How Does Adiabatic Quantum Computation Fit into Quantum Automata Theory?
Tomoyuki Yamakami
TL;DR
This work introduces adiabatic evolutionary quantum systems (AEQS) as a bridge between adiabatic quantum computation and quantum automata theory, incorporating constant-memory (finite automata) models to generate the Hamiltonians governing adiabatic evolution. By allowing nonuniform, resource-bounded Hamiltonian construction via quantum quasi-automata (1qqaf/1moqqaf/2qqaf families), the paper demonstrates how a broad class of decision problems can be solved with controlled system size, spectral gaps, and energies, and it shows concrete simulations for 1moqfa, garbage-tape 1qfa, unambiguous pushdown automata, and polynomial-size 2qfa languages. It develops a formal framework for conditional AEQSs AEQS(F), analyzes closure properties, and connects AEQS classes to existing complexity classes (e.g., ptime-BQL/poly), while outlining future research directions in uniformity, optimal condition sets, and broader machine-model integrations. The results provide a structured, automata-theoretic lens on adiabatic quantum computing, enabling principled resource-bounded realizations of quantum computations with potential practical impact on near-term devices and theoretical insights into quantum computational models. All mathematical constructs are given with explicit Hamiltonians, spectral gaps, and ground-state proximity criteria to support rigorous classification of languages under AEQS-based complexity.
Abstract
Quantum computation has emerged as a powerful computational medium of our time, having demonstrated the remarkable efficiency in factoring a positive integer and searching databases faster than any currently known classical computing algorithm. Adiabatic evolution of quantum systems have been studied as a potential means that physically realizes quantum computation. Up to now, all the research on adiabatic quantum systems has dealt with polynomial time-bounded computation and little attention has been paid to, for instance, adiabatic quantum systems consuming only constant memory space. Such quantum systems can be modeled in a form similar to quantum finite automata. This exposition dares to ask a bold question of how to make adiabatic quantum computation fit into the rapidly progressing framework of quantum automata theory. As our answer to this eminent but profound question, we first lay out a fundamental platform to carry out adiabatic evolutionary quantum systems (AEQSs) with limited computational resources (in size, energy, spectral gap, etc.) and then establish how to construct such AEQSs by operating suitable families of quantum finite automata. We further explore fundamental structural properties of decision problems (as well as promise problems) solved quickly by the appropriately constructed AEQSs.