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Variational Circuit

Variational circuits are also known as "parameterized quantum circuits"

Adaptable quantum circuits

Variational quantum circuit are "quantum algorithm" that depend on free parameters. Like standard quantum circuits, they consist of three ingredients:

  • Preparation of fixed initial state.
  • A quantum circuit U(\theta), parameterized by a set of free parameters \theta.
  • Measurement of an observable \widehat{B} at the output.

The expectation values f(\theta) = \langle 0|U^{\dagger}(\theta)\widehat{B}U(\theta)|0\rangle defines a scalar cost for a given task. The free parameters \theta = (\theta_{1}, \theta_{2}, \cdots) of the citcuit(s) are tuned to optimzed this cost function.

Variational circuits are trained by a classical optimization algorithm that makes queries to the quantum device.

Variational circuits have become popular as a way to think about quantum algorithms for near-term quantum devices. Such devices can only run short gate sequences, since without fault tolerance every gate increases the error in the output. Usually, a quantum algorithm is decomposed into a set of standard elementary operations, which are in turn implemented by the quantum hardware.

The intriguing idea of variational circuit for near-term devices is to merge this two-step procedure into a single step by "learning" the circuit on the noisy device for a given task. This way, the "natural" tunable gates of a device can be used to formulate the algorithm, without the detour via a fixed elementary gate set. Furthermore, systematic errors can automatically be corrected during optimization.

Building circuit

The variational parameters with a set of non-adaptable parameters x = (x_{1},x_{2},\cdots) enter the quantum circuit as arguments for the circuit gates.

  • This allows to convert "classical information (\theta and x)" into quantum informtaion (|U(x;\theta)|0\rangle).
  • The non-adaptable parameter usually plays a role of data inputs in quantum machine learning.

"Quantum information" is turned "back" into classical information by evaluating the expectation value of the observable \widehat{B}.

\begin{array}{ll} f(x;\theta) & = \langle\widehat{B}\rangle\\ \ & = \langle 0|U^{\dagger}(x;\theta)\widehat{B}U(x;\theta)|0\rangle \end{array}

Reference

  1. PENNYLANE - Variational Circuit: https://pennylane.ai/qml/glossary/variational_circuit