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 spin-1/2 particles, using a variant of the spin-network 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 approximate certain transition amplitudes from the Ponzano-Regge spin foam model of quantum gravity. No polynomial time classical algorithms for these problems are known.
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 spin-1/2 particles, using a variant of the spin-network 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 approximate certain transition amplitudes from the Ponzano-Regge spin foam model of quantum gravity. No polynomial time classical algorithms for these problems are known.
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 spin-1/2 particles, using a variant of the spin-network 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 Ponzano-Regge spin foam model of quantum gravity. No polynomial time classical algorithms for these problems are known.
Inspired by connections to two dimensional quantum theory, we define several models of computation based on permuting distinguishable particles (which we call balls), and characterize their computational complexity. In the quantum setting, we find that the computational power of this model depends on the initial input states. More precisely, with a standard basis input state, we show how to approximate the amplitudes of this model within additive error using the model DQC1 (the class of problems solvable with one clean qubit), providing evidence that the model in this case is weaker than universal quantum computing. However, for specific choices of input states, the model is shown to be universal for BQP in an encoded sense. We use representation theory of the symmetric group to partially classify the computational complexity of this model for arbitrary input states. Interestingly, we find some input states which yield a model intermediate between DQC1 and BQP. Furthermore, we consider a restricted version of this model based on an integrable scattering problem in 1+1 dimensions. We show it is universal under postselection, if we allow intermediate destructive measurements and specific input states. Therefore, the existence of any classical procedure to sample from the output distribution of this model within multiplicative error implies collapse of polynomial hierarchy to its third level. Finally, we define a classical version of this model in which one can probabilistically permute balls. We find this yields a complexity class which is intermediate between L and BPP. Moreover, we find a nondeterministic version of this model is NP-complete.
We define several models of computation based on permuting distinguishable particles (which we call balls) and characterize their computational complexity. In the quantum setting, we use the representation theory of the symmetric group to find variants of this model which are intermediate between BPP and DQC1 (the class of problems solvable with one clean qubit) and between DQC1 and BQP. Furthermore, we consider a restricted version of this model based on an exactly solvable scattering problem of particles moving on a line. Despite the simplicity of this model from the perspective of mathematical physics, we show that if we allow intermediate destructive measurements and specific input states, then the model cannot be efficiently simulated classically up to multiplicative error unless the polynomial hierarchy collapses. Finally, we define a classical version of this model in which one can probabilistically permute balls. We find this yields a complexity class which is intermediate between L and BPP, and that a nondeterministic version of this model is NP-complete.
Abstract Topological quantum computers promise a fault tolerant means to perform quantum computation. Topological quantum computers use particles with exotic exchange statistics called non-Abelian anyons, and the simplest anyon model which allows for universal quantum computation by particle exchange or braiding alone is the Fibonacci anyon model. One classically hard problem that can be solved efficiently using quantum computation is finding the value of the Jones polynomial of knots at roots of unity. We aim to provide a pedagogical, self-contained, review of topological quantum computation with Fibonacci anyons, from the braiding statistics and matrices to the layout of such a computer and the compiling of braids to perform specific operations. Then we use a simulation of a topological quantum computer to explicitly demonstrate a quantum computation using Fibonacci anyons, evaluating the Jones polynomial of a selection of simple knots. In addition to simulating a modular circuit-style quantum algorithm, we also show how the magnitude of the Jones polynomial at specific points could be obtained exactly using Fibonacci or Ising anyons. Such an exact algorithm seems ideally suited for a proof of concept demonstration of a topological quantum computer.
Certain exotic states of matter, so-called non-Abelian states, have the potential to provide a natural medium for the storage and manipulation of quantum information. In these states, localized particle-like excitations (quasiparticles) possess quantum numbers which are in many ways analogous to ordinary spin quantum numbers. However, unlike ordinary spins, the quantum information associated with these quantum numbers is stored globally, throughout the entire system, and so is intrinsically protected against decoherence. Quantum computation can then be carried out by dragging these quasiparticles around one another so that their trajectories sweep out world-lines in 2+1-dimensional space-time. The resulting computation depends only on the topology of the braids formed by these world-lines, and thus is robust against error. In these lectures I will review the theory of non-Abelian states, including the necessary mathematical background for describing the braiding of their quasiparticles. I will then introduce the basic ideas behind topological quantum computation and demonstrate explicitly that certain non-Abelian quasiparticles can indeed by used for universal quantum computation by showing how any quantum algorithm can be "compiled" into a braiding pattern for them. I will also discuss the most promising experimental systems for realizing non-Abelian quasiparticles, focusing primarily on fractional quantum Hall states.
Topological quantum computers promise a fault tolerant means to perform quantum computation. Topological quantum computers use particles with exotic exchange statistics called non-Abelian anyons, and the simplest anyon model which allows for universal quantum computation by particle exchange or braiding alone is the Fibonacci anyon model. One classically hard problem that can be solved efficiently using quantum computation is finding the value of the Jones polynomial of knots at roots of unity. We aim to provide a pedagogical, self-contained, review of topological quantum computation with Fibonacci anyons, from the braiding statistics and matrices to the layout of such a computer and the compiling of braids to perform specific operations. Then we use a simulation of a topological quantum computer to explicitly demonstrate a quantum computation using Fibonacci anyons, evaluating the Jones polynomial of a selection of simple knots.