Optimal query error of quantum approximation on some Sobolev classes

We study the approximation of the imbedding of functions from anisotropic and generalized Sobolev classes into L q ([0, 1]d) space in the quantum model of computation. Based on the quantum algorithms for approximation of finite imbedding from L p N to L q N , we develop quantum algorithms for approximating the imbedding from anisotropic Sobolev classes B(W p r ([0, 1] d )) to L q ([0, 1] d ) space for all 1 ? q,p ? ∞ and prove their optimality. Our results show that for p < q the quantum model of computation can bring a speedup roughly up to a squaring of the rate in the classical deterministic and randomized settings.

QUANTUM COMPLEXITY OF THE APPROXIMATION FOR THE CLASSES B(W_p^r([0,1]~d)) AND B(H_p^r([0,1]~d))

We study the approximation of functions from anisotropic Sobolev classes b（WpR（[0, 1]d）） and HSlder-Nikolskii classes B（HPr（[0, 1]d）） in the Lq （[0, 1]d） norm with q 〈 p in the quantum model of computation. We determine the quantum query complexity of this problem up to logarithmic factors. It shows that the quantum algorithms are significantly better than the classical deterministic or randomized algorithms.

The Quantum Approximate Algorithm for Solving Traveling Salesman Problem

The Quantum Approximate Optimization Algorithm(QAOA)is an algorithmic framework for finding approximate solutions to combinatorial optimization problems.It consists of interleaved unitary transformations induced by two operators labelled the mixing and problem Hamiltonians.To fit this framework,one needs to transform the original problem into a suitable form and embed it into these two Hamiltonians.In this paper,for the well-known NP-hard Traveling Salesman Problem(TSP),we encode its constraints into the mixing Hamiltonian rather than the conventional approach of adding penalty terms to the problem Hamiltonian.Moreover,we map edges(routes)connecting each pair of cities to qubits,which decreases the search space significantly in comparison to other approaches.As a result,our method can achieve a higher probability for the shortest round-trip route with only half the number of qubits consumed compared to IBM Q’s approach.We argue the formalization approach presented in this paper would lead to a generalized framework for finding,in the context of QAOA,high-quality approximate solutions to NP optimization problems.

An upper bound for the generalized adiabatic approximation error with a superposition initial state

The classical adiabatic approximation theory gives an adiabatic approximate solution to the Schr6dinger equation （SE） by choosing a single eigenstate of the Hamiltonian as the initial state. The superposition principle of quantum states enables us to mathematically discuss the exact solution to the SE starting from a superposition of two different eigenstates of the time-dependent Hamiltonian H（0）. Also, we can construct an approximate solution to the SE in terms of the corresponding instantaneous eigenstates of H（t）. On the other hand, any physical experiment may bring errors so that the initial state （input state） may be a superposition of different eigenstates, not just at the desired eigenstate. In this paper, we consider the generalized adiabatic evolution of a quantum system starting from a superposition of two different eigenstates of the Hamiltonian at t = 0. A generalized adiabatic approximate solution （GAAS） is constructed and an upper bound for the generalized adiabatic approximation error is given. As an application, the fidelity of the exact solution and the GAAS is estimated.

Investigation on tunneling in optoelectronic devices with consideration of subwaves

Since novel optoelectronic devices based on the peculiar behaviors of the tunneling probability, e.g., resonant tunneling devices (RTD) and band-pass filter, are steadily proposed, the analytic transfer matrix (ATM) method is extended to study these devices. For several examples, we explore the effect of the scattered subwaves on tunneling; it is shown that the resonant or band-pass structures in tunneling probability are determined by the phase shift results from the scattered subwaves.