Elsevier

Annals of Physics

Volume 318, Issue 2, August 2005, Pages 345-407
Annals of Physics

Computing spin networks

https://doi.org/10.1016/j.aop.2005.01.005Get rights and content

Abstract

We expand a set of notions recently introduced providing the general setting for a universal representation of the quantum structure on which quantum information stands. The dynamical evolution process associated with generic quantum information manipulation is based on the (re)coupling theory of SU (2) angular momenta. Such scheme automatically incorporates all the essential features that make quantum information encoding much more efficient than classical: it is fully discrete; it deals with inherently entangled states, naturally endowed with a tensor product structure; it allows for generic encoding patterns. The model proposed can be thought of as the non-Boolean generalization of the quantum circuit model, with unitary gates expressed in terms of 3nj coefficients connecting inequivalent binary coupling schemes of n + 1 angular momentum variables, as well as Wigner rotations in the eigenspace of the total angular momentum. A crucial role is played by elementary j-gates (6j symbols) which satisfy algebraic identities that make the structure of the model similar to “state sum models” employed in discretizing topological quantum field theories and quantum gravity. The spin network simulator can thus be viewed also as a Combinatorial QFT model for computation. The semiclassical limit (large j) is discussed.

Introduction

It is by-now a generally accepted fact that the laws of quantum theory provide in principle a radically novel, and more powerful way to process information than any classically operated device [1]. In the past few years a big deal of activity has been devoted to devise and to implement schemes for taking actual advantage from such quantum extra power. In particular in quantum computation (QC) the states of a quantum system S are used for encoding information in such a way that the final state, obtained by the appropriate unitary time evolution of S, encodes the solution to a given computational problem. A system S with state-space H (the Quantum Computer) supports universal QC if any unitary transformation UU(H) can be approximated with arbitrarily high accuracy by a sequence (the network) of simple unitaries (the gates) that the experimenter is supposed to be able to implement. The case in which S is a multi-partite system is the most relevant, as it allows for entanglement, a unique quantum feature that is generally believed to be one of the crucial elements from which quantum speed-up (polynomial or possibly exponential) is generated [2].

In the above picture of QC the realization of the quantum network is achieved at the physical level by turning on and off external fields coupled to S as well as local interactions among the subsystems of S. In other words the experimenter “owns” a basic set of time–dependent Hamiltonians that she/he activates at will to perform the necessary sequences of quantum logic gates.

At variance with such a standard dynamical view of QC, more recently several authors considered geometrical and topological approaches [3], [4], [5], [6], [7]. The peculiarity of these proposals is far reaching: over the manifold C of quantum codewords one can have a trivial Hamiltonian, for example H|C=0, yet obtain nevertheless a non-trivial quantum evolution due to the existence of an underlying geometrical/topological global structure. The quantum gates—or parts thereof—in this latter case are realized in terms of operations having a purely geometrical/topological nature. Besides being conceptually intriguing on their own, these schemes have some built-in fault-tolerant features. This latter attractive characteristic stems out of the fact that often certain topological as well as geometrical quantities are inherently stable against local perturbations. This in turn allows for quantum information processing inherently stable against special classes of computational errors.

There has been a number of proposals suggesting general conceptual schemes of interpretation of quantum computation. Most of them are indeed based on topological notions, even though this is not always explicitly stated. Among these, anyonic quantum computation [3], fermionic quantum computation [8], localized modular functor quantum field computation [4], holonomic quantum computation [9], [10] have mostly attracted attention. However, such models appear to be simply different realizations of a unique conceptual scheme that incorporates all of them as particular instances (once one focuses on their suitable “discretized” counterparts).

We propose here, expanding a set of notions introduced in [11], a general setting for a universal representation of the very quantum structure on which quantum information stands. The associated dynamical evolution process, giving rise to information manipulation, is based on the (re)coupling theory of SU (2) angular momenta (see [12], [13], [14]). The scheme automatically incorporates all the essential features that make quantum information encoding so much more efficient than classical: it is fully discrete (both for space-like and time-like variables); it deals with inherently entangled states, naturally endowed with a (non-associative) tensor product structure; it allows for generic encoding patterns. The minimal set of requirements listed by Feynman [15] as essential for the proper characterization of an efficient quantum simulator is automatically satisfied: (i) locality of interactions; (ii) number of “computer” elements proportional to a function which is at most polynomial in the space-time volume of the physical system; (iii) time discreteness (time is itself “simulated” in the computer by computational steps).

Key element of our argument is the fact that all such basic features are typical of spin networks. It should be emphasized that by spin networks we mean here—contrary to what happens in solid state physics, but somewhat in the spirit of combinatorial approach to quantum space-time representation [16]—graphs the node and edge sets of which can be labelled by quantum numbers associated with SU (2) irreducible representations and by SU (2) recoupling coefficients, respectively. For this reason spin networks can be thought of as an ideal candidate conceptual framework for dealing with tensorial transformations and topological effects in groups of observables. The idea is to exploit to their full extent the discreteness hypotheses (ii) and (iii), by modelling the computational space in terms of a set of combinatorial and topological rules that mimic space-time features in a way that automatically includes quantum mechanics.

The model proposed can be thought of as a non-Boolean generalization of the quantum circuit model, with unitary gates expressed in terms of: (a) recoupling coefficients (3nj symbols) between inequivalent binary coupling schemes of N = n + 1 SU(2)-angular momentum variables (j-gates); (b) Wigner rotations in the eigenspace of the total angular momentum (M-gates). These basic ingredients of the spin network simulator, namely computational Hilbert spaces and admissible elementary gates, are discussed in details in Sections 2 Computational Hilbert spaces, 3 Gates, respectively. The picture does contain the Boolean case as the particular case when all N angular momenta are spin 12.

In Section 4, both the architecture and the computational capabilities of the simulator are described in full extent. On the kinematical side (Section 4.1), the computational space is shown to be modeled as an SU (2)-decorated graph or, more precisely, as a fiber space structure over a discrete base space—the Rotation graph—which encodes all possible computational Hilbert spaces as well as gates for any fixed number N of incoming angular momenta. A crucial role is played by elementary j-gates (Racah transforms, related to 6j symbols of SU (2)) which satisfy suitable algebraic identities [14] and make the structure of the model similar to “state sum models” employed in discretizing topological quantum field theories (TQFT) and quantum gravity (see Section 5). In Section 4.2, after a discussion of the hypotheses of Feynman [15], general circuit-type computation processes on the spin network are described and classified into computing classes. Virtual, polylocal Hamiltonians are generated by the simulator which evolves in an intrinsic discrete time variable. Section 4.3 deals with questions in (quantum) computational complexity which turn out to be closely related to graph combinatorics. We argue that our new framework, when implemented on the basis of explicit encoding schemes [17], could be suitable to handle “combinatorially hard” problems more efficiently than any classical machine.

The key ingredient of Section 5 is the Ponzano–Regge asymptotic formula [18] for the 6j symbol which plays a twofold role. On the one hand, it gives a precise meaning to the semiclassical analog of a j-gate providing, together with the asymptotics of Wigner rotation matrices [13, Topic 9], the notion of “approaching a classical simulator” out of a quantum one. Moreover, as stated in [11], the whole conceptual scheme of spin network computing can be reformulated in terms of density matrix formalism, namely resorting no longer to sharp eigenstates of angular momenta but rather to generalized multipole moments (see e.g. [19], Ch. 7.7). Such generalizations can be summarized in the following diagram:

On the other hand, the Ponzano–Regge asymptotic formula opens the intriguing possibility of bridging the quantum theory of angular momenta to Euclidean gravity in dimension three. More precisely, a state sum functional for triangulated 3-dimensional space-time manifolds, built up by associating a 6j symbol with each tetrahedron, is shown to correspond, in the asymptotic limit, to the semiclassical partition function of gravity [18] with a classical action representing the discretized counterpart of the Einstein–Hilbert action of general relativity [20]. Since quantum gravity in dimension three is strictly related to a TQFT with an SU (2) Chern–Simons–type action (see references quoted at the end of Section 5 and at the beginning of Section 6) we present in this part of the paper some known results concerning spin networks viewed as “Combinatorial Quantum Field Theories,” to be interpreted as discretized versions of TQFTs based on SU (2)-decorated triangulations. At the end of Section 5 we compare the discrete partition functions of Ponzano–Regge gravity with the partition functions introduced in Section 4.2 in connection with the simulator’s dynamics. We conclude that a spin network simulator working by switching on j-gates acting on states with N incoming spins is able to simulate some subclasses of triangulated surfaces and possibly some subclasses of triangulated 3-manifolds but, since partition functions for quantum field theories must be “sums over all configurations” (apart from regularization), we cannot infer the possibility of fully simulating Combinatorial QFT (unless we take some sort of thermodynamical limit for N  ∞ which does not sound good when dealing with quantum circuit schemes for computation).

We start Section 6 (Spin network and topological quantum computation) by reviewing some basic definitions on TQFTs. Section 6.1 addresses holonomic quantum computation and we give indications that the discrete setting developed in Section 4 could support also such kind of computational processes. In Section 6.2, we compare the spin network approach with the approach of Freedman et al. [4]. We provide a (not unique) mapping between the spin network and the modular functor approach by introducing combinatorial marked 2-disks which display localized interactions between spins. The algebraic structures of the two approaches, summarized in the Yang–Baxter identity for the standard topological one and in the (hexagon + pentagon) identities for the spin network, suggest that the partition functions of the two models are related to each other in the same way as the regularized version of Ponzano–Regge functional corresponds to a double Chern–Simons partition function.

In Appendix A, we present the graph-theoretical rationale underlying spin network combinatorics by collecting results spread over a number of references in discrete mathematics, binary couplings and recoupling theory of angular momenta, complexity theory. Binary coupling trees are defined in Appendix A.1, Twist-Rotation and Rotation graphs in Appendix A.2, and some results in (classical) combinatorial complexity theory are summarized in Appendix A.3. Appendix B contains (standard) technical results concerning the composition of Wigner rotation matrices (B.1) and U-rotation matrices (B.2) which are employed in Section 3.2 (M-gates).

The twofold possible interpretation of the deliberately ambiguous title we have chosen for the paper should have become clear at this point:

  • on the one hand, spin networks are computing devices supporting simulations of the dynamical behavior of composite quantum systems described in terms of pure angular momentum eigenstates;

  • such computing devices, on the other hand, are able to simulate classes of extended geometrical objects modeled as spin networks.

We may summarize the content of the paper in the following diagram, where the spin network simulator may be viewed both as a generalized quantum circuit and as a Combinatorial QFT model for computation. The standard Boolean quantum circuit (shown to be equivalent to the topological approach [4]) is a particular case of this general scheme for computation. To complete the picture, the combinatorial approach can be suitably mapped into the purely topological one as discussed in Section 6.2.

We plan to develop in the next future the upper connection which points toward “quantum automata” since our framework seems quite promising to address such issues like quantum languages and grammars, quantum encoding [17] and quantum complexity classes of algorithms, naturally related here to enumerative combinatorics of graphs.

Section snippets

Computational Hilbert spaces

Following [13, Topic 12] let us consider N = n + 1 mutually commuting angular momentum operators of the algebra of SU (2)J1,J2,J3,,Jn+1{Ji}and the corresponding components{Ji(z)}i=1,2,,n+1along the quantization axis. For each i = 1, 2,  , n + 1 the simultaneous eigenstates of the complete sets Ji2 and Ji (z) are:Ji2|jimi=ji(ji+1)|jimi,Ji(z)|jimi=mi|jimi,where we set  = 1 and the eigenvalues range overji=0,12,1,32,;-jimiji(integersteps).Denoting byHjispan{|jimi}the (2ji + 1)-dimensional Hilbert

j-gates

By j-gates we mean unitary transformations on the computational Hilbert spaces (15) which act on the set of the spin variables {j1, j2, j3,  , jn+1, k1, k2, k3,  , kn−1} of the eigenstates without changing the quantum numbers J and M.

According to the recoupling theory of angular momenta [12], [13], [14] (see also [22], [23]) the most general unitary transformation between two computational states characterized by different binary coupling schemes b and b|[j1,j2,j3,,jn+1]b;k1b,k2b,,kn-1b;JM|[j1,j2,j3

Spin network quantum circuit

By exploiting the basic ingredients introduced in the previous sections (computational Hilbert spaces, j-gates and M-gates) we present here the structural setting of a quantum simulator M—the spin network simulator– modeled as a generalized (i.e., not Boolean) quantum circuit model. In a broader sense such a computing machine could be reinterpreted as a concrete realization of what should be a Quantum Automaton (see e.g. [25], [26]), namely a theoretical framework able to deal consistently with

Semiclassical simulator and SU (2) state sum models

According to the Bohr correspondence principle, classical concepts become increasingly valid in the regime where quantum numbers are large. In handling with angular momenta variables measured in units of , the classical limit   0 implies that, for finite angular momenta, both the j-quantum numbers and the magnetic ones are much bigger than one. For what concerns pure angular momentum states—and in particular the computational Hilbert spaces introduced in Section 2 and involved in dynamical

Spin network and topological quantum computation

We begin this section by introducing some basic ingredients of Chern–Simons-type topological quantum field theories (CS TQFTs) to deal with the topological approach to quantum computation. Our presentation will be necessarily sketchy, and we refer the reader to [40], [41], [42] for general reviews on TQFTs, while the 3-dimensional CS case is extensively addressed in [32], [33], [39].

TQFTs are particular types of gauge theories, namely field theories quantized through the (Euclidean) path

Acknowledgments

We thank Vincenzo Aquilanti, Mauro Carfora, Silvano Garnerone, and Tullio Regge for interesting discussions. We are in debt with M. Carfora also for his unexhaustible enthusiasm in helping us with the preparation of all figures, and with V. Aquilanti and C. Coletti for the permission to reproduce Fig. 21 from [27].

References (66)

  • E. Majorana

    Nuovo Cimento

    (1932)
  • A. Steane

    Rep. Prog. Phys.

    (1998)
    Yu.I. Manin, Classical computing, quantum computing, and Shor’s factoring algorithm. Available from preprint:...D.P. DiVincenzo et al.

    Nature

    (2000)
  • R. Jozsa, Entanglement and quantum computation. Available from preprint:...
  • A. Kitaev

    Ann. Phys.

    (2003)
  • M.H. Freedman et al.

    Commun. Math. Phys.

    (2002)
    M.H. Freedman et al.

    Commun. Math. Phys.

    (2002)
    M.H. Freedman et al.

    Bull. Am. Math. Soc.

    (2002)
  • S. Lloyd, Quantum computation with Abelian anyons. Available from preprint:...
  • E. Dennis et al.

    J. Math. Phys.

    (2002)
  • S.B. Bravyi, A.Yu. Kitaev, Fermionic quantum computation. Available from preprint:...
  • P. Zanardi et al.

    Phys. Lett. A

    (1999)
  • J. Pachos et al.

    Phys. Rev. A

    (2000)
  • A. Marzuoli et al.

    Phys. Lett. A

    (2002)
  • A.P. Yutsis, I.B. Levinson, V.V. Vanagas, The Mathematical Apparatus of the Theory of Angular Momentum, Israel Program...
  • L.C. Biedenharn, J.D. Louck, G.-C. Rota (Ed.), The Racah–Wigner Algebra in Quantum Theory, Encyclopedia of Mathematics...
  • R.P. Feynman

    Int. J. Theor. Phys.

    (1982)
  • S. Garnerone, A. Marzuoli, M. Rasetti, in...
  • G. Ponzano et al.

    Semiclassical limit of Racah coefficients

  • L.C. Biedenharn et al.

    Angular momentum in quantum physics, theory and applications

  • T. Regge

    Nuovo Cimento

    (1961)
  • J. Kempe et al.

    Phys. Rev. A

    (2001)
  • V. Fack et al.

    Comp. Phys. Commun.

    (1999)
  • V. Fack et al.

    Discr. Math.

    (2002)
  • A. Barenco et al.

    Phys. Rev. A

    (1995)
  • E. Bernstein et al.

    SIAM J. Comput.

    (1997)
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