Efficient verification of anticoncentrated quantum states

I present a method for estimating the fidelity F(μ, τ) between a preparable quantum state μ and a classically specified pure target state τ= ∣ τ⟩ ⟨ τ∣ , using simple quantum circuits and on-the-fly classical calculation (or lookup) of selected amplitudes of ∣ τ⟩. The method is sample efficient for anticoncentrated states (including many states that are hard to simulate classically), with approximate cost 4ϵ⁻²(1 − F)dp_coll where ϵ is the desired precision of the estimate, d is the dimension of the Hilbert space, and p_coll is the collision probability of the target distribution. This scaling is exponentially better than that of any method based on classical sampling. I also present a more sophisticated version of the method that uses any efficiently preparable and well-characterized quantum state as an importance sampler to further reduce the number of copies of μ needed. Though some challenges remain, this work takes a significant step toward scalable verification of complex states produced by quantum processors.