SIAM Journal on Computing
On the Power of Quantum Computation
SIAM Journal on Computing
Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer
SIAM Journal on Computing
Quantum algorithms for solvable groups
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
A polynomial-time approximation algorithm for the permanent of a matrix with non-negative entries
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
Polynomial-time quantum algorithms for Pell's equation and the principal ideal problem
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
Quantum computation and quantum information
Quantum computation and quantum information
Exponential algorithmic speedup by a quantum walk
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
Approximate Counting and Quantum Computation
Combinatorics, Probability and Computing
On the impossibility of a quantum sieve algorithm for graph isomorphism
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Quantum Information Processing
BQP and the polynomial hierarchy
Proceedings of the forty-second ACM symposium on Theory of computing
Measurement-based and universal blind quantum computation
SFM'10 Proceedings of the Formal methods for quantitative aspects of programming languages, and 10th international conference on School on formal methods for the design of computer, communication and software systems
On the Impossibility of a Quantum Sieve Algorithm for Graph Isomorphism
SIAM Journal on Computing
A promiseBQP-complete string rewriting problem
Quantum Information & Computation
Permutational quantum computing
Quantum Information & Computation
Quantum algorithm design using dynamic learning
Quantum Information & Computation
The Jones polynomial: quantum algorithms and applications in quantum complexity theory
Quantum Information & Computation
Efficient quantum circuits for approximating the Jones polynomial
Quantum Information & Computation
Estimating Jones and Homfly polynomials with one clean qubit
Quantum Information & Computation
Quantum automata, braid group and link polynomials
Quantum Information & Computation
Estimating Jones polynomials is a complete problem for one clean qubit
Quantum Information & Computation
Gravitational topological quantum computation
UC'07 Proceedings of the 6th international conference on Unconventional Computation
Quantum algorithms for invariants of triangulated manifolds
Quantum Information & Computation
Exponential quantum speed-ups are generic
Quantum Information & Computation
On upper bounds for toroidal mosaic numbers
Quantum Information Processing
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The Jones polynomial, discovered in 1984 [18], is an important knot invariant in topology. Among its many connections to various mathematical and physical areas, it is known (due to Witten [32]) to be intimately connected to Topological Quantum Field Theory (TQFT). The works of Freedman, Kitaev, Larsen and Wang [13, 14] provide an efficient simulation of TQFT by a quantum computer, and vice versa. These results implicitly imply the existence of an efficient quantum algorithm that provides a certain additive approximation of the Jones polynomial at the fifth root of unity, e2π i/5, and moreover, that this problem is BQP-complete. Unfortunately, this important algorithm was never explicitly formulated. Moreover, the results in [13, 14] are heavily based on TQFT, which makes the algorithm essentially inaccessible to computer scientists.We provide an explicit and simple polynomial quantum algorithm to approximate the Jones polynomial of an n strands braid with m crossings at any primitive root of unity e2π i/k, where the running time of the algorithm is polynomial in m,n and k. Our algorithm is based, rather than on TQFT, on well known mathematical results (specifically, the path model representation of the braid group and the uniqueness of the Markov trace for the Temperly Lieb algebra). By the results of [14], our algorithm solves a BQP complete problem.The algorithm we provide exhibits a structure which we hope is generalizable to other quantum algorithmic problems. Candidates of particular interest are the approximations of other downwards self-reducible P-hard problems, most notably, the Potts model.