Sharp Error Bounds on Quantum Boolean Summation in Various Settings

We study the quantum summation (QS) algorithm of Brassard, Hoyer, Mosca and Tapp, that approximates the arithmetic mean of a Boolean function defined on N elements. We improve error bounds presented in [1] in the worst-probabilistic setting, and present new error bounds in the average-probabilistic setting. In particular, in the worst-probabilistic setting, we prove that the error of the QS algorithm using $M - 1$ queries is $3\pi /(4M)$ with probability $8/\pi^2$, which improves the error bound $\pi M^{-1} + \pi^2 M^{-2}$ of Brassard et al. We also present bounds with probabilities $p\in (1/2, 8/\pi^2]$ and show they are sharp for large $M$ and $NM^{-1}$. In the average-probabilistic setting, we prove that the QS algorithm has error of order $\min\{M^{-1}, N^{-1/2}\}$ if $M$ is divisible by 4. This bound is optimal, as recently shown in [10]. For M not divisible by 4, the QS algorithm is far from being optimal if $M \ll N^{1/2}$ since its error is proportional to M^{-1}^.Comment: 32 pages, 2 figure

Quantum Algorithms and Complexity for Certain Continuous and Related Discrete Problems

The thesis contains an analysis of two computational problems. The first problem is discrete quantum Boolean summation. This problem is a building block of quantum algorithms for many continuous problems, such as integration, approximation, differential equations and path integration. The second problem is continuous multivariate Feynman-Kac path integration, which is a special case of path integration. The quantum Boolean summation problem can be solved by the quantum summation (QS) algorithm of Brassard, HíŸyer, Mosca and Tapp, which approximates the arithmetic mean of a Boolean function. We improve the error bound of Brassard et al. for the worst-probabilistic setting. Our error bound is sharp. We also present new sharp error bounds in the average-probabilistic and worst-average settings. Our average-probabilistic error bounds prove the optimality of the QS algorithm for a certain choice of its parameters. The study of the worst-average error shows that the QS algorithm is not optimal in this setting; we need to use a certain number of repetitions to regain its optimality. The multivariate Feynman-Kac path integration problem for smooth multivariate functions suffers from the provable curse of dimensionality in the worst-case deterministic setting, i.e., the minimal number of function evaluations needed to compute an approximation depends exponentially on the number of variables. We show that in both the randomized and quantum settings the curse of dimensionality is vanquished, i.e., the minimal number of function evaluations and/or quantum queries required to compute an approximation depends only polynomially on the reciprocal of the desired accuracy and has a bound independent of the number of variables. The exponents of these polynomials are 2 in the randomized setting and 1 in the quantum setting. These exponents can be lowered at the expense of the dependence on the number of variables. Hence, the quantum setting yields exponential speedup over the worst-case deterministic setting, and quadratic speedup over the randomized setting

Robust and Efficient Hamiltonian Learning

With the fast development of quantum technology, the sizes of both digital and analog quantum systems increase drastically. In order to have better control and understanding of the quantum hardware, an important task is to characterize the interaction, i.e., to learn the Hamiltonian, which determines both static and dynamic properties of the system. Conventional Hamiltonian learning methods either require costly process tomography or adopt impractical assumptions, such as prior information on the Hamiltonian structure and the ground or thermal states of the system. In this work, we present a robust and efficient Hamiltonian learning method that circumvents these limitations based only on mild assumptions. The proposed method can efficiently learn any Hamiltonian that is sparse on the Pauli basis using only short-time dynamics and local operations without any information on the Hamiltonian or preparing any eigenstates or thermal states. The method has a scalable complexity and a vanishing failure probability regarding the qubit number. Meanwhile, it performs robustly given the presence of state preparation and measurement errors and resiliently against a certain amount of circuit and shot noise. We numerically test the scaling and the estimation accuracy of the method for transverse field Ising Hamiltonian with random interaction strengths and molecular Hamiltonians, both with varying sizes and manually added noise. All these results verify the robustness and efficacy of the method, paving the way for a systematic understanding of the dynamics of large quantum systems.Comment: 41 pages, 6 figures, Open source implementation available at https://github.com/zyHan2077/HamiltonianLearnin