Presented is a topological representation of quantum logic that views entangled qubit spacetime histories (or qubit world lines) as a generalized braid, referred to as a superbraid. The crossing of world lines can be quantum-mechanical in nature, most conveniently expressed analytically with ladder-operator-based quantum gates. At a crossing, independent world lines can become entangled. Complicated superbraids are systematically reduced by recursively applying quantum skein relations. If the superbraid is closed (e.g., representing quantum circuits with closed-loop feedback, quantum lattice gas algorithms, loop or vacuum diagrams in quantum field theory), then one can decompose the resulting superlink into an entangled superposition of classical links. Thus, one can compute a superlink invariant, for example, the Jones polynomial for the square root of a classical knot.
Presented is a topological representation of quantum logic that views entangled qubit spacetime histories (or qubit world lines) as a generalized braid, referred to as a superbraid. The crossing of world lines can be quantum-mechanical in nature, most conveniently expressed analytically with ladder-operator-based quantum gates. At a crossing, independent world lines can become entangled. Complicated superbraids are systematically reduced by recursively applying quantum skein relations. If the superbraid is closed (e.g., representing quantum circuits with closed-loop feedback, quantum lattice gas algorithms, loop or vacuum diagrams in quantum field theory), then one can decompose the resulting superlink into an entangled superposition of classical links. Thus, one can compute a superlink invariant, for example, the Jones polynomial for the square root of a classical knot.
Presented is a study of quantum entanglement from the perspective of the theory of quantum information dynamics. We consider pairwise entanglement and present an analytical development using joint ladder operators, the sum of two single-particle fermionic ladder operators. This approach allows us to write down analytical representations of quantum algorithms and to explore quantum entanglement as it is manifested in a system of qubits.;We present a topological representation of quantum logic that views entangled qubit spacetime histories (or qubit world lines) as a generalized braid, referred to as a super-braid. The crossing of world lines may be either classical or quantum mechanical in nature, and in the latter case most conveniently expressed with our analytical expressions for entangling quantum gates. at a quantum mechanical crossing, independent world lines can become entangled. We present quantum skein relations that allow complicated superbraids to be recursively reduced to alternate classical histories. If the superbraid is closed, then one can decompose the resulting superlink into an entangled superposition of classical links. Also, one can compute a superlink invariant, for example the Jones polynomial for the square root of a knot.;We present measurement-based quantum computing based on our joint number operators. We take expectation values of the joint number operators to determine kinetic-level variables describing the quantum information dynamics in the qubit system at the mesoscopic scale. We explore the issue of reversibility in quantum maps at this scale using a quantum Boltzmann equation. We then present an example of quantum information processing using a qubit system comprised of nuclear spins. We also discuss quantum propositions cast in terms of joint number operators.;We review the well known dynamical equations governing superfluidity, with a focus on self-consistent dynamics supporting quantum vortices in a Bose-Einstein condensate (BEC). Furthermore, we review the mutual vortex-vortex interaction and the consequent Kelvin wave instability. We derive an effective equation of motion for a Fermi condensate that is the basis of our qubit representation of superfluidity.;We then present our quantum lattice gas representation of a superfluid. We explore aspects of our model with two qubits per point, referred to as a Q2 model, particularly its usefulness for carrying out practical quantum fluid simulations. We find that it is perhaps the simplest yet most comprehensive model of superfluid dynamics. as a prime application of Q2, we explore the power-law regions in the energy spectrum of a condensate in the low-temperature limit. We achieved the largest quantum simulations to date of a BEC and, for the first time, Kolmogorov scaling in superfluids, a flow regime heretofore only obtainably by classical turbulence models.;Finally, we address the subject of turbulence regarding information conservation on the small scales (both mesoscopic and microscopic) underlying the flow dynamics on the large hydrodynamic (macroscopic) scale. We present a hydrodynamic-level momentum equation, in the form of a Navier-Stokes equation, as the basis for the energy spectrum of quantum turbulence at large scales. Quantum turbulence, in particular the representation of fluid eddies in terms of a coherent structure of polarized quantum vortices, offers a unique window into the heretofore intractable subject of energy cascades.
Abstract This paper is an introduction to relationships between quantum topology and quantum computing. We discuss unitary solutions to the YangBaxter equation that are universal quantum gates, quantum entanglement and topological entanglement, and we give an exposition of knot-theoretic recoupling theory, its relationship with topological quantum field theory and apply these methods to produce unitary representations of the braid groups that are dense in the unitary groups. Our methods are rooted in the bracket state sum model for the Jones polynomial. We give our results for a large class of representations based on values for the bracket polynomial that are roots of unity. We make a separate and self-contained study of the quantum universal Fibonacci model in this framework. We apply our results to give quantum algorithms for the computation of the colored Jones polynomials for knots and links, and the Witten-Reshetikhin-Turaev invariant of three manifolds.
Kauffman and Lomonaco (New J Phys 4:73.1–73.18, 2002. arXiv:quant-ph/0401090 , New J Phys 6:134.1–134.40, 2004) explored the idea of understanding quantum entanglement (the non-local correlation of certain properties of particles) topologically by viewing unitary entangling operators as braiding operators. In Alagic et al. (Yang–Baxter operators need quantum entanglement to distinguish knots, 2015. arXiv:1507.05979v1 ), it is shown that entanglement is a necessary condition for forming non-trivial invariants of knots from braid closures via solutions to the Yang–Baxter equation. We show that the arguments used by Alagic et al. (2015) generalize to essentially the same results for quantum invariant state summation models of knots. In one case (the unoriented swap case) we give an example of a Yang–Baxter operator, and associated quantum invariant, that can detect the Hopf link. Again this is analogous to the results of Alagic et al. (2015). We also give a class of R matrices that are entangling and are weak invariants of classical knots and links yet strong invariants of virtual knots and links. We also give an example of an SU(2) representation of the three-strand braid group that models the Jones polynomial for closures of three-strand braids. This invariant is a quantum model for the Jones polynomial restricted to three-strand braids, and it does not involve quantum entanglement. These relationships between topological braiding and quantum entanglement can be used as a framework for future work in understanding the properties of entangling gates in topological quantum computing. The paper ends with a discussion of the Aravind hypothesis about the direct relationship of knots and quantum entanglement and the ER=EPR E R = E P R hypothesis about the relationship of quantum entanglement with the connectivity of space. We describe how, given a background space and a quantum tensor network, to construct a new topological space that welds the network and the background space together. This construction embodies the principle that quantum entanglement and topological connectivity are intimately related.
. This paper is an introduction to relationships between quantum topology and quantum computing. We show how knots are related not just to braiding and quantum operators, but to quantum set theoretical foundations, algebras of fermions, and we show how the operation of negation in logic, seen as both a value and an operator, can generate the fusion algebra for a Majorana fermion. We call negation in this mode the mark, as it operates on itself to change from marked to unmarked states. The mark viewed recursively as a simplest discrete dynamical system naturally generates the fermion algebra, the quaternions and the braid group representations related to Majorana fermions. The paper begins with these fundamentals. It then discusses unitary solutions to the Yang-Baxter equation that are universal quantum gates, quantum entanglement and topological entanglement, and gives an exposition of knot-theoretic recoupling theory, its relationship with topological quantum field theory and applies these methods to produce unitary representations of the braid groups that are dense in the unitary groups. These methods are rooted in the bracket state sum model for the Jones polynomial. A self-contained study of the quantum universal Fibonacci model is given. Results are applied to give quantum algorithms for the computation of the colored Jones polynomials for knots and links, and theWitten-Reshetikhin-Turaev invariant of three manifolds. Two constructions are given for the Fibonacci model, one based in Temperley-Lieb recoupling theory, the other quite elementary and also based on the Temperley-Lieb algebra. This paper begins an exploration of quantum epistemology in relation to the structure of discrimination as the underpinning of basic logic, perception and measurement. Introduction. This paper is an introduction to relationships between quantum topology and quantum computing. We take a foundational approach, showing how knots are related not just to braiding and quantum operators, but to quantum set theoretical foundations and algebras of fermions. We show how the operation of negation in logic, seen as both a value and an operator, can generate the fusion algebra for a Majorana fermion, a particle that is its own anti-particle and interacts with itself either to annihilate itself or to produce itself. We call negation in this mode the mark , as it operates on itself to change from marked to unmarked states.
We analyze the connections between the mathematical theory of knots and quantum physics by addressing a number of algorithmic questions related to both knots and braid groups. Knots can be distinguished by means of “knot invariants,” among which the Jones polynomial plays a prominent role, since it can be associated with observables in topological quantum field theory. Although the problem of computing the Jones polynomial is intractable in the framework of classical complexity theory, it has been recently recognized that a quantum computer is capable of approximating it in an efficient way. The quantum algorithms discussed here represent a breakthrough for quantum computation, since approximating the Jones polynomial is actually a “universal problem,” namely, the hardest problem that a quantum computer can efficiently handle.
This paper is an exploration of the role of unitary braiding operators in quantum computing. We show that a single specific solution, R, of the Yang-Baxter Equation is a universal gate for quantum computing, in the presence of local unitary transformations. We show that this same R generates a new non-trivial invariant of braids, knots, and links. The paper discusses these results in the context of comparing quantum and topological points of view. In particular, we discuss quantum computation of link invariants, the relationship between quantum entanglement and topological entanglement, and the structure of braiding in a topological quantum field theory.