigenvalues and eigenstates of H(t) may be achieved analytically, but here we only focus on calculation of \(g_{\min }\) and \(D_{\max }\) numerically. Therefore, we draw all eigenvalues of H(t) in Fig. 4 and find two lowest eigenvalues of H(t). Then, we define two quantity \(g(s)=E_1(s)-E_0(s)\) and \(D(s)=\langle E_1|\frac{\mathrm{d}H(s)}{\mathrm{d}t}|E_0\rangle =\frac{1}{T}\langle E_1|\frac{\mathrm{d}H(s)}{\mathrm{d}s}|E_0\rangle \) to calculate adiabatic condition (11). The minimum of g(s) occur at \(s=0.381607\)

$$\begin{aligned} g_{\min }=0.593428, \end{aligned}$$
(25)

and the maximum value of \(\big |\big \langle \frac{\mathrm{d}H(s)}{\mathrm{d}s}\big \rangle \big |\) is attained as follows

$$\begin{aligned} \max _{0<t<T}\big |\big \langle \frac{\mathrm{d}H(s)}{\mathrm{d}s}\big \rangle \big |_{s=0.392477}=1.82461. \end{aligned}$$
(26)

Now we can determine lower bound of runtime for AQC form Eq. (11)

$$\begin{aligned} T\ge \frac{1.82461}{(0.593428)^2}\frac{1}{\varepsilon }, \end{aligned}$$
(27)

which gives an estimate of the time to evolve \(|\psi _0\rangle \) via adiabatic Hamiltonian (8) to attain an accuracy of order \(\varepsilon \) of the final result. As an example, for accuracy of \( \%90\), the lower bound of runtime should be on the order of

$$\begin{aligned} T_{\mathrm{low}}\approx \frac{1.82461}{(0.593428)^2}\frac{1}{\sqrt{1 - 0.9}}\approx 16.3845 \end{aligned}$$
(28)

in units of \(\bar{T}\). Although this runtime is obtained for the input pattern \((1,1,1,1)^t\), for each pattern in its own subspace is valid.

The \(\Gamma \) coefficient in the classic neural network is called learning rate and interpreted as the association between the stimulus (input) and response (output). By increasing the association between the stimulus and response, the network learns faster than before. The result of minimum evolution time for different Hamiltonian shows that similar to classic network, for the same percent of accuracy of the final state (%90), by decreasing the \(\Gamma \) (decreasing the association) in Eq. (8) \(T_{\mathrm{low}}\) will increase gradually (see Table 7).

Table 7 Influence of decreasing the coefficient \(\Gamma \) on \(T_{\mathrm{low}}\) when input patterns are partially known

The runtime of evolution to recall pattern in Sects. 4.2 and 4.3 is calculated numerically and their result presented in Table 8 for \(\%90\) percent of accuracy.

Table 8 The order of minimum evolution time (\(T_{\mathrm{low}}\)) for accuracy of \(\%90\) to recall pattern in three different cases in Sects. 4.1, 4.2 and 4.3

Runtime of AQC for partial patterns

In this case, due to the presence of unknown bits in the input pattern, the runtime of AQC is different from previous case. Here, we must repeat all calculation and finally obtain the quantities \(g_{\min }\) and \(D_{\max }\). The numerical results show that the Hamiltonians in Sects. 4.1 and 4.2 for all partial patterns have a specific minimum runtime (see Table 9). But as already outlined in Sect. 4.3, the QNN converges to a superposition state for partial patterns of the fourth model in Fig. 1b. The minimum energy for these patterns is zero, and thus, the quantum system cannot evolve adiabatically (see Fig. 5). Although for other models in Fig. 1b, \(g_{min}\) is nonzero and the quantum system adiabatically evolve which minimum runtime (\(T_{\mathrm{low}}\)) for all partial patterns is 21.769 in units of \(\bar{T}\) for accuracy of \(\%90\) and \(\Gamma =1/2\).

Table 9 The order of minimum evolution time (\(T_{\mathrm{low}}\)) for accuracy of \(\%90\) to reterive partial pattern in Sects. 4.1, 4.2
Fig. 5
figure5

The energy gap between two lowest eigenvalues is not suitable for adiabatic evolution. Therefore, system cannot converge to retrieved pattern which is obtained in theoretical result as a superposition state

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Fard, E.R., Aghayar, K. & Amniat-Talab, M. Quantum pattern recognition with multi-neuron interactions. Quantum Inf Process 17, 42 (2018). https://doi.org/10.1007/s11128-018-1816-y

Download citation

Keywords

  • Quantum neural network
  • Pattern recognition
  • Pattern association
  • Pattern clustering
  • Multi-neuron interaction