D 1.4 Black-box complexity: limits of evolution

D 1.4 Black-box complexity: limits of evolution

Black-Box Complexity: Breaking the O(n logn) Barrier of LeadingOnes

We show that the unrestricted black-box complexity of the n-dimensional XORand permutation-invariant LeadingOnes function class is O(n log(n)/ log logn). This shows that the recent natural looking O(n logn) bound is not tight. The black-box optimization algorithm leading to this bound can be implemented in a way that only 3-ary unbiased variation operators are used. Hence our bound is also vali...

Black-box hamiltonian simulation and unitary implementation

We present general methods for simulating black-box Hamiltonians using quantum walks. These techniques have two main applications: simulating sparse Hamiltonians and implementing black-box unitary operations. In particular, we give the best known simulation of sparse Hamiltonians with constant precision. Our method has complexity linear in both the sparseness D (the maximum number of nonzero el...

Complexity Theory for Discrete Black-Box Optimization Heuristics

A predominant topic in the theory of evolutionary algorithms and, more generally, theory of randomized black-box optimization techniques is running time analysis. Running time analysis aims at understanding the performance of a given heuristic on a given problem by bounding the number of function evaluations that are needed by the heuristic to identify a solution of a desired quality. As in gen...

The computational power of normalizer circuits over black-box groups

This work presents a precise connection between Clifford circuits, Shor’s factoring algorithm and several other famous quantum algorithms with exponential quantum speed-ups for solving Abelian hidden subgroup problems. We show that all these different forms of quantum computation belong to a common new restricted model of quantum operations that we call black-box normalizer circuits. To define ...