Quantum Error Correction and Quantum Metrology with Non-Markovian Noise

Abstract

Quantum technologies have the potential to solve many important problems across science and industry. An important example is quantum computation. Quantum simulators promise to better model chemistry. Further, Shor’s factoring algorithm solves a problem in exponentially less time than what it would take now on our classical computers. This has led many to believe that quantum computers could bring exponential speedups to other difficult, real-world problems, such as optimization. Another example is quantum sensing, where quantum mechanical effects can be leveraged to increase measurement precision beyond the classical state of the art. This has many applications in both fundamental science, such as the LIGO experiment, and in industry, such as Nitrogen vacancy magnetometers. For these quantum technologies to reach their full potential, however, the barrier of noise must be overcome. Quantum effects usually live at very small system sizes or very cold temperatures, making them extra sensitive to thermal noise or small perturbations of the environment. A proposed solution to this problem, for both computation and sensing, is to use quantum error correction. Quantum error correction encodes a few quantum degrees of freedom into many physical degrees of freedom, building in redundancy. This redundancy allows for the detection and correction of unwanted errors in our protocol. Most of the literature on quantum error correction focuses on Markovian noise models, i.e., models where the noise is not temporally correlated. The temporally correlated, or non-Markovian, regime remains relatively unexplored. In this thesis, we explore quantum error correction for non-Markovian noise models. We first present a few of the many definitions and models for quantum non-Markovian phenomena present in the literature. We then generalize the Knill-Laflamme quantum error conditions to the hidden Markov model, an experimentally motivated model of non-Markovian noise. These conditions allow one to guarantee that a quantum error-correcting code will still do its job for more realistic noise models. Finally, we apply our notion of non-Markovian error correction to quantum sensing. We generalize previous Markovian results and derive conditions for guaranteeing Heisenberg limited precision scaling in the presence of temporally correlated noise using quantum error correction. The Heisenberg limit is the fundamental precision limit allowed by quantum mechanics for parameter estimation in a physical system. We also study the next-best achievable precision scaling when the Heisenberg limit is unattainable.