Approximate Counting and Quantum Computation

Bordewich, Magnus; Freedman, Michael; Lovász, László; Welsh, Dominic

doi:10.1017/s0963548305007005

Cited by 51 publications

(77 citation statements)

References 16 publications

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“…This of course is much weaker than a multiplicative approximation, which is what one might desire (see discussion in [7]). One might wonder if under such weak requirements, the problem remains meaningful at all.…”

Section: Theorem 12mentioning

confidence: 91%

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A Polynomial Quantum Algorithm for Approximating the Jones Polynomial

2008

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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 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 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 #P-hard 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].

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Section: Theorem 12mentioning

confidence: 91%

“…This connection is also discussed, from the point of view of TQFT, in Preskill's notes [25]. Unfortunately, the important quantum algorithm implied by these intriguing results, though referred to in [7], was never explicitly formulated.…”

Section: Introductionmentioning

confidence: 97%

A Polynomial Quantum Algorithm for Approximating the Jones Polynomial

2008

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“…Motivated by the Jones polynomial result, the computational complexity of additive approximations has been further investigated in [BFLW05].…”

Section: Vid07 Hkh + 08])mentioning

confidence: 99%

Quantum Computation and the Evaluation of Tensor Networks

Arad¹,

Landau²

2010

SIAM J. Comput.

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We present a quantum algorithm that additively approximates the value of a tensor network to a certain scale. When combined with existing results, this provides a complete problem for quantum computation. The result is a simple new way of looking at quantum computation in which unitary gates are replaced by tensors and time is replaced by the order in which the tensornetwork is "swallowed". We use this result to derive new quantum algorithms that approximate the partition function of a variety of classical statistical mechanics models, including the Potts model.

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“…The interest in this problem stems from the fact that an additive approximation of the Jones polynomial is sufficient to simulate any polynomial quantum computation [36]. The construction of the algorithm involves three different contexts:…”

Section: A Quantum Algorithm That Approximates the Colored Jones Polymentioning

confidence: 99%

“…The latter was applied for the first time in [2] dealing just with the problem of evaluating the Jones polynomial. We further recall that the notion of approximation used in the present context, formalized in [36], is that of additive approximation, which has the following meaning: given a normalized function f (x), where x denotes an instance of the problem in the selected coding, we have an additive approximation of its value for each instance x if we can associate to f (x) a random variable Z such that Pr {|f (x) − Z| ≤ δ} ≥ 3/4 , for any δ ≥ 0. The time needed to achieve the approximation must be polynomial in the size of the problem and in δ −1 .…”

Section: The Algorithmmentioning

confidence: 99%

Quantum geometry and quantum algorithms

Garnerone¹,

Marzuoli²,

Rasetti³

2007

J. Phys. A: Math. Theor.

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Abstract. Motivated by algorithmic problems arising in quantum field theories whose dynamical variables are geometric in nature, we provide a quantum algorithm that efficiently approximates the colored Jones polynomial. The construction is based on the complete solution of Chern-Simons topological quantum field theory and its connection to Wess-Zumino-Witten conformal field theory. The colored Jones polynomial is expressed as the expectation value of the evolution of the q-deformed spin-network quantum automaton. A quantum circuit is constructed capable of simulating the automaton and hence of computing such expectation value. The latter is efficiently approximated using a standard sampling procedure in quantum computation.

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Approximate Counting and Quantum Computation

Cited by 51 publications

References 16 publications

A Polynomial Quantum Algorithm for Approximating the Jones Polynomial

A Polynomial Quantum Algorithm for Approximating the Jones Polynomial

Quantum Computation and the Evaluation of Tensor Networks

Quantum geometry and quantum algorithms

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