The Toffoli gate is a three-quantum-bit (three-qubit) operation that inverts the state of a target qubit conditioned on the state of two control qubits. It makes universal reversible classical computation1 possible and, together with a Hadamard gate2, forms a universal set of gates in quantum computation. It is also a key element in quantum error correction schemes3,4,5,6,7. The Toffoli gate has been implemented in nuclear magnetic resonance3, linear optics8 and ion trap systems9. Experiments with superconducting qubits have also shown significant progress recently: two-qubit algorithms10 and two-qubit process tomography have been implemented11, three-qubit entangled states have been prepared12,13, first steps towards quantum teleportation have been taken14 and work on quantum computing architectures has been done15. Implementation of the Toffoli gate with only single- and two-qubit gates requires six controlled-NOT gates and ten single-qubit operations16, and has not been realized in any system owing to current limits on coherence. Here we implement a Toffoli gate with three superconducting transmon qubits coupled to a microwave resonator. By exploiting the third energy level of the transmon qubits, we have significantly reduced the number of elementary gates needed for the implementation of the Toffoli gate, relative to that required in theoretical proposals using only two-level systems. Using full process tomography and Monte Carlo process certification, we completely characterize the Toffoli gate acting on three independent qubits, measuring a fidelity of 68.5 ± 0.5 per cent. A similar approach15 to realizing characteristic features of a Toffoli-class gate has been demonstrated with two qubits and a resonator and achieved a limited characterization considering only the phase fidelity. Our results reinforce the potential of macroscopic superconducting qubits for the implementation of complex quantum operations with the possibility of quantum error correction17.
Your institute does not have access to this article
Open Access articles citing this article.
Scientific Reports Open Access 17 June 2021
npj Quantum Information Open Access 26 February 2021
Subscribe to Journal
Get full journal access for 1 year
only $3.90 per issue
All prices are NET prices.
VAT will be added later in the checkout.
Tax calculation will be finalised during checkout.
Get time limited or full article access on ReadCube.
All prices are NET prices.
Toffoli, T. Reversible Computing (Lect. Notes Computer Sci. 85, Springer, 1980)
Shi, Y. Both Toffoli and controlled-NOT need little help to do universal quantum computation. Quantum Inf. Comput. 3, 84–92 (2003)
Cory, D. G. et al. Experimental quantum error correction. Phys. Rev. Lett. 81, 2152–2155 (1998)
Knill, E., Laflamme, R., Martinez, R. & Negrevergne, C. Benchmarking quantum computers: the five-qubit error correcting code. Phys. Rev. Lett. 86, 5811–5814 (2001)
Chiaverini, J. et al. Realization of quantum error correction. Nature 432, 602–605 (2004)
Pittman, T. B., Jacobs, B. C. & Franson, J. D. Demonstration of quantum error correction using linear optics. Phys. Rev. A 71, 052332 (2005)
Aoki, T. et al. Quantum error correction beyond qubits. Nature Phys. 5, 541–546 (2009)
Lanyon, B. P. et al. Simplifying quantum logic using higher-dimensional Hilbert spaces. Nature Phys. 5, 134–140 (2009)
Monz, T. et al. Realization of the quantum Toffoli gate with trapped ions. Phys. Rev. Lett. 102, 040501 (2009)
DiCarlo, L. et al. Demonstration of two-qubit algorithms with a superconducting quantum processor. Nature 460, 240–244 (2009)
Yamamoto, T. et al. Quantum process tomography of two-qubit controlled-z and controlled-not gates using superconducting phase qubits. Phys. Rev. B 82, 184515 (2010)
DiCarlo, L. et al. Preparation and measurement of three-qubit entanglement in a superconducting circuit. Nature 467, 574–578 (2010)
Neeley, M. et al. Generation of three-qubit entangled states using superconducting phase qubits. Nature 467, 570–573 (2010)
Baur, M. et al. Benchmarking a teleportation protocol realized in superconducting circuits. Preprint at 〈http://arxiv.org/abs/1107.4774〉 (2011)
Mariantoni, M. et al. Implementing the quantum Von Neumann architecture with superconducting circuits. Science 334, 61–65 (2011)
Barenco, A. et al. Elementary gates for quantum computation. Phys. Rev. A 52, 3457–3467 (1995)
Reed, M. D. et al. Realization of three-qubit quantum error correction with superconducting circuits. Preprint at 〈http://arxiv.org/abs/1109.4948〉 (2011)
Filipp, S. et al. Two-qubit state tomography using a joint dispersive readout. Phys. Rev. Lett. 102, 200402 (2009)
Majer, J. et al. Coupling superconducting qubits via a cavity bus. Nature 449, 443–447 (2007)
Ralph, T. C., Resch, K. J. & Gilchrist, A. Efficient Toffoli gates using qudits. Phys. Rev. A 75, 022313 (2007)
Borrelli, M., Mazzola, L., Paternostro, M. & Maniscalco, S. Simple trapped-ion architecture for high-fidelity Toffoli gates. Phys. Rev. A 84, 012314 (2011)
Spörl, A. et al. Optimal control of coupled Josephson qubits. Phys. Rev. A 75, 012302 (2007)
Stojanovic, V. M., Fedorov, A., Bruder, C. & Wallraff, A. Quantum-control approach to realizing a Toffoli gate in circuit QED. Preprint at 〈http://arxiv.org/abs/1108.3442〉 (2011)
Gambetta, J. M., Motzoi, F., Merkel, S. T. & Wilhelm, F. K. Analytic control methods for high-fidelity unitary operations in a weakly nonlinear oscillator. Phys. Rev. A 83, 012308 (2011)
Motzoi, F., Gambetta, J. M., Rebentrost, P. & Wilhelm, F. K. Simple pulses for elimination of leakage in weakly nonlinear qubits. Phys. Rev. Lett. 103, 110501 (2009)
Strauch, F. W. et al. Quantum logic gates for coupled superconducting phase qubits. Phys. Rev. Lett. 91, 167005 (2003)
Haack, G., Helmer, F., Mariantoni, M., Marquardt, F. & Solano, E. Resonant quantum gates in circuit quantum electrodynamics. Phys. Rev. B 82, 024514 (2010)
Chuang, I. L. & Nielsen, M. A. Prescription for experimental determination of the dynamics of a quantum black box. J. Mod. Opt. 44, 2455–2467 (1997)
da Silva, M. P., Landon-Cardinal, O. & Poulin, D. Practical characterization of quantum devices without tomography. Phys. Rev. Lett. 107, 210404 (2011)
Flammia, S. T. & Liu, Y.-K. Direct fidelity estimation from few Pauli measurements. Phys. Rev. Lett. 106, 230501 (2011)
Ježek, M., Fiurášek, J., Hradil, Z. & v Quantum inference of states and processes. Phys. Rev. A 68, 012305 (2003)
Bylander, J. et al. Noise spectroscopy through dynamical decoupling with a superconducting flux qubit. Nature Phys. 7, 565–570 (2011)
Paik, H. et al. How coherent are Josephson junctions? Preprint at 〈http://arxiv.org/abs/1105.4652v1〉 (2011)
We thank S. Filipp, A. Blais for useful discussions and K. Pakrouski for his contributions in early stages of the experimental work. This work was supported by the Swiss National Science Foundation, the EU IP SOLID and ETH Zurich.
The authors declare no competing financial interests.
About this article
Cite this article
Fedorov, A., Steffen, L., Baur, M. et al. Implementation of a Toffoli gate with superconducting circuits. Nature 481, 170–172 (2012). https://doi.org/10.1038/nature10713
Nature Physics (2022)
Efficient scheme for realizing a multiplex-controlled phase gate with photonic qubits in circuit quantum electrodynamics
Frontiers of Physics (2022)
Frontiers of Physics (2022)
Scientific Reports (2021)
npj Quantum Information (2021)