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34
A Polynomial Quantum Algorithm for Approximating the Jones Polynomial
, 2008
"... 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, Lar ..."
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Cited by 71 (3 self)
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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, e 2πi/5, and moreover, that this problem is BQPcomplete. 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 e 2πi/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 selfreducible #Phard problems, most notably, the important open problem of efficient approximation of the partition function of the Potts model, a model which is known to be tightly connected to the Jones polynomial [33].
Quantum algorithms for algebraic problems
, 2008
"... Quantum computers can execute algorithms that dramatically outperform classical computation. As the bestknown example, Shor discovered an efficient quantum algorithm for factoring integers, whereas factoring appears to be difficult for classical computers. Understanding what other computational pro ..."
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Cited by 23 (1 self)
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Quantum computers can execute algorithms that dramatically outperform classical computation. As the bestknown example, Shor discovered an efficient quantum algorithm for factoring integers, whereas factoring appears to be difficult for classical computers. Understanding what other computational problems can be solved significantly faster using quantum algorithms is one of the major challenges in the theory of quantum
Estimating Jones polynomials is a complete problem for one clean qubit
, 2007
"... It is known that evaluating a certain approximation to the Jones polynomial for the plat closure of a braid is a BQPcomplete problem. That is, this problem exactly captures the power of the quantum circuit model[12, 3, 1]. The one clean qubit model is a model of quantum computation in which all but ..."
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Cited by 22 (4 self)
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It is known that evaluating a certain approximation to the Jones polynomial for the plat closure of a braid is a BQPcomplete problem. That is, this problem exactly captures the power of the quantum circuit model[12, 3, 1]. The one clean qubit model is a model of quantum computation in which all but one qubit starts in the maximally mixed state. One clean qubit computers are believed to be strictly weaker than standard quantum computers, but still capable of solving some classically intractable problems [20]. Here we show that evaluating a certain approximation to the Jones polynomial at a fifth root of unity for the trace closure of a braid is a complete problem for the one clean qubit complexity class. That is, a one clean qubit computer can approximate these Jones polynomials in time polynomial in both the number of strands and number of crossings, and the problem of simulating a one clean qubit computer is reducible to approximating the Jones polynomial of the trace closure of a braid.
qdeformed spin networks, knot polynomials and anyonic topological . . .
, 2006
"... We review the qdeformed spin network approach to Topological Quantum Field Theory and apply these methods to produce unitary representations of the braid groups that are dense in the unitary groups. Our methods are rooted in the bracket state sum model for the Jones polynomial. We give our results ..."
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Cited by 22 (9 self)
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We review the qdeformed spin network approach to Topological Quantum Field Theory and apply these methods to produce unitary representations of the braid groups that are dense in the unitary groups. Our methods are rooted in the bracket state sum model for the Jones polynomial. We give our results for a large class of representations based on values for the bracket polynomial that are roots of unity. We make a separate and selfcontained study of the quantum universal Fibonacci model in this framework. We apply our results to give quantum algorithms for the computation of the colored Jones polynomials for knots and links, and the WittenReshetikhinTuraev invariant of three manifolds.
The BQPhardness of approximating the Jones Polynomial
"... Following the work by Kitaev, Freedman and Wang [1], Aharonov, Jones and Landau [3] recently gave an explicit and efficient quantum algorithm for approximating the Jones polynomial of the plat closure of a braid, at the kth root of unity, for constant k. The universality proof of Freedman, Larsen an ..."
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Cited by 11 (2 self)
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Following the work by Kitaev, Freedman and Wang [1], Aharonov, Jones and Landau [3] recently gave an explicit and efficient quantum algorithm for approximating the Jones polynomial of the plat closure of a braid, at the kth root of unity, for constant k. The universality proof of Freedman, Larsen and Wang [2] implies that the problem which these algorithms solve is BQPhard. The fact that this is the only nontrivial BQPcomplete problem known today motivates a deep investigation of this topic. A natural question which was raised in [3] is the following. The results of [3] actually gave efficient algorithms also in the case of asymptotically growing k’s up to k which is polynomial in the size of the braid. However, the results of [2] only imply universality in the case of constant k, via the SolovayKitaev theorem; The application of this theorem relies heavily on the fact that the generators of the groups in question are fixed. The question of the complexity of the problems with asymptotically growing k was thus left open. In this paper we resolve this question and prove that the Jones polynomial approximation problem is BQPcomplete also for asymptotically growing k (bounded by a polynomial). To do this we introduce some new techniques for analyzing universality in quantum computation, which enable us to apply SolovayKitaev indirectly. As a side benefit, we reprove the density theorem of [2] using quite elementary arguments; this hopefully sheds light on the reason that these problems are indeed quantumhard.
Multiparty Quantum Computation
 MASTER'S THESIS, MIT
, 2001
"... We investigate definitions of and protocols for multiparty quantum computing in the scenario where the secret data are quantum systems. We work in the quantum informationtheoretic model, where no assumptions are made on the computational power of the adversary. For the slightly weaker task of veri ..."
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Cited by 7 (1 self)
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We investigate definitions of and protocols for multiparty quantum computing in the scenario where the secret data are quantum systems. We work in the quantum informationtheoretic model, where no assumptions are made on the computational power of the adversary. For the slightly weaker task of verifiable quantum secret sharing, we give a protocol which tolerates any t < n/4 cheating parties (out of n). This is shown to be optimal. We use this new tool to establish that any multiparty quantum computation can be securely performed as long as the number of dishonest players is less than n/6.
Topological quantum computing and the Jones polynomial
, 2006
"... In this paper, we give a description of a recent quantum algorithm created by Aharonov, Jones, and Landau for approximating the values of the Jones polynomial at roots of unity of the form e 2πi/k. This description is given with two objectives in mind. The first is to describe the algorithm in such ..."
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Cited by 6 (2 self)
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In this paper, we give a description of a recent quantum algorithm created by Aharonov, Jones, and Landau for approximating the values of the Jones polynomial at roots of unity of the form e 2πi/k. This description is given with two objectives in mind. The first is to describe the algorithm in such a way as to make explicit the underlying and inherent control structure. The second is to make this algorithm accessible to a larger audience.
On the Exact Evaluation of Certain Instances of the Potts Partition Function by Quantum Computers
 COMMUN. MATH. PHYS.
, 2007
"... We present an efficient quantum algorithm for the exact evaluation of either the fully ferromagnetic or antiferromagnetic qstate Potts partition function Z for a family of graphs related to irreducible cyclic codes. This problem is related to the evaluation of the Jones and Tutte polynomials. We ..."
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Cited by 5 (0 self)
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We present an efficient quantum algorithm for the exact evaluation of either the fully ferromagnetic or antiferromagnetic qstate Potts partition function Z for a family of graphs related to irreducible cyclic codes. This problem is related to the evaluation of the Jones and Tutte polynomials. We consider the connection between the weight enumerator polynomial from coding theory and Z and exploit the fact that there exists a quantum algorithm for efficiently estimating Gauss sums in order to obtain the weight enumerator for a certain class of linear codes. In this way we demonstrate that for a certain class of sparse graphs, which we call Irreducible Cyclic Cocycle Code (ICCCɛ) graphs, quantum computers provide a polynomial speed up in the difference between the number of edges and vertices of the graph, and an exponential speed up in q, over the best classical algorithms known to date.
Efficient quantum processing of 3–manifold topological invariants. arXiv: quantph/0703037
"... A quantum algorithm for approximating efficiently 3–manifold topological invariants in the framework of SU(2) Chern–Simons–Witten (CSW) topological quantum field theory at finite values of the coupling constant k is provided. The model of computation adopted is the q–deformed spin network model view ..."
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Cited by 4 (3 self)
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A quantum algorithm for approximating efficiently 3–manifold topological invariants in the framework of SU(2) Chern–Simons–Witten (CSW) topological quantum field theory at finite values of the coupling constant k is provided. The model of computation adopted is the q–deformed spin network model viewed as a quantum recognizer in the sense of [1], where each basic unitary transition function can be efficiently processed by a standard quantum circuit. This achievement is an extension of the algorithm for approximating polynomial invariants of colored oriented links found in [2, 3]. Thus all the significant quantities –partition functions and observables – of quantum CSW theory can be processed efficiently on a quantum computer, reflecting the intrinsic, field– theoretic solvability of such theory at finite k. The paper is supplemented by a critical overview of the basic conceptual tools underlying the construction of quantum invariants of links and 3–manifolds and connections with algorithmic questions that arise in geometry and quantum gravity models are discussed.
BQPcomplete problems concerning mixing properties of classical random walks on sparse graphs. arXiv:quantph/0610235
"... We describe two BQPcomplete problems concerning properties of sparse graphs having a certain symmetry. The graphs are specified by efficiently computable functions which output the adjacent vertices for each vertex. Let i and j be two given vertices. The first problem consists in estimating the dif ..."
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Cited by 4 (1 self)
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We describe two BQPcomplete problems concerning properties of sparse graphs having a certain symmetry. The graphs are specified by efficiently computable functions which output the adjacent vertices for each vertex. Let i and j be two given vertices. The first problem consists in estimating the difference between the number of paths of length m from j to j and those which from i to j, where m is polylogarithmic in the number of vertices. The scale of the estimation accuracy is specified by some a priori known upper bound on the growth of these differences with increasing m. The problem remains BQPhard for regular graphs with degree 4. The second problem is related to continuoustime classical random walks. The walk starts at some vertex j. The promise is that the difference of the probabilities of being at j and at i, respectively, decays