Fermionic Partial Tomography via Classical Shadows.

Area of Science:

• Quantum Information Science
• Computational Physics
• Quantum Chemistry

Background:

• Estimating k-body reduced density matrices (k-RDMs) is crucial for quantum simulation of many-body systems.
• Near-term quantum algorithms require efficient methods for learning properties of fermionic states.

Purpose of the Study:

• To propose a novel tomographic protocol for estimating any k-RDM of an n-mode fermionic state.
• To extend the classical shadows framework to the fermionic setting for quantum state learning.

Main Methods:

• Utilizing a sampling protocol with randomized measurement settings generated by fermionic Gaussian unitaries.
• Implementing circuits with linear depth for efficient state preparation and measurement.
• Adapting the method to incorporate particle-number symmetry for potential circuit depth reduction.

Main Results:

• Proving an optimal scaling of (n/k)k^{3/2}log(n)/ϵ^{2} for state preparations to estimate k-RDM elements.
• Demonstrating substantial improvement in constant overheads for k≥2 compared to deterministic strategies via numerical calculations.
• Showing that particle-number symmetry adaptation can halve circuit depth at the cost of increased repetitions.

Conclusions:

• The proposed protocol provides an efficient and scalable method for fermionic quantum state tomography.
• This work offers a significant advancement for near-term quantum algorithms in simulating complex quantum systems.
• The protocol's adaptability and proven optimality make it a valuable tool for quantum computation applications.