Quantum Control and Quantum Simulation of Many-body Systems

量子计算机将使我们能够解决某些经典计算机难以解决的问题。本论文提出了几种提高含噪声中等规模量子处理器性能的新方法，以及当前量子处理器作为量子模拟器在多体量子物理研究中的应用。

首先，我们简要回顾了超导电路，介绍了超导量子比特及其耦合的基本电路结构，以及超导量子比特的标定和量子控制。我们还简要回顾了两能级系统的量子动力学，包括几个解析可解模型和周期驱动系统的形式理论，它们在很多量子控制方案中扮演着重要角色。

在第四章，我们讨论了固态多比特量子芯片中实现高精度量子控制的主要障碍之一：由相互作用引起的剩余耦合。我们提出了一个几何参数化下的容错量子门方案，在微扰区间展示了高保真度、高容错性的通用量子门集合；并开发了一种实用的脉冲优化算法，借助自动微分进行梯度求解，并将脉冲约束整合进优化过程中，实现了给定噪声参数下的高保真度量子门。

在第五章中，我们研究了一种量子干涉效应启发的测量Renyi-2纠缠熵的新方法及其在多体量子模拟中的应用。最后，我们在第六章和第七章介绍了我们在非平衡量子物态和非厄米量子临界性的量子模拟方案。我们提出了Floquet伊辛模型的数字量子模拟方案，从动力学的角度探测了该模型在时间、空间上的非平庸量子序；探索了非幺正哈密顿量演化和一般非幺正操作在量子处理器中的实现，并提出了在开放量子系统中探测杨-李量子临界性的实验方案。

Quantum computers would enable us to tackle certain problems that are intractable on their classical counterparts. This thesis presents several new approaches to improve the performance of noisy intermediate-scale quantum processors and the application of the current quantum processors as quantum simulators in the study of many-body quantum physics.

First, we make a brief review of superconducting circuit--one of the most promising platforms for quantum information processing. We summarize the basic architecture for superconducting qubits and their coupling, as well as calibration and quantum control of superconducting qubits. We also make a short review of quantum dynamics of two-level systems, including several analytically solvable models and a formalism for periodically driven systems, which are important in many quantum control protocols.

We then address the issue of residual zz-coupling caused by unwanted interaction—a major obstacle to high-precision quantum control in solid-state multi-qubit systems in Chapter 4. We provide a geometrical parametrized error-robust quantum gate and demonstrate a high-fidelity universal gate-set in the perturbative region, and develop a practical constrained optimization algorithm assisted by auto-differentiation to implement targeted-correction gates with few parameters in given noise strengths, offering a promising quantum optimal control approach via pulse engineering.

In Chapter 5, we study a novel approach to measuring Renyi-2 entanglement entropy inspired by quantum interferometry and its application in the study of many-body quantum dynamics. Finally, we highlight our studies in quantum simulation of non-equilibrium quantum phases of matter and non-Hermitian quantum criticality, in Chapter 6 and Chapter 7. We propose a digital quantum simulation of the Floquet Ising model and probe its spatial-temporal order dynamically. We then study the realization of non-unitary Hamiltonian evolution and general non-unitary operation in quantum processors and propose an experimental study of Yang-Lee quantum criticality embedded in an open quantum system.

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