This paper initiates a systematic study of quantum functions, which are (partial) functions defined in terms of quantum mechanical computations. Of all quantum functions, we focus on resource-bounded quantum functions whose inputs are classical bit strings. We prove complexity-theoretical properties and unique characteristics of these quantum functions by recent techniques developed for the analysis of quantum computations. We also discuss relativized quantum functions that make adaptive and nonadaptive oracle queries.
This thesis focuses on two different aspects of quantum computation: quantum complexity and quantum semantics. In the first part, I study the quantum complexity mainly within the quantum query model. Together with my colleagues, I introduce a new framework for quantum query complexity, phrased in terms of the minimal oracle and analyse limits and strengths of this new model in comparison to their classical and quantum counterparts. Working within the query model I study quantum one-way functions. I show that in the quantum setting, the problem of the existence of a quantum one-way permutation can be reduced to the problem of constructing polynomial size networks for performing the specific task of the reflection about a sequence of states. Furthermore, I extend these results to the domain of the state and operator complexity. I show that if a quantum one-way function exists, then we can construct a sequence of so called “hard” states with the property that the reflection operators about those states are efficiently implementable. In the second part, I study the extension of domain theory to the quantum setting and develop the semantics of quantum computation. By defining a quantum domain I introduce a rigourous definition of quantum computability for quantum states and operators. Furthermore I show that the denotational semantics of quantum computation has the same semantical structure as the denotational semantics of classical probabilistic computation introduced by Kozen. This could be considered as a foundation for designing functional programming languages for quantum computation. Finally, I continue with an abstract mathematical approach to study a general formalism for describing entanglement manipulation and introduce a new approach to derive a unique measure of entanglement for bipartite quantum pure states.
The last decade witnessed a great surge of fruitful studies in the new paradigm of quantum computing. Remarkable progress has been made in all the areas including quantum algorithms, quantum complexity theory, quantum error-correcting code, quantum information theory, quantum cryptography and physical implementations of quantum computers. However, quantum algorithm design, probably the most important task in quantum computing, did not make as much progress as people expected. This thesis aims at both designing new quantum algorithms and studying why efficient quantum algorithms are so hard to design by proving many lower bounds on quantum query complexity for various problems.
We first review (and generalize) known techniques for designing quantum algorithms and for proving quantum lower bounds. Using these techniques, we then study the quantum query complexity for many problems, including some specific problems such as Graph Matching and other natural graph properties, the general tuple search, and local search. We also study the lowest possible quantum query complexity for a class of functions as a whole, such as graph properties, circular functions, and functions invariant of a transitive permutation group, where tight results are given in all these cases.
Finally for the quantum lower bounds techniques themselves, we study the main two techniques and show that they are incomparable in power. For the most widely used one, which has a lot of parameters to choose, we show limitations for it in terms of certificate complexity. This implies that we cannot use the method to prove better lower bounds for almost all known open problems in this area.
We present several families of total boolean functions which have exact quantum query com-plexity which is a constant multiple (between 1/2 and 2/3) of their classical query complexity, and show that optimal quantum algorithms for these functions cannot be obtained by simply computing parities of pairs of bits. We also characterise the model of nonadaptive exact quantum query complexity in terms of coding theory and completely characterise the query complexity of symmetric boolean functions in this context. These results were originally inspired by nu-merically solving the semidefinite programs characterising quantum query complexity for small problem sizes. We include numerical results giving the optimal success probabilities achievable by quantum algorithms computing all boolean functions on up to 4 bits, and all symmetric boolean functions on up to 6 bits. 1
We study quantum query complexity under several models of imperfect oracle access, and develop lower bounds through a common framework based on fractional block sensitivity (
fbs).
For a negligent oracle that applies the correct query with probability
1−p, we prove a lower bound in terms of
fbs(f,0n). We also show, perhaps surprisingly, that negligence need not destroy quantum speedups: any function
f can be transformed into a partial function
f′ whose negligent query complexity essentially preserves the quantum query complexity of
f. This gives partial functions with exponential quantum speedups even under negligent queries, and in particular rules out a general lower bound in terms of
fbs(f) for partial functions in this model.
For two other models, we obtain general lower bounds in terms of
fbs(f) for all Boolean functions. For hybrid algorithms using
Q coherent and
C classical queries, we prove the tradeoff
C+Q2=Ω(fbs(f)). For an IID dephasing noisy model where each query dephases the query-index register at rate
p independently, we prove
Ω(pfbs(f)) queries are necessary.
Finally, we introduce a broader family of time-varying dephasing models and identify a variational resource that is always lower bounded by
fbs(f). Computing this resource reduces to a convex optimization problem, providing a simple way to derive lower bounds for new noise schedules. As applications, we recover the hybrid and IID dephasing bounds and determine the query complexity of unstructured search when the dephasing rate grows over time.