Computational Problems Related to Open Quantum Systems

Abstract

A quantum system that has interaction with external resources, such as probability distribution, dissipation, and noise, is referred to as an open quantum system. Not only do open quantum systems play a vital role in the field of quantum physics, but they are also fundamental objects in quantum information and quantum computing. In this thesis, we focus on computational problems related to open quantum systems. In particular, we study efficient constructions of open quantum systems and their algorithmic applications.

A unitary 2-design is a quantum analogue of universal 2-hash functions. It is an example of open quantum systems in the sense that it is a probability distribution of unitaries. As unitary 2-designs inherit many properties of the Haar randomness on the unitary group, they have many applications in quantum information, such as benchmarking and decoupling. We study the structures of unitary 2-designs and present efficient methods for their constructions.

The continuous-time evolution of a closed quantum system can be described by the Schrödinger equation. A natural generalization of the Schrödinger equation to Markovian open quantum systems, in the sense of generating dynamical semigroups, is called the Lindblad equation. We show that it is impossible for a simple reductionist approach to simulate Lindblad evolution with gate complexity that has linear dependence in evolution time. Moreover, we use a novel variation of the "linear combination of unitaries" construction that pertains to quantum channels to achieve the desired linear dependence in evolution time and poly-logarithmic dependence in precision.

Open quantum systems can also be used as building blocks of quantum algorithms. We present a dissipative query model, which is based on the amplitude damping process. With this dissipative query model, we provide a quantum algorithm that performs a fixed-point quantum search while preserving the quadratic speedup against classical algorithms.