The computational complexity of simultaneous diophantine approximation problems
SIAM Journal on Computing
Theory of linear and integer programming
Theory of linear and integer programming
Computerized patient information system in a psychiatric unit: five-year experience
Journal of Medical Systems
Fault-tolerant quantum computation with constant error
STOC '97 Proceedings of the twenty-ninth annual ACM symposium on Theory of computing
Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer
SIAM Journal on Computing
The art of computer programming, volume 2 (3rd ed.): seminumerical algorithms
The art of computer programming, volume 2 (3rd ed.): seminumerical algorithms
A method for obtaining digital signatures and public-key cryptosystems
Communications of the ACM
Approximating Good Simultaneous Diophantine Approximations Is Almost NP-Hard
MFCS '96 Proceedings of the 21st International Symposium on Mathematical Foundations of Computer Science
The Hidden Subgroup Problem and Eigenvalue Estimation on a Quantum Computer
QCQC '98 Selected papers from the First NASA International Conference on Quantum Computing and Quantum Communications
Using the inhomogeneous simultaneous approximation problem for cryptographic design
AFRICACRYPT'11 Proceedings of the 4th international conference on Progress in cryptology in Africa
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While quantum computers might speed up in principle certain computations dramatically, in practice, though quantum computing technology is still in its infancy. Even we cannot clearly envision at present what the hardware of that machine will be like. Nevertheless, we can be quite confident that it will be much easier to build any practical quantum computer operating on a few number of quantum bits rather than one operating on a huge number of quantum bits. It is therefore of big practical impact to use the resource of quantum bits very spare, i.e., to find quantum algorithms which use as few as possible quantum bits. Here, we present a method to reduce the number of actually needed qubits in Shor's algorithm to factor a composite number N. Exploiting the inherent probabilism of quantum computation we are able to substitute the continued fraction algorithm to find a certain unknown fraction by a simultaneous Diophantine approximation. While the continued fraction algorithm is able to find a Diophantine approximation to a single known fraction with a denominator greater than N2, our simultaneous Diophantine approximation method computes in polynomial time unusually good approximations to known fractions with a denominator of size N1+驴, where 驴 is allowed to be an arbitrarily small positive constant. As these unusually good approximations are almost unique we are able to recover an unknown denominator using fewer qubits in the quantum part of our algorithm.