Computational complexity of certain quantum theories in 1+1 dimensions

Abstract

While physical theories attempt to break down the observed structure and behavior of possibly large and complex systems to short descriptive axioms, the perspective of a computer scientist is to start with simple and believable set of rules to discover their large scale behaviors. Computer science and physics, however, can be combined into a new framework, wherein structures can be compared with each other according to scalar observables like mass and temperature, and also complexity at the same time. For example, similar to saying that one object is heavier than the other, we can discuss which system is more complex. According to this point of view, a more complex system can be interpreted as the one which can be programmed to encode and simulate the behavior of the others within its own degrees of freedom. Within this framework, at the most fundamental layer, physical systems are related to languages. In this thesis, I try to exemplify this point of view through an analysis of certain quantum theories in two dimensional space-time. In simple words, these models are the quantum analogues of elastic scattering of colored balls moving on a line. The models are closely related to each other in both relativistic and non-relativistic regimes. Physical examples that motivate this are the factorized scattering matrix of quantum field theory, and the repulsive delta interactions of quantum mechanics, in 1+1 dimensions. In classical mechanics, when two hard balls collide, they bounce off and remain in the same order. However, in the quantum setting, during a collision, either the balls bounce off, or otherwise they tunnel through each other, and exchange their configurations. Each event occurs with a certain probability. As a result, moving balls are put in a superposition of being in different color configurations. Thereby, if we consider n distinct balls, the state space is according to their n! possible arrangements, and their collisions act as quantum transpositions. Quantum transpositions can then be viewed as local quantum gates. I therefore consider the general Hilbert space of permutations, and study the space of unitary operators that can be generated by the local permuting gates. I first show that all of the discussed quantum theories can be programmed into an idealized model, the quantum ball permuting model, and then I will try to pin down the language of this model within the already known complexity classes. The main approach is to consider a series of models, as the variations of the ball scattering problem, and then to relate them to each other, using tools of computational complexity and quantum complexity theory. I find that the computational complexity of the ball permuting model depends on the initial superposition of the balls. More precisely, if the balls start out from the identity permutation, the model can be simulated in a one clean qubit, which is believed to be strictly weaker than the standard model of quantum computing. Given this upper-bound on the ball permuting model, the result is that the model of ball scatterings can be simulated within a one clean qubit, if they start out from an identity permutation. Furthermore, I will demonstrate that if special superpositions are allowed in the initial state, then the ball permuting model can efficiently simulate and sample from the output distribution of standard quantum computers. Next, I show how to use intermediate demolition ball detections to simulate the ball permuting model nondeterministically. According to this result, using post-selection on the outcome of these measurements, one obtains the original ball permuting model. Therefore, the post-selected analogue of ball scattering model can efficiently simulate standard quantum computers, when arbitrary initial superpositions are allowed. In the end, I formalize a quantum computer based on ball collisions and intermediate ball detections, and then I prove that the possibility of efficient simulation of this model on a classical computer is ruled out, unless the polynomial hierarchy collapses to its third level.