Towards Entropic Constraints on Quantum Speedups

TL;DR

This paper explores whether entropic measures can explain quantum speedups by analyzing entanglement entropy in quantum algorithms, but finds current entropy inequalities insufficient to distinguish quantum advantage.

Contribution

It develops a framework for entropic analysis of quantum algorithms and applies it to known algorithms, revealing limitations of existing entropy inequalities in explaining quantum speedups.

Findings

Known entropy inequalities do not identify quantum speedups.

Ingleton inequality is not violated in analyzed examples.

Monogamy of mutual information fails at some points in quantum circuits.

Abstract

Some quantum algorithms have "quantum speedups": improved time complexity as compared with the best-known classical algorithms for solving the same tasks. Can we understand what fuels these speedups from an entropic perspective? Information theory gives us a multitude of metrics we might choose from to measure how fundamentally 'quantum' is the behavior of a quantum computer running an algorithm. The entanglement entropies for subsystems of a quantum state can be analyzed for subsystems of qubits in a quantum computer throughout the running of an algorithm. Here, a framework for making this entropic analysis is constructed, and performed on a selection of quantum circuits implementing known fast quantum algorithms and subroutines: Grover search, the quantum Fourier transform, and phase estimation. Our results are largely unsatisfactory: known entropy inequalities do not suffice to identify the presence or absence of quantum speedups. Although we know our algorithms must have quantum "magic", the Ingleton inequality, which holds for all entropies of subsystems of stabilizer states, is not violated in any of our examples. In some cases, however, monogamy of mutual information, which is obeyed for product states but violated for highly entangled bipartite states such as the $GHZ$ states, fails at some point in the course of our quantum circuits.