Many Known Quantum Algorithms Are Optimal: Symmetry-Based Proofs

Many quantum algorithms have been proposed which are drastically more efficient that the best of the non-quantum algorithms for solving the same problems. A natural question is: are these quantum algorithms already optimal – in some reasonable sense – or they can be further improved? In this paper, we review recent results showing that many known quantum algorithms are actually optimal. Several of these results are based on appropriate invariances (symmetries). Specifically, we show that the following algorithms are optimal: Grover’s algorithm for fast search in an unsorted array, teleportation algorithm – which is important for parallel quantum computations, and quantum annealing optimization algorithm. This covers many algorithms related to quantum computing. We also mention that algorithms for quantum communication and Deutsch-Josza algorithm – for fast checking whether a bit affect computation results – are optimal. In all these cases, optimality is shown not just for one specific optimality criterion, but for all possible optimality criteria that satisfy the natural invariance requirement.