Celebratio Mathematica

article M. H. Freedman, M. J. Larsen, and Z. Wang: “The two-eigenvalue problem and density of Jones representation of braid groups,” Comm. Math. Phys. 228 : 1 (2002), pp. 177–199. MR 1911253 Zbl 1045.20027

Zhenghan Wang	Related
Michael J. Larsen	Related

article M. H. Freedman, M. Larsen, and Z. Wang: “A modular functor which is universal for quantum computation,” Comm. Math. Phys. 227 : 3 (2002), pp. 605–622. MR 1910833 Zbl 1012.81007 ArXiv quant-ph/0001108

Zhenghan Wang	Related
Michael J. Larsen	Related

article M. H. Freedman, A. Kitaev, and Z. Wang: “Simulation of topological field theories by quantum computers,” Comm. Math. Phys. 227 : 3 (2002), pp. 587–603. MR 1910832 Zbl 1014.81006 ArXiv quant-ph/0001071

Quantum computers will work by evolving a high tensor power of a small (e.g. two) dimensional Hilbert space by local gates, which can be implemented by applying a local Hamiltonian \( H \) for a time \( t \). In contrast to this quantum engineering, the most abstract reaches of theoretical physics has spawned “topological models” having a finite dimensional internal state space with no natural tensor product structure and in which the evolution of the state is discrete, \( H \equiv 0 \). These are called topological quantum field theories (TQFTs). These exotic physical systems are proved to be efficiently simulated on a quantum computer. The conclusion is two-fold:

TQFTs cannot be used to define a model of computation stronger than the usual quantum model “BQP”.
TQFTs provide a radically different way of looking at quantum computation. The rich mathematical structure of TQFTs might suggest a new quantum algorithm.

Zhenghan Wang	Related
Alexei Kitaev	Related

article M. H. Freedman, A. Kitaev, M. J. Larsen, and Z. Wang: “Topological quantum computation,” Bull. Amer. Math. Soc. (N.S.) 40 : 1 (2003), pp. 31–38. MR 1943131 Zbl 1019.81008 ArXiv quant-ph/0101025

The theory of quantum computation can be constructed from the abstract study of anyonic systems. In mathematical terms, these are unitary topological modular functors. They underlie the Jones polynomial and arise in Witten–Chern–Simons theory. The braiding and fusion of anyonic excitations in quantum Hall electron liquids and \( 2D \)-magnets are modeled by modular functors, opening a new possibility for the realization of quantum computers. The chief advantage of anyonic computation would be physical error correction: An error rate scaling like \( e^{-\alpha l} \), where \( l \) is a length scale, and \( \alpha \) is some positive constant. In contrast, the “presumptive” qubit-model of quantum computation, which repairs errors combinatorically, requires a fantastically low initial error rate (about \( 10^{-4} \)) before computation can be stabilized.

Zhenghan Wang	Related
Michael J. Larsen	Related
Alexei Kitaev	Related

article M. Freedman, C. Nayak, K. Shtengel, K. Walker, and Z. Wang: “A class of \( P,T \)-invariant topological phases of interacting electrons,” Ann. Physics 310 : 2 (2004), pp. 428–492. MR 2044743 Zbl 1057.81053

We describe a class of parity- and time-reversal-invariant topological states of matter which can arise in correlated electron systems in \( 2+1 \)-dimensions. These states are characterized by particle-like excitations exhibiting exotic braiding statistics. \( P \) and \( T \) invariance are maintained by a ‘doubling’ of the low-energy degrees of freedom which occurs naturally without doubling the underlying microscopic degrees of freedom. The simplest examples have been the subject of considerable interest as proposed mechanisms for high-\( T_c \) superconductivity. One is the ‘doubled’ version of the chiral spin liquid. The chiral spin liquid gives rise to anyon superconductivity at finite doping and the corresponding field theory is \( U(1) \) Chern–Simons theory at coupling constant \( m=2 \). The ‘doubled’ theory is two copies of this theory, one with \( m=2 \) the other with \( m=-2 \). The second example corresponds to \( Z_2 \) gauge theory, which describes a scenario for spin-charge separation. Our main concern, with an eye towards applications to quantum computation, are richer models which support non-Abelian statistics. All of these models, richer or poorer, lie in a tightly organized discrete family indexed by the Baraha numbers, \( 2\cos(\pi/(k+2)) \), for positive integer \( k \). The physical inference is that a material manifesting the \( Z_2 \) gauge theory or a doubled chiral spin liquid might be easily altered to one capable of universal quantum computation. These phases of matter have a field-theoretic description in terms of gauge theories which, in their infrared limits, are topological field theories. We motivate these gauge theories using a parton model or slave-fermion construction and show how they can be solved exactly. The structure of the resulting Hilbert spaces can be understood in purely combinatorial terms. The highly constrained nature of this combinatorial construction, phrased in the language of the topology of curves on surfaces, lays the groundwork for a strategy for constructing microscopic lattice models which give rise to these phases.

Kirill Shtengel	Related
Zhenghan Wang	Related
Kevin Walker	Related
Chetan Nayak	Related

article M. Freedman, A. Feiguin, S. Trebst, A. Ludwig, M. Troyer, A. Kitaev, and Z. Wang: “Interacting anyons in topological quantum liquids: The golden chain,” Phys. Rev. Lett. 98 (2007), pp. 160409. ArXiv cond-mat/0612341

We discuss generalizations of quantum spin Hamiltonians using anyonic degrees of freedom. The simplest model for interacting anyons energetically favors neighboring anyons to fuse into the trivial (“identity”) channel, similar to the quantum Heisenberg model favoring neighboring spins to form spin singlets. Numerical simulations of a chain of Fibonacci anyons show that the model is critical with a dynamical critical exponent \( z=1 \), and described by a two-dimensional (\( 2D \)) conformal field theory with central charge \( c=7/10 \). An exact mapping of the anyonic chain onto the \( 2D \) tricritical Ising model is given using the restricted-solid-on-solid representation of the Temperley–Lieb algebra. The gaplessness of the chain is shown to have topological origin.

Zhenghan Wang	Related
Adrian Esteban Feiguin	Related
A. Ludwig	Related
Simon Trebst	Related
Matthias Troyer	Related
Alexei Kitaev	Related

article M. H. Freedman and Z. Wang: “Large quantum Fourier transforms are never exactly realized by braiding conformal blocks,” Phys. Rev. A (3) 75 : 3 (2007), pp. 032322. MR 2312110 ArXiv cond-mat/0609411

Fourier transform is an essential ingredient in Shor’s factoring algorithm. In the standard quantum circuit model with the gate set \( \{U(2) \), controlled-NOT\( \} \), the discrete Fourier transforms \( F_N=(\omega^{ij})_{N\times N} \) for \( i,j=0,1,\dots,N{-}1 \) and \( \omega=e^{2\pi i/N} \) can be realized exactly by quantum circuits of size \( O(n^2) \) with \( n=\ln N \), and so can the discrete sine or cosine transforms. In topological quantum computing, the simplest universal topological quantum computer is based on the Fibonacci \( (2+1) \)-topological quantum field theory (TQFT), where the standard quantum circuits are replaced by unitary transformations realized by braiding conformal blocks. We report here that the large Fourier transforms \( F_N \) and the discrete sine or cosine transforms can never be realized exactly by braiding conformal blocks for a fixed TQFT. It follows that an approximation is unavoidable in the implementation of Fourier transforms by braiding conformal blocks.

incollection M. Freedman, C. Nayak, K. Walker, and Z. Wang: “On picture \( (2+1) \)-TQFTs,” pp. 19–106 in Topology and physics (Tianjin, China, 27–31 July 2007). Edited by K. Lin, Z. Weng, and W. Zhang. Nankai Tracts in Mathematics 12. World Scientific (Hackensack, NJ), 2008. MR 2503392 Zbl 1168.81024 ArXiv 0806.1926

Zhenghan Wang	Related
Kevin Lin	Related
Kevin Walker	Related
Chetan Nayak	Related
Weiping Zhang	Related
Xiao-Song Lin	Related

article L. Fidkowski, M. Freedman, C. Nayak, K. Walker, and Z. Wang: “From string nets to nonabelions,” Comm. Math. Phys. 287 : 3 (2009), pp. 805–827. MR 2486662 Zbl 1196.82072 ArXiv cond-mat/0610583

We discuss Hilbert spaces spanned by the set of string nets, i.e. trivalent graphs, on a lattice. We suggest some routes by which such a Hilbert space could be the low-energy subspace of a model of quantum spins on a lattice with short-ranged interactions. We then explain conditions which a Hamiltonian acting on this string net Hilbert space must satisfy in order for the system to be in the DFib (Doubled Fibonacci) topological phase, that is, be described at low energy by an \( \mathit{SO}(3)_3\times\mathit{SO}(3)_3 \) doubled Chern–Simons theory, with the appropriate non-abelian statistics governing the braiding of the low-lying quasiparticle excitations (nonabelions). Using the string net wavefunction, we describe the properties of this phase. Our discussion is informed by mappings of string net wavefunctions to the chromatic polynomial and the Potts model.

Zhenghan Wang	Related
Lukasz Fidkowski	Related
Kevin Walker	Related
Chetan Nayak	Related

article M. Freedman, M. B. Hastings, C. Nayak, X.-L. Qi, K. Walker, and Z. Wang: “Projective ribbon permutation statistics: A remnant of non-Abelian braiding in higher dimensions,” Phys. Rev. B 83 : 11 (2011), pp. 115132. ArXiv 1005.0583

In a recent paper, Teo and Kane Phys. Rev. Lett. 104 046401 (2010) proposed a three-dimensional (3D) model in which the defects support Majorana fermion zero modes. They argued that exchanging and twisting these defects would implement a set \( R \) of unitary transformations on the zero-mode Hilbert space which is a “ghostly” recollection of the action of the braid group on Ising anyons in two dimensions. In this paper, we find the group \( T_{2n} \), which governs the statistics of these defects by analyzing the topology of the space \( K_{2n} \) of configurations of \( 2n \) defects in a slowly spatially varying gapped free-fermion Hamiltonian: \( T_{2n}\equiv \pi_1(K_{2n}) \). We find that the group \( T_{2n}=\mathbb{Z}\times T_{2n}^r \), where the “ribbon permutation group” \( T_{2n}^r \) is a mild enhancement of the permutation group \[ S_{2n}: T_{2n}^r\equiv \mathbb{Z}_2\rtimes E((\mathbb{Z}_2)^{2n}\rtimes S_{2n}) .\] Here, \( E((\mathbb{Z}_2)^{2n}\rtimes S_{2n}) \) is the “even part” of \( (\mathbb{Z}_2)^{2n}\rtimes S_{2n} \), namely, those elements for which the total parity of the element in \( (\mathbb{Z}_2)^{2n} \) added to the parity of the permutation is even. Surprisingly, \( R \) is only a projective representation of \( T_{2n} \), a possibility proposed by Wilczek [hep-th/9806228]. Thus, Teo and Kane’s defects realize projective ribbon permutation statistics, which we show to be consistent with locality. We extend this phenomenon to other dimensions, codimensions, and symmetry classes. We note that our analysis applies to 3D networks of quantum wires supporting Majorana fermions; thus, these networks are not required to be planar. Because it is an essential input for our calculation, we review the topological classification of gapped free-fermion systems and its relation to Bott periodicity.

Zhenghan Wang	Related
Kevin Walker	Related
Matthew B. Hastings	Related
Chetan Nayak	Related
Xiao-Liang Qi	Related

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Michael H. Freedman

Quantum computing

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