Calculating unknown eigenvalues with a quantum algorithm

Journal name:
Nature Photonics
Year published:
Published online


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.

At a glance


  1. The phase estimation algorithm.
    Figure 1: The phase estimation algorithm.

    a, The standard quantum circuit for phase estimation, where H is the standard Hadamard gate and QFT−1 is the inverse quantum Fourier transform. The measurements are all implemented in computational basis. b, The kth iteration of the iterative phase estimation algorithm (IPEA). The algorithm is iterated m times to get an m-bit (equation (4)), which is the approximation to the phase of the eigenstate ϕ (equation (1)). The measurement is implemented in +/− basis, where . Each iteration obtains one estimated bit ; starting from the least significant ( ), k is iterated backwards from m to 1. The feedback angle ωk depends on the previously measured bits as ωk = −2πξk, where in binary expansion and ωm = 0.

  2. Optical implementation of the phase estimation algorithm.
    Figure 2: Optical implementation of the phase estimation algorithm.

    a, Simplified entanglement-based circuit for C − U2k−1 gate. The initial input state is , where |ψright fence is a multi-qubit polarization-encoded state and red r and blue b denote different spatial modes of the photons. After the blue mode s passes through the unitary gate U2k−1, which is realized by cascading 2k−1 copies of U, the red and blue modes of each target qubit are mixed on beamsplitters (BS). By retaining the case where an even number of target photons arrives in lower spatial modes, C − U2k−1 is realized for the input state |+right fence circle plus ψright fence, where . The rotation Rz(ωk) and the measurement in +/− basis are then used to extract —the estimate of the kth bit of the phase ϕ. b, Experimental set-up for the kth iteration of the two-qubit iterative phase estimation algorithm (IPEA). A 60 mW continuous-wave laser beam with a central wavelength of 404 nm is focused onto a type-II BBO crystal to create the polarization entangled photon pairs. The PBS part of the BS/PBS cube and the following waveplates convert the two photons to the desired polarization-spatial entangled state (equation (7)). Based on this state, the C − U2k−1 gate is effectively realized, where U is the unitary whose eigenvalue is to be estimated. The rotation gate Rz(ωk) (for the value of ωk, see the caption of Fig. 1b) is implemented by three waveplates—two quarter-waveplates with a HWP in between. The displaced-Sagnac structure makes the phase between modes 2r and 2b inherently stable.

  3. Phase estimation data for 12 different Us.
    Figure 3: Phase estimation data for 12 different Us.

    al, Each U is composed of two HWPs. The first is set to 0°, the second HWP is oriented at 0° (a), 15° (b), 30° (c), 45° (d), 60° (e), 75° (f), 90° (g), 105° (h), 120° (i), 135° (j), 150° (k) and 165° (l). For each U, three iterations of the algorithm are implemented and thus a three-digit estimated phase is obtained. Compared with the phase ϕ the error in is always less then 0.0001 in binary, which is consistent with theoretical prediction.

  4. Using the phase estimation algorithm to generate the eigenstates of U.
    Figure 4: Using the phase estimation algorithm to generate the eigenstates of U.

    ai, The density matrix of the target output: (L) experimental, (R) ideal. Unitaries U1, U2 and U3 are implemented with a single HWP set to 30°, 45° and 67.5°, respectively. ac, The case where the unitary is U1, U2, U3, the initial target state is |Hright fence and the measurement outcome of the control qubit is 0. df, The case where the unitary is U1, U2, U3, the initial target state is |Hright fence and the measurement result of the control qubit is 1. gi, The case where the unitary is U1, U2, U3, the initial target state is |Vright fence and the measurement result of the control qubit is 0. The output target state is determined only by the measurement result of the control qubit and not affected by changing the initial target state, as verified by the similarity between matrices a and g, b and h, c and i. State tomography and maximum-likelihood are used for the reconstruction of the density matrices. The fidelities of the reconstructed density matrices with the ideal case are shown. The error estimates are obtained by performing many reconstructions with random noise added to the raw data in each case.


  1. Nielsen, M. A. & Chuang, I. L. Quantum Computation and Quantum Information (Cambridge Univ. Press, 2000).
  2. Vandersypen, L. M. K. et al. Experimental realization of Shor's quantum factoring algorithm using nuclear magnetic resonance. Nature 414, 883887 (2001).
  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).
  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).
  5. Politi, A., Matthews, J. C. F. & O'Brien, J. L. Shor's quantum factoring algorithm on a photonic chip. Science 325, 1221 (2009).
  6. Lanyon, B. P. et al. Towards quantum chemistry on a quantum computer. Nature Chem. 2, 106111 (2010).
  7. Du, J. et al. NMR implementation of a molecular hydrogen quantum simulation with adiabatic state preparation. Phys. Rev. Lett. 104, 030502 (2010).
  8. Li, Z. et al. Solving quantum ground-state problems with nuclear magnetic resonance. Sci. Rep. 1, 88 (2011).
  9. Griffiths, R. B. & Niu, C.-S. Semiclassical Fourier transform for quantum computation. Phys. Rev. Lett. 76, 32283231 (1996).
  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).
  11. Ralph, T. C., Resch, K. J. & Gilchrist, A. Efficient Toffoli gates using qudits. Phys. Rev. A 75, 022313 (2007).
  12. Lanyon, B. P. et al. Simplifying quantum logic using higher-dimensional Hilbert spaces. Nature Phys. 5, 134140 (2009).
  13. Zhou, X.-Q. et al. Adding control to arbitrary unknown quantum operations. Nature Commun. 2, 413 (2011).
  14. Higgins, B. L., Berry, D. W., Bartlett, S. D., Wiseman, H. M. & Pryde, G. J. Entanglement-free Heisenberg-limited phase estimation. Nature 450, 393396 (2007).
  15. Gao, W. et al. Experimental demonstration of a hyper-entangled ten-qubit Schrödinger cat state. Nature Phys. 6, 331335 (2010).
  16. Abrams, D. S. & Lloyd, S. Quantum algorithm providing exponential speed increase for finding eigenvalues and eigenvectors. Phys. Rev. Lett. 83, 51625165 (1999).
  17. Politi, A., Cryan, M. J., Rarity, J. G., Yu, S., & O'Brien, J. L. Silica-on-silicon waveguide quantum circuits. Science 320, 646649 (2008).
  18. Matthews, J. C. F., Politi, A., Stefanov, A., & O'Brien, J. L. Manipulation of multiphoton entanglement in waveguide quantum circuits. Nature Photon. 3, 346350 (2009).
  19. Shadbolt, P. et al. Generating, manipulating and measuring entanglement and mixture with a reconfigurable photonic circuit. Nature Photon. 6, 4549 (2011).
  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).
  21. Engin, E. et al. Photon pair generation in silicon microring resonator and enhancement via reverse bias. Preprint at (2012).
  22. Parker, S., & Plenio, M. B. Efficient factorization with a single pure qubit and logN mixed qubits. Phys. Rev. Lett. 85, 30493052 (2000).
  23. Shields, A. J. Semiconductor quantum light sources. Nature Photon. 1, 215223 (2007).
  24. Hadfield, R. Single-photon detectors for optical quantum information applications. Nature Photon. 3, 696705 (2009).
  25. Knill, E., Laflamme, R., & Milburn, G. J. A scheme for efficient quantum computation with linear optics. Nature 409, 4652 (2001).

Download references

Author information


  1. Centre for Quantum Photonics, H. H. Wills Physics Laboratory & Department of Electrical and Electronic Engineering, University of Bristol, Bristol BS8 1UB, UK

    • Xiao-Qi Zhou,
    • Pruet Kalasuwan &
    • Jeremy L. O'Brien
  2. Department of Materials Science and Technology, Faculty of Science, Prince of Songkla University, Hat-Yai, Songkla 90112, Thailand

    • Pruet Kalasuwan
  3. Centre for Quantum Computation and Communication Technology, School of Mathematics and Physics, University of Queensland, Brisbane 4072, Australia

    • Timothy C. Ralph


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.

Competing financial interests

The authors declare no competing financial interests.

Corresponding author

Correspondence to:

Author details

Supplementary information

PDF files

  1. Supplementary information (428 KB)

    Supplementary information

Additional data