Sampling and the complexity of nature

TL;DR

This thesis investigates the complexity and verification of quantum sampling algorithms, focusing on the quantum sign problem as a key factor in distinguishing classical and quantum computational capabilities.

Contribution

It develops measures for the quantum sign problem, analyzes their practicality, and explores the boundary between classical and quantum computing through sampling complexity.

Findings

Quantum sign problem underpins the intractability of quantum output probabilities.

Verifying quantum sampling distributions from few samples is computationally hard.

Assessing the quantum sign problem itself is an intractable computational task.

Abstract

Randomness is an intrinsic feature of quantum theory. The outcome of any quantum measurement will be random, sampled from a probability distribution that is defined by the measured quantum state. The task of sampling from a prescribed probability distribution is therefore a natural technological application of quantum devices. In the research presented in this thesis, I investigate the complexity-theoretic and physical foundations of quantum sampling algorithms. I assess the computational power of natural quantum simulators and close loopholes in the complexity-theoretic argument for the classical intractability of quantum samplers (Part I). I shed light on how and under which conditions quantum sampling devices can be tested or verified in regimes that are not simulable on classical computers (Part II). Finally, I explore the computational boundary between classical and quantum computing devices (Part III). In particular, I develop efficiently computable measures of the infamous Monte Carlo sign problem and assess those measures both in terms of their practicability as a tool for alleviating or easing the sign problem and the computational complexity of this task. An overarching theme of the thesis is the quantum sign problem which arises due to destructive interference between paths -- an intrinsically quantum effect. The (non-)existence of a sign problem takes on the role as a criterion which delineates the boundary between classical and quantum computing devices. I begin the thesis by identifying the quantum sign problem as a root of the computational intractability of quantum output probabilities. It turns out that the intricate structure of the probability distributions the sign problem gives rise to, prohibits their verification from few samples. In an ironic twist, I show that assessing the intrinsic sign problem of a quantum system is again an intractable problem.