2105.00080v2_Eq-GAN

Last updated Dec 29, 2024

• Literature

• Certain distributions are hard to sample from clasically (1608.00263)

• Also hard to learn clasically (2010.11983)

• QuGAN does not converge (mode collapse)

• ? Entagle true and fake data ?

• “Perfect” swap test $\equiv$ supervised learning, but: Less robust to noise

• Classical GAN:

  • Generator $G(\Theta_g,z)$
  • Discriminator $D(\Theta_d, z)$

• Optimization: Min Max game $$V(\Theta_g, \Theta_d) =E_{x~p_{data}(x)}[\log D(\Theta_d,x)] + E_{z~p_0(z)}[\log (1-D(\Theta_d,G(\Theta_g,z)))]$$

• Solve $\min_{\Theta_g}\max_{\Theta_d} V(\Theta_g,\Theta_d)$

• If $D$ and $G$ are arbitrary functions (e.g. Deep nets with enough capacity) it is proven that unique optimum exists!

• Classical data encoded as $\sigma = \sum_i p_i \ket{\psi_i}\bra{\psi_i}$

In QuGAN:

• Generator output: $\rho = U(\Theta_g)\rho_0 U^{\dagger}(\Theta_g)$
• Discriminator operator $T$ measures probability of data beeing true
• Training by $\min_{\Theta_g}\min_T (\text{Tr}[T\sigma] - \text{Tr}[T\rho(\Theta_g)])$

Discriminator $D_\sigma(\Theta_d, \rho(\Theta_g))$ gives Probability of measuring $\ket{0}$.

• perfect swap test would give fidelity between data $\sigma$ and generator $\rho(\Theta_g)$.
• If there is some parameters $\Theta^{\text{opt}}d$ so that the discriminator performs perfect swap test $D\sigma(\Theta^{\text{opt}}d, \rho(\Theta_g))=\frac{1}{2} + \frac{1}{2}D^{\text{fid}}\sigma(\rho(\Theta_g))$, this can reach optimum
• Real swap test not good on hardware due to deep circuits and swap gates
• With this discriminator we have the cost function $$\min_{\Theta_g}\max_{\Theta_d}V(\Theta_g,\Theta_d) = \min_{\Theta_g}\max_{\Theta_d}[1-D_\sigma(\Theta_d,\rho(\Theta_g)]$$
• Different discriminator architectures possible
• poor choice can yield non-convex loss function landscape

• tldr: parametrized ansatz better at lerning state than perfect swap test

• Dataset ${i}$ with probabilites ${p_i}$ into quantum state $\sum_i \sqrt{p_i}\ket{\psi_i}$ where $\ket{\psi_i}$ is basis of states representing the different classical data entries
• Preparing this state is expensive, so try to learn it.