Embedding quantum simulators

Résumé

Quantum simulations consist in the reproduction of the dynamics of a quantum systemon a controllable platform, with the goal of capturing an interesting feature of theconsidered model. It is broadly believed that the advent of quantum simulators willrepresent a technological revolution, as they promise to solve several problems whichare considered intractable in a classical computer. Although there are strong theoreticalbases confirming this claim, several aspects of quantum simulators have still to bestudied, in order to faithfully prove their feasibility. Moreover, the general question onwhich features of the considered models are simulatable is an attractive research topic,whose study would help to define the limits of a quantum simulator.In this Thesis, we develop several algorithms, which are able to catch relevant propertiesof the simulated quantum model. The proposed protocols follow a new conceptnamed embedding quantum simulator, in which the simulated Schr¿odinger equation ismapped onto an enlarged Hilbert space in a nontrivial way. Via this embedding, weare able to retrieve, by measuring few observables, quantities that generally require fulltomography in order to be evaluated. Moreover, we pay a special attention to the experimentalfeasibility, defining mappings which are space efficient, and do not requirethe implementation of challenging Hamiltonians. The presented algorithms are general,and they may be implemented in several quantum platforms, e.g. photonics, trappedions, circuit QED, among others.First, we propose a protocol which simulate the dynamics of an embedded Hamiltonian,allowing for the efficient extraction of a class of entanglement monotones. Thisis done using an embedding that is able to implement unphysical operations, as is thecase of complex conjugation. The analysis is accompanied with a study of feasibilityin a trapped-ion setup, which can be generalised to other platforms following similarcomputational models. Second, we propose an algorithm to measure n-time correlationfunctions of spinorial, fermionic, and bosonic operators, by considerably improving previousversions of the same result. We apply this protocol to the computation of magneticsusceptibilities, as well as to the simulation of Markovian and non-Markovian dissipativeprocesses in a novel way, without the necessity of engineering any bath. All the proposedprotocols are designed with a single ancillary qubit, minimising the needed experimentalresources.We believe that embedding quantum simulators have a potential to become a powerfultool in the quantum simulation theory, since they pave the way for improving theflexibility of a quantum simulator in di¿erent experimental contexts.