Complexity: knots, colourings and counting
Complexity: knots, colourings and counting
SIAM Journal on Computing
Relationships Between Quantum and Classical Space-Bounded Complexity Classes
COCO '98 Proceedings of the Thirteenth Annual IEEE Conference on Computational Complexity
Measuring 4-local n-qubit observables could probablistically solve PSPACE
WISICT '04 Proceedings of the winter international synposium on Information and communication technologies
Approximate Counting and Quantum Computation
Combinatorics, Probability and Computing
A polynomial quantum algorithm for approximating the Jones polynomial
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Computational Complexity
Quantum computing and polynomial equations over the finite field Z2
Quantum Information & Computation
Quantum Information & Computation
Quantum automata, braid group and link polynomials
Quantum Information & Computation
Quantum Computation and the Evaluation of Tensor Networks
SIAM Journal on Computing
A promiseBQP-complete string rewriting problem
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 Information & Computation
Quantum algorithms for invariants of triangulated manifolds
Quantum Information & Computation
QMA variants with polynomially many provers
Quantum Information & Computation
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We analyze relationships between quantum computation and a family of generalizations of the Jones polynomial. Extending recent work by Aharonov et al., we give efficient quantum circuits for implementing the unitary Jones-Wenzl representations of the braid group. We use these to provide new quantum algorithms for approximately evaluating a family of specializations of the HOMFLYPT two-variable polynomial of trace closures of braids. We also give algorithms for approximating the Jones polynomial of a general class of closures of braids at roots of unity. Next we provide a self-contained proof of a result of Freedman et al. that any quantum computation can be replaced by an additive approximation of the Jones polynomial, evaluated at almost any primitive root of unity. Our proof encodes two-qubit unitaries into the rectangular representation of the eightstrand braid group. We then give QCMA-complete and PSPACE-complete problems which are based on braids. We conclude with direct proofs that evaluating the Jones polynomial of the plat closure at most primitive roots of unity is a #P-hard problem, while learning its most significant bit is PP-hard, circumventing the usual route through the Tutte polynomial and graph coloring.