Digitized adiabatic quantum computing with a superconducting circuit

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Quantum mechanics can help to solve complex problems in physics1 and chemistry2, provided they can be programmed in a physical device. In adiabatic quantum computing3, 4, 5, a system is slowly evolved from the ground state of a simple initial Hamiltonian to a final Hamiltonian that encodes a computational problem. The appeal of this approach lies in the combination of simplicity and generality; in principle, any problem can be encoded. In practice, applications are restricted by limited connectivity, available interactions and noise. A complementary approach is digital quantum computing6, which enables the construction of arbitrary interactions and is compatible with error correction7, 8, but uses quantum circuit algorithms that are problem-specific. Here we combine the advantages of both approaches by implementing digitized adiabatic quantum computing in a superconducting system. We tomographically probe the system during the digitized evolution and explore the scaling of errors with system size. We then let the full system find the solution to random instances of the one-dimensional Ising problem as well as problem Hamiltonians that involve more complex interactions. This digital quantum simulation9, 10, 11, 12 of the adiabatic algorithm consists of up to nine qubits and up to 1,000 quantum logic gates. The demonstration of digitized adiabatic quantum computing in the solid state opens a path to synthesizing long-range correlations and solving complex computational problems. When combined with fault-tolerance, our approach becomes a general-purpose algorithm that is scalable.

At a glance


  1. Spin-chain problem and device.
    Figure 1: Spin-chain problem and device.

    a, We implement one-dimensional spin problems with variable local fields and couplings between adjacent spins. An example of a stoquastic problem Hamiltonian with local x and z fields, indicated by the gold arrows in the spheres, and σzσz couplings, whose strength is indicated by the radius of the links, is shown. Red denotes a ferromagnetic (J = +1) and blue an antiferromagnetic (J = −1) link. The problem Hamiltonian is for the instance shown in Fig. 4c. b, Optical picture of the superconducting quantum device with nine Xmon22 qubits Q0–Q8 (false-coloured cross-shaped structures), made from aluminium (light) on a sapphire substrate (dark). Connections to read-out resonators are at the top; control wiring is at the bottom. Scale bar, 200 μm.

  2. Quantum state tomography of the digital evolution into a Greenberger–Horne–Zeilinger state.
    Figure 2: Quantum state tomography of the digital evolution into a Greenberger–Horne–Zeilinger state.

    A four-qubit system is adiabatically evolved from an initial Hamiltonian in which all spins are aligned along the x axis to a problem Hamiltonian with equal ferromagnetic couplings between adjacent qubits (Jzz = 2). a, Real part of the experimental density matrix ρ at the start (left-most panel) and after each Trotter step, showing the growth of the major elements on the four corners, measured using quantum state tomography. The target state is shown with a black outline in the right-most panel. The final state has a fidelity of 0.55. Coloured squares surrounding the left-most panel indicate qubit indices: for example, Q0 being excited is indicated by a red square. Black arrows indicate notable elements for states that differ from the target state by a single kink. b, As in a, but for the ideal digitized evolution, showing major elements on the four corners as well as other populations and correlations. c, Hamiltonian at different s, showing the vanishing transversal field and increasing coupling strength; arrows and links as in Fig. 1a. d, Gate sequence showing initialization and the five Trotter steps. e, Pulse sequence, showing the single-qubit microwave gates (wave-like pulses) and frequency detuning (rectangular-like) pulses. Corresponding interactions and local field terms are highlighted. The displayed five-step algorithm is 2.1-μs long. Colours correspond to the physical qubits in Fig. 1b. Implementations of σzσz coupling and local x-fields are highlighted. Angles of rotation are denoted by ϕ and θ. See Supplementary Information for imaginary parts of the density matrices and the ideal continuous evolution.

  3. Kink errors, residual energy and scaling with system size.
    Figure 3: Kink errors, residual energy and scaling with system size.

    a, Kink likelihood for the four-qubit configuration. Solid lines, experiment; dashed lines, ideal digital evolution; dotted lines, ideal continuous time evolution. b, Residual energy in the adiabatic evolutions of ferromagnetic chains (Jzz = 2) in configurations with two to nine qubits (as indicated by the coloured-coded numerals). The green solid line shows the ideal square-root trend for the large-scale limit (Supplementary Information). Distinct contributions to error are highlighted.

  4. Digital evolutions of random stoquastic and non-stoquastic problems.
    Figure 4: Digital evolutions of random stoquastic and non-stoquastic problems.

    As stoquastic problems we use frustrated Ising Hamiltonians, with random local x and z fields, and random σzσz couplings. ac, Stoquastic results for three, six and nine qubits. a, For three qubits we have done tomography. An example instance is provided on the left, where we show the real part of the density matrix ρ. Coloured bars denote the experimental data, and black and grey outlined bars show the ideal digital evolution and the target state, respectively. The diagonals of the experiment (colour) and the target state (grey) are shown in the bottom right panel (as indicated by the dashed arrow), sorted by ideal target state results. The fidelity results for all 100 instances are summarized in the histogram (top right), where ratio denotes the normalized occurrence; coloured bars, fidelities of experimental results with respect to the target state; grey bars, fidelities of the ideal digital evolution with respect to the target state. b, c, The correlated probabilities for six (b) and nine (c) qubits, sorted by target state results. Experimental data are in colour, the target state is in grey. The results for all 250 instances are summarized in the insets. For the nine-qubit instance (c), the first 100 elements are shown. In ac, the coloured squares surrounding or below the plots indicate qubit indices, as in Fig. 2. df, As in ac, but for non-stoquastic problems, which have additional random σxσx couplings. Here we plot the data for three, six and seven qubits, for which the average measure of success is above the random baseline (not shown; see text). The results show that the system can find the ground states of stoquastic and non-stoquastic Hamiltonians with similar performance.


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Author information

  1. Present address: IBM T. J. Watson Research Center, Yorktown Heights, New York 10598, USA.

    • A. Mezzacapo


  1. Google Inc., Santa Barbara, California 93117, USA

    • R. Barends,
    • J. Kelly,
    • A. G. Fowler,
    • Yu Chen,
    • E. Jeffrey,
    • E. Lucero,
    • J. Y. Mutus,
    • M. Neeley,
    • P. Roushan,
    • D. Sank &
    • John M. Martinis
  2. Google Inc., Venice, California 90291, USA

    • A. Shabani,
    • R. Babbush &
    • H. Neven
  3. Department of Physical Chemistry, University of the Basque Country UPV/EHU, Apartado 644, E-48080 Bilbao, Spain

    • L. Lamata,
    • A. Mezzacapo,
    • U. Las Heras &
    • E. Solano
  4. Department of Physics, University of California, Santa Barbara, California 93106, USA

    • B. Campbell,
    • Z. Chen,
    • B. Chiaro,
    • A. Dunsworth,
    • A. Megrant,
    • C. Neill,
    • P. J. J. O’Malley,
    • C. Quintana,
    • A. Vainsencher,
    • J. Wenner,
    • T. C. White &
    • John M. Martinis
  5. IKERBASQUE, Basque Foundation for Science, Maria Diaz de Haro 3, 48013 Bilbao, Spain

    • E. Solano


R. Barends, A.S. and L.L. designed the experiment, with E.S., H.N. and J.M.M. providing supervision and A. Mezzacapo, U.L.H. and R. Babbush providing additional theoretical support. R. Barends, A.S., L.L. and R. Babbush co-wrote the manuscript with E.S., H.N. and J.M.M. R. Barends, A.S. and L.L. performed the experiment and analysed the data. The device was designed by R. Barends and J.K. All authors contributed to the fabrication process, experimental set-up and manuscript preparation.

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The authors declare no competing financial interests.

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Supplementary information

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  1. Supplementary Information (2.8 MB)

    This fie contains Supplementary Text and Data, Supplementary Figures 1-10, Supplementary Tables 1-10 and additional references.

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