In the first part of this paper, we provide polynomial quantum algorithms for additive approximations of the Tutte polynomial, at any point in the Tutte plane, for any planar graph. This includes an additive approximation of the partition function of the Potts model for any weighted planer graph at any temperature, as well as approximations to many other combinatorial graph properties described by the (multivariate or not) Tutte polynomial. To achieve these algorithms, we generalize the Temperley Lieb algebra representations, used in [6], to apply for any graph (not necessarily coming from a braid). Moreover, our representations are non-unitary, as are all representations of the Temperley Lieb algebra not corresponding to Jones polynomial related parameters. It might seem at first sight that this makes it impossible to apply them by a quantum circuit. We show how to do this nevertheless. The approximation window size turns out to be inverse polynomial in |G| times the product of the norms of the operators we apply. Additive approximations are tricky; the range of the possible outcomes, might be smaller than the size of the approximation window, in which case the outcome is meaningless. Unfortunately, ruling out this possibility is difficult: If we want to argue that our a are meaningful, we have to provide an estimate of the scale of the problem, which is difficult here exactly because no efficient algorithm for the problem exists! In the second part of the paper we provide an indirect but very convincing proof that our approximation is meaningful for a large range of parameters, by showing that in those cases, the problems
In the first part of this paper, we provide polynomial quantum algorithms for additive approximations of the Tutte polynomial, at any point in the Tutte plane, for any planar graph. This includes an additive approximation of the partition function of the Potts model for any weighted planer graph at any temperature, as well as approximations to many other combinatorial graph properties described by the (multivariate or not) Tutte polynomial. To achieve these algorithms, we generalize the Temperley Lieb algebra representations, used in [6], to apply for any graph (not necessarily coming from a braid). Moreover, our representations are non-unitary, as are all representations of the Temperley Lieb algebra not corresponding to Jones polynomial related parameters. It might seem at first sight that this makes it impossible to apply them by a quantum circuit. We show how to do this nevertheless. The approximation window size turns out to be inverse polynomial in |G| times the product of the norms of the operators we apply. Additive approximations are tricky; the range of the possible outcomes, might be smaller than the size of the approximation window, in which case the outcome is meaningless. Unfortunately, ruling out this possibility is difficult: If we want to argue that our a are meaningful, we have to provide an estimate of the scale of the problem, which is difficult here exactly because no efficient algorithm for the problem exists! In the second part of the paper we provide an indirect but very convincing proof that our approximation is meaningful for a large range of parameters, by showing that in those cases, the problems
In the first 36 pages of this paper, we provide polynomial quantum algorithms for additive approximations of the Tutte polynomial, at any point in the Tutte plane, for any planar graph. This includes as special cases the AJL algorithm for the Jones polynomial, the partition function of the Potts model for any weighted planer graph at any temperature, and many other combinatorial graph properties. In the second part of the paper we prove the quantum universality of many of the problems for which we provide an algorithm, thus providing a large set of new quantum-complete problems. Unfortunately, we do not know that this holds for the Potts model case; this is left as an important open problem. The main progress in this work is in our ability to handle non-unitary representations of the Temperley Lieb algebra, both when applying them in the algorithm, and, more importantly, in the proof of universality, when encoding quantum circuits using non-unitary operators. To this end we develop many new tools, that allow proving density and applying the Solovay Kitaev theorem in the case of non-unitary matrices. We hope that these tools will open up new possibilities of using non-unitary reps in other quantum computation contexts.
The quantum algorithm of AJL [3] (following the work of Freedman et al. [10]) to approximate the Jones polynomial, is of a new type: rather than using the quantum Fourier transform, it encodes the local combinatorial structure of the problem by the relations of an algebra, called the Temperley-Lieb algebra, whose matrix representation is then applied, using a quantum computer. By the results of [11], the problems this algorithm solves are the first non trivial BQP-complete problems. A natural question is whether it is possible to generalize this algorithm to other points of the famous Tutte polynomial, of which the Jones polynomial is a special case. However, the only points for which
Abstract We study the complexity of approximating the partition function of the q-state Potts model and the closely related Tutte polynomial for complex values of the underlying parameters. Apart from the classical connections with quantum computing and phase transitions in statistical physics, recent work in approximate counting has shown that the behaviour in the complex plane, and more precisely the location of zeros, is strongly connected with the complexity of the approximation problem, even for positive real-valued parameters. Previous work in the complex plane by Goldberg and Guo focused on q = 2, which corresponds to the case of the Ising model; for q > 2, the behaviour in the complex plane is not as well understood and most work applies only to the real-valued Tutte plane. Our main result is a complete classification of the complexity of the approximation problems for all non-real values of the parameters, by establishing #P-hardness results that apply even when restricted to planar graphs. Our techniques apply to all q ≥ ≥ 2 and further complement/refine previous results both for the Ising model and the Tutte plane, answering in particular a question raised by Bordewich, Freedman, Lovász and Welsh in the context of quantum computations.
We show that the problem of approximately evaluating the Tutte polynomial of triangular graphs at the points (q, 1/q) of the Tutte plane is BQP-complete for (most) roots of unity q. We also consider circular graphs and show that the problem of approximately evaluating the Tutte polynomial of these graphs at the point (e2πi/5, e-2πi/5) is DQC1-complete and at points [Formula: see text] for some integer k is in BQP. To show that these problems can be solved by a quantum computer, we rely on the relation of the Tutte polynomial of a planar G graph with the Jones and HOMFLY polynomial of the alternating link D(G) given by the medial graph of G. In the case of our graphs the corresponding links are equal to the plat and trace closures of braids. It is known how to evaluate the Jones and HOMFLY polynomial for closures of braids. To establish the hardness results, we use the property that the images of the generators of the braid group under the irreducible Jones–Wenzl representations of the Hecke algebra have finite order. We show that for each braid b we can efficiently construct a braid [Formula: see text] such that the evaluation of the Jones and HOMFLY polynomials of their closures at a fixed root of unity leads to the same value and that the closures of [Formula: see text] are alternating links.
In this dissertation, we investigate three different problems in the field of Quantum computation. First, we discuss the quantum complexity of evaluating the Tutte polynomial of a planar graph. Furthermore, we devise a new quantum algorithm for approximating the phase of a unitary matrix. Finally, we provide quantum tools that can be utilized to extract the structure of black-box modules and algebras. While quantum phase estimation (QPE) is at the core of many quantum algorithms known to date, its physical implementation (algorithms based on quantum Fourier transform (QFT) ) is highly constrained by the requirement of high-precision controlled phase shift operators, which remain difficult to realize. In the second part of this dissertation, we introduce an alternative approach to approximately implement QPE with arbitrary constantprecision controlled phase shift operators. The new quantum algorithm bridges the gap between QPE algorithms based on QFT and Kitaev’s original approach. For approximating the eigenphase precise to the nth bit, Kitaev’s original approach does not require any controlled phase shift operator. In contrast, QPE algorithms based on QFT or approximate QFT require controlled phase shift operators with precision of at least Pi/2n. The new approach fills the gap and requires only arbitrary constant-precision controlled phase shift operators. From a physical implementation viewpoint, the new algorithm outperforms Kitaev’s approach. iii The other problem we investigate relates to approximating the Tutte polynomial. We show that the problem of approximately evaluating the Tutte polynomial of triangular graphs at the points (q, 1/q) of the Tutte plane is BQP-complete for (most) roots of unity q. We also consider circular graphs and show that the problem of approximately evaluating the Tutte polynomial of these graphs at the point (e 2πi/5 ,e−2πi/5 ) is DQC1-complete and at points (q k , 1 + 1−q−k (q 1/2−q−1/2) 2 ) for some integer k is in BQP. To show that these problems can be solved by a quantum computer, we rely on the relation of the Tutte polynomial of a planar G graph with the Jones and HOMFLY polynomial of the alternating link D(G) given by the medial graph of G. In the case of our graphs the corresponding links are equal to the plat and trace closures of braids. It is known how to evaluate the Jones and HOMFLY polynomial for closures of braids. To establish the hardness results, we use the property that the images of the generators of the braid group under the irreducible Jones-Wenzl representations of the Hecke algebra have finite order. We show that for each braid b we can efficiently construct a braid ˜b such that the evaluation of the Jones and HOMFLY polynomials of their closures at a fixed root of unity leads to the same value and that the closures of ˜b are alternating links. The final part of the dissertation focuses on finding the structure of a black-box module or algebra. Suppose we are given black-box access to a finite module M or algebra over a finite ring R, and a list of generators for M and R. We show how to find a linear basis and structure constants for M in quantum poly(log |M|) time. This generalizes a recent quantum algorithm of Arvind et al. which finds a basis representation for rings. We then show that iv our algorithm is a useful primitive allowing quantum computers to determine the structure of a finite associative algebra as a direct sum of simple algebras. Moreover, it solves a wide variety of problems regarding finite modules and rings. Although our quantum algorithm is based on Abelian Fourier transforms, it solves problems regarding the multiplicative structure of modules and algebras, which need not be commutative. Examples include finding the intersection and quotient of two modules, finding the additive and multiplicative identities in a module, computing the order of an module, solving linear equations over modules, deciding whether an ideal is maximal, finding annihilators, and testing the injectivity and surjectivity of ring homomorphisms. These problems appear to be exponentially hard classically.
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.