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We investigate the notion of quantum chromatic number of a graph, which is the minimal number of colours necessary in a protocol in which two separated provers can convince an interrogator with certainty that they have a colouring of the graph. After discussing this notion from first principles, we go on to establish relations with the clique number and… (More)

We present several families of total boolean functions which have exact quantum query complexity 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… (More)

The calculation of ground-state energies of physical systems can be formalised as the k-local Hamiltonian problem, which is the natural quantum analogue of classical constraint satisfaction problems. One way of making the problem more physically meaningful is to restrict the Hamiltonian in question by picking its terms from a fixed set S. Examples of such… (More)

We study the complexity of the popular one player combinatorial game known as Flood-It. In this game the player is given an n×n board of tiles where each tile is allocated one of c colours. The goal is to make the colours of all tiles equal via the shortest possible sequence of flooding operations. In the standard version, a flooding operation consists of… (More)

We consider two combinatorial problems. The first we call " search with wildcards " : given an unknown n-bit string x, and the ability to check whether any subset of the bits of x is equal to a provided query string, the goal is to output x. We give an optimal O(√ n) quantum query algorithm for search with wildcards. Rather than using amplitude… (More)

An XOR function is a function of the form g(x, y) = f (x ⊕ y), for some boolean function f on n bits. We study the quantum and classical communication complexity of XOR functions. In the case of exact protocols, we completely characterise one-way communication complexity for all f. We also show that, when f is monotone, g's quantum and classical… (More)

We give a test that can distinguish efficiently between product states of <i>n</i> quantum systems and states that are far from product. If applied to a state |<i>ψ</i>⟩ whose maximum overlap with a product state is 1 − <i>ε</i>, the test passes with probability 1 − <i>Θ</i>(<i>ε</i>), regardless of <i>n</i> or the local… (More)

We give a test that can distinguish efficiently between product states of n quantum systems and states which are far from product. If applied to a state psi whose maximum overlap with a product state is 1-epsilon, the test passes with probability 1-Theta(epsilon), regardless of n or the local dimensions of the individual systems. The test uses two copies of… (More)

In this paper we introduce the study of quantum boolean functions, which are unitary operators f whose square is the identity: f 2 = I. We describe several generalisations of well-known results in the theory of boolean functions, including quantum property testing; a quantum version of the Goldreich-Levin algorithm for finding the large Fourier coefficients… (More)

This work studies the quantum query complexity of Boolean functions in a scenario where it is only required that the query algorithm succeeds with a probability strictly greater than 1/2. We show that, just as in the communication complexity model, the unbounded error quantum query complexity is exactly half of its classical counterpart for any (partial or… (More)