Deterministic Preparation of Dicke States

  • Andreas BärtschiEmail author
  • Stephan Eidenbenz
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11651)


The Dicke state Open image in new window is an equal-weight superposition of all n-qubit states with Hamming Weight k (i.e. all strings of length n with exactly k ones over a binary alphabet). Dicke states are an important class of entangled quantum states that among other things serve as starting states for combinatorial optimization quantum algorithms.

We present a deterministic quantum algorithm for the preparation of Dicke states. Implemented as a quantum circuit, our scheme uses \(\mathcal {O}(kn)\) gates, has depth \(\mathcal {O}(n)\) and needs no ancilla qubits. The inductive nature of our approach allows for linear-depth preparation of arbitrary symmetric pure states and – used in reverse – yields a quasilinear-depth circuit for efficient compression of quantum information in the form of symmetric pure states, improving on existing work requiring quadratic depth. All of these properties even hold for Linear Nearest Neighbor architectures.



We thank Yiğit Subaşı for helpful discussions.


  1. 1.
    Bacon, D., Chuang, I.L., Harrow, A.W.: Efficient quantum circuits for Schur and Clebsch-Gordan transforms. Phys. Rev. Lett. 97(17), 170502 (2006). Scholar
  2. 2.
    Barenco, A., et al.: Elementary gates for quantum computation. Phys. Rev. A 52(5), 3457–3467 (1995). Scholar
  3. 3.
    Bärtschi, A., Eidenbenz, S.: Deterministic Preparation of Dicke States. arXiv e-prints, April 2019.
  4. 4.
    Bastin, T., Thiel, C., von Zanthier, J., Lamata, L., Solano, E., Agarwal, G.S.: Operational determination of multiqubit entanglement classes via tuning of local operations. Phys. Rev. Lett. 102(5), 053601 (2009). Scholar
  5. 5.
    Chakraborty, K., Choi, B.S., Maitra, A., Maitra, S.: Efficient quantum algorithms to construct arbitrary dicke states. Quantum Inf. Process. 13(9), 2049–2069 (2014). Scholar
  6. 6.
    Childs, A.M., Farhi, E., Goldstone, J., Gutmann, S.: Finding cliques by quantum adiabatic evolution. Quantum Inf. Comput. 2(3), 181–191 (2002). Scholar
  7. 7.
    Chuang, I.L., Modha, D.S.: Reversible arithmetic coding for quantum data compression. IEEE Trans. Inf. Theory 46(3), 1104–1116 (2000). Scholar
  8. 8.
    Dicke, R.H.: Coherence in spontaneous radiation processes. Phys. Rev. 93(1), 99–110 (1954). Scholar
  9. 9.
    Diker, F.: Deterministic construction of arbitrary \(W\) states with quadratically increasing number of two-qubit gates. arXiv e-prints, June 2016. arXiv:1606.09290
  10. 10.
    Dür, W., Vidal, G., Cirac, J.I.: Three qubits can be entangled in two inequivalent ways. Phys. Rev. A 62(6), 062314 (2000). Scholar
  11. 11.
    Farhi, E., Goldstone, J., Gutmann, S.: A Quantum Approximate Optimization Algorithm. arXiv e-prints, November 2014. arXiv:1411.4028
  12. 12.
    Gidney, C.: Quirk: Quantum Circuit Simulator. A drag-and-drop quantum circuit simulator.
  13. 13.
    Gidney, C.: Constructing large controlled nots/Constructing large increment gates/Using quantum gates instead of ancilla bits, June 2015.
  14. 14.
    Hadfield, S., Wang, Z., O’Gorman, B., Rieffel, E.G., Venturelli, D., Biswas, R.: From the quantum approximate optimization algorithm to a quantum alternating operator ansatz. Algorithms 12(2), 34 (2019). Scholar
  15. 15.
    Hume, D.B., Chou, C.W., Rosenband, T., Wineland, D.J.: Preparation of dicke states in an ion chain. Phys. Rev. A 80(5), 052302 (2009). Scholar
  16. 16.
    Ionicioiu, R., Popescu, A.E., Munro, W.J., Spiller, T.P.: Generalized parity measurements. Phys. Rev. A 78(5), 052326 (2008). Scholar
  17. 17.
    Ivanov, S.S., Vitanov, N.V., Korolkova, N.V.: Creation of arbitrary dicke and NOON states of trapped-ion qubits by global addressing with composite pulses. New J. Phys. 15(2), 023039 (2013). Scholar
  18. 18.
    Kay, A.: Quantikz: A tikz library to typeset quantum circuit diagrams. Tutorial on the Quantikz Package.
  19. 19.
    Kiesel, N., Schmid, C., Tóth, G., Solano, E., Weinfurter, H.: Experimental observation of four-photon entangled dicke state with high fidelity. Phys. Rev. Lett. 98(6), 063604 (2007). Scholar
  20. 20.
    Lamata, L., López, C.E., Lanyon, B.P., Bastin, T., Retamal, J.C., Solano, E.: Deterministic generation of arbitrary symmetric states and entanglement classes. Phys. Rev. A 87(3), 032325 (2013). Scholar
  21. 21.
    Microsoft: Quantum Katas/Superposition, March 2019. Programming exercises for learning Q# and quantum computing.
  22. 22.
    Moreno, M.G.M., Parisio, F.: All bipartitions of arbitrary Dicke states. arXiv e-prints, January 2018.
  23. 23.
    Mosca, M., Kaye, P.: Quantum networks for generating arbitrary quantum states. In: Optical Fiber Communication Conference and International Conference on Quantum Information ICQI, p. PB28, June 2001.
  24. 24.
    Plesch, M., Bužek, V.: Efficient compression of quantum information. Phys. Rev. A 81(3), 032317 (2010). Scholar
  25. 25.
    Prevedel, R., et al.: Experimental realization of dicke states of up to six qubits for multiparty quantum networking. Phys. Rev. Lett. 103(2), 020503 (2009). Scholar
  26. 26.
    Rozema, L.A., Mahler, D.H., Hayat, A., Turner, P.S., Steinberg, A.M.: Quantum data compression of a qubit ensemble. Phys. Rev. Lett. 113(16), 160504 (2014). Scholar
  27. 27.
    Shao, X.Q.S., Chen, L., Zhang, S., Zhao, Y.F., Yeon, K.H.: Deterministic generation of arbitrary multi-atom symmetric Dicke states by a combination of quantum Zeno dynamics and adiabatic passage. EPL (Europhys. Lett.) 90(5), 50003 (2010). Scholar
  28. 28.
    Shende, V.V., Bullock, S.S., Markov, I.L.: Synthesis of quantum-logic circuits. IEEE Trans. Comput.-Aided Des. Integr. Circ. Syst. 25(6), 1000–1010 (2006). Scholar
  29. 29.
    Shende, V.V., Markov, I.L.: On the CNOT-cost of TOFFOLI gates. Quantum Inf. Comput. 9(5), 461–486 (2009). Scholar
  30. 30.
    Stockton, J.K., van Handel, R., Mabuchi, H.: Deterministic Dicke-state preparation with continuous measurement and control. Phys. Rev. A 70(2), 022106 (2004). Scholar
  31. 31.
    Tóth, G.: Multipartite entanglement and high-precision metrology. Phys. Rev. A 85(2), 022322 (2012). Scholar
  32. 32.
    Wieczorek, W., Krischek, R., Kiesel, N., Michelberger, P., Tóth, G., Weinfurter, H.: Experimental entanglement of a six-photon symmetric dicke state. Phys. Rev. Lett. 103(2), 020504 (2009). Scholar
  33. 33.
    Wu, C., Guo, C., Wang, Y., Wang, G., Feng, X.L., Chen, J.L.: Generation of Dicke states in the ultrastrong-coupling regime of circuit QED systems. Phys. Rev. A 95(1), 013845 (2017). Scholar
  34. 34.
    Xiao, Y.F., Zou, X.B., Guo, G.C.: Generation of atomic entangled states with selective resonant interaction in cavity quantum electrodynamics. Phys. Rev. A 75(1), 012310 (2007). Scholar
  35. 35.
    Özdemir, S.K., Shimamura, J., Imoto, N.: A necessary and sufficient condition to play games in quantum mechanical settings. New J. Phys. 9(2), 43–43 (2007). Scholar

Copyright information

© This is a U.S. government work and not under copyright protection in the U.S.; foreign copyright protection may apply 2019

Authors and Affiliations

  1. 1.Center for Nonlinear StudiesLos Alamos National LaboratoryLos AlamosUSA
  2. 2.National Security Education CenterLos Alamos National LaboratoryLos AlamosUSA

Personalised recommendations