Asymptotic enumeration methods
Handbook of combinatorics (vol. 2)
Quantum lower bounds by polynomials
Journal of the ACM (JACM)
Quantum lower bounds by quantum arguments
Journal of Computer and System Sciences - Special issue on STOC 2000
Adiabatic quantum state generation and statistical zero knowledge
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
Exponential algorithmic speedup by a quantum walk
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
Quantum information processing in continuous time
Quantum information processing in continuous time
Quantum Algorithms for Hidden Nonlinear Structures
FOCS '07 Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science
Simulating sparse Hamiltonians with star decompositions
TQC'10 Proceedings of the 5th conference on Theory of quantum computation, communication, and cryptography
Black-box hamiltonian simulation and unitary implementation
Quantum Information & Computation
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The problem of simulating sparse Hamiltonians on quantum computers is well studied. The evolution of a sparse N × N Hamiltonian H for time t can be simulated using O(||Ht|| poly(logN)) operations, which is essentially optimal due to a no-fast-forwarding theorem. Here, we consider non-sparse Hamiltonians and show significant limitations ontheir simulation. We generalize the no-fast-forwarding theorem to dense Hamiltonians, ruling out generic simulations taking time o(||Ht||), even though ||H|| is not a uniquemeasure of the size of a dense Hamiltonian H. We also present a stronger limitationruling out the possibility of generic simulations taking time poly(||Ht||; logN), showingthat known simulations based on discrete-time quantum walk cannot be dramatically improved in general. On the positive side, we show that some non-sparse Hamiltonianscan be simulated efficiently, such as those with graphs of small arboricity.