Core Concepts
Quantum circuits can sample from distributions that classical circuits cannot, even with shallow depths.
Abstract
Recent research shows that constant-depth quantum circuits can sample from distributions that are hard for classical circuits to reproduce. The study focuses on input-independent sampling tasks and provides unconditional proofs of quantum advantage over classical methods. By utilizing GHZ states and non-unitary gates, the research demonstrates the ability of quantum circuits to approximate complex distributions efficiently. The results highlight the potential of quantum computing in solving sampling problems beyond the capabilities of classical systems.
Stats
For each δ < 1, there exists a family of distributions {Dn} such that Cn samples from Dn within total variation distance at most 1/6 + O(n^-c).
Any classical circuit with fan-in 2 taking n + nδ random bits has depth Ω(log log n) to produce output close to Dn.
The distribution (X, majmodp(X)) is hard to sample from for low-depth classical circuits with bounded fan-in and limited inputs.
Viola's techniques prove lower bounds for locality functions related to circuit depth.
A Poor Man’s GHZ state allows for a constant-depth quantum circuit sampling distribution close to (X, majmodp(X) ⊕ parity(X)).
Quotes
"Quantum devices can process information in ways that classical devices cannot."
"Constant-depth quantum circuits can sample from distributions that can't be reproduced by constant-depth bounded fan-in classical circuits."
"The complexity of state preparation has implications in quantum cryptography and physics."