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Permutational Quantum Computing
, 906
"... 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. 1
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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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.
Estimating Jones and HOMFLY polynomials with One Clean Qubit
, 807
"... 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 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
unknown title
, 811
"... Fast quantum algorithms for approximating the irreducible representations of groups ..."
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Fast quantum algorithms for approximating the irreducible representations of groups
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, 811
"... Fast quantum algorithms for approximating some irreducible representations of groups ..."
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Fast quantum algorithms for approximating some irreducible representations of groups
Dedication To Maa, for everything. vAcknowledgments
"... vi The most significant character in any doctoral career is that of the advisor. Carl Caves, my advisor, has been more. Working with and under him has been a brilliant and fascinating experience. His strategy of offering ample freedom in choosing problems and the pace of solving them, and collaborat ..."
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vi The most significant character in any doctoral career is that of the advisor. Carl Caves, my advisor, has been more. Working with and under him has been a brilliant and fascinating experience. His strategy of offering ample freedom in choosing problems and the pace of solving them, and collaborating on them has been his single greatest contribution to my research. Encouraging oneliners like ‘The truth shall set you free ’ when I’ve gotten negative results and confidence boosters like ‘A lot of people say a lot of things ’ have been priceless; so has been his dictum of writing papers with meticulous precision. His ability to guide and advise has been phenomenal and a source of constant inspiration. The contribution of the other members of the Information Physics group have been no less. Both Ivan Deutsch and Andrew Landahl have always been enthusiastic and encouraging of my research, asking penetrating question in group meeting presentations and discussing such topics on the outside. I also thank them for being on my dissertation committee. Ivan is also to be thanked for teaching fabulous courses on quantum optics and atomic physics. I’m infinitely thankful to Anil Shaji for being my surrogate advisor, with whom I have discussed almost all my projects, and he’s always been there to put me right and prevented me from going awry so often. Other members of the group with whom I’ve collaborated have taught me a lot as well. Steve Flammia, with his random math problems and refusal ever to be frightened by formidable mathematics has been a motivating experience. Discussions with Sergio Boixo have also been fruitful. Other student members of the group have been no less important. Seth has always been there to chat about a nagging physics, math or programming problem.
The Fibonacci Model and the TemperleyLieb Algebra
, 804
"... We give an elementary construction of the Fibonacci model, a unitary braid group representation that is universal for quantum computation. This paper is dedicated to Professor C. N. Yang, on his 85th birthday. ..."
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We give an elementary construction of the Fibonacci model, a unitary braid group representation that is universal for quantum computation. This paper is dedicated to Professor C. N. Yang, on his 85th birthday.
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.