Analog quantum algorithms are formulated in terms of Hamiltonians rather than unitary gates and include quantum adiabatic computing, quantum annealing, and the Quantum Approximate Optimization Algorithm (QAOA).These algorithms are promising candidates for near-term quantum applications, but they often require fine tuning via the annealing schedule or variational parameters.In this work we connect all these algorithms to the optimal analog procedure.Notably, we explore how the optimal procedure approaches a smooth adiabatic procedure but with a superposed oscillatory pattern that can be explained in terms of the interactions between the ground state and first excited state.Furthermore, we provide numeric and analytic evidence that QAOA emulates this optimal procedure with the length of each QAOA layer equal to the period of the oscillatory pattern.Furthermore, the ratios of the QAOA bangs is determined by the smooth, non-oscillatory part of the optimal procedure.We provide arguments for these phenomena in terms of the product formula expansion of the optimal procedure.All of this shows that these analog algorithms are all different limits and approximations of the same optimal quantum protocol.In the appendix, we present a new algorithm for better approximating the optimal protocol using the insights from the main body.
Grover's search algorithm is the optimal quantum algorithm that can search an unstructured database quadratically faster than any known classical algorithm. The role of entanglement and correlations in the search algorithm have been studied in great detail and it is known that entanglement between the qubits is necessary to gain a quadratic speedup, for pure state implementation of the Grover search algorithm. Here, we systematically investigate the behavior of quantum coherence and monogamy of entanglement in the discrete analogue of the
analog analogue of Grover search algorithm. The analog analogue of Grover search is a continuous time quantum algorithm based on the adiabatic Hamiltonian evolution that gives a quadratic speedup, similar to the original Grover search algorithm. We show that the decrease of quantum coherence, quantified using various coherence monotones, is a clear signature of attaining the maximum success probability in the analog Grover search. We also show that for any two qubit reduced density matrix of the system, the concurrence evolves in close vicinity to the increasing rate of success probability. Furthermore, we show that the system satisfies a
n-party monogamy inequality for arbitrary times, hence bounding the amount of
n-qubit entanglement during the quantum search.
Abstract We propose a digital-analog quantum algorithm for simulating the Hubbard-Holstein model, describing strongly-correlated fermion-boson interactions, in a suitable architecture with superconducting circuits. It comprises a linear chain of qubits connected by resonators, emulating electron-electron (e-e) and electron-phonon (e-p) interactions, as well as fermion tunneling. Our approach is adequate for a digital-analog quantum computing (DAQC) of fermion-boson models including those described by the Hubbard-Holstein model. We show the reduction in the circuit depth of the DAQC algorithm, a sequence of digital steps and analog blocks, outperforming the purely digital approach. We exemplify the quantum simulation of a half-filling two-site Hubbard-Holstein model. In such example we obtain fidelities larger than 0.98, showing that our proposal is suitable to study the dynamical behavior of solid-state systems. Our proposal opens the door to computing complex systems for chemistry, materials, and high-energy physics.