Abstract. The permanent of a square matrix is defined in a way similar to the determinant, but without using signs. The exact computation of the permanent is hard, but there are Monte-Carlo algorithms that can estimate general permanents. Given a planar diagram of a link L with n crossings, we define a 7n×7n matrix whose permanent equals to the Jones polynomial of L. This result accompanied with recent work of Freedman, Kitaev, Larson and Wang ([FKLW]) provides a Monte-Carlo algorithm to any decision problem belonging to the class BQP, i.e. such that it can be computed with bounded error in polynomial time using quantum resources. 1.
Abstract. The permanent of a square matrix is defined in a way similar to the determinant, but without using signs. The exact computation of the permanent is hard, but there are Monte-Carlo algorithms that can estimate general permanents. Given a planar diagram of a link L with n crossings, we define a 7n×7n matrix whose permanent equals to the Jones polynomial of L. This result accompanied with recent work of Freedman, Kitaev, Larson and Wang ([FKLW]) provides a Monte-Carlo algorithm to any decision problem belonging to the class BQP, i.e. such that it can be computed with bounded error in polynomial time using quantum resources. 1.
Freedman, Kitaev, and Wang [arXiv:quant-ph/0001071], and later Aharonov, Jones, and Landau [arXiv:quant-ph/0511096], established a quantum algorithm to "additively" approximate the Jones polynomial V(L,t) at any principal root of unity t. The strength of this additive approximation depends exponentially on the bridge number of the link presentation. Freedman, Larsen, and Wang [arXiv:math/0103200] established that the approximation is universal for quantum computation at a non-lattice, principal root of unity; and Aharonov and Arad [arXiv:quant-ph/0605181] established a uniform version of this result. In this article, we show that any value-dependent approximation of the Jones polynomial at these non-lattice roots of unity is #P-hard. If given the power to decide whether |V(L,t)| > a or |V(L,t)| < b for fixed constants a > b > 0, there is a polynomial-time algorithm to exactly count the solutions to arbitrary combinatorial equations. In our argument, the result follows fairly directly from the universality result and Aaronson's theorem that PostBQP = PP [arXiv:quant-ph/0412187].
A celebrated important result due to Freedman et al (2002 Commun. Math. Phys. 227 605–22) states that providing additive approximations of the Jones polynomial at the k th root of unity, for constant k =5 and k ⩾7, is BQP-hard. Together with the algorithmic results of Aharonov et al (2005) and Freedman et al (2002 Commun. Math. Phys. 227 587–603), this gives perhaps the most natural BQP-complete problem known today and motivates further study of the topic. In this paper, we focus on the universality proof; we extend the result of Freedman et al (2002) to k s that grow polynomially with the number of strands and crossings in the link, thus extending the BQP-hardness of Jones polynomial approximations to all values to which the AJL algorithm applies (Aharonov et al 2005), proving that for all those values, the problems are BQP-complete. As a side benefit, we derive a fairly elementary proof of the Freedman et al density result, without referring to advanced results from Lie algebra representation theory, making this important result accessible to a wider audience in the computer science research community. We make use of two general lemmas we prove, the bridge lemma and the decoupling lemma , which provide tools for establishing the density of subgroups in SU( n ). Those tools seem to be of independent interest in more general contexts of proving the quantum universality. Our result also implies a completely classical statement, that the multiplicative approximations of the Jones polynomial, at exactly the same values, are #P-hard, via a recent result due to Kuperberg (2009 arXiv:0908.0512). Since the first publication of those results in their preliminary form (Aharonov and Arad 2006 arXiv:quant-ph/0605181), the methods we present here have been used in several other contexts (Aharonov and Arad 2007 arXiv:quant-ph/0702008; Peter and Stephen 2008 Quantum Inf. Comput. 8 681). The present paper is an improved and extended version of the results presented by Aharonov and Arad (2006) and includes discussions of the developments since then.
Let A be an n×n matrix with 0-1 valued entries, and let per(A) be the permanent of A. This paper describes a Monte-Carlo algorithm that produces a “good in the relative sense” estimate of per(A) and has running time ${\operatorname{poly}}(n)2^{{n / 2}} $, where poly(n) denotes a function that grows polynomially with n.
The study of anyonic systems as computational means has led to the exciting discovery of a new quantum algorithm. This algorithm provides a novel paradigm that fundamentally differs from searching (Grover, 1996) and factoring (Shor, 1997) algorithms. It is based on the particular behaviour of anyons and its goal is to evaluate Jones polynomials (Jones, 1985, 2005). These polynomials are topological invariants of knots and links, i.e., they depend on the global characteristics of their strands and not on their local geometry. The Jones polynomials were first connected to topological quantum field theories by Edward Witten (1989). Since then they have found applications in various areas of research, such as biology for DNA reconstruction (Nechaev, 1996) and statistical physics (Kauffman, 1991). The best known classical algorithm for the exact evaluation of the Jones polynomial demands exponential resources (Jaeger et al. , 1990). Employing anyons involves only a polynomial number of resources to produce an approximate answer to this problem (Freedman et al. , 2003b). Evaluating Jones polynomials by manipulating anyons resembles an analogue computer. Indeed, the idea is equivalent to the classical setup, where a wire is wrapped several times around a solenoid that confines magnetic flux. By measuring the current that runs through the wire one can obtain the number of times the wire was wrapped around the solenoid, i.e., their linking number. Similarly, by creating anyons and spanning links with their worldlines we are able to extract the Jones polynomials of these links (Kauffman and Lomanaco, 2006).
The Jones polynomial, discovered in 1984, is an important knot invariant in topology. Among its many connections to various mathematical and physical areas, it is known (due to Witten) to be intimately connected to Topological Quantum Field Theory (TQFT). The works of Freedman, Kitaev, Larsen and Wang 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 mentioned 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. By the results of Freedman et. al., 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 #P-hard problems, most notably, the partition function of the Potts model, a model which is known to be tightly connected to the Jones polynomial.
In this work, we study the computational complexity of quantum determinants, a q-deformation of matrix permanents: Given a complex number q on the unit circle in the complex plane and an n×n matrix X, the q-permanent of X is defined as Perq(X)=∑σ∈Snqℓ(σ)X1,σ(1)…Xn,σ(n), where ℓ(σ) is the inversion number of permutation σ in the symmetric group Sn on n elements. The function family generalizes determinant and permanent, which correspond to the cases q=−1 and q=1 respectively. For worst-case hardness, by Liouville's approximation theorem and facts from algebraic number theory, we show that for primitive m-th root of unity q for odd prime power m=pk, exactly computing q-permanent is ModpP-hard. This implies that an efficient algorithm for computing q-permanent results in a collapse of the polynomial hierarchy. Next, we show that computing q-permanent can be achieved using an oracle that approximates to within a polynomial multiplicative error and a membership oracle for a finite set of algebraic integers. From this, an efficient approximation algorithm would also imply a collapse of the polynomial hierarchy. By random self-reducibility, computing q-permanent remains to be hard for a wide range of distributions satisfying a property called the strong autocorrelation property. Specifically, this is proved via a reduction from 1-permanent to q-permanent for O(1/n2) points z on the unit circle. Since the family of permanent functions shares common algebraic structure, various techniques developed for the hardness of permanent can be generalized to q-permanents.