# Quantum Complexity of Integration

@article{Novak2001QuantumCO, title={Quantum Complexity of Integration}, author={Erich Novak}, journal={J. Complex.}, year={2001}, volume={17}, pages={2-16} }

It is known that quantum computers yield a speed-up for certain discrete problems. Here we want to know whether quantum computers are useful for continuous problems. We study the computation of the integral of functions from the classical Holder classes Fk, ?d on 0, 1]d and define ? by ?=(k+?)/d. The known optimal orders for the complexity of deterministic and (general) randomized methods are comp(Fk, ?d, ?)???1/? and comprandom(Fk, ?d, ?)???2/(1+2?). For a quantum computer we prove…

## 82 Citations

### Improved bounds on the randomized and quantum complexity of initial-value problems

- Computer Science, MathematicsJ. Complex.
- 2005

### The Quantum Setting with Randomized Queries for Continuous Problems

- Computer Science, MathematicsQuantum Inf. Process.
- 2006

It is proved that for path integration the authors have an exponential improvement for the qubit complexity over the quantum setting with deterministic queries, which limits the power of quantum computation for continuous problems.

### Sharp error bounds on quantum Boolean summation in various settings

- Computer ScienceJ. Complex.
- 2004

### Quantum algorithms and complexity for certain continuous and related discrete problems

- Computer Science, Mathematics
- 2005

The thesis shows that in both the randomized and quantum settings the curse of dimensionality is vanquished, i.e., the minimal number of function evaluations and/or quantum queries required to compute an approximation depends only polynomially on the reciprocal of the desired accuracy and has a bound independent of the number of variables.

### On the quantum and randomized approximation of linear functionals on function spaces

- Mathematics, Computer ScienceQuantum Inf. Process.
- 2011

Lower bounds are provided on the power of quantum, randomized and deterministic algorithms for the exemplary problems, and some cases sharpness of the obtained results is compared.

### Path Integration on a Quantum Computer

- Computer ScienceQuantum Inf. Process.
- 2002

A lower bound is obtained for the minimal number of quantum queries which shows that this bound cannot be significantly improved, and it is proved that path integration on a quantum computer is tractable.

### Improved Upper Bounds on the Randomized and Quantum Complexity of Initial-Value Problems 1

- Computer Science, Mathematics
- 2004

This paper gives up the deterministic optimality of the basic algorithm, defining a new integral algorithm that is better suited for randomization and implementation of a quantum computer, and applies the optimal algorithms for summation of real numbers.

### Randomized and quantum algorithms yield a speed-up for initial-value problems

- Computer Science, MathematicsJ. Complex.
- 2004

### Randomized and quantum algorithms for solving initial-value problems in ordinary differential equations of order k

- Mathematics, Computer Science
- 2008

This paper considers two models of computation, the randomized model and the quantum model, and constructs almost optimal algorithms adjusted to scalar equations of higher order, without passing to systems of first order equations.

### Ju n 20 05 Improved Bounds on the Randomized and Quantum Complexity of Initial-Value Problems 1

- Computer Science, Mathematics
- 2005

It is proved, by defining new algorithms, that further improvement in upper bounds on the randomized and quantum complexity can be achieved, and in the Hölder class of right-hand side functions with r continuous bounded partial derivatives, the ε-complexity is shown to be O ( (1/ε) ) in the randomized setting, and O (1 /ε) on a quantum computer (up to logarithmic factors).

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