Eigenstates of an operating quantum computer: hypersensitivity to static imperfections

We study the properties of eigenstates of an operating quantum computer which simulates the dynamical evolution in the regime of quantum chaos. Even if the quantum algorithm is polynomial in number of qubits n_q, it is shown that the ideal eigenstates become mixed and strongly modified by static imperfections above a certain threshold which drops exponentially with n_q. Above this threshold the quantum eigenstate entropy grows linearly with n_q but the computation remains reliable during a time scale which is polynomial in the imperfection strength and in n_q. Received 7 March 2002/ Received in final form 3 May 2002 Published online 19 July 2002     

Exact Results on Potts Model Partition Functions in a Generalized External Field and Weighted-Set Graph Colorings

We present exact results on the partition function of the q-state Potts model on various families of graphs G in a generalized external magnetic field that favors or disfavors spin values in a subset I _s ={1,…,s} of the total set of possible spin values, Z(G,q,s,v,w), where v and w are temperature- and field-dependent Boltzmann variables. We remark on differences in thermodynamic behavior between our model with a generalized external magnetic field and the Potts model with a conventional magnetic field that favors or disfavors a single spin value. Exact results are also given for the interesting special case of the zero-temperature Potts antiferromagnet, corresponding to a set-weighted chromatic polynomial Ph(G,q,s,w) that counts the number of colorings of the vertices of G subject to the condition that colors of adjacent vertices are different, with a weighting w that favors or disfavors colors in the interval I _s . We derive powerful new upper and lower bounds on Z(G,q,s,v,w) for the ferromagnetic case in terms of zero-field Potts partition functions with certain transformed arguments. We also prove general inequalities for Z(G,q,s,v,w) on different families of tree graphs. As part of our analysis, we elucidate how the field-dependent Potts partition function and weighted-set chromatic polynomial distinguish, respectively, between Tutte-equivalent and chromatically equivalent pairs of graphs.     

Phase Space Analysis of the Two-mode Binomial State Produced by Quantum Entanglement in a Beamsplitter

Phase space analysis of quantum states is a newly developed topic in quantum optics. In this work we present Wigner phase space distributions for the two-mode binomial state produced by quantum entanglement between a vacuum state and a number state in a beamsplitter. By using two new binomial formulas involving two-variable Hermite polynomials and the so-called entangled Wigner operator, we find that the analytical Wigner function for the binomial state |ξ〉_q ≡ D(ξ) |q, 0〉 is related to a Laguerre polynomial, i.e.,

$ W\left (\sigma _{,}\gamma \right ) =\frac {(-1)^{q}e^{-\left \vert \gamma \right \vert ^{2}-\left \vert \sigma \right \vert ^{2}}}{\pi ^{2}}L_{q}\left (\left \vert \frac {-\varsigma (\sigma -\gamma )+\sigma ^{\ast }+\gamma ^{\ast }} {\sqrt {1+|\varsigma |^{2}}}\right \vert ^{2}\right ) $

and its marginal distributions are proportional to the module-square of a single-variable Hermite polynomial. Also, the numerical results show that the larger number sum q of two modes lead to the stronger interference effect and the nonclassicality of the states |ξ〉_q is stronger for odd q than for even q.

     

量子Generalized Reed-Solomon码

构造出了一族量子纠错码,这族码具有参数［［n,n-2k,k+1］］_q,是q维量子系统上的码,q是任意素数的幂.这族码的最小距离达到了理论上限,因此,以码距来说,它是最优的.证明了当2≤n≤q或者q²-q+2≤n≤q²时,码都是存在的. 关键词： 量子Generalized Reed-Solomon码 量子MDS码 量子纠错码 量子信息     

A Hopf Bundle Over a Quantum Four-Sphere from the Symplectic Group

We construct a quantum version of the SU(2) Hopf bundle S⁷→S⁴. The quantum sphere S⁷_q arises from the symplectic group Sp_q(2) and a quantum 4-sphere S⁴_q is obtained via a suitable self-adjoint idempotent p whose entries generate the algebra A(S⁴_q) of polynomial functions over it. This projection determines a deformation of an (anti-)instanton bundle over the classical sphere S⁴. We compute the fundamental K-homology class of S⁴_q and pair it with the class of p in the K-theory getting the value −1 for the topological charge. There is a right coaction of SU_q(2) on S⁷_q such that the algebra A(S⁷_q) is a non-trivial quantum principal bundle over A(S⁴_q) with structure quantum group A(SU_q(2)).     

Weighted Graph Colorings

We study two weighted graph coloring problems, in which one assigns q colors to the vertices of a graph such that adjacent vertices have different colors, with a vertex weighting w that either disfavors or favors a given color. We exhibit a weighted chromatic polynomial Ph(G,q,w) associated with this problem that generalizes the chromatic polynomial P(G,q). General properties of this polynomial are proved, and illustrative calculations for various families of graphs are presented. We show that the weighted chromatic polynomial is able to distinguish between certain graphs that yield the same chromatic polynomial. We give a general structural formula for Ph(G,q,w) for lattice strip graphs G with periodic longitudinal boundary conditions. The zeros of Ph(G,q,w) in the q and w planes and their accumulation sets in the limit of infinitely many vertices of G are analyzed. Finally, some related weighted graph coloring problems are mentioned.     

Strong connections on quantum principal bundles

A gauge invariant notion of a strong connection is presented and characterized. It is then used to justify the way in which a global curvature form is defined. Strong connections are interpreted as those that are induced from the base space of a quantum bundle. Examples of both strong and non-strong connections are provided. In particular, such connections are constructed on a quantum deformation of the two-sphere fibrationS ²→RP ². A certain class of strongU _q (2)-connections on a trivial quantum principal bundle is shown to be equivalent to the class of connections on a free module that are compatible with theq-dependent hermitian metric. A particular form of the Yang-Mills action on a trivialU _q (2)-bundle is investigated. It is proved to coincide with the Yang-Mills action constructed by A. Connes and M. Rieffel. Furthermore, it is shown that the moduli space of critical points of this action functional is independent ofq. This work was in part supported by the NSF grant 1-443964-21858-2. Writing up the revised version was partially supported by the KBN grant 2 P301 020 07 and by a visiting fellowship at the International Centre for Theoretical Physics in Trieste.     

The quantum super-Yangian and Casimir operators of Uq(gl(M |N))

TheZ ₂ graded Yangian Yq(gl(M |N)) associated with the Perk-SchultzR matrix is introduced. Its structural properties, the central algebra in particular, are studied. AZ ₂-graded associative algebra epimorphism Yq(gl(M |N)) Uq (gl(M |N)) is obtained in explicit form. Images of central elements of the quantum super-Yangian under this epimorphism yield the Casimir operators of the quantum supergroup Uq(gl(M |N)) constructed in an earlier publication.     

Finite-dimensional representations of the quantum superalgebraU q[gl(n/m)] and relatedq-identities

Explicit expressions for the generators of the quantum superalgebraU _q [gl(n/m)] acting on a class of irreducible representations are given. The class under consideration consists of all essentially typical representations: for these a Gel'fand-Zetlin basis is known. The verification of the quantum superalgebra relations to be satisfied is shown to reduce to a set ofq-number identities.     

An extension of the Borel-Weil construction to the quantum groupU q (n)

The Borel-Weil (BW) construction for unitary irreps of a compact Lie group is extended to a construction of all unitary irreps of the quantum groupU _q(n). Thisq-BW construction uses a recursion procedure forU _q(n) in which the fiber of the bundle carries an irrep ofU _q(n–1)×U _q(1) with sections that are holomorphic functions in the homogeneous spaceU _q(n)/U _q(n–1)×U _q(1). Explicit results are obtained for theU _q(n) irreps and for the related isomorphism of quantum group algebras.Supported in part by the National Science Foundation, No. PHY-9008007     

第Ⅱ类两态叠加多模叠加态光场的非线性高阶压缩特性研究

本文根据量子力学中的线性叠加原理,构造了由多模(即q模)虚相干态|{iZ_j}>_q及其相反态|{-iZ_j}>_q的线性叠加所组成的第Ⅱ类两态叠加多模叠加态光场|ψ₂⁽²⁾>_q.利用新近建立的有关双模及多模辐射场的非线性高阶压缩理论,首次对态|ψ₂⁽²⁾>_q的两种非线性高阶压缩(即N次方Y压缩和N次方H压缩)效应进行了详细研究。结果表明:1)当N为奇数时,如果各模的初始相位满足一定的量子化条件,而态间的初始相位差在其态间压缩区内连续变化;或者态间的初始相位差取某一特定值,而各模的初始相位在其腔模压缩区内连续变化时,态|ψ₂⁽²⁾>_q就会呈现出周期性变化的、任意阶的N次方Y压缩效应。2)当q·N为奇数时,如果各模的初始相位和满足一定的量子化条件,而态间的初始相位差在上述的态间压缩区内连续变化;或者态间的初始相位差取上述的某一固定值,而各模的初始相位和在其腔模和压缩区内连续变化时,态|ψ₂⁽²⁾>_q就呈现出周期性变化的、任意阶的N次方H压缩效应。3)与文献10相比,态|ψ₂⁽²⁾>_q与态|ψ(2)1>_q这两者之间的压缩情况正好相反。     

第Ⅰ种强度不对称三态叠加多模叠加态光场N次方Y压缩—第一正交相位分量的压缩情况

根据量子力学中的线性叠加原理,构造了由三个强度不等的多模相干态光场|{Z_j^(A)}>_q、|{Z_j^(B)}>_q和|{Z_j^(C)}>_q的线性叠加所组成的第Ⅰ种强度不对称三态叠加多模叠加态光场|ψ_l^(ABC)>_q.利用多模压缩态理论,研究了态|ψ_l^(ABC)>_q的第一正交方分量(即磁场分量)的广义非线性等幂次N次方Y压缩特性.结果发现:①在上述各多模相干态光场中各模的强度和各模的初始相位各不相等的情况下,态|ψ_l^(ABC)>_q的第一正交分量-磁场分量在一定的条件下,总可呈现出周期性变化的、任意等幂次的N次方Y压缩效应;②当上述各多模相干态光场的强度和各模的初始相位相等时,态|ψ_l^(ABC)>_q的磁场分量的N次方Y压缩现象消失,态|ψ_l^(ABC)>_q可恒处于等幂次N-Y最小测不准态.     

Free fermions and the Alexander-Conway polynomial

We show how the Conway Alexander polynomial arises from theq deformation of (Z ₂ graded)sl(n, n) algebras. In the simplestsl(1, 1) case we then establish connection between classical knot theory and its modern versions based on quantum groups. We first shown how the crystal and the fundamental group of the complement of a knot give rise naturally to the Burau representation of the braid group. The Burau matrix is then transformed into theU _q sl(1, 1) R matrix by going to the exterior power algebra. Using a det=str identity, this allows us to recover the state model of [K2, 89] as well. We also show how theU _q> sl(1, 1) algebra describes free fermions propagating on the knot diagram. We rewrite the Conway Alexander polynomial as a Berezin integral, and thus as an apparently new determinant.Work supported in part by NSF grant no. DMS-8822602Work supported in part by the NSF: grant nos. PYI PHY 86-57788 and PHY 90-00386 and by CNRS, France     

A Locally Compact Quantum Group Analogue of the Normalizer of SU(1,1) in SL(2, ℂ)

 S.L. Woronowicz proved in 1991 that quantum SU(1,1) does not exist as a locally compact quantum group. Results by L.I. Korogodsky in 1994 and more recently by Woronowicz gave strong indications that the normalizer of SU(1,1) in SL(2,ℂ) is a much better quantization candidate than SU(1,1) itself. In this paper we show that this is indeed the case by constructing , a new example of a unimodular locally compact quantum group (depending on a parameter 0<q<1) that is a deformation of . After defining the underlying von Neumann algebra of we use a certain class of q-hypergeometric functions and their orthogonality relations to construct the comultiplication. The coassociativity of this comultiplication is the hardest result to establish. We define the Haar weight and obtain simple formulas for the antipode and its polar decomposition. As a final result we produce the underlying C ^* -algebra of . The proofs of all these results depend on various properties of q-hypergeometric ₁ϕ₁ functions. Received: 28 June 2001 / Accepted: 25 July 2002 Published online: 10 December 2002 RID="*" ID="*" Post-doctoral researcher of the Fund for Scientific Research – Flanders (Belgium) (F.W.O.) Communicated by L. Takhtajan     

Exact Potts Model Partition Functions for Strips of the Triangular Lattice

We present exact calculations of the Potts model partition function Z(G,q,v) for arbitrary q and temperature-like variable v on n-vertex strip graphs G of the triangular lattice for a variety of transverse widths equal to L vertices and for arbitrarily great length equal to m vertices, with free longitudinal boundary conditions and free and periodic transverse boundary conditions. These partition functions have the form Z(G,q,v)= $\sum _{j = 1}^{N_{Z,G,\lambda } }$ c _z,G,j (λ _z,G,j ) ^m-1. We give general formulas for N _Z,G,j and its specialization to v=?1 for arbitrary L. The free energy is calculated exactly for the infinite-length limit of the graphs, and the thermodynamics is discussed. It is shown how the internal energy calculated for the case of cylindrical boundary conditions is connected with critical quantities for the Potts model on the infinite triangular lattice. Considering the full generalization to arbitrary complex q and v, we determine the singular locus ${\mathcal{B}}$ , arising as the accumulation set of partition function zeros as m→∞, in the q plane for fixed v and in the v plane for fixed q. Explicit results for partition functions are given in the text for L=3 (free) and L=3, 4 (cylindrical), and plots of partition function zeros and their asymptotic accumulation sets are given for L up to 5. A new estimate for the phase transition temperature of the q=3 Potts antiferromagnet on the 2D triangular lattice is given.     

多模虚共轭相干态与多模真空态的叠加态光场的等阶N次方Y压缩

利用多模压缩态理论,研究了多模虚共轭相干态|{iZ_j^*}〉q与多模真空态|{O_j}〉q的叠加态|{Ψ_p⁽²⁾}〉q的广义非线性等阶N次方Y压缩特性.结果发现态|{Ψ_p⁽²⁾}〉q是一种典型的多模非经典光场,它在一定的条件下,可呈现出周期性变化的、任意奇数阶和任意偶数阶的等阶N次方Y压缩效应,而在另外的条件下,则可呈现出等阶N次方Y相似压缩现象.     

The 50% Advanced Information Rule of?the?Quantum Algorithms

The oracle chooses a function out of a known set of functions and gives to the player a black box that, given an argument, evaluates the function. The player should find out a certain character of the function (e.g. its period) through function evaluation. This is the typical problem addressed by the quantum algorithms. In former theoretical work, we showed that a quantum algorithm requires the number of function evaluations of a classical algorithm that knows in advance 50% of the information that specifies the solution of the problem. This requires representing physically, besides the solution algorithm, the possible choices of the oracle. Here we check that this 50% rule holds for the main quantum algorithms. In structured problems, a classical algorithm with the advanced information, to identify the missing information should perform one function evaluation. The speed up is exponential since a classical algorithm without advanced information should perform an exponential number of function evaluations. In unstructured database search, a classical algorithm that knows in advance n/2 bits of the database location, to identify the n/2 missing bits should perform O(2 ^n/2) function evaluations. The speed up is quadratic since a classical algorithm without advanced information should perform O(2 ⁿ ) function evaluations. The 50% rule allows to identify in an entirely classical way the problems solvable with a quantum sped up. The advanced information classical algorithm also defines the quantum algorithm that solves the problem. Each classical history, corresponding to a possible way of getting the advanced information and a possible result of computing the missing information, is represented in quantum notation as a sequence of sharp states. The sum of the histories yields the function evaluation stage of the quantum algorithm. Function evaluation entangles the oracle’s choice register (containing the function chosen by the oracle) and the solution register (in which to read the solution at the end of the algorithm). Information about the oracle’s choice propagates from the former to the latter register. Then the basis of the solution register should be rotated to make this information readable. This defines the quantum algorithm, or its iterate and the number of iterations.     

Graph invariants of Vassiliev type and application to 4D quantum gravity

We consider graph invariants of Vassiliev type extended by the quantum group link invariants. When they are expanded byx whereq=e ^x , the expansion coefficients are known as the Vassiliev invariants of finite type. In the present paper, we define tangle operators of graphs given by a functor from a category of colored and oriented graphs embedded into a 3-space to a category of representations of the quasi-triangular ribbon Hopf algebra extended byU _q (sl(2),C)), which are subject to a quantum group analog of the spinor identity. In terms of them, we obtain the graph invariants of Vassiliev type expressed to be identified with Chern Simons vacuum expectation values of Wilson loops including intersection points. We also consider the 4d canonical quantum gravity of Ashtekar. It is verified that the graph invariants of Vassiliev type satisfy constraints of the quantum gravity in the loop space representation of Rovelli and Smolin.This is not the author's present address.