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22
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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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
Quantum knots and mosaics
 QUANTUM INF PROCESS
"... In this paper, we give a precise and workable definition of a quantum knot system, the states of which are called quantum knots. This definition can be viewed as a blueprint for the construction of an actual physical quantum system. Moreover, this definition of a quantum knot system is intended to ..."
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Cited by 17 (7 self)
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In this paper, we give a precise and workable definition of a quantum knot system, the states of which are called quantum knots. This definition can be viewed as a blueprint for the construction of an actual physical quantum system. Moreover, this definition of a quantum knot system is intended to represent the “quantum embodiment” of a closed knotted physical piece of rope. A quantum knot, as a state of this system, represents the state of such a knotted closed piece of rope, i.e., the particular spatial configuration of the knot tied in the rope. Associated with a quantum knot system is a group of unitary transformations, called the ambient group, which represents all possible ways of moving the rope around (without cutting the rope, and without letting the rope pass through itself.) Of course, unlike a classical closed piece of rope, a quantum knot can exhibit nonclassical behavior, such as quantum superposition and quantum entanglement. This raises some interesting and puzzling questions about the relation between topological and quantum entanglement. The knot type of a quantum knot is simply the orbit of the quantum knot under the action of the ambient group. We investigate quantum observables which are invariants of quantum knot type. We also study the Hamiltonians associated with the generators of the ambient group, and briefly look at the quantum tunneling of overcrossings into undercrossings. A basic building block in this paper is a mosaic system which is a formal (rewriting) system of symbol strings. We conjecture that this formal system fully captures in an axiomatic way all of the properties of tame knot theory.
Permutational Quantum Computing
, 2009
"... In topological quantum computation the geometric details of a particle trajectory are irrelevant; only the topology matters. Taking this one step further, we consider a model of computation that disregards even the topology of the particle trajectory, and computes by permuting particles. Whereas top ..."
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In topological quantum computation the geometric details of a particle trajectory are irrelevant; only the topology matters. Taking this one step further, we consider a model of computation that disregards even the topology of the particle trajectory, and computes by permuting particles. Whereas topological quantum computation requires anyons, permutational quantum computation can be performed with ordinary spin1/2 particles, using a variant of the spinnetwork scheme of Marzuoli and Rasetti. We do not know whether permutational computation is universal. It may represent a new complexity class within BQP. Nevertheless, permutational quantum computers can in polynomial time approximate matrix elements of certain irreducible representations of the symmetric group and simulate certain processes in the PonzanoRegge spin foam model of quantum gravity. No polynomial time classical algorithms for these problems are known.
Quantum Knots and Lattices, or a Blueprint for Quantum Systems that Do Rope Tricks
"... Abstract. Within the framework of the cubic honeycomb (cubic tessellation) of Euclidean 3space, we define a quantum system whose states, called quantum knots, represent a closed knotted piece of rope, i.e., represent the particular spatial configuration of a knot tied in a rope in 3space. This qua ..."
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Abstract. Within the framework of the cubic honeycomb (cubic tessellation) of Euclidean 3space, we define a quantum system whose states, called quantum knots, represent a closed knotted piece of rope, i.e., represent the particular spatial configuration of a knot tied in a rope in 3space. This quantum system, called a quantum knot system, is physically implementable in the same sense as Shor’s quantum factoring algorithm is implementable. To define a quantum knot system, we replace the standard three Reidemeister knot moves with an equivalent set of three moves, called respectively wiggle, wag, and tug, so named because they mimic how a dog might wag its tail. We argue that these moves are in fact more ”physics friendly ” than the Reidemeister moves because, unlike the Reidemeister moves, they respect the differential geometry of 3space, and moreover they can be transformed into infinitesimal moves. These three moves wiggle, wag, and tug generate a unitary group, called the lattice ambient group, which acts on the state space of the quantum system. The lattice ambient group represents all possible ways of moving a rope around in 3space without cutting the rope, and without letting the rope pass through itself. We then investigate those quantum observables of the quantum knot system which are knot invariants. We also study Hamiltonians associated with the generators of the lattice ambient group. We conclude with a list of open questions. Contents
Estimating Jones and HOMFLY polynomials with One Clean Qubit
, 2008
"... The Jones and HOMFLY polynomials are link invariants with close connections to quantum computing. It was recently shown that finding a certain approximation to the Jones polynomial of the trace closure of a braid at the fifth root of unity is a complete problem for the one clean qubit complexity cla ..."
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The Jones and HOMFLY polynomials are link invariants with close connections to quantum computing. It was recently shown that finding a certain approximation to the Jones polynomial of the trace closure of a braid at the fifth root of unity is a complete problem for the one clean qubit complexity class[18]. This is the class of problems solvable in polynomial time on a quantum computer acting on an initial state in which one qubit is pure and the rest are maximally mixed. Here we generalize this result by showing that one clean qubit computers can efficiently approximate the Jones and singlevariable HOMFLY polynomials of the trace closure of a braid at any root of unity.
1 QUANTUM COMPUTING AND THE ENTANGLEMENT FRONTIER
, 2014
"... Quantum information science explores the frontier of highly complex quantum states, the “entanglement frontier. ” This study is motivated by the observation (widely believed but unproven) that classical systems cannot simulate highly entangled quantum systems efficiently, and we hope to hasten the d ..."
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Quantum information science explores the frontier of highly complex quantum states, the “entanglement frontier. ” This study is motivated by the observation (widely believed but unproven) that classical systems cannot simulate highly entangled quantum systems efficiently, and we hope to hasten the day when well controlled quantum systems can perform tasks surpassing what can be done in the classical world. One way to achieve such “quantum supremacy ” would be to run an algorithm on a quantum computer which solves a problem with a superpolynomial speedup relative to classical computers, but there may be other ways that can be achieved sooner, such as simulating exotic quantum states of strongly correlated matter. To operate a large scale quantum computer reliably we will need to overcome the debilitating effects of decoherence, which might be done using “standard ” quantum hardware protected by quantum errorcorrecting codes, or by exploiting the nonabelian quantum statistics of anyons realized in solid state systems, or by combining both methods. Only by challenging the entanglement frontier will we learn whether Nature provides extravagant resources far beyond what the classical world would allow. Rapporteur talk at the 25th Solvay Conference on Physics
Fast quantum algorithms for approximating some irreducible representations of groups
, 2009
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Dequantizing Readonce Quantum Formulas
"... Quantum formulas, defined by Yao [FOCS’93], are the quantum analogs of classical formulas, i.e., classical circuits in which all gates have fanout one. We show that any readonce quantum formula over a gate set that contains all singlequbit gates is equivalent to a readonce classical formula of th ..."
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Quantum formulas, defined by Yao [FOCS’93], are the quantum analogs of classical formulas, i.e., classical circuits in which all gates have fanout one. We show that any readonce quantum formula over a gate set that contains all singlequbit gates is equivalent to a readonce classical formula of the same size and depth over an analogous classical gate set. For example, any readonce quantum formula over Toffoli and singlequbit gates is equivalent to a readonce classical formula over Toffoli and not gates. We then show that the equivalence does not hold if the readonce restriction is removed. To show the power of quantum formulas without the readonce restriction, we define a new model of computation called the onequbit model and show that it can compute all boolean functions. This model may also be of independent interest.