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Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer
- P. Shor
- Computer ScienceSIAM Rev.
- 30 August 1995
Efficient randomized algorithms are given for factoring integers and finding discrete logarithms, two problems that are generally thought to be hard on classical computers and that have been used as the basis of several proposed cryptosystems.
Algorithms for quantum computation: discrete logarithms and factoring
- P. Shor
- Computer ScienceProceedings 35th Annual Symposium on Foundations…
- 20 November 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.
Quantum error correction via codes over GF(4)
- A. Calderbank, E. Rains, P. Shor, N. Sloane
- Computer Science, PhysicsProceedings of IEEE International Symposium on…
- 6 August 1996
In the present paper the problem of finding quantum-error-correcting codes is transformed into one of finding additive codes over the field GF(4) which are self-orthogonal with respect to a certain trace inner product.
Applications of random sampling in computational geometry, II
Asymptotically tight bounds for (≤k)-sets are given, which are certain halfspace partitions of point sets, and a simple proof of Lee's bounds for high-order Voronoi diagrams is given.
Quantum Error Correction and Orthogonal Geometry
- A. Calderbank, E. Rains, P. Shor, N. J. A. S. A. Research, Institute for Defense Analysis
- 9 May 1996
A group theoretic framework is introduced that simplifies the description of known quantum error-correcting codes and greatly facilitates the construction of new examples. Codes are given which map 3…
The Capacity of a Quantum Channel for Simultaneous Transmission of Classical and Quantum Information
An expression is derived characterizing the set of admissible rate pairs for simultaneous transmission of classical and quantum information over a given quantum channel, generalizing both the…
Quantum nonlocality without entanglement
We exhibit an orthogonal set of product states of two three-state particles that nevertheless cannot be reliably distinguished by a pair of separated observers ignorant of which of the states has…
Entanglement-assisted capacity of a quantum channel and the reverse Shannon theorem
- C. H. Bennett, P. Shor, J. Smolin, Ashish V. Thapliyal
- Computer ScienceIEEE Trans. Inf. Theory
- 8 June 2001
In the classical analog of entanglement-assisted communication - communication over a discrete memoryless channel (DMC) between parties who share prior random information - one parameter is sufficient, i.e., that in the presence of prior shared random information, all DMCs of equal capacity can simulate one another with unit asymptotic efficiency.
Geometric applications of a matrix-searching algorithm
The Θ(m) bound on finding the maxima of wide totally monotone matrices is used to speed up several geometric algorithms by a factor of logn.
Unextendible product bases and bound entanglement
An unextendible product basis( UPB) for a multipartite quantum system is an incomplete orthogonal product basis whose complementary subspace contains no product state. We give examples of UPBs, and…