## Topological quantum computation

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Venue: | Bull. Amer. Math. Soc. (N.S |

Citations: | 112 - 15 self |

### BibTeX

@ARTICLE{Freedman_topologicalquantum,

author = {Michael H. Freedman and Alexei Kitaev and Michael J. Larsen and Zhenghan Wang},

title = {Topological quantum computation},

journal = {Bull. Amer. Math. Soc. (N.S},

year = {},

pages = {31--38}

}

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### Abstract

Abstract. 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−αℓ, where ℓ is a length scale, and α 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. Quantum computation is a catch-all for several models of computation based on a theoretical ability to manufacture, manipulate and measure quantum states. In this context, there are three areas where remarkable algorithms have been found: searching a data base [15], abelian groups (factoring and discrete logarithm) [19],

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Citation Context ..., BQP (functions computable with bounded error, given quantum resources, in polynomial time), has been defined in three distinct but equivalent ways: via quantum Turing machines [2], quantum circuits =-=[3]-=-, [6], and modular functors [7], [8]. The last is the subject of this article. We may now propose a “thesis” in the spirit of Alonzo Church: all “reasonable” computational models which add the resourc... |

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Citation Context ...rs whose index is within ɛ〉0 ofthe i-th point in the geometry of the surface). Local Hamiltonians H have been found [10], [20] with highly d-degenerate ground states corresponding to modular functors =-=[31]-=-, [32] (and thus braid group representation [16] and link polynomials [18]). In √these cases, the ground state G of H will be k-code for k ∼ injectivity radius of T ∼ area T. The topological degrees o... |

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Citation Context ... “errors” which can be quantified by a fidelity distance [17] or a super-operator norm [19]. A crucial step in the theory of quantum computing has been the discovery of error-correcting quantum codes =-=[28]-=- and fault-tolerant quantum computation [25], [29]. These techniques cope with sufficiently small errors. However, the error magnitude must be smaller than some constant (called an accuracy threshold)... |

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Citation Context ...stance [17] or a super-operator norm [19]. A crucial step in the theory of quantum computing has been the discovery of error-correcting quantum codes [28] and fault-tolerant quantum computation [25], =-=[29]-=-. These techniques cope with sufficiently small errors. However, the error magnitude must be smaller than some constant (called an accuracy threshold) for these methods to work. According to rather op... |

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Citation Context ...ity distance [17] or a super-operator norm [19]. A crucial step in the theory of quantum computing has been the discovery of error-correcting quantum codes [28] and fault-tolerant quantum computation =-=[25]-=-, [29]. These techniques cope with sufficiently small errors. However, the error magnitude must be smaller than some constant (called an accuracy threshold) for these methods to work. According to rat... |

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Citation Context ...e Microsoft Way, Redmond, WA 98052 U.S.A Abstract A polynomial depth quantum circuit effects, by definition a polylocal unitary transformation of tensor product state space. It is a reasonable belief =-=[Fy]-=-[L][FKW] that, at a fine scale, these are precisely the transformations which will be available from physics to solve computational problems. The poly-locality of discrete Fourier transform on cyclic ... |

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Citation Context ...prate superconductors above Tc [12], [23]. Though much studied since the mid-1980’s the connection between fractional quantum Hall effect and quantum computation has only recently been realized [11], =-=[20]-=-. It was shown by [4] that the ground state of the ν = 1 3 electron liquid on the torus is 3-fold degenerate. This follows from the fact that excitations in this system are abelian anyons: moving one ... |

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Citation Context ...nded error, given quantum resources, in polynomial time), has been defined in three distinct but equivalent ways: via quantum Turing machines [2], quantum circuits [3], [6], and modular functors [7], =-=[8]-=-. The last is the subject of this article. We may now propose a “thesis” in the spirit of Alonzo Church: all “reasonable” computational models which add the resources of quantum mechanics (or quantum ... |

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Citation Context ...crosoft Way, Redmond, WA 98052 U.S.A Abstract A polynomial depth quantum circuit effects, by definition a polylocal unitary transformation of tensor product state space. It is a reasonable belief [Fy]=-=[L]-=-[FKW] that, at a fine scale, these are precisely the transformations which will be available from physics to solve computational problems. The poly-locality of discrete Fourier transform on cyclic gro... |

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Citation Context ...n-Chern-Simons theory for SU(2) at level=k [33]. Anyons in these models behave as topological defects of a geometric construction [10] and their braiding matrices have been shown to be universal [8], =-=[9]-=- for k ≥ 3,k �=4. Code spaces and quantum media Even after the particle types and positions of anyons are specified, there is an exponentially large (but finite dimensional) Hilbert space describing t... |

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Citation Context ...e. A more flexible and controllable way of storing quantum information is based on nonabelian anyons. These are believed to exist in the ν = 5 2 fractional quantum Hall state. According to the theory =-=[22]-=-, [24], there should be charge 1 anyonic particles 4 and some other excitations. The quantum state of the system with 2n charge 1 4 particles on the plane is 2n−1-degenerate. The degeneracy is gradual... |

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Citation Context ...onstruction is direct - no ancilla qubits are used. As an application, we prove (Corollary 1) that all the standard orthogonal discrete wavelet transforms, in particular the Daubechies transforms D2n =-=[D]-=-, are poly-local. I thank C. Williams for pointing out that he and A. Fijany previously obtained an explicit factoring of D2( = Haar) and D4 into quantum circuits [AW]. Our alternative approach has th... |

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Citation Context ...se index is within ɛ〉0 ofthe i-th point in the geometry of the surface). Local Hamiltonians H have been found [10], [20] with highly d-degenerate ground states corresponding to modular functors [31], =-=[32]-=- (and thus braid group representation [16] and link polynomials [18]). In √these cases, the ground state G of H will be k-code for k ∼ injectivity radius of T ∼ area T. The topological degrees of free... |

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Citation Context ...to local error. However, there is another type of macroscopic quantum degree of freedom. It is related to topology and arises in collective electronic systems, e.g. the fractional quantum Hall effect =-=[13]-=- and most recently in 2D cuprate superconductors above Tc [12], [23]. Though much studied since the mid-1980’s the connection between fractional quantum Hall effect and quantum computation has only re... |

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Citation Context ... level which appears to represent the same universality class as Witten-Chern-Simons theory for SU(2) at level=k [33]. Anyons in these models behave as topological defects of a geometric construction =-=[10]-=- and their braiding matrices have been shown to be universal [8], [9] for k ≥ 3,k �=4. Code spaces and quantum media Even after the particle types and positions of anyons are specified, there is an ex... |

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Citation Context ... 2D cuprate superconductors above Tc [12], [23]. Though much studied since the mid-1980’s the connection between fractional quantum Hall effect and quantum computation has only recently been realized =-=[11]-=-, [20]. It was shown by [4] that the ground state of the ν = 1 3 electron liquid on the torus is 3-fold degenerate. This follows from the fact that excitations in this system are abelian anyons: movin... |

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Citation Context ...lar the Daubechies transforms D2n [D], are poly-local. I thank C. Williams for pointing out that he and A. Fijany previously obtained an explicit factoring of D2( = Haar) and D4 into quantum circuits =-=[AW]-=-. Our alternative approach has the usual virtues and demerits of abstraction. Some perspective into the limitations poly-locality imposes on the polytime computational class of the QCM, BQP, can be ac... |

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Citation Context ... (functions computable with bounded error, given quantum resources, in polynomial time), has been defined in three distinct but equivalent ways: via quantum Turing machines [2], quantum circuits [3], =-=[6]-=-, and modular functors [7], [8]. The last is the subject of this article. We may now propose a “thesis” in the spirit of Alonzo Church: all “reasonable” computational models which add the resources of... |

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Citation Context ...n outcome effects a measurement on the Hilbert space Hn. This model supports adaptive quantum computation when surfaces of high genus are included in the theory and admits a combinatorial description =-=[1]-=- apparently in the same universal class as the fractional quantum Hall fluid. Beyond this, a discrete family of quantum Hall models exists [26] based on k+1fold hard-core interaction between electrons... |

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Citation Context ...m degree of freedom. It is related to topology and arises in collective electronic systems, e.g. the fractional quantum Hall effect [13] and most recently in 2D cuprate superconductors above Tc [12], =-=[23]-=-. Though much studied since the mid-1980’s the connection between fractional quantum Hall effect and quantum computation has only recently been realized [11], [20]. It was shown by [4] that the ground... |

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4 |
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Citation Context ...instance (F,y) and builds a sequence of “gates”, but this time the gates are braid generators (right half twist between adjacent anyons) σi , 1 ≤ i ≤ 2n −1, and a powerful approximation theorem [19], =-=[30]-=- is used to select the braid sequence which approximates the more traditional quantum circuit solving (F,y). So the topological model may be described as: (1) Initialization of a known state in the mo... |

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