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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)
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 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)
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 use the class of commuting quantum computations known as IQP (instantaneous quantum polynomial time) to strengthen the conjecture that quantum computers are hard to simulate classically. We show that, if either of two plausible average-case hardness conjectures holds, then IQP computations are hard to simulate classically up to constant additive error.(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)
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>&psi;</i>&rang; whose maximum overlap with a product state is 1 &minus; <i>&#949;</i>, the test passes with probability 1 &minus; <i>&Theta;</i>(<i>&#949;</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)