Parameterized quantum circuits as universal generative models for continuous multivariate distributions

Area of Science:

• Quantum Computing
• Quantum Machine Learning

Background:

• Parameterized quantum circuits are crucial for quantum machine learning tasks.
• Quantum Circuit Born machines generate discrete distributions, limiting their application to continuous variables.
• Existing models upload classical randomness to quantum circuits for continuous distributions, but their expressivity is underexplored.

Purpose of the Study:

• To formalize and establish the theoretical foundation for quantum circuit models generating continuous multivariate distributions.
• To prove the universality of variational quantum circuit architectures for this task.
• To derive resource bounds for achieving universality and explore practical applications.

Main Methods:

• Proving the universality of specific variational quantum circuit architectures.
• Utilizing tools related to the Holevo bound to derive tight resource bounds.
• Analyzing the trade-offs between the number of qubits and measurements required.

Main Results:

• Demonstrated the universality of several variational circuit architectures for generating continuous multivariate distributions.
• Derived tight resource bounds for achieving universality, highlighting a trade-off between qubit count and measurement requirements.
• Identified potential domains for quantum advantage through relaxed notions of universality and a practical use case.

Conclusions:

• The formalized theoretical foundation supports the use of quantum circuits for continuous distributions in quantum machine learning.
• Resource bounds provide guidance on optimizing quantum circuit design for generating complex probability distributions.
• This work opens avenues for exploring quantum advantage in generative modeling and other continuous variable tasks.