A fast quantum mechanical algorithm for database search
STOC '96 Proceedings of the twenty-eighth annual ACM symposium on Theory of computing
A Course in Digital Signal Processing
A Course in Digital Signal Processing
Adiabatic quantum state generation and statistical zero knowledge
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
Quantum Speed-Up of Markov Chain Based Algorithms
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
An Introduction to Quantum Computing
An Introduction to Quantum Computing
Adiabatic Quantum Computation is Equivalent to Standard Quantum Computation
SIAM Journal on Computing
Quantum Walk Algorithm for Element Distinctness
SIAM Journal on Computing
Efficient discrete-time simulations of continuous-time quantum query algorithms
Proceedings of the forty-first annual ACM symposium on Theory of computing
Analysis and synthesis of attractive quantum Markovian dynamics
Automatica (Journal of IFAC)
SIAM Journal on Computing
Proceedings of the 3rd Innovations in Theoretical Computer Science Conference
ACM Transactions on Computation Theory (TOCT) - Special issue on innovations in theoretical computer science 2012
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A computation in adiabatic quantum computing is implemented by traversing a path of nondegenerate eigenstates of a continuous family of Hamiltonians. We introduce a method that traverses a discretized form of the path: At each step we apply the instantaneous Hamiltonian for a random time. The resulting decoherence approximates a projective measurement onto the desired eigenstate, achieving a version of the quantum Zeno effect. If negative evolution times can be implemented with constant overhead, then the average absolute evolution time required by our method is O(L2/Δ) for constant error probability, where L is the length of the path of eigenstates and Δ is the minimum spectral gap of the Hamiltonian. The dependence of the cost on Δ is optimal. Making explicit the dependence on the path length is useful for cases where L is much less than the general bound. The complexity of our method has a logarithmic improvement over previous algorithms of this type. The same cost applies to the discrete-time case, where a family of unitary operators is given and each unitary and its inverse can be used. Restriction to positive evolution times incurs an error that decreases exponentially with the cost. Applications of this method to unstructured search and quantum sampling are considered. In particular, we discuss the quantum simulated annealing algorithm for solving combinatorial optimization problems. This algorithm provides a quadratic speed-up in the gap of the stochastic matrix over its classical counterpart implemented via Markov chain Monte Carlo.