Information-based complexity
The computational complexity of differential and integral equations: an information-based approach
The computational complexity of differential and integral equations: an information-based approach
Random number generation and quasi-Monte Carlo methods
Random number generation and quasi-Monte Carlo methods
Optimal linear randomized methods for linear operators in Hilbert spaces
Journal of Complexity
There exists a linear problem with infinite combinatory complexity
Journal of Complexity
SIAM Journal on Computing
Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer
SIAM Journal on Computing
The quantum query complexity of approximating the median and related statistics
STOC '99 Proceedings of the thirty-first annual ACM symposium on Theory of computing
Complexity and information
Delayed curse of dimension for Gaussian integration
Journal of Complexity
Quantum complexity of integration
Journal of Complexity
Quantum computation and quantum information
Quantum computation and quantum information
Quantum summation with an application to integration
Journal of Complexity
Quantum integration in Sobolev classes
Journal of Complexity
Path Integration on a Quantum Computer
Quantum Information Processing
Quantum Lower Bounds by Polynomials
FOCS '98 Proceedings of the 39th Annual Symposium on Foundations of Computer Science
Quantum approximation I. Embeddings of finite-dimensional Lp spaces
Journal of Complexity
Quantum approximation II. Sobolev embeddings
Journal of Complexity
Sharp error bounds on quantum Boolean summation in various settings
Journal of Complexity
Average case quantum lower bounds for computing the Boolean mean
Journal of Complexity
Classical and Quantum Complexity of the Sturm--Liouville Eigenvalue Problem
Quantum Information Processing
Improved bounds on the randomized and quantum complexity of initial-value problems
Journal of Complexity
On the complexity of the multivariate Sturm--Liouville eigenvalue problem
Journal of Complexity
Hi-index | 0.00 |
The standard setting of quantum computation for continuous problems uses deterministic queries and the only source of randomness for quantum algorithms is through measurement. Without loss of generality we may consider quantum algorithms which use only one measurement. This setting is related to the worst case setting on a classical computer in the sense that the number of qubits needed to solve a continuous problem must be at least equal to the logarithm of the worst case information complexity of this problem. Since the number of qubits must be finite, we cannot solve continuous problems on a quantum computer with infinite worst case information complexity. This can even happen for continuous problems with small randomized complexity on a classical computer. A simple example is integration of bounded continuous functions. To overcome this bad property that limits the power of quantum computation for continuous problems, we study the quantum setting in which randomized queries are allowed. This type of query is used in Shor's algorithm. The quantum setting with randomized queries is related to the randomized classical setting in the sense that the number of qubits needed to solve a continuous problem must be at least equal to the logarithm of the randomized information complexity of this problem. Hence, there is also a limit to the power of the quantum setting with randomized queries since we cannot solve continuous problems with infinite randomized information complexity. An example is approximation of bounded continuous functions. We study the quantum setting with randomized queries for a number of problems in terms of the query and qubit complexities defined as the minimal number of queries/qubits needed to solve the problem to within 驴 by a quantum algorithm. We prove that for path integration we have an exponential improvement for the qubit complexity over the quantum setting with deterministic queries.