Quantum Generative Adversarial Networks

A gentle introduction to QGANs

What are GANs in the first place?

Generator

Discriminator

The fake artist & art expert

This person does not exist.

QGANs

Source: Dallaire-Demers et al. (2018)

An aside on variational quantum algorithms

Variational Quantum Algorithms — How do they work?

Here’s the gist

The general structure of QGANs. The real source R or the parametrized generator G(theta_G) is applied on an initial state |0⟩, and each outputs a mixed or pure quantum state described by a density matrix. The discriminator is denoted as D(theta_D). We will discuss details on each register later.

A quick note before we jump into the fun stuff

Timeline of the development of quantum generative adversarial network models (source). The implementation we’ll be covering (circled in red) was first proposed by Dallaire-Demers et al. (source), and several concepts in this article are drawn from that original paper.

Let’s dive a bit deeper.

On the discriminator

The cost function — for the mathematically motivated

Eq 1
Eq 2
Quantum adversarial learning strategy — Eq 3
The simplified cost function for the generator
Classical adversarial learning strategy — log-likelihood functions
The general form of Z operator
Z = |0⟩⟨0| - |1⟩⟨1|  # when we label |real⟩ as |0⟩ and |fake⟩ as |1⟩ 〈real|Z|real〉= 1  # Expvals of statevectors with respect to Z are
〈fake|Z|fake〉= −1 proportional to the correct classification
The real source R or G(θ_G) is applied on the initial state |0,z⟩ respectively defined on the Out R|G and Bath R|G registers. The discriminator uses the outputted state from the source and an initial state |0,0⟩ defined on the Out D, and Bath D registers to output its’ answer |real⟩ or |fake⟩ in the Out D register. Bath D and Bath R|G are workspaces for the discriminator and generator, respectively.
Eq 4 — recall that the expectation value of a density matrix with respect to an observable Z is the trace of that Z operator applied to the density matrix. Read here to learn more
The final quantum optimization problem for QGANs

Now what? Update rule.

Convergence!

The cross-entropy between R and G

The algorithmic flow summed up

The algorithmic flow of a QGAN, source: Dallaire-Demers et al. (2018)

Some thoughts to keep in mind

Evaluating the D and G cost on quantum circuits

The final quantum optimization problem for QGANs
Simplified individual cost functions for D and G

The D and G training tradeoff

Thinking on G and |z⟩

Beware: classical data encoding

An ansatz suggestion

A practical universal circuit ansatz for either G or D. Each layer is composed of parameterized single-qubit X rotations followed by parameterized Z rotations. A layer of staggered sets of parameterized nearest-neighbour ZZ rotations follows the single-qubit rotations.

Why quantum?

Wrapping it up

Resources

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