Research ArticlePHYSICS

Training of quantum circuits on a hybrid quantum computer

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Science Advances  18 Oct 2019:
Vol. 5, no. 10, eaaw9918
DOI: 10.1126/sciadv.aaw9918

Figures

  • Fig. 1 DDQCL is a hybrid quantum algorithm scheme that can be used for generative modeling, illustrated here by the example of two by two BAS data.

    From top left, clockwise: A parametrized circuit is initialized at random. Then, at each iteration, the circuit is executed on a trapped ion quantum computer. The probability distribution of measurement is compared on a classical computer against the BAS target dataset. Next, the quantified difference is used to optimize the parametrized circuit. This learning process is iterated until convergence.

  • Fig. 2 Connectivity graphs and corresponding training circuits.

    Top: Fully connected training circuit layer, with layers of rotations (square boxes) and entanglement gates (rounded boxes) between any pair of the four qubits. Bottom: Star-connected training circuit layer, with restricted entangling gates. In either case, each rotation (denoted by X or Z) and each entanglement gate (denoted by XX) include a distinct control parameter, for a total of 18 parameters for the fully connected circuit layer and 15 parameters for the star-connected circuit layer. We remove the first Z rotation (dashed square boxes) acting on the initial state ∣0>, resulting in 14 and 11 parameters. The connectivity figures on the left define the mapping between the four qubits and the pixels of the BAS images (see Fig. 1).

  • Fig. 3 Quantum circuit training results with PSO, with simulations (orange) and trapped ion quantum computer results (blue).

    Column (A) corresponds to a circuit with one layer of single qubit rotations (square boxes) and one layer of entanglement gates (rounded boxes) of all-to-all connectivity. The circuit converges well to produce the BAS distribution. Columns (B) and (C) correspond to a circuit with two and four layers and star connectivity, respectively. In (B), the simulation shows imperfect convergence with two extra state components (6 and 9), due to the limited connectivity, and the experimental results follow the simulation. In (C), the simulation shows convergence to the BAS distribution, but the experiment fails to converge despite performing 1400 quantum circuits. The optimization is sensitive to the choice of initialization seeds. To illustrate the convergence behavior, the shaded regions span the 5th to 95th percentile range of random seeds [500 for (A) and (B), 1000 for (C)], and the orange curve shows the median. The two-layer circuits have 14 and 11 parameters for (A) all-to-all and (B) star connectivity, respectively while the (C) star-connectivity circuit with four layers has 26 parameters. The number of PSO particles used is twice the number of parameters, and each training sample is repeated 5000 times. Including circuit compilation, controller-upload time, and classical PSO optimization, each circuit instance takes about 1 min to be processed, in addition to periodic interruptions for the recalibration of gates.

  • Fig. 4 Quantum circuit training results with BO, with simulations (orange) and trapped ion quantum computer results (blue).

    Column (A) corresponds to a circuit with two layers of gates and all-to-all connectivity. Columns (B) and (C) correspond to a circuit with two and four layers and star connectivity, respectively. Convergence is much faster than with PSO (Fig. 3). Unlike the PSO results, the four-layer star-connected circuit in (C) is trained successfully, and no prior knowledge enters BO process. As before, the two-layer circuits have 14 and 11 parameters for (A) all-to-all and (B) star connectivity, respectively while the (C) star-connectivity circuit with four layers has 26 parameters. We used a batch of five circuits per iteration, and each training sample was repeated 5000 times. Including circuit compilation, controller-upload time, and BO classical optimization, each circuit instance takes 2 to 5 min, depending on the amount of accumulated data.

Tables

  • Table 1

    DKL (see Materials and Methods), qBAS score, and entanglement entropy (S) for the state obtained at the end of each of the DDQCL training on hardware for various circuits and classical optimizers used.


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