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Simplifying quantum logic using higher-dimensional Hilbert spaces

Nature Physics volume 5, pages 134–140 (2009)Cite this article

Abstract

Quantum computation promises to solve fundamental, yet otherwise intractable, problems across a range of active fields of research. Recently, universal quantum logic-gate sets—the elemental building blocks for a quantum computer—have been demonstrated in several physical architectures. A serious obstacle to a full-scale implementation is the large number of these gates required to build even small quantum circuits. Here, we present and demonstrate a general technique that harnesses multi-level information carriers to significantly reduce this number, enabling the construction of key quantum circuits with existing technology. We present implementations of two key quantum circuits: the three-qubit Toffoli gate and the general two-qubit controlled-unitary gate. Although our experiment is carried out in a photonic architecture, the technique is independent of the particular physical encoding of quantum information, and has the potential for wider application.

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Figure 1: Simplifying the Toffoli gate.
Figure 2: Simplifying higher-order Toffoli gates.
Figure 3: Simplifying controlled-unitary gates.
Figure 4: Efficiently adding control qubits to an arbitrary controlled circuit.
Figure 5: Toffoli and controlled-unitary experimental layout.
Figure 6: Experimentally constructed Toffoli logical truth table.
Figure 7: Experimentally reconstructed Toffoli output density matrices.
Figure 8: Experimentally reconstructed controlled-unitary gate process matrices.

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References

  1. Schmidt-Kaler, F. et al. Realization of the Cirac–Zoller controlled-NOT quantum gate. Nature 422, 408–411 (2003).

    Article  ADS  Google Scholar 

  2. Leibfried, D. et al. Experimental demonstration of a robust, high-fidelity geometric two ion-qubit phase gate. Nature 422, 412–415 (2003).

    Article  ADS  Google Scholar 

  3. O’Brien, J. L., Pryde, G. J., White, A. G., Ralph, T. C. & Branning, D. Demonstration of an all-optical quantum controlled-NOT gate. Nature 426, 264–267 (2003).

    Article  ADS  Google Scholar 

  4. Gasparoni, S., Pan, J.-W., Walther, P., Rudolph, T. & Zeilinger, A. Realization of a photonic controlled-NOT gate sufficient for quantum computation. Phys. Rev. Lett. 93, 020504 (2004).

    Article  ADS  Google Scholar 

  5. Pittman, T. B., Fitch, M. J., Jacobs, B. C. & Franson, J. D. Experimental controlled-NOT logic gate for single photons in the coincidence basis. Phys. Rev. A 68, 032316 (2003).

    Article  ADS  Google Scholar 

  6. Bao, X.-H. et al. Optical nondestructive controlled-NOT gate without using entangled photons. Phys. Rev. Lett. 98, 170502 (2007).

    Article  ADS  Google Scholar 

  7. Steffen, M. et al. Measurement of the entanglement of two superconducting qubits via state tomography. Science 313, 1423–1425 (2006).

    Article  ADS  MathSciNet  Google Scholar 

  8. Plantenberg, J. H., de Groot, P. C., Harmans, C. J. P. M. & Mooij, J. E. Demonstration of controlled-NOT quantum gates on a pair of superconducting quantum bits. Nature 447, 836–839 (2007).

    Article  ADS  Google Scholar 

  9. Mandel, O. et al. Controlled collisions for multi-particle entanglement of optically trapped atoms. Nature 425, 937–940 (2003).

    Article  ADS  Google Scholar 

  10. Anderlini, M. et al. Controlled exchange interaction between pairs of neutral atoms in an optical lattice. Nature 448, 452–456 (2007).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    MATH  Google Scholar 

  13. Cory, D. G. et al. Experimental quantum error correction. Phys. Rev. Lett. 81, 2152–2155 (1998).

    Article  ADS  Google Scholar 

  14. Dennis, E. Toward fault-tolerant quantum computation without concatenation. Phys. Rev. A 63, 052314 (2001).

    Article  ADS  Google Scholar 

  15. Shi, Y. Both Toffoli and controlled-NOT need little help to do universal quantum computation. Quantum Inform. Comput. 3, 84–92 (2003).

    MATH  Google Scholar 

  16. Aspuru-Guzik, A., Dutoi, A. D., Love, P. J. & Head-Gordon, M. Simulated quantum computation of molecular energies. Science 309, 1704–1707 (2005).

    Article  ADS  Google Scholar 

  17. Shor, P. Proc. 35th Ann. Symp. Found. Comp. Sci. 124–134 (IEEE Comp. Soc. Press, 1994).

    Book  Google Scholar 

  18. Kwiat, P., Mitchell, J. R., Schwindt, P. & White, A. Grover’s search algorithm: An optical approach. J. Mod. Opt. 47, 257–266 (2000).

    Article  ADS  MathSciNet  Google Scholar 

  19. Schuck, C., Huber, G., Kurtsiefer, C. & Weinfurter, H. Complete deterministic linear optics Bell state analysis. Phys. Rev. Lett. 96, 190501 (2006).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  21. Roos, C. F. et al. Control and measurement of three-qubit entangled states. Science 304, 1478–1480 (2004).

    Article  ADS  Google Scholar 

  22. Cirac, J. I. & Zoller, P. Quantum computations with cold trapped ions. Phys. Rev. Lett. 74, 4091–4094 (1995).

    Article  ADS  Google Scholar 

  23. Cory, D. G., Price, M. D. & Havel, T. F. Nuclear magnetic resonance spectroscopy: An experimentally accessible paradigm for quantum computing. Physica D 120, 82–101 (1998).

    Article  ADS  Google Scholar 

  24. Braunstein, S. L. et al. Separability of very noisy mixed states and implications for NMR quantum computing. Phys. Rev. Lett. 83, 1054–1057 (1999).

    Article  ADS  Google Scholar 

  25. Menicucci, N. C. & Caves, C. M. Local realistic model for the dynamics of bulk-ensemble NMR information processing. Phys. Rev. Lett. 88, 167901 (2002).

    Article  ADS  Google Scholar 

  26. Jones, J. NMR quantum computation: A critical evaluation. Fortschritte der Physik 48, 909–924 (2000).

    ADS  Google Scholar 

  27. O’Brien, J. L. et al. Quantum process tomography of a controlled-NOT gate. Phys. Rev. Lett. 93, 080502 (2004).

    Article  ADS  Google Scholar 

  28. Langford, N. K. et al. Demonstration of a simple entangling optical gate and its use in Bell-state analysis. Phys. Rev. Lett. 95, 210504 (2005).

    Article  ADS  Google Scholar 

  29. Kiesel, N., Schmid, C., Weber, U., Ursin, R. & Weinfurter, H. Linear optics controlled-phase gate made simple. Phys. Rev. Lett. 95, 210505 (2005).

    Article  ADS  Google Scholar 

  30. Okamoto, R., Hofmann, H. F., Takeuchi, S. & Sasaki, K. Demonstration of an optical quantum controlled-NOT gate without path interference. Phys. Rev. Lett. 95, 210506 (2005).

    Article  ADS  Google Scholar 

  31. O’Brien, J. L. Optical quantum computing. Science 318, 1567–1570 (2007).

    Article  ADS  Google Scholar 

  32. Kok, P. et al. Linear optical quantum computing with photonic qubits. Rev. Mod. Phys. 79, 135–174 (2007).

    ADS  Google Scholar 

  33. Nielsen, M. A. Optical quantum computation using cluster states. Phys. Rev. Lett. 93, 040503 (2004).

    Article  ADS  Google Scholar 

  34. Yoran, N. & Reznik, B. Deterministic linear optics quantum computation with single photon qubits. Phys. Rev. Lett. 91, 037903 (2003).

    Article  ADS  Google Scholar 

  35. Ralph, T. C., Hayes, A. J. F. & Gilchrist, A. Loss-tolerant optical qubits. Phys. Rev. Lett. 95, 100501 (2005).

    Article  ADS  Google Scholar 

  36. Browne, D. E. & Rudolph, T. Resource-efficient linear optical quantum computation. Phys. Rev. Lett. 95, 010501 (2005).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  38. Ralph, T. C. Scaling of multiple postselected quantum gates in optics. Phys. Rev. A 70, 012312 (2004).

    Article  ADS  Google Scholar 

  39. White, A. G. et al. Measuring two-qubit gates. J. Opt. Soc. Am. B 24, 172–183 (2007).

    Article  ADS  Google Scholar 

  40. Vidal, G. Efficient classical simulation of slightly entangled quantum computations. Phys. Rev. Lett. 91, 147902 (2003).

    Article  ADS  Google Scholar 

  41. Rarity, J., Tapster, P. & Loudon, R. Quantum Interferometry (VCH, 1996).

    Google Scholar 

  42. Weinhold, T. J. et al. Understanding photonic quantum-logic gates: the road to fault tolerance. Preprint at <http://arxiv.org/abs/0808.0794> (2008).

  43. Monz, T. et al. Realization of the quantum Toffoli gate with trapped ions. Preprint at <http://arxiv.org/abs/0804.0082> (2008).

  44. James, D. F. V., Kwiat, P. G., Munro, W. J. & White, A. G. Measurement of qubits. Phys. Rev. A 64, 052312 (2001).

    Article  ADS  Google Scholar 

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Acknowledgements

We acknowledge discussions with W. Munro and D. Kielpinski, and financial support from the Australian Research Council Discovery and Federation Fellow programmes, the DEST Endeavour Europe and International Linkage programmes, and an IARPA-funded US Army Research Office Contract.

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Authors and Affiliations

  1. Department of Physics and Centre for Quantum Computer Technology, University of Queensland, Brisbane 4072, Australia

    Benjamin P. Lanyon, Marco Barbieri, Marcelo P. Almeida, Thomas Jennewein, Timothy C. Ralph, Kevin J. Resch, Geoff J. Pryde, Jeremy L. O’Brien, Alexei Gilchrist & Andrew G. White

  2. Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences, Boltzmanng. 3, A-1090 Vienna, Austria

    Thomas Jennewein

  3. Institute for Quantum Computing and Department of Physics & Astronomy, University of Waterloo, N2L 3G1, Canada

    Kevin J. Resch

  4. Centre for Quantum Dynamics, Griffith University, Brisbane 4111, Australia

    Geoff J. Pryde

  5. H. H. Wills Physics Laboratory and Department of Electrical and Electronic Engineering, Centre for Quantum Photonics, University of Bristol, Merchant Venturers Building, Woodland Road, Bristol BS8 1UB, UK

    Jeremy L. O’Brien

  6. Physics Department, Macquarie University, Sydney 2109, Australia

    Alexei Gilchrist

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  1. Benjamin P. Lanyon
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Correspondence to Benjamin P. Lanyon.

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Lanyon, B., Barbieri, M., Almeida, M. et al. Simplifying quantum logic using higher-dimensional Hilbert spaces. Nature Phys 5, 134–140 (2009). https://doi.org/10.1038/nphys1150

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