Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

Calculating unknown eigenvalues with a quantum algorithm

Subjects

Abstract

A quantum algorithm solves computational tasks using fewer physical resources than the best-known classical algorithm. Of most interest are those for which an exponential reduction is achieved. The key example is the phase estimation algorithm, which provides the quantum speedup in Shor's factoring algorithm and quantum simulation algorithms. To date, fully quantum experiments of this type have demonstrated only the read-out stage of quantum algorithms, but not the steps in which input data is read in and processed to calculate the final quantum state. Indeed, knowing the answer beforehand was essential. We present a photonic demonstration of a full quantum algorithm—the iterative phase estimation algorithm (IPEA)—without knowing the answer in advance. This result suggests practical applications of the phase estimation algorithm, including quantum simulations and quantum metrology in the near term, and factoring in the long term.

Access options

Rent or Buy article

Get time limited or full article access on ReadCube.

from$8.99

All prices are NET prices.

Figure 1: The phase estimation algorithm.
Figure 2: Optical implementation of the phase estimation algorithm.
Figure 3: Phase estimation data for 12 different Us.
Figure 4: Using the phase estimation algorithm to generate the eigenstates of U.

References

  1. 1

    Nielsen, M. A. & Chuang, I. L. Quantum Computation and Quantum Information (Cambridge Univ. Press, 2000).

    MATH  Google Scholar 

  2. 2

    Vandersypen, L. M. K. et al. Experimental realization of Shor's quantum factoring algorithm using nuclear magnetic resonance. Nature 414, 883–887 (2001).

    ADS  Article  Google Scholar 

  3. 3

    Lu, C.-Y., Browne, D. E., Yang, T., & Pan, J.-W. Demonstration of a compiled version of Shor's quantum factoring algorithm using photonic qubits. Phys. Rev. Lett. 99, 250504 (2007).

    ADS  Article  Google Scholar 

  4. 4

    Lanyon, B. P. et al. Experimental demonstration of a compiled version of Shor's algorithm with quantum entanglement. Phys. Rev. Lett. 99, 250505 (2007).

    ADS  Article  Google Scholar 

  5. 5

    Politi, A., Matthews, J. C. F. & O'Brien, J. L. Shor's quantum factoring algorithm on a photonic chip. Science 325, 1221 (2009).

    ADS  MathSciNet  Article  Google Scholar 

  6. 6

    Lanyon, B. P. et al. Towards quantum chemistry on a quantum computer. Nature Chem. 2, 106–111 (2010).

    ADS  Article  Google Scholar 

  7. 7

    Du, J. et al. NMR implementation of a molecular hydrogen quantum simulation with adiabatic state preparation. Phys. Rev. Lett. 104, 030502 (2010).

    ADS  Article  Google Scholar 

  8. 8

    Li, Z. et al. Solving quantum ground-state problems with nuclear magnetic resonance. Sci. Rep. 1, 88 (2011).

    Article  Google Scholar 

  9. 9

    Griffiths, R. B. & Niu, C.-S. Semiclassical Fourier transform for quantum computation. Phys. Rev. Lett. 76, 3228–3231 (1996).

    ADS  Article  Google Scholar 

  10. 10

    Dobŝíĉek, M., Johansson, G., Shumeiko, V., & Wendin, G. Arbitrary accuracy iterative quantum phase estimation algorithm using a single ancillary qubit: a two-qubit benchmark. Phys. Rev. A 76, 030306 (2007).

    ADS  Article  Google Scholar 

  11. 11

    Ralph, T. C., Resch, K. J. & Gilchrist, A. Efficient Toffoli gates using qudits. Phys. Rev. A 75, 022313 (2007).

    ADS  Article  Google Scholar 

  12. 12

    Lanyon, B. P. et al. Simplifying quantum logic using higher-dimensional Hilbert spaces. Nature Phys. 5, 134–140 (2009).

    ADS  Article  Google Scholar 

  13. 13

    Zhou, X.-Q. et al. Adding control to arbitrary unknown quantum operations. Nature Commun. 2, 413 (2011).

    ADS  Article  Google Scholar 

  14. 14

    Higgins, B. L., Berry, D. W., Bartlett, S. D., Wiseman, H. M. & Pryde, G. J. Entanglement-free Heisenberg-limited phase estimation. Nature 450, 393–396 (2007).

    ADS  Article  Google Scholar 

  15. 15

    Gao, W. et al. Experimental demonstration of a hyper-entangled ten-qubit Schrödinger cat state. Nature Phys. 6, 331–335 (2010).

    ADS  Article  Google Scholar 

  16. 16

    Abrams, D. S. & Lloyd, S. Quantum algorithm providing exponential speed increase for finding eigenvalues and eigenvectors. Phys. Rev. Lett. 83, 5162–5165 (1999).

    ADS  Article  Google Scholar 

  17. 17

    Politi, A., Cryan, M. J., Rarity, J. G., Yu, S., & O'Brien, J. L. Silica-on-silicon waveguide quantum circuits. Science 320, 646–649 (2008).

    ADS  Article  Google Scholar 

  18. 18

    Matthews, J. C. F., Politi, A., Stefanov, A., & O'Brien, J. L. Manipulation of multiphoton entanglement in waveguide quantum circuits. Nature Photon. 3, 346–350 (2009).

    ADS  Article  Google Scholar 

  19. 19

    Shadbolt, P. et al. Generating, manipulating and measuring entanglement and mixture with a reconfigurable photonic circuit. Nature Photon. 6, 45–49 (2011).

    ADS  Article  Google Scholar 

  20. 20

    Lobino, M. et al. Correlated photon-pair generation in a periodically poled MgO doped stoichiometric lithium tantalate reverse proton exchanged waveguide. Appl. Phys. Lett. 99, 081110 (2011).

    ADS  Article  Google Scholar 

  21. 21

    Engin, E. et al. Photon pair generation in silicon microring resonator and enhancement via reverse bias. Preprint at http://arXiv.org/abs/1204.4922 (2012).

  22. 22

    Parker, S., & Plenio, M. B. Efficient factorization with a single pure qubit and logN mixed qubits. Phys. Rev. Lett. 85, 3049–3052 (2000).

    ADS  Article  Google Scholar 

  23. 23

    Shields, A. J. Semiconductor quantum light sources. Nature Photon. 1, 215–223 (2007).

    ADS  Article  Google Scholar 

  24. 24

    Hadfield, R. Single-photon detectors for optical quantum information applications. Nature Photon. 3, 696–705 (2009).

    ADS  Article  Google Scholar 

  25. 25

    Knill, E., Laflamme, R., & Milburn, G. J. A scheme for efficient quantum computation with linear optics. Nature 409, 46–52 (2001).

    ADS  Article  Google Scholar 

Download references

Acknowledgements

The authors thank P.J. Shadbolt for writing the quantum process tomography code and J.C.F. Matthews, A. Peruzzo, G.J. Pryde and P. Zhang for helpful discussions. This work was supported by the Engineering and Physical Sciences Research Council (EPSRC), the European Research Council (ERC), PHORBITECH, Quantum Interfaces, Sensors, and Communication based on Entanglement (QESSENCE) and the Centre for Nanoscience and Quantum Information (NSQI). J.O'B. acknowledges a Royal Society Wolfson Merit Award.

Author information

Affiliations

Authors

Contributions

The theory was developed by X.-Q.Z. and T.C.R. The theory was mapped to the experimental circuit by X.-Q.Z., P.K., T.C.R. and J.O.B. Experiments were performed by X.-Q.Z. and P.K. Data were analysed by X.-Q.Z., P.K., T.C.R. and J.O.B. The manuscript was written by X.-Q.Z., P.K., T.C.R. and J.O.B. The project was supervised by X.-Q.Z. and J.O.B.

Corresponding author

Correspondence to Jeremy L. O'Brien.

Ethics declarations

Competing interests

The authors declare no competing financial interests.

Supplementary information

Supplementary information

Supplementary information (PDF 424 kb)

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Zhou, XQ., Kalasuwan, P., Ralph, T. et al. Calculating unknown eigenvalues with a quantum algorithm. Nature Photon 7, 223–228 (2013). https://doi.org/10.1038/nphoton.2012.360

Download citation

Further reading

Search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing