Pseudoentanglement Ain't Cheap

TL;DR

This paper establishes a lower bound on the non-Clifford gates needed to prepare pseudoentangled states with a certain entropy gap, and provides an efficient algorithm to estimate entanglement entropy in quantum states.

Contribution

It introduces a tight bound on gate complexity for pseudoentangled states and presents a polynomial-time algorithm for entanglement entropy estimation.

Findings

Any pseudoentangled state with entropy gap t requires Ω(t) non-Clifford gates.

The entropy estimation algorithm is accurate within t/2 bits for certain stabilizer states.

The bound is tight assuming the existence of quantum-secure pseudorandom functions.

Abstract

We show that any pseudoentangled state ensemble with a gap of $t$ bits of entropy requires $\Omega(t)$ non-Clifford gates to prepare. This bound is tight up to polylogarithmic factors if linear-time quantum-secure pseudorandom functions exist. Our result follows from a polynomial-time algorithm to estimate the entanglement entropy of a quantum state across any cut of qubits. When run on an $n$-qubit state that is stabilized by at least $2^{n-t}$ Pauli operators, our algorithm produces an estimate that is within an additive factor of $\frac{t}{2}$ bits of the true entanglement entropy.