English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

In this work, we present a polynomial-time quantum algorithm for solving theground states of a class of classically hard Hamiltonians. The mechanism of theexponential speedup that appeared in our algorithm is different from allexisting quantum algorithms. The idea is to introduce a mapping $f:\text{}\rho\rightarrow |\rho\rangle$ to use density matrices to represent purestates. We show that this mapping makes sense by giving an efficient method toobtain the information of $|\rho\rangle$ from measurements on $\rho$. Underthis mapping, the Lindblad master equation (LME) becomes a Schr\"odingerequation with non-Hermitian Hamiltonian which contains natural imaginary timeevolution. The steady state of the LME, therefore, corresponds to the groundstate of $L^\dag L$ with $L$ the Liouvillian operator of the LME. We show theruntime of the LME has the $\mathcal{O}(log(\zeta^{-1}))$ scaling with $\zeta$the overlap between the initial state and the ground state compared with the$\mathcal{O}(poly(\zeta^{-1}))$ scaling in other algorithms. The Hamiltonians$L^\dag L$ are guaranteed to be difficult for classical computers if we believethe simulation of LME is difficult. Further, for any given local Hamiltonian$H$ with known ground energy $E_0$, we give a polynomial-time classicalprocedure to judge and solve whether there exists $L$ such that $H-E_0=L^\dagL$. Later, We discuss and analyze several important aspects of the algorithmincluding the non-linear dynamics that appeared in the algorithm.

A polynomial-time quantum algorithm for solving the ground states of a class of classically hard Hamiltonians