# The Sturm-Liouville Eigenvalue Problem and NP-Complete Problems in the Quantum Setting with Queries

@article{Papageorgiou2007TheSE, title={The Sturm-Liouville Eigenvalue Problem and NP-Complete Problems in the Quantum Setting with Queries}, author={Anargyros Papageorgiou and Henryk Wozniakowski}, journal={Quantum Information Processing}, year={2007}, volume={6}, pages={101-120} }

We show how a number of NP-complete as well as NP-hard problems can be reduced to the Sturm-Liouville eigenvalue problem in the quantum setting with queries. We consider power queries which are derived from the propagator of a system evolving with a Hamiltonian obtained from the discretization of the Sturm-Liouville operator. We use results of our earlier paper concering the complexity of the Sturm-Liouville eigenvalue problem. We show that the number of power queries as well the number of…

## 8 Citations

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## References

SHOWING 1-10 OF 27 REFERENCES

### 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.

### 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.

### Quantum Summation with an Application to Integration

- Computer ScienceJ. Complex.
- 2002

Developing 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 proving lower bounds showing that the proposed algorithms are, in many cases, optimal within the setting of quantum computing.

### Tractability of Approximation for Weighted Korobov Spaces on Classical and Quantum Computers

- Mathematics, Computer ScienceFound. Comput. Math.
- 2004

The worst case, randomized, and quantum settings are considered and it is proved that strong tractability and tractability in the class $\lall$ are equivalent and this holds under the same assumption as for the class £lall in the worst case setting.

### Quantum complexity theory

- Computer ScienceSTOC
- 1993

This paper gives the first formal evidence that quantum Turing machines violate the modern (complexity theoretic) formulation of the Church--Turing thesis, and proves that bits of precision suffice to support a step computation.

### Eigenvector approximation leading to exponential speedup of quantum eigenvalue calculation.

- Computer SciencePhysical review letters
- 2003

The method can be applied when solving continuous Hermitian eigenproblems, e.g., the Schrödinger equation, on a discrete grid, and efficiently construct an approximation of the same eigenvector on a fine grid.

### Strengths and Weaknesses of Quantum Computing

- Computer ScienceSIAM J. Comput.
- 1997

It is proved that relative to an oracle chosen uniformly at random with probability 1 the class $\NP$ cannot be solved on a quantum Turing machine (QTM) in time $o(2^{n/2})$.

### Quantum Algorithm Providing Exponential Speed Increase for Finding Eigenvalues and Eigenvectors

- Physics, Computer Science
- 1999

A new polynomial time quantum algorithm is described that uses the quantum fast Fourier transform to find eigenvalues and eigenvectors of a local Hamiltonian and that can be applied in cases for which all known classical algorithms require exponential time.

### Quantum approximation I. Embeddings of finite-dimensional Lp spaces

- Mathematics, Computer ScienceJ. Complex.
- 2004

### 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…