Quantum Computation and Isomorphism Testing
Rosenbaum, David Jeffrey
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In this thesis, we study quantum computation and algorithms for isomorphism problems. Some of the problems that we cover are fundamentally quantum and therefore require quantum techniques. For other problems, classical approaches are more appropriate; however, we often give quantum variants of our classical algorithms as well. The field of quantum computation aims to accelerate algorithms by exploiting the laws of quantum mechanics. Several quantum algorithms are known that are exponentially faster than the best classical algorithms known for the same problems, including several which are relevant to cryptography. Quantum computing is therefore of great importance. On the theory side, it has already transformed our notions of which problems are tractable and which are not. On the practical side, the construction of a quantum computer would render many popular techniques for encryption obsolete. We derive several results in quantum computation. Quantum algorithms are normally formulated in an abstract way that ignores practical details such as where different parts of the computation are located physically. Using naive methods for translating these algorithms into practical quantum computing technologies increases the time complexity by a linear factor, while previous work reduced this factor to a square root. We further reduce this factor to a constant at the cost of requiring additional space. This retroactively justifies an important assumption made in many quantum algorithms. Another interesting problem is to determine the query complexity of extracting different types of information from physical processes. In the case of deterministic processes, it is well known that quantum algorithms cannot outperform deterministic algorithms by more than an exponential factor. We show --- somewhat surprisingly --- that when the process is allowed to be randomized, there are problems that have a constant quantum query complexity but cannot be solved classically no matter how many queries are made. In fact, we show how to construct such an infinity-vs-one separation from any weaker separation between the classical and quantum query complexities. In an isomorphism problem, we seek to determine if two algebraic or combinatorial objects have the same structure. One of the most well-known of these is the graph isomorphism problem, which is interesting since it is suspected to be of complexity intermediate between P and the NP-complete problems. We transition from quantum computation to isomorphism testing by considering the tree isomorphism problem. While linear-time classical algorithms are known for this problem, we show that a promising framework for developing efficient quantum algorithms for graph isomorphism, known as the state preparation approach, can also solve tree isomorphism. While this result might seem modest, it is important to know that the state preparation approach works for trees, since otherwise there would be little hope of using it to obtain efficient algorithms for more interesting classes of graphs. We also derive a powerful primitive along the way that is of independent interest. Next, we study the group isomorphism problem, which is a special case of graph isomorphism. Group isomorphism is suspected to be significantly easier than graph isomorphism, but still has unknown complexity. While group isomorphism is already of interest since it is a fundamental computational question about groups, searching for faster algorithms for group isomorphism is therefore also one way of approaching the graph isomorphism problem. We use collision detection methods to give a classical algorithm that obtains a square-root speedup over the best algorithm previously known for the general group isomorphism problem. For the solvable groups (which are conjectured to be difficult and contain almost all groups), we combine our collision detection techniques with group-theoretic methods to obtain a classical fourth-root speedup over the best algorithm known previously. Prior to this work, it was a longstanding open problem to obtain any improvement for either of these classes of groups. Finally, we give a general framework that yields speedups for many isomorphism problems. In particular, it yields the square-root speedup for testing isomorphism of general groups mentioned above. All of our group isomorphism-testing algorithms also have quantum variants that are slightly faster than their classical counterparts.