Quantum phase processing and its applications in estimating phase and entropies

Resource-efficient quantum principal component analysis

“…QPS is a hybrid quantum-classical framework for extracting spectral information about unitary operators. In the procedure, phases are classified by using the so-called 'quantum phase processing' (QPP) circuit [32], and the binary search is employed to locate phases. A diagram for illustrating QPP is shown in figure 1.…”

“…Conversely, the rotation parameters ϕ, θ, α can be classically determined in advance, given a trigonometric polynomial. The calculation method can be found in [32].…”

“…Since the QPP circuit has the ability to classify eigenvalues, the search process consists of two main steps. The first step narrates the eigenvalue region to about 2∆ (see the phase interval search algorithm in [32]). The second step updates the phases of the unitary operator.…”

“…Theorem 1 (quantum phase search [32]). Given a unitary U and an eigenvector |χ⟩ such that U |χ⟩ = e iτ |χ⟩, QPS can estimate τ up to precision ϵ with a success probability at least 1 − δ by querying conditional U and U † at most O(ϵ −1 log δ −1 log ϵ −1 ) times.…”

“…This improvement makes our algorithm friendly to near-term quantum computers and scalable to large-scale systems. The main idea of our methods is to use one ancillary qubit to probe the spectrum of the target state and then extract principal components through classical post-processing, which is achieved by adopting the quantum phase search (QPS) algorithm [32]. In order to handle quantum data in state form or unitary operator form, we consider the sampling access and purified quantum query access models as input.…”