A fast quantum mechanical algorithm for database search
STOC '96 Proceedings of the twenty-eighth annual ACM symposium on Theory of computing
Strengths and Weaknesses of Quantum Computing
SIAM Journal on Computing
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
An Exact Quantum Polynomial-Time Algorithm for Simon's Problem
ISTCS '97 Proceedings of the Fifth Israel Symposium on the Theory of Computing Systems (ISTCS '97)
Classical and Quantum Computation
Classical and Quantum Computation
Quantum Speed-Up of Markov Chain Based Algorithms
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
Quantum information processing in continuous time
Quantum information processing in continuous time
Simulating sparse Hamiltonians with star decompositions
TQC'10 Proceedings of the 5th conference on Theory of quantum computation, communication, and cryptography
Limitations on the simulation of non-sparse hamiltonians
Quantum Information & Computation
Hamiltonian simulation using linear combinations of unitary operations
Quantum Information & Computation
Efficient algorithms for universal quantum simulation
RC'13 Proceedings of the 5th international conference on Reversible Computation
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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 elements in a column) and the evolution time t, whereas previous methods had complexity scaling as D4 and were superlinear in t. We also consider the task of implementing an arbitrary unitary operation given a black-box description of its matrix elements. Whereas standard methods for performing an explicitly specified N × N unitary operation use Õ(N2) elementary gates, we show that a black-box unitary can be performed with bounded error using O(N2/3(log logN)4/3) queries to its matrix elements. In fact, except for pathological cases, it appears that most unitaries can be performed with only Õ(√N) queries, which is optimal.