Pseudoentanglement Ain't Cheap
For quantum computing theorists, this establishes a fundamental resource lower bound for pseudoentanglement, showing that it is not cheap to generate.
The paper proves that preparing pseudoentangled states with a t-bit entropy gap requires Ω(t) non-Clifford gates, a bound that is tight up to polylog factors assuming linear-time quantum-secure pseudorandom functions. This follows from a polynomial-time algorithm that estimates entanglement entropy within t/2 bits for states stabilized by at least 2^{n-t} Pauli operators.
We show that any pseudoentangled state ensemble with a gap of $t$ bits of entropy requires $Ω(t)$ 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.