Celebratio Mathematica

[1] article M. H. Freedman: “P/NP, and the quantum field computer,” Proc. Natl. Acad. Sci. USA 95 : 1 (1998), pp. 98–101. MR 1612425 Zbl 0895.68053

The central problem in computer science is the conjecture that two complexity classes, \( P \) (polynomial) and \( NP \) (nondeterministic polynomial time — roughly those decision problems for which a proposed solution can be checked in polynomial time), are distinct in the standard Turing model of computation: \( P\neq NP \). As a generality, we propose that each physical theory supports computational models whose power is limited by the physical theory. It is well known that classical physics supports a multitude of implementation of the Turing machine. Non-Abelian topological quantum field theories exhibit the mathematical features necessary to support a model capable of solving all \( \#P \) problems, a computationally intractable class, in polynomial time. Specifically [Witten 1989] has identified expectation values in a certain \( \mathit{SU}(2) \)-field theory with values of the Jones polynomial [Jones 1985] that are \( \#P \)-hard [Jaeger, Vertigen and Welsh 1990]. This suggests that some physical system whose effective Lagrangian contrains a non-Abelian topological term might be manipulated to serve as an analog computer capable of solving \( NP \) or even \( \#P \)-hard problems in polynomial time. Defining such a system and addressing the accuracy issues inherent in preparation and measurement is a major unsolved problem.

[2] incollection M. H. Freedman: “Topological views on computational complexity,” pp. 453–464 in Proceedings of the International Congress of Mathematicians (Berlin, 1998), published as Doc. Math. Extra II. Fakultät für Mathematik, Universität Bielefeld (Bielefeld), 1998. MR 1648095 Zbl 0967.68520

[3] incollection M. H. Freedman: “\( K \)-sat on groups and undecidability,” pp. 572–576 in Proceedings of the thirtieth annual ACM symposium on theory of computing (Dallas, TX, May 23–26, 1998). Edited by Association for Computing Machinery. Association for Computing Machinery (New York), 1998. MR 1715605 Zbl 1028.68068

The general Boolean formula can be quickly converted into a normal form, \( N_3 \), the conjunction of triple disjunctions of literals, which is satisfiable iff the original is. On the other hand, formulae in \( N_2 \), the conjunction of 2-fold disjunctions of literals, can be checked for a satisfaction in polynomial time. Thus these two satisfaction problems, “2-sat” and “3-sat,” have been considered as an interesting boundary point between \( P \) and \( NP \). We define an infinite generalization of 2-sat and 3-sat which are respectively algorithmic and undecidable. As a corollary it is noted that the 3-colorability of doubly-periodic planar graphs is undecidable. It was suggested in [Freedman 1998] that a general approach to proving \( P\neq NP \) would be to construct some infinitary limit of decision problems with the property that those admitting polynomial time algorithms would be decidable in this limit. The hope here is to exploit the strong connection between polynomial growth and finite dimensionality. Since logic has a method — self-reference — for establishing problems as undecidable, this technique applied to the limit could potentially show that the finite-decision problem lies outside of \( P \). This paper supplies one way of extending \( k \)-sat to an infinite context in which decidability distinguishes 2-sat from 3-sat.

[4] article M. H. Freedman and D. A. Meyer: “Projective plane and planar quantum codes,” Found. Comput. Math. 1 : 3 (2001), pp. 325–332. MR 1838758 Zbl 0995.94037 ArXiv quant-ph/9810055

[5] article M. H. Freedman: “Quantum computation and the localization of modular functors,” Found. Comput. Math. 1 : 2 (2001), pp. 183–204. MR 1830035 Zbl 1004.57026 ArXiv quant-ph/0003128

[6] 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

[7] 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

[8] 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

[9] article M. H. Freedman: “Poly-locality in quantum computing,” Found. Comput. Math. 2 : 2 (2002), pp. 145–154. MR 1894373 Zbl 1075.81507 ArXiv quant-ph/0001077

A polynomial depth quantum circuit affects, by definition, a poly-local unitary transformation of a tensor product state space. It is a reasonable belief [Feynman 1982; Lloyd 1996; Freedman, Kitaev and Wang 2002] that, at a fine scale, these are precisely the transformations which will be available from physics to solve computational problems. The poly-locality of a discrete Fourier transform on cyclic groups is at the heart of Shor’s factoring algorithm. We describe a class of poly-local transformations, which include the discrete orthogonal wavelet transforms, in the hope that these may be helpful in constructing new quantum algorithms. We also observe that even a rather mild violation of poly-locality leads to a model without one-way functions, giving further evidence that poly-locality is an essential concept.

[10] incollection M. H. Freedman, D. A. Meyer, and F. Luo: “\( Z_2 \)-systolic freedom and quantum codes,” pp. 287–320 in Mathematics of quantum computation. Edited by R. K. Brylinski and G. Chen. Computational Mathematics 3. Chapman & Hall/CRC (Boca Raton, FL), 2002. MR 2007952 Zbl 1075.81508

A closely coupled pair of conjectures/questions — one in differential geometry (by M. Gromov), the other in quantum information theory — are both answered in the negative. The answer derives from a certain metrical flexibility of manifolds and a corresponding improvement to the theoretical efficiency of existing local quantum codes. We exhibit this effect by constructing a family of metrics on \( \mathbb{S}^2\times\mathbb{S}^1 \), and other three and four dimensional manifolds. Quantitatively, the explicit “freedom” exhibited is too weak (a \( \log^{1/2} \) factor in the natural scaling) to yield practical codes but we cannot rule out the possibility of other families of geometries with more dramatic freedom.

Goong Chen	Related
Feng Luo	Related
David A. Meyer	Related
Ranee K. Brylinski	Related

[11] article M. H. Freedman: “A magnetic model with a possible Chern–Simons phase,” Comm. Math. Phys. 234 : 1 (2003), pp. 129–183. With an appendix by F. Goodman and H. Wenzl. MR 1961959 Zbl 1060.81054 ArXiv quant-ph/0110060

An elementary family of local Hamiltonians \( H_{\circ,\ell} \), \( \ell = 1, 2, 3,\dots\, \), is described for a 2-dimensional quantum mechanical system of spin \( = 1/2 \) particles. On the torus, the ground state space \( G_{\circ,\ell} \) is (log) extensively degenerate but should collapse under “perturbation” to an anyonic system with a complete mathematical description: the quantum double of the \( \mathit{SO}(3) \)-Chern–Simons modular functor at \( q = e^{2\pi i/\ell+2} \) which we call \( \mathit{DE}\ell \). The Hamiltonian \( H_{\circ,\ell} \) defines a quantum loop gas. We argue that for \( \ell = 1 \) and 2, \( G_{\circ,\ell} \) is unstable and the collapse to \( G_{\varepsilon,\ell} \cong \mathit{DE}\ell \) can occur truly by perturbation. For \( \ell\geq 3 \), \( G_{\circ,\ell} \) is stable and in this case finding \( G_{\varepsilon,\ell} \cong \mathit{DE}\ell \) must require either \( \varepsilon > \varepsilon_\ell > 0 \), help from finite system size, surface roughening (see section 3), or some other trick, hence the initial use of quotes “ ”. A hypothetical phase diagram is included in the introduction.

The effect of perturbation is studied algebraically: the ground state space \( G_{\circ,\ell} \) of \( H_{\circ,\ell} \) is described as a surface algebra and our ansatz is that perturbation should respect this structure yielding a perturbed ground state \( G_{\varepsilon,\ell} \) described by a quotient algebra. By classification, this implies \( G_{\varepsilon,\ell} \cong \mathit{DE}\ell \). The fundamental point is that nonlinear structures may be present on degenerate eigenspaces of an initial \( H_{\circ} \) which constrain the possible effective action of a perturbation.

There is no reason to expect that a physical implementation of \( G_{\varepsilon,\ell} \cong \mathit{DE}\ell \) as an anyonic system would require the low temperatures and time asymmetry intrinsic to Fractional Quantum Hall Effect (FQHE) systems or rotating Bose–Einstein condensates — the currently known physical systems modeled by topological modular functors. A solid state realization of \( \mathit{DE}3 \), perhaps even one at a room temperature, might be found by building and studying systems, “quantum loop gases,” whose main term is \( H_{\circ,3} \). This is a challenge for solid state physicists of the present decade. For \( l\geq 3 \), \( \ell\neq 2\mod 4 \), a physical implementation of \( \mathit{DE}\ell \) would yield an inherently fault-tolerant universal quantum computer. But a warning must be posted, the theory at \( \ell = 2 \) is not computationally universal and the first universal theory at \( \ell = 3 \) seems somewhat harder to locate because of the stability of the corresponding loop gas. Does nature abhor a quantum computer?

Frederick Michael Goodman	Related
Hans G. Wenzl	Related

[12] 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

[13] techreport M. H. Freedman, C. Nayak, and K. Shtengel: Non-Abelian topological phases in an extended Hubbard model. Preprint, September 2003. ArXiv cond-mat/0309120

We describe four closely related Hubbard-like models (models A, B, C and D) of particles that can hop on a \( 2D \) Kagome lattice interacting via Coulomb repulsion. The particles can be either bosons (models A and B) or (spinless) fermions (models C and D). Models A and C also include a ring exchange term. In all four cases we solve equations in the model parameters to arrive at an exactly soluble point whose ground state manifold is the extensively degenerate “\( d \)-isotopy space” \( \bar{V}_d \), \( 0 < d < 2 \). Near the “special” values, \( d = 2 \cos \pi/k+2 \), \( \bar{V}_d \) should collapse to a stable topological phase with anyonic excitations closely related to \( \mathit{SU}(2) \) Chern–Simons theory at level \( k \). We mention simplified models \( A^- \) and \( C^- \) which may also lead to these topological phases.

Kirill Shtengel	Related
Chetan Nayak	Related

[14] 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

[15] article M. Bordewich, M. Freedman, L. Lovász, and D. Welsh: “Approximate counting and quantum computation,” Combin. Probab. Comput. 14 : 5–6 (2005), pp. 737–754. MR 2174653 Zbl 1089.68040

Motivated by the result that an ‘approximate’ evaluation of the Jones polynomial of a braid at a 5th root of unity can be used to simulate the quantum part of any algorithm in the quantum complexity class BQP, and results relating BQP to the counting class GapP, we introduce a form of additive approximation which can be used to simulate a function in BQP. We show that all functions in the classes \( \# \)P and GapP have such an approximation scheme under certain natural normalizations. However, we are unable to determine whether the particular functions we are motivated by, such as the above evaluation of the Jones polynomial, can be approximated in this way. We close with some open problems motivated by this work.

D. Welsh	Related
Magnus Bordewich	Related
László Lovász	Related

[16]M. H. Freedman, C. Nayak, and K. Shtengel: “Line of critical points in \( 2+1 \) dimensions: Quantum critical loop gases and non-abelian gauge theory,” Phys. Rev. Lett. 94 : 14 (2005), pp. 147205.

In this Letter, we
(1) construct a one-parameter family of lattice models of interacting spins;
(2) obtain their exact ground states;
(3) derive a statistical-mechanical analogy which relates their ground states to \( O(n) \) loop gases;
(4) show that the models are critical for \( d\leq\sqrt{2} \), where \( d \) parametrizes the models;
(5) note that, for the special values \( d=2\cos[\pi/(k+2)] \), they are related to doubled level-\( k \) \( \mathit{SU}(2) \) Chern–Simons theory;
(6) conjecture that they are in the universality class of a nonrelativistic \( \mathit{SU}(2) \) gauge theory; and
(7) show that its one-loop \( \beta \) function vanishes for all values of the coupling constant, implying that it is also on a critical line.

Kirill Shtengel	Related
Chetan Nayak	Related

[17] article D. Das Sarma, M. H. Freedman, and C. Nayak: “Topologically-protected qubits from a possible non-abelian fractional quantum Hall state,” Phys. Rev. Lett. 94 : 6 (2005), pp. 166802. ArXiv cond-mat/0412343

The Pfaffian state is an attractive candidate for the observed quantized Hall plateau at a Landau-level filling fraction \( \nu = 5/2 \). This is particularly intriguing because this state has unusual topological properties, including quasiparticle excitations with non-Abelian braiding statistics. In order to determine the nature of the \( \nu = 5/2 \) state, one must measure the quasiparticle braiding statistics. Here, we propose an experiment which can simultaneously determine the braiding statistics of quasiparticle excitations and, if they prove to be non-Abelian, produce a topologically protected qubit on which a logical Not operation is performed by quasiparticle braiding. Using the measured excitation gap at \( \nu = 5/2 \), we estimate the error rate to be \( 10^{-30} \) or lower.

Sankar Das Sarma	Related
Chetan Nayak	Related

[18]M. Freedman, C. Nayak, and K. Shtengel: “An extended Hubbard model with ring exchange: A route to a non-abelian topological phase,” Phys. Rev. Lett. 94 : 6 (2005), pp. 066401.

Kirill Shtengel	Related
Chetan Nayak	Related

[19] techreport M. Freedman, C. Nayak, and K. Walker: Tilted interferometry realizes universal quantum computation in the Ising TQFT without overpasses. Preprint, December 2005. ArXiv cond-mat/0512072

Kevin Walker	Related
Chetan Nayak	Related

[20]M. Freedman, C. Nayak, and K. Walker: “Towards universal topological quantum computation in the \( \nu=5/2 \) fractional quantum Hall state,” Phys. Rev. B 73 : 24 (2006), pp. 245307.

The Pfaffian state, which may describe the quantized Hall plateau observed at Landau level filling fraction \( \nu = 5/2 \), can support topologically-protected qubits with extremely low error rates. Braiding operations also allow perfect implementation of certain unitary transformations of these qubits. However, in the case of the Pfaffian state, this set of unitary operations is not quite sufficient for universal quantum computation (i.e. is not dense in the unitary group). If some topologically unprotected operations are also used, then the Pfaffian state supports universal quantum computation, albeit with some operations which require error correction. On the other hand, if certain topology-changing operations can be implemented, then fully topologically-protected universal quantum computation is possible. In order to accomplish this, it is necessary to measure the interference between quasiparticle trajectories which encircle other moving trajectories in a time-dependent Hall droplet geometry [cond-mat/0512072].

Kevin Walker	Related
Chetan Nayak	Related

[21]M. Freedman, S. Das Sarma, and C. Nayak: “Topological quantum computation,” Physics Today 59 : 7 (July 2006), pp. 32–38.

Sankar Das Sarma	Related
Chetan Nayak	Related

[22] article S. H. Simon, N. E. Bonesteel, M. H. Freedman, N. Petrovic, and L. Hormozi: “Topological quantum computing with only one mobile quasiparticle,” Phys. Rev. Lett. 96 : 7 (2006), pp. 070503. MR 2205654 ArXiv quant-ph/0509175

Steven H. Simon	Related
Nicholas E. Bonesteel	Related
Layla Hormozi	Related
Nikola Petrovic	Related

[23] 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

[24] 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.

[25] article P. Bonderson, M. Freedman, and C. Nayak: “Measurement-only topological quantum computation,” Phys. Rev. Lett. 101 : 1 (2008), pp. 010501. MR 2429542 Zbl 1228.81121 ArXiv 0802.0279

Parsa Bonderson	Related
Chetan Nayak	Related

[26] article C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma: “Non-abelian anyons and topological quantum computation,” Rev. Modern Phys. 80 : 3 (2008), pp. 1083–1159. MR 2443722 Zbl 1205.81062 ArXiv 0707.1889

Topological quantum computation has emerged as one of the most exciting approaches to constructing a fault-tolerant quantum computer. The proposal relies on the existence of topological states of matter whose quasiparticle excitations are neither bosons nor fermions, but are particles known as non-Abelian anyons, meaning that they obey non-Abelian braiding statistics. Quantum information is stored in states with multiple quasiparticles, which have a topological degeneracy. The unitary gate operations that are necessary for quantum computation are carried out by braiding quasiparticles and then measuring the multiquasiparticle states. The fault tolerance of a topological quantum computer arises from the nonlocal encoding of the quasiparticle states, which makes them immune to errors caused by local perturbations. To date, the only such topological states thought to have been found in nature are fractional quantum Hall states, most prominently the \( \nu =5/2 \) state, although several other prospective candidates have been proposed in systems as disparate as ultracold atoms in optical lattices and thin-film superconductors. In this review article, current research in this field is described, focusing on the general theoretical concepts of non-Abelian statistics as it relates to topological quantum computation, on understanding non-Abelian quantum Hall states, on proposed experiments to detect non-Abelian anyons, and on proposed architectures for a topological quantum computer. Both the mathematical underpinnings of topological quantum computation and the physics of the subject are addressed, using the \( \nu =5/2 \) fractional quantum Hall state as the archetype of a non-Abelian topological state enabling fault-tolerant quantum computation.

Steven H. Simon	Related
Ady Stern	Related
Chetan Nayak	Related
Sankar Das Sarma	Related

[27]M. Freedman, C. Nayak, and K. Shtengel: “Lieb–Schultz–Mattis theorem for quasitopological systems,” Phys. Rev. B 78 (2008), pp. 174411.

In this paper we address the question of the existence of a spectral gap in a class of local Hamiltonians. These Hamiltonians have the following properties: their ground states are known exactly; all equal-time correlation functions of local operators are short-ranged; and correlation functions of certain nonlocal operators are critical. A variational argument shows gaplessness with \( \omega \propto k^2 \) at critical points defined by the absence of certain terms in the Hamiltonian, which is remarkable because equal-time correlation functions of local operators remain short ranged. We call such critical points, in which spatial and temporal scaling are radically different, quasitopological. When these terms are present in the Hamiltonian, the models are in gapped topological phases which are of special interest in the context of topological quantum computation.

Kirill Shtengel	Related
Chetan Nayak	Related

[28] 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

[29] techreport M. H. Freedman: A topological phase in a quantum gravity model. Preprint, December 2008. A talk at Solvay conference, October 2008. ArXiv 0812.2278

I would claim that we do not have a suitably general definition of what a topological phase is, or more importantly, any robust understanding of how to enter one even in the world of mathematical models. The latter is, of course, the more important issue and the main subject of this note. But a good definition can sharpen our thinking and a poor definition can misdirect us. I will not attempt a final answer here but merely comment on the strengths and weaknesses of possible definitions and argue for some flexibility. In particular, I describe a rather simple class of “quantum gravity” models which are neither lattice nor field theoretic but appear to contain strong candidates for topological phases.

[30] article P. Bonderson, M. Freedman, and C. Nayak: “Measurement-only topological quantum computation via anyonic interferometry,” Ann. Physics 324 : 4 (2009), pp. 787–826. MR 2508474 Zbl 1171.81004 ArXiv 0808.1933

We describe measurement-only topological quantum computation using both projective and interferometrical measurement of topological charge. We demonstrate how anyonic teleportation can be achieved using “forced measurement” protocols for both types of measurement. Using this, it is shown how topological charge measurements can be used to generate the braiding transformations used in topological quantum computation, and hence that the physical transportation of computational anyons is unnecessary. We give a detailed discussion of the anyonics for implementation of topological quantum computation (particularly, using the measurement-only approach) in fractional quantum Hall systems.

Parsa Bonderson	Related
Chetan Nayak	Related

[31] 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

[32] techreport P. Bonderson, S. Das Sarma, M. Freedman, and C. Nayak: A blueprint for a topologically fault-tolerant quantum computer. Preprint, March 2010. ArXiv 1003.2856

The advancement of information processing into the realm of quantum mechanics promises a transcendence in computational power that will enable problems to be solved which are completely beyond the known abilities of any “classical” computer, including any potential non-quantum technologies the future may bring. However, the fragility of quantum states poses a challenging obstacle for realization of a fault-tolerant quantum computer. The topological approach to quantum computation proposes to surmount this obstacle by using special physical systems — non-Abelian topologically ordered phases of matter — that would provide intrinsic fault-tolerance at the hardware level. The so-called “Ising-type” non-Abelian topological order is likely to be physically realized in a number of systems, but it can only provide a universal gate set (a requisite for quantum computation) if one has the ability to perform certain dynamical topology-changing operations on the system. Until now, practical methods of implementing these operations were unknown. Here we show how the necessary operations can be physically implemented for Ising-type systems realized in the recently proposed superconductor-semiconductor and superconductor-topological insulator heterostructures. Furthermore, we specify routines employing these methods to generate a computationally universal gate set. We are consequently able to provide a schematic blueprint for a fully topologically-protected Ising based quantum computer using currently available materials and techniques. This may serve as a starting point for attempts to construct a fault-tolerant quantum computer, which will have applications to cryptanalysis, drug design, efficient simulation of quantum many-body systems, solution of large systems of linear equations, searching large databases, engineering future quantum computers, and — most importantly — those applications which no one in our classical era has the prescience to foresee.

Parsa Bonderson	Related
Sankar Das Sarma	Related
Chetan Nayak	Related

[33] article M. H. Freedman, L. Gamper, C. Gils, S. V. Isakov, S. Trebst, and M. Troyer: “Topological phases: An expedition off lattice,” Ann. Physics 326 : 8 (2011), pp. 2108–2137. MR 2812881 Zbl 1221.81219 ArXiv 1102.0270

Motivated by the goal to give the simplest possible microscopic foundation for a broad class of topological phases, we study quantum mechanical lattice models where the topology of the lattice is one of the dynamical variables. However, a fluctuating geometry can remove the separation between the system size and the range of local interactions, which is important for topological protection and ultimately the stability of a topological phase. In particular, it can open the door to a pathology, which has been studied in the context of quantum gravity and goes by the name of ‘baby universe’, Here we discuss three distinct approaches to suppressing these pathological fluctuations. We complement this discussion by applying Cheeger’s theory relating the geometry of manifolds to their vibrational modes to study the spectra of Hamiltonians. In particular, we present a detailed study of the statistical properties of loop gas and string net models on fluctuating lattices, both analytically and numerically.

Lukas Gamper	Related
Simon Trebst	Related
Matthias Troyer	Related
Charlotte Gils	Related
Sergei V. Isakov	Related

[34] techreport S. J. Yamamoto, M. Freedman, and K. Yang: 3D non-abelian anyons: Degeneracy splitting and detection by adiabatic cooling. Preprint, February 2011. ArXiv 1102.5742

3D non-abelian anyons have been theoretically proposed to exist in heterostructures composed of type II superconductors and topological insulators. We use realistic material parameters for a device derived from \( \mathrm{Bi}_2\mathrm{Se}_3 \) to quantitatively predict the temperature and magnetic field regimes where an experiment might detect the presence of these exotic states by means of a cooling effect. Within the appropriate parameter regime, an adiabatic increase of the magnetic field will result in a decrease of system temperature when anyons are present. If anyons are not present, the same experiment would result in heating.

Kun Yang	Related
S. J. Yamamoto	Related

[35] 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

Michael H. Freedman

Quantum computing