Lower Bounds on Quantum Query and Learning Graph Complexities

In this thesis we study the power of quantum query algorithms and learning graphs; the latter essentially being very specialized quantum query algorithms themselves. We almost exclusively focus on proving lower bounds for these computational models. First, we study lower bounds on learning graph complexity. We consider two types of learning graphs: adaptive and, more restricted, non-adaptive learning graphs. We express both adaptive and non-adaptive learning graph complexities of Boolean-valued functions (i.e., decision problems) as semidefinite minimization problems, and derive their dual problems. For various functions, we construct feasible solutions to these dual problems, thereby obtaining lower bounds on the learning graph complexity of the functions. Most notably, we prove an almost optimal Omega(n^(9/7)/sqrt(log n)) lower bound on the non-adaptive learning graph complexity of the Triangle problem. We also prove an Omega(n^(1-2^(k-2)/(2^k-1))) lower bound on the adaptive learning graph complexity of the k-Distinctness problem, which matches the complexity of the best known quantum query algorithm for this problem. Second, we construct optimal adversary lower bounds for various decision problems. Our main procedure for constructing them is to embed the adversary matrix into a larger matrix whose properties are easier to analyze. This embedding procedure imposes certain requirements on the size of the input alphabet. We prove optimal Omega(n^(1/3)) adversary lower bounds for the Collision and Set Equality problems, provided that the alphabet size is at least Omega(n^2). An optimal lower bound for Collision was previously proven using the polynomial method, while our lower bound for Set Equality is new. (An optimal lower bound for Set Equality was also independently and at about the same time proven by Zhandry using the polynomial method [arXiv, 2013].) We compare the power of non-adaptive learning graphs and quantum query algorithms that only utilize the knowledge on the possible positions of certificates in the input string. To do that, we introduce a notion of a certificate structure of a decision problem. Using the adversary method and the dual formulation of the learning graph complexity, we show that, for every certificate structure, there exists a decision problem possessing this certificate structure such that its non-adaptive learning graph and quantum query complexities differ by at most a constant multiplicative factor. For a special case of certificate structures, we construct a relatively general class of problems having this property. This construction generalizes the adversary lower bound for the k-Sum problem derived recently by Belovs and Spalek [ACM ITCS, 2013]. We also construct an optimal Omega(n^(2/3)) adversary lower bound for the Element Distinctness problem with minimal non-trivial alphabet size, which equals the length of the input. Due to the strict requirement on the alphabet size, here we cannot use the embedding procedure, and the construction of the adversary matrix heavily relies on the representation theory of the symmetric group. While an optimal lower bound for Element Distinctness using the polynomial method had been proven for any input alphabet, an optimal adversary construction was previously only known for alphabets of size at least Omega(n^2). Finally, we introduce the Enhanced Find-Two problem and we study its query complexity. The Enhanced Find-Two problem is, given n elements such that exactly k of them are marked, find two distinct marked elements using the following resources: (1) one initial copy of the uniform superposition over all marked elements, (2) an oracle that reflects across this superposition, and (3) an oracle that tests if an element is marked. This relational problem arises in the study of quantum proofs of knowledge. We prove that its query complexity is Theta(min{sqrt(n/k),sqrt(k)})

Adversary Lower Bound for Element Distinctness with Small Range

The Element Distinctness problem is to decide whether each character of an input string is unique. The quantum query complexity of Element Distinctness is known to be $\Theta(N^{2/3})$; the polynomial method gives a tight lower bound for any input alphabet, while a tight adversary construction was only known for alphabets of size $\Omega(N^2)$. We construct a tight $\Omega(N^{2/3})$ adversary lower bound for Element Distinctness with minimal non-trivial alphabet size, which equals the length of the input. This result may help to improve lower bounds for other related query problems.Comment: 22 pages. v2: one figure added, updated references, and minor typos fixed. v3: minor typos fixe

Symmetry-assisted adversaries for quantum state generation

We introduce a new quantum adversary method to prove lower bounds on the query complexity of the quantum state generation problem. This problem encompasses both, the computation of partial or total functions and the preparation of target quantum states. There has been hope for quite some time that quantum state generation might be a route to tackle the {\sc Graph Isomorphism} problem. We show that for the related problem of {\sc Index Erasure} our method leads to a lower bound of $\Omega(\sqrt N)$ which matches an upper bound obtained via reduction to quantum search on $N$ elements. This closes an open problem first raised by Shi [FOCS'02]. Our approach is based on two ideas: (i) on the one hand we generalize the known additive and multiplicative adversary methods to the case of quantum state generation, (ii) on the other hand we show how the symmetries of the underlying problem can be leveraged for the design of optimal adversary matrices and dramatically simplify the computation of adversary bounds. Taken together, these two ideas give the new result for {\sc Index Erasure} by using the representation theory of the symmetric group. Also, the method can lead to lower bounds even for small success probability, contrary to the standard adversary method. Furthermore, we answer an open question due to \v{S}palek [CCC'08] by showing that the multiplicative version of the adversary method is stronger than the additive one for any problem. Finally, we prove that the multiplicative bound satisfies a strong direct product theorem, extending a result by \v{S}palek to quantum state generation problems.Comment: 35 pages, 5 figure

Negative weights make adversaries stronger

The quantum adversary method is one of the most successful techniques for proving lower bounds on quantum query complexity. It gives optimal lower bounds for many problems, has application to classical complexity in formula size lower bounds, and is versatile with equivalent formulations in terms of weight schemes, eigenvalues, and Kolmogorov complexity. All these formulations rely on the principle that if an algorithm successfully computes a function then, in particular, it is able to distinguish between inputs which map to different values. We present a stronger version of the adversary method which goes beyond this principle to make explicit use of the stronger condition that the algorithm actually computes the function. This new method, which we call ADV+-, has all the advantages of the old: it is a lower bound on bounded-error quantum query complexity, its square is a lower bound on formula size, and it behaves well with respect to function composition. Moreover ADV+- is always at least as large as the adversary method ADV, and we show an example of a monotone function for which ADV+-(f)=Omega(ADV(f)^1.098). We also give examples showing that ADV+- does not face limitations of ADV like the certificate complexity barrier and the property testing barrier.Comment: 29 pages, v2: added automorphism principle, extended to non-boolean functions, simplified examples, added matching upper bound for AD