# Quantum Summation with an Application to Integration

@article{Heinrich2002QuantumSW, title={Quantum Summation with an Application to Integration}, author={Stefan Heinrich}, journal={J. Complex.}, year={2002}, volume={18}, pages={1-50} }

We study summation of sequences and integration in the quantum model of computation. We develop quantum algorithms for computing the mean of sequences that satisfy a p-summability condition and for integration of functions from Lebesgue spaces Lp(0, 1]d), and analyze their convergence rates. We also prove lower bounds showing that the proposed algorithms are, in many cases, optimal within the setting of quantum computing. This extends recent results of G. Brassard et al. (2000, “Quantum…

## 119 Citations

### 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.

### Interner Bericht Optimal Summation and Integration by Deterministic , Randomized , and Quantum Algorithms UNIVERSITÄT KAISERSLAUTERN

- Mathematics
- 2017

We survey old and new results about optimal algorithms for summation of finite sequences and for integration of functions from Hölder or Sobolev spaces. First we discuss optimal deterministic and…

### Optimal Summation and Integration by Deterministic, Randomized, and Quantum Algorithms

- Mathematics
- 2002

We survey old and new results about optimal algorithms for summation of finite sequences and for integration of functions from Holder or Sobolev spaces. First we discuss optimal deterministic and…

### 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.

### Complexity of multivariate Feynman-Kac path integration in randomized and quantum settings

- Computer Science, Mathematics
- 2004

It is shown that in both the randomized and quantum settings the curse of dimensionality is vanquished, i.e., the number of function evaluations and/or quantum queries required to compute an e-approximation has a bound independent of d and depending polynomially on 1/e.

### Quantum Algorithms and Complexity for Numerical Problems

- Computer Science
- 2011

This thesis designs an adiabatic quantum algorithm for the counting problem, and derives the optimal order of convergence, given e and the cost of the resulting algorithm, which is close to the best lower bound on query complexity known for the classical PAC learning model.

### Quantum Sub-Gaussian Mean Estimator

- Mathematics, Computer ScienceESA
- 2021

We present a new quantum algorithm for estimating the mean of a real-valued random variable obtained as the output of a quantum computation. Our estimator achieves a nearly-optimal quadratic speedup…

### Classical and Quantum Complexity of the Sturm–Liouville Eigenvalue Problem

- Computer Science, MathematicsQuantum Inf. Process.
- 2005

A formula is derived that relates the Sturm–Liouville eigenvalue problem to a weighted integration problem, which allows us to solve them with a polylog number of power queries and the lower bound on the number of quantum queries is proven.

### A Survey of Quantum Computing for Finance

- Computer Science
- 2022

A comprehensive summary of the state of the art of quantum computing for financial applications, with particular emphasis on stochastic modeling, optimization, and machine learning, describing how these solutions, adapted to work on a quantum computer, can potentially help to solve financial problems more efficiently and accurately.

## References

SHOWING 1-10 OF 24 REFERENCES

### Optimal Summation and Integration by Deterministic, Randomized, and Quantum Algorithms

- Mathematics
- 2002

We survey old and new results about optimal algorithms for summation of finite sequences and for integration of functions from Holder or Sobolev spaces. First we discuss optimal deterministic and…

### An Introduction to Quantum Computing Algorithms

- Computer Science
- 2000

This monograph is a good self-contained introductory resource for newcomers to the field of quantum computing algorithms, as well as a useful self-study guide for the more specialized scientist, mathematician, graduate student, or engineer.

### Quantum lower bounds by polynomials

- Computer ScienceProceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280)
- 1998

This work examines the number T of queries that a quantum network requires to compute several Boolean functions on {0,1}/sup N/ in the black-box model and gives asymptotically tight characterizations of T for all symmetric f in the exact, zero-error, and bounded-error settings.

### INTRODUCTION TO QUANTUM ALGORITHMS

- Computer Science, Physics
- 2000

These notes discuss the quantum algorithms we know of that can solve problems significantly faster than the corresponding classical algorithms. So far, we have only discovered a few techniques which…

### Tight bounds on quantum searching

- Computer Science
- 1996

A lower bound on the efficiency of any possible quantum database searching algorithm is provided and it is shown that Grover''s algorithm nearly comes within a factor 2 of being optimal in terms of the number of probes required in the table.

### Quantum Counting

- PhysicsICALP
- 1998

This work generalizes the Grover iteration in the light of a concept called amplitude amplification, and shows that the quadratic speedup obtained by the quantum searching algorithm over classical brute force can still be obtained for a large family of search problems for which good classical heuristics exist.

### Quantum Complexity of Integration

- Computer Science, MathematicsJ. Complex.
- 2001

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…

### A framework for fast quantum mechanical algorithms

- Computer Science, PhysicsSTOC '98
- 1998

The sqrt(N) step quantum search algorithm is an immediate consequence of a framework for the design and analysis of quantum mechanical algorithms, and several other search-type applications are presented.

### Algorithms for quantum computation: discrete logarithms and factoring

- Computer ScienceProceedings 35th Annual Symposium on Foundations of Computer Science
- 1994

Las Vegas algorithms for finding discrete logarithms and factoring integers on a quantum computer that take a number of steps which is polynomial in the input size, e.g., the number of digits of the integer to be factored are given.