Essays on Labor Economics

Quantum annealing (QA) uses the principles of quantum mechanics for solving unconstrained optimization problems. Since its initial proposal, there has been much interest in the search for practical problems for which QA has advantage over classical algorithms. Although it is believed that quantum computers cannot in general solve NP-complete problems effciently, there has been evidence suggesting that effects such as quantum tunneling can bring quantum speedup over classical computation for some problems in NP. In this dissertation, we consider two problems that have both theoretical and practical importance. One is the Set Cover with Pairs problem. The other is the Factorization problem. The Set Cover with Pairs (SCP) problem is a generalization of the Set Cover problem. It requires each element in a set to be covered by at least two objects instead of one as in the Set Cover problem. The SCP problem is believed to be not only NP-hard, but also hard to approximate. In this dissertation, we explicitly construct the Ising Hamiltonians whose ground states encode the solutions of SCP instances. The resulting Ising Hamiltonian has the appealing property that the control precision required for encoding an SCP instance scales only linearly with the number of objects in the covering set. The Integer Factorization problem is of great interest in the quantum community because it is diffcult to solve using classical methods and its importance in cryptography. In this dissertation, we propose a general framework for solving factorization problems using quantum annealing, by mapping the framework to an Ising Hamiltonian. We develop several methods to consider the specifc requirements such as control precision, spin-spin connection in the current Ising machines. We test the methods on the latest D-Wave 2000Q machine

On Efficiently Solvable Cases of Quantum k-SAT

The constraint satisfaction problems k-SAT and Quantum k-SAT (k-QSAT) are canonical NP-complete and QMA_1-complete problems (for k >= 3), respectively, where QMA_1 is a quantum generalization of NP with one-sided error. Whereas k-SAT has been well-studied for special tractable cases, as well as from a parameterized complexity perspective, much less is known in similar settings for k-QSAT. Here, we study the open problem of computing satisfying assignments to k-QSAT instances which have a "matching" or "dimer covering"; this is an NP problem whose decision variant is trivial, but whose search complexity remains open. Our results fall into three directions, all of which relate to the "matching" setting: (1) We give a polynomial-time classical algorithm for k-QSAT when all qubits occur in at most two clauses. (2) We give a parameterized algorithm for k-QSAT instances from a certain non-trivial class, which allows us to obtain exponential speedups over brute force methods in some cases by reducing the problem to solving for a single root of a single univariate polynomial. (3) We conduct a structural graph theoretic study of 3-QSAT interaction graphs which have a "matching". We remark that the results of (2), in particular, introduce a number of new tools to the study of Quantum SAT, including graph theoretic concepts such as transfer filtrations and blow-ups from algebraic geometry; we hope these prove useful elsewhere

NP-complete Problems and Physical Reality

Can NP-complete problems be solved efficiently in the physical universe? I survey proposals including soap bubbles, protein folding, quantum computing, quantum advice, quantum adiabatic algorithms, quantum-mechanical nonlinearities, hidden variables, relativistic time dilation, analog computing, Malament-Hogarth spacetimes, quantum gravity, closed timelike curves, and "anthropic computing." The section on soap bubbles even includes some "experimental" results. While I do not believe that any of the proposals will let us solve NP-complete problems efficiently, I argue that by studying them, we can learn something not only about computation but also about physics.Comment: 23 pages, minor correction