Constraint Inequalities from Hilbert Space Geometry & Efficient Quantum Computation

TL;DR

This paper reviews how geometric constraints in Hilbert space can inform quantum system parameters and discusses their relevance for quantum simulation and parameterized quantum circuits on NISQ devices.

Contribution

It highlights the connection between Hilbert space geometry and physical constraints, proposing ways to enhance quantum circuit design and simulation techniques.

Findings

Relations from Hilbert space geometry aid quantum parameter estimation

Insights applicable to quantum simulation of many-body systems

Guidance for improving parameterized quantum circuits

Abstract

Useful relations describing arbitrary parameters of given quantum systems can be derived from simple physical constraints imposed on the vectors in the corresponding Hilbert space. This is well known and it usually proceeds by partitioning the large dimensional Hilbert space into relevant sub spaces and relating points in the Hilbert space to the expectation values of physical observables. The aim of this note is quite modest. We describe the procedure and point out that this parallels the necessary considerations that make Quantum Simulation of quantum fields and interacting many body quantum systems on Noisy Intermediate Scale Quantum (NISQ) devices possible. We conclude by pointing out relevant parts of Quantum Computing where these ideas could be useful. This work proceeds in density matrix formalism and is a review of materials found in references. We enrich the literature by suggesting how to use these ideas to guide and improve parameterized quantum circuits.