Research ArticlePHYSICS

Quantum generative adversarial learning in a superconducting quantum circuit

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Science Advances  25 Jan 2019:
Vol. 5, no. 1, eaav2761
DOI: 10.1126/sciadv.aav2761
  • Fig. 1 The QGAN.

    (A) The basic components of the QGAN, including a black box quantum process for the quantum true data σ, the generator (G) that produces an ensemble of pure quantum states (ρ), and the discriminator (D) that performs projective measurements M. (B) The process of the QGAN with the quantum states and the measurement basis on a Bloch sphere, where {|g〉, |e〉} are the ground and excited state of a qubit. D and G play the adversarial game alternatively, in which D optimizes the measurement strategy to discriminate ρ from σ, while G optimizes the generation strategy to fool D.

  • Fig. 2 The experimental protocol of the QGAN algorithm.

    (A) The experiment starts with a state σ as the quantum true data and a randomly generated state ρ(r0, θ0, ϕ0) from the generator. Then, the discriminator (D) and generator (G) optimize their strategies to compete against each other alternatively and repetitively. The stop condition of the game is either D fails to distinguish ρ from σ (the measurement output difference d < dB, a preset bound) or the step count cstep reaches the limit cB. (B) Procedure of optimizing D and G using the gradient descent method. The initial measurement axis M(β0, γ0) for D is randomly chosen. The parameters β and γ are updated in the process of optimizing D, while r, θ, and ϕ are updated in the process of optimizing G. The measurement and control of the quantum system are realized through field programmable gate arrays (FPGAs), while the estimations of the gradients are performed on a classical computer.

  • Fig. 3 Tracking of the QGAN.

    (A to C) Experimental results for selecting σ = |g〉〈g| as the quantum true data. (A) The snapshots of the system at the particular steps indicated by the black vertical arrows in (B) (from left to right in the same order), in the Bloch sphere representation. (B) The tracking of pσ, pρ, d, and F during the quantum adversarial learning process. The gray shadow regions are the processes of optimizing D, while the rests are those for optimizing G. The horizontal color arrows indicate the vertical axis that each curve with the same color corresponds to. Since the convergence condition dB for the case of pure states is small (see the Supplementary Materials) and there is inevitable measurement imprecision, the optimized M is difficult to obtain or could even be randomized in certain trials. In this particular trial, M ends up nearly antiparallel with both ρ and σ, resulting in trMσ ≈ trMρ ≈ 0 and d ≈ 0. (C) The measured state tomography of the experimental σe and final pf with a state fidelity F = 0.991, demonstrating a successful QGAN that G can fool D by generating quantum data highly similar to the true data. (D to F) Typical experimental results for σ in an arbitrary mixed state with each panel being the counterpart of (A) to (C), respectively.

  • Fig. 4 Statistics of the QGAN performance.

    (A) The cumulative probability of the total step count to finish the adversarial learning process. The QGAN is implemented for two different cases, with a pure (|g〉〈g|) and an arbitrary mixed state as the quantum true data, respectively. The count limit cB for these two cases is 500 and 300, respectively. The obtained average cstep for these two types of adversarial learning process is 243 and 170, respectively. Exp., experimental; sim., simulation. (B) The cumulative probability of final state fidelity F. The average fidelities for the pure state and the mixed state are both 98.8%. For comparison, the noiseless numerical simulations of the adversarial learning process are also performed 100 times, and their distributions are shown as solid lines.

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/5/1/eaav2761/DC1

    Section S1. Tracking of ρ − σ and M during the QGAN process

    Section S2. Comparisons between the experimental and the numerical simulation results

    Section S3. Algorithm and numerical results

    Fig. S1. Tracking of ρ − σ and M and the comparison between experiments and numerical simulations based on the recorded parameters ξ in the QGAN process as shown in Fig. 3.

    Fig. S2. Statistics of the QGAN performance based on numerical simulations.

    Fig. S3. Influence of δ and sd in the QGAN performance and count of steps when the QGAN converges to the same dB for different γ and p.

    Fig. S4. Performance of the QGAN algorithm for multipartite quantum systems with the quantum true data being a random mixed state.

    Fig. S5. Performance of the QGAN algorithm with Greenberger-Horne-Zeilinger and W states as the quantum true data.

  • Supplementary Materials

    This PDF file includes:

    • Section S1. Tracking of ρ − σ and M during the QGAN process
    • Section S2. Comparisons between the experimental and the numerical simulation results
    • Section S3. Algorithm and numerical results
    • Fig. S1. Tracking of ρ − σ and M and the comparison between experiments and numerical simulations based on the recorded parameters ξ in the QGAN process as shown in Fig. 3.
    • Fig. S2. Statistics of the QGAN performance based on numerical simulations.
    • Fig. S3. Influence of δ and sd in the QGAN performance and count of steps when the QGAN converges to the same dB for different γ and p.
    • Fig. S4. Performance of the QGAN algorithm for multipartite quantum systems with the quantum true data being a random mixed state.
    • Fig. S5. Performance of the QGAN algorithm with Greenberger-Horne-Zeilinger and W states as the quantum true data.

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