In this thesis the problem of computing the sum of a string of arbitrary finite length drawn from an arbitrary finite alphabet is treated. The resource considered is the number of queries to some oracle which hides the string and gives access to each of its digits. The sum of a string is defined as adding all of the string's digits together modulo the alphabet size. \n \nClassically, this problem is straightforward as any less than a number of queries equal to the string length reveals no useful information about the sum of the string. In the quantum information setting however, things are not so clear. When the alphabet size is equal to two, the problem becomes finding the parity of a bit string. This is a seminal result in quantum computation that allows a correct answer with certainty by making only half the queries that are classically needed. As the alphabet size increases beyond two however, less is known. There is an algorithm by Meyer and Pommersheim which computes the sum in this general setting with probability of success: $\\min\\left\\lbrace\\frac{\\floor{\\frac{n}{n-q}}}{k}, 1\\right\\rbrace$ where $n$ is the string length, $k$ is the alphabet size, and $q$ is the number of queries made. This algorithm has probability of success slightly above guessing when the number of queries are half the string length, and perfect probability of success when $n-1$ queries are made. The question dealt with in this thesis is whether this algorithm is optimal for the general sum case. \n \nThe problem is expressed as a semidefinite program, given for all instances. The instance for strings of length two and algorithms making a single query is solved and a proof is given. Significant insight into the multi-query case is also provided.
In this thesis the problem of computing the sum of a string of arbitrary finite length drawn from an arbitrary finite alphabet is treated. The resource considered is the number of queries to some oracle which hides the string and gives access to each of its digits. The sum of a string is defined as adding all of the string's digits together modulo the alphabet size. \n \nClassically, this problem is straightforward as any less than a number of queries equal to the string length reveals no useful information about the sum of the string. In the quantum information setting however, things are not so clear. When the alphabet size is equal to two, the problem becomes finding the parity of a bit string. This is a seminal result in quantum computation that allows a correct answer with certainty by making only half the queries that are classically needed. As the alphabet size increases beyond two however, less is known. There is an algorithm by Meyer and Pommersheim which computes the sum in this general setting with probability of success: minleftlbracefracfloorfracnn−qk,1rightrbrace where n is the string length, k is the alphabet size, and q is the number of queries made. This algorithm has probability of success slightly above guessing when the number of queries are half the string length, and perfect probability of success when n−1 queries are made. The question dealt with in this thesis is whether this algorithm is optimal for the general sum case. \n \nThe problem is expressed as a semidefinite program, given for all instances. The instance for strings of length two and algorithms making a single query is solved and a proof is given. Significant insight into the multi-query case is also provided.
Consider a quantum computer in combination with a binary oracle of domain size N. It is shown how N/2+/spl radic/N calls to the oracle are sufficient to guess the whole content of the oracle (being an N bit string) with probability greater than 95%. This contrasts the power of classical computers which would require N calls to achieve the same task. From this result it follows that any function with the N bits of the oracle as input can be calculated using N/2+/spl radic/N queries if we allow a small probability of error. It is also shown that this error probability can be made arbitrary small by using N/2+O(/spl radic/N) oracle queries. In the second part of the article 'approximate interrogation' is considered. This is when only a certain fraction of the N oracle bits are requested. Also for this scenario does the quantum algorithm outperform the classical protocols. An example is given where a quantum procedure with N/10 queries returns a string of which 80% of the bits are correct. Any classical protocol would need 6N/10 queries to establish such a correctness ratio.
In the search with wildcards problem [Ambainis, Montanaro, Quantum Inf.~Comput.'14], one's goal is to learn an unknown bit-string x∈{−1,1}n. An algorithm may, at unit cost, test equality of any subset of the hidden string with a string of its choice. Ambainis and Montanaro showed a quantum algorithm of cost O(nlogn) and a near-matching lower bound of Ω(n). Belovs [Comput.~Comp.'15] subsequently showed a tight O(n) upper bound.
We consider a natural generalization of this problem, parametrized by a subset Q⊆2[n], where an algorithm may test whether xS=b for an arbitrary S∈Q and b∈{−1,1}S of its choice, at unit cost. We show near-tight bounds when Q is any of the following collections: bounded-size sets, contiguous blocks, prefixes, and only the full set.
All of these results are derived using a framework that we develop. Using symmetries of the task at hand we show that the quantum query complexity of learning x is characterized, up to a constant factor, by an optimization program, which is succinctly described as follows: `maximize over all odd functions f:{−1,1}n→R the ratio of the maximum value of f to the maximum (over T∈Q) standard deviation of f on a subcube whose free variables are exactly T.'
To the best of our knowledge, ours is the first work to use the primal version of the negative-weight adversary bound (which is a maximization program typically used to show lower bounds) to show new quantum query upper bounds without explicitly resorting to SDP duality.
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 investigate the query complexity of quantum biased oracles.Supposϵ that the biased oracles answer queries correctly with probability at least 1/2+ϵ .Given such an oracle, we present an algorithm to simulate a single query to an oracle that answers queries COαecuy with probability at least 2/3, using 0(1/ϵ) queries to the given oracle.For searching problems, combining the algorithm with a known result.we can obtain an optimal algorithm.The simulating algorithm works effectively when we know the value of ϵ .We also consider the situation where no knowledge about ϵ is given.
We study the query complexity of Weak Parity: the problem of computing the parity of an n-bit input string, where one only has to succeed on a 1/2+eps fraction of input strings, but must do so with high probability on those inputs where one does succeed. It is well-known that n randomized queries and n/2 quantum queries are needed to compute parity on all inputs. But surprisingly, we give a randomized algorithm for Weak Parity that makes only O(n/log^0.246(1/eps)) queries, as well as a quantum algorithm that makes only O(n/sqrt(log(1/eps))) queries. We also prove a lower bound of Omega(n/log(1/eps)) in both cases; and using extremal combinatorics, prove lower bounds of Omega(log n) in the randomized case and Omega(sqrt(log n)) in the quantum case for any eps>0. We show that improving our lower bounds is intimately related to two longstanding open problems about Boolean functions: the Sensitivity Conjecture, and the relationships between query complexity and polynomial degree.
In the first part of the thesis we construct a new, simple framework which amplifies to a constant the success probability of any abstract search algorithm. The total query complexity is given by the quantum hitting time of the resulting operator, which we show that it is of the same order as the quantum hitting time of the original algorithm. As a major application of our framework, we show that for any reversible walk P and a single marked state, the quantum walk corresponding to P can find the solution using a number of queries that is quadratically smaller than the classical hitting time of P. Our algorithm is more general and simpler to implement than the solution known previously in the literature (Krovi, Magniez, Ozols, and Roland, 2015), which was developed specifically for quantum walks; we also prove that, for the particular case of quantum walks, we can embed their algorithm into our framework, thus simulating it exactly. Finally, we show that we can implement amplitude amplification using our tool. In the second part of the thesis we give a new lower bound in the query model which proves that Grover's algorithm for unordered searching is exactly optimal. Similar to existing methods for proving lower bounds, we bound the amount of information we can gain from a single oracle query, but we bound this information in terms of angles. This allows our proof to be simple, self-contained, based on only elementary mathematics, capturing our intuition, while obtaining at the same an exact bound. We then turn our attention to non-adaptive algorithms for the same problem of searching an unordered set. In this model, we obtain a lower bound and we give an algorithm which matches the lower bound, thus showing that the lower bound is exactly optimal.