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Experimental realization of non-Abelian non-adiabatic geometric gates


The geometric aspects of quantum mechanics are emphasized most prominently by the concept of geometric phases, which are acquired whenever a quantum system evolves along a path in Hilbert space, that is, the space of quantum states of the system. The geometric phase is determined only by the shape of this path1,2,3 and is, in its simplest form, a real number. However, if the system has degenerate energy levels, then matrix-valued geometric state transformations, known as non-Abelian holonomies—the effect of which depends on the order of two consecutive paths—can be obtained4. They are important, for example, for the creation of synthetic gauge fields in cold atomic gases5 or the description of non-Abelian anyon statistics6,7. Moreover, there are proposals8,9 to exploit non-Abelian holonomic gates for the purposes of noise-resilient quantum computation. In contrast to Abelian geometric operations10, non-Abelian ones have been observed only in nuclear quadrupole resonance experiments with a large number of spins, and without full characterization of the geometric process and its non-commutative nature11,12. Here we realize non-Abelian non-adiabatic holonomic quantum operations13,14 on a single, superconducting, artificial three-level atom15 by applying a well-controlled, two-tone microwave drive. Using quantum process tomography, we determine fidelities of the resulting non-commuting gates that exceed 95 per cent. We show that two different quantum gates, originating from two distinct paths in Hilbert space, yield non-equivalent transformations when applied in different orders. This provides evidence for the non-Abelian character of the implemented holonomic quantum operations. In combination with a non-trivial two-quantum-bit gate, our method suggests a way to universal holonomic quantum computing.

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Figure 1: Geometric gate operation on a three-level system.
Figure 2: Transmon qubit in a cavity resonator.
Figure 3: Process tomography of holonomic gates.
Figure 4: Non-commutativity of holonomic gates.


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We are grateful to E. Sjöqvist and members of the EU-funded project GEOMDISS for discussions. We acknowledge feedback on the manuscript from P. Zanardi and I. Cirac. This work is supported financially by GEOMDISS, the Swiss National Science Foundation and ETH Zurich.

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Authors and Affiliations



A.A.A. and S.F. developed the idea for the experiment; A.A.A. performed the measurements and analysed the data; J.M.F. designed and fabricated the sample, J.M.F. and K.J. set up the experimental hardware and characterized the sample, M.P. and S.B. contributed to the experiment; A.A.A. and S.F. wrote the manuscript; A.W. and S.F. planned the project; all authors commented on the manuscript.

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Correspondence to A. A. Abdumalikov Jr.

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The authors declare no competing financial interests.

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Abdumalikov Jr, A., Fink, J., Juliusson, K. et al. Experimental realization of non-Abelian non-adiabatic geometric gates. Nature 496, 482–485 (2013).

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