The representation of numbers by tensor product states of composite quantum
systems is examined. Consideration is limited to k-ary representations of
length L and arithmetic modulo k^{L}. An abstract representation on an L fold
tensor product Hilbert space H^{arith} of number states and operators for the
basic arithmetic operations is described. Unitary maps onto a physical
parameter based tensor product space H^{phy} are defined and the relations
between these two spaces and the dependence of algorithm dynamics on the
unitary maps is discussed. The important condition of efficient implementation
by physically realizable Hamiltonians of the basic arithmetic operations is
also discussed.Comment: Paper, 8 pages, for Proceedings, QCM&C 3, O Hirota and P. Tombesi,
  Editors, Kluver/Plenum, publisher

Benioff, Paul

English

arXiv

Benioff, P.

UNT Digital Library

. . .THE REPRESENTATION OF NUMBERS BYSTATES IN QUANTUM MECHANICSPaul BenioffPhgsics Division, Argonne National LaboratoryArgonne, IL 60439pbenioff@anl.govKeywords: Number represent ation, quantum states, quantum computersAbstract The representation of numbers by tensor product states of compositequantum systems is examined. Numbering maps are introduced thatinterpolate between number string states and physical parameter basedstates. The main condition that the basic arithmetic operations beefficiently implement able is discussed.INTRODUCTIONThe use of numbers and their representation by states of physical sys-tems is basic and widespread in the computational aspects of science.However, it is not clear exactly what is meant by thk representation.Here this question is examined for the nonnegative integers. Consider-ation wiU be limited to k — w-y representations of length L of numbersby tensor product states and to arithmetic modulo kL.Based on the universality of quantum mechanics, all physical systemsof interest here are quantum systems described by pure or mixed quan-tum states. This is the case whether the systems are microscopic, as isthe case for quantum computers, or macroscopic, as is the case for allpreseptly existing computers. Microscopic systems are those for whichthe ratio t&C/t.W >> 1 where t~,C and t,w are the decoherence andswitching times [1]. In this case the system remains isolated from theenvironment for a time duration of many switthing steps. If -t~,C/t~W<1then the system is macroscopic. Here environmental interactions stabi-lize some system states (the pointer states) [2] for a time duration ofmany switthing steps.1The submitted manuscript has been created’by the University of Chicago as Operator ofArgonne National Laboratory (“Argonne”)under Contract No. W-31 -109-ENG-38 withthe U.S. Department of Energy. The U.S.Government retains for itself, and others act- ;ing on its behalf, a paid-up, nonexclusive,irrevocable worldwide license in said articleto reproduce, prepare derivative works, dis-tribute copies to the public, and perform pub-licly and display publicly, by or on behalf ofthe Government.DISCIJUMERThis repofi was prepared as an account of work sponsoredbyanagency of the United States Government. Neitherthe United States Government nor any agency thereof, norany of their employees, make any warranty, express orimplied, or assumes any Iegai liability or responsibility forthe accuracy, completeness, or usefulness of anyinformation, apparatus, product, or process disclosed, orrepresents that its use would not infringe privately ownedrights. Reference herein to any specific commercialproduct, process, or service by trade name, trademark,manufacturer, or otherwise does not necessarily constituteor imply its endorsement, recommendation, or favoring bythe United States Government or any agency thereof. Theviews and opinions of authors expressed herein do notnecessarily state or reflect those of the United StatesGovernment or any agency thereof.DISCLAIMERPortions of this document may be illegiblein electronic image products. Images areproduced from the best available originaldocument.2Composite systems are represented by sums over tensor product statesof the form Is) = 6$=1 Igj) for pure states or pg = ‘8;=lPsj for mixedstates. These states represent the number s wheres = ~sjkj-’,j=l(1.1)Sj=Oj -”-, k – 1 is the number corresponding to l~j) or p~l, and the jindices in the tensor product states and Eq. 1.1 are identified.One route to the exact meaning of the representation of numbersby states of physical systems begins with pure mathematics. A set ofmathematical elements is a set of numbers if and only if it satisfies theaxioms of number theory or arithmetic [3]. Here these axioms needto be modified by inclusion of relevant axioms for a ring as the setof numbers modulo kL is a ring [4]. Details of the modifications arenot important here. However, the definitions and properties of basicarithmetic operations, successor (or +1), + and x, given by the axioms,are relevant.If the axioms are consistent then there are many mathematical modelsof the axioms. Included are models containing tensor product states andunitary operators on a product Hilbert space of states. Here the unitaryoperators V2+1 for addition of 1#–1 mod kL for j = 1,. ... L, +, and xneed to be defined. The reason for L successors, one for each j, willbecome clear later.The connection to physics is made by the requirement that there existphysical models of the axioms in which the basic arithmetic operationsare efficiently implementable. The widespread existence of computersshows that this existence requirement is satisfied, at least for macroscopicsystems.In the following, physical models with composite quantum systemswill be investigated in more detaiL The emphasis will be on microscopicsystems although much is also valid for macroscopic systems. The roleof numbering maps that associate numbers to physical parameters ofthe quantum systems will be demonstrated. The importance of themain condition that the basic arithmetic operations can be efficientlyimplemented by a physically realizable dynamics will be discussed. Dueto space limitations, the main ideas are summarized here. Details andproofs are provided elsewhere [5].1. THE ROLE OF NUMBERING MAPSLet It) = @a,Alt(a), a) be a tensor product state of a composite sys-tem where ~ is a function from A to S. A and S are sets of L and kThe Representation ofIVurnbersbyStates in QuantumMechanics 3physical parameters for physical observable A, S for the system. Theparameters in A distinguish the L component systems from one anotherand the parameters of S refer to k different internal physical states of thecomponent systems. Examples of A include a set of locations in space oia set of excitation energies, as is used in NMR quantum computers [6].Examples of S include spin projections along an axis or energy levels ofa particle in a potential well.The presence of the a component of the state It(a), a) is essential asit is the property used by a physical process or algorithm to distinguishthe different qubytes. The presence of this property in the state is es-sential to the extent that the state of the composite quantum systemcontains all the quant urn information available to the algorithm. Thisis especially the case for any algorithm whose dynamics is describedby a Hamiltonian that is selfadjoint and time independent. This is anexample of Landauer’s dictum “Information is Physical” [7].It is clear that product states of the form 1~)= 8.,Al~, a) do notrepresent numbers. To see this consider the state It) = f t, z) @ I $, y) @ I t, Z) of 3 spin 1/2 systems for k = 2. Here the arrows {~, $}denote spin up, spin down and Z, y, z denote space locations. This statedoes not represent a number because there is no association of {fi, ~}with the numbers O,1. Also there is no association of the locations Z, y, zwith numbers j = 1,2,3 corresponding to powers of 2 in Eq. 1.1.Toremedy this letd:{O,.. -,l}a}a Sand g:{l,2, -.., L} aAbe numberings of S and A.l These maps convert Ij) to the A labelednumber state 1~~)and 1~~)to the number string state Is$):(1.2)where 1~~) = @acA&(a), a). The state lg~) can also be written as aproduct over A: ]s$) = lt~,g–l) = @a6A\&((Z),g–l (a)). Here & = d–itis a function from A to O,-.. , k–land2~ =&g = d–l~ is a functionfrom l,..., Lto Oj... , k – 1. d-ltg denotes a combination d-l(~(g(–)))of the three functions d– 1,~,g. The state Is$) corresponds to a numberS$ if the j indices in Eqs. 1.2 and 1.1 are identified.The maps shown by Eq. 1.2 can be represented by a unitary transfor-mation W~l~) = 1~~). This form shows that the number represented by aphysical state depends on the maps d, g. If g, h and d, e are respectivelytwo of the L! different A maps and two of the k! different S maps, thenthe states W~li) and FV{l~) represent different numbers in general. As1A numbering map is ~ bijection Orone-one onto map from a Set of numbers onto a set ‘fphysical parameters. As bisections these maps all have inverses.4an example of this for the A maps let 1~~)= [0, z) @ 11,g) @ 10,z) be athree qubit state where Z, y, z are as before. For the two maps that give1 + y the state gives the number 1. For the two maps with 2 ~ y thestate gives the number 2, and for 3 + y the maps give the number 4.Yet the state is unchanged physically by any of these 6 maps.Let ~, h be two A numbering maps and Wf,~ be a unitary permutationoperator that carries out the permutation ~h- 1 of .4. ThenNote that W~,~ = 1 if and only if f relative to h is a product of pairwiseexchanges with no common indices as ~h– 1 = hf’1.The maps W$ andWf,k can be combined to give(1.4)One can also work with the inverse of Eq. 1.2 by starting with modelsof the axioms of arithmetic or number theory [3] with the basic arith-met ic operations as unitary operators, Vj+l, +, x, acting on tensor prod-uct states 1~)= @~_l \&i), j). Here the maps d, g interpret these as statesand operators based on sets A, S of physical parameters. One has/s)7’pd)?l!Lt9) = C+=,lwd(j),9 j)) (1.5)where l@ = ld~) = @~=ll@(j), j). Here Wd is a function from 1, ..-, Lto S. One can also express 1~~,g) as a product over A: Iwd, g) = It$ =8aEAlt$(a), a) where ~~= d~–l.Eq. 1.5 can also be expressed using the adjoint of W$ as Is) =(W$t lw~,g). It follows that the physical state representing a numberI&)depends on the numbering maps. If g # h and d # e then in general(W$)u)# P’w)u).One also sees that the dependence, on numbering maps, of dynamicalprocesses for implementing quantum algorithms depends on the algo-rithm. This can be seen by a comparison of Grover’s [8] and Shor’s [9]Algorithms. Grover’s Algorithm [8] is a quantum version of a search ofa completely unstructured data set. As such it is independent dynam-ically of the numbering maps and can be implemented on states of theform It) = @.,Alt(a), a) without any reference to the numbering.Shor’s algorithm [9] is quite different in that the dynamics of anyimplementation depends on the numbers represented by the physicalproduct states and must be different for different maps. This followsfrom the fact that (W~)t 1~) # (Wf)tl~) and the requirement that theThe Representation of Numbers by States in QuantumMechanics 5arithmetic function (factoring &f) determined by the algorithm must beinvariant under any change in numbering maps.These maps also apply to \+l, +, x to give Vj+l~~d’+l ~Vg~;+l andsimilarly for + and x. V9~;+1, +d~,h, and ~ $,h are unitary operators forthe basic arithmetic operations on states based on physical parametersin A,S rather than on numbers in 11. . . ,L and O,. . . ,k —1.The full connection to physics has not yet been established. Thisis provided by the condition of effective implementation of the basicarithmetic operations.2. EFFICIENT IMPLEMENTABILITY OFTHE BASIC ARITHMETIC OPERATIONSThe meaning of this important requirement is that it must be possibleto physically implement the basic operations and that the implementa-tion must be efficient. That is for each j there must exist a physicallyrealizable Hamiltonian H~,j and a time tj such thate—iH:,j tj = Vd,.+ 1913 (1.6)This definition is quite general in that the Hamiltonian can depend onj. However for many systems and dynamics a Hamiltonian H; thatimplements Vg~;+l and is independent of j is realizable.The requirement of efficiency means that both the space-time and thethermodynamic resources required to physically implement the opera-tions must be at most polynomial in L, k. For quantum computers theadditional condition that tj << tdec[1] must be satisfied. This condi-tion excludes k = 1 or unary representations as all arithmetic operationsare exponentially hard in this case. Large values of k are also excludedas distinguishing among a large set of symbols and carrying out sim-ple arithmetic operations becomes thermodynamically expensive. Alsothere are physical limitations on the amount of information that can bereliably stored and distinguished per unit space time volume [10].Thermodynamic resources are needed to protect the system from er-rors resulting from operation in a noisy environment. Microscopic sys-tems also need protection from decoherence [11]. Methods include theuse of quant urn error correction codes [12], EPR pairs [13], and decoher-ence free subspaces [14]. Protection of macroscopic systems is less diffi-cult since one takes advantage of decoherence to give stabilized” pointer”[2] states that represent numbers in a macroscopic computer.The reason for separate definitions of the V9~;+1for each j is that therequirement means that each of these operators for j = 1, . . . . L must be6efficiently implemented. If the V~~;+1 were defined in terms of iterationsof V9~1+1,then implementation of V9~;+1would require k~–1 iterations ofVg$l+l. This is not efficient even if V9~1+1can be efficiently implemented. ISince the Vg~j+l, +, and x operators are many system nonlocal oper-ators, many dynamical steps would be needed to implement these oper-ators by a realizable two particle local Hamiltonian, Eq. 1.6. As is wellknown, there are many methods of efficiently implementing these opera-tors, at least in macroscopic computers. For the V~~;+lmethods includetranslation along the path g in A to the site g(j) of the procedure ford,+lefficiently implementing V9,1 .The thermodynamic resources required to physically implement theVg~l and other arithmetic operations also depend on the path g. Ingeneral paths are chosen that respect the topological or neighborhoodproperties of A as this reduces the resources required. But in principleany path is possible as the resource dependence on the path choice isnot exponent ial.3. DEFINITIONS OF THE BASICARITHMETIC OPERATIONSThe operators can be defined either as Vj‘1, +, x or as V9,1“+1 , +;,h , Xj,h.Here to emphasize the physical aspects the latter will be chosen. Toreduce clutter the superscript d will be suppressed on the operators.Definitions of the v“: and +9,~ are given. x is defined in [5]. One hasVg>l =, s~9(d’#+v9(d ‘ii4(@w)>9(4n=j e+L+ H %(#’d(k-l),9(q (1.7)[=jHere ~ti(~-l),g(j)= ld(~ – l),W)(W – l),d~)l @I&j1 @J19(1) is theprojection operator for finding a qubyte in state Id(k – 1), g(j)) andin anY of the k S states for any A value # g(j). AlSO P#d(k_l),g(j) =i – ‘d(k-l),g(j).The operator ug(j) = u 8 lg(j) @f~j (1@ 19(1)) where u is a cyclic shiftof period k (i.e. (u~ = l))defined by uld(k)) = Id(k + 1) mod k~). Sincethe operators ug(~)~~(Pl,g~n) and ug(~)~~(~),g(n) commute for any m # nandanyp, q= O,..., k – 1, the unordered product JJl=j has been usedin Eq. 1.7.The Representation ofNumbersbyStates in Quantum Mechanics 7Summaries of several interesting properties of these operators follow.Proofs are provided elsewhere [5]. They are unitary, which is requiredby Eq. 1.6. They are cyclic shifts of period k~–~+l on states of the form@,9), Eq.~ 1.5. Also—(1.8)This shows the exponential dependence on j and the need for separatedefinitions and efiicient implementation of each of these operators.Transformation properties of the V9]l under the action of the W~,h,Eq. 1.3, are given by(1.9)This shows that the arithmetic action of the Vg>l is preserved by Wf,~.V~~,j carries out the same addition on @, rng) with m = fh-l as V~~ldoes on l~~g-lmg, g).The operator +g,~ is defined on pairs of states lad,g)ld, ~) by+g,hkd, g) Id, h) = l&, g) [sd +~,h Wd, h) where+@)v+s@-l)Isd +g,h Wd, h) = v~,~-, Vh+:g(l)h,L–1 ~“ “ , Iud, h) (1.10)+1:(~)and Vh,j = (V;;)sw.Note that for pairs of product qubyte states, which are first introducedhere, the range sets A and B of physical parameters for the maps g andh respectively are different. This follows from the requirement thatall qubytes in @d, h) must be physically distinguishable from those inlad,9).4. SUMMARY AND CONCLUSIONHere some steps were taken in making clear the meaning of repre-senting numbers by states of quant urn systems. The role of numbermaps in connecting basic arithmetic operators and tensor product statesrepresenting number strings to states and operators based on physicalobservable was shown. Definitions and properties of these operatorswere given. The main condition that these operators be efficiently im-plementable was discussed. This condition connects physical and math-emat ical aspects of the representation.Much remains to be done. Future work includes dropping the modulolimitation and considering other types of numbers. The use of anni-hilation and creation operators to represent states needs examination,8especially regarding number representation states in second quantiza-tion. Also physical limitations on methods of communicantion, by symbolstrings represented by tensor product states, may be relevant.,AcknowledgmentsThis work is supported by the LT.S.Department of Energy, Nuclear Physics Divi-sion, under contract W-31 -109-ENG-38.References[1][2][3][4][5][6][7][8][9]DiVincenzo, D. P. Science 270: 255, (1995); Los Alamos Archivesquant-ph/OO02077.W. H. Zurek, Phys. Rev. D 24:1516, (1981); 26:1862 (1982); E.Joos and H. D. Zeh, Z. Phys. B 59: 23, (1985); H. D. Zeh quant-ph/9905004; E Joos, quant-ph/9808008.J. R. Shoenfield, Mathematical Logic (Addison-Weseley, Reading,MA 1967); R. Smullyan, Godei’s Incompleteness Theorems (OxfordUniversity Press, Oxford, 1992).I. T. Adamson, Introduction to Field Theo~y, 2nd. Edition, Cam-bridge University Press, London, 1982.P. Benioff, Los Alamos Archives Preprint quant-ph/0003063.N.A. Gershenfeld, Science 275:350 (1997); D.G. Cory, A.F. Fahmy,and T.F. Havel, Proc. Natl. Acad. Sci. 94:1634 (1997).R. Landauer, Physics Today 44: No 5, 23, (1991) ;Physics LettersA 217: 188, (1996); in Feynman and Computation, Exploring theLimits of Computers, A. J.G.Hey, Ed., (Perseus Books, Reading MA,1998).L.K.Grover, in Proceedings of 28th Annual A CM Symposium on The-ory of Computing ACM Press New York 1996, p. 212; Phys. Rev.Letters, 79:325 (1997); Phys. Rev. Letters, 80:4329 (1998).P. W. Shor, in Proceedings, 35th Annual Symposium on the Foun-dations of Computer Science, S. Goldwasser (Ed), IEEE ComputerSociety Press, Los Alamitos, CA, 1994, pp 124-134; SIAM J. Com-puting, 26:1481 (1997).[10] S. Lloyd, Los Alamos Archives preprint quant-ph/9908043; Y. J.Ng, Los Alamos Archives Preprint quant-ph/0006105.[11] W. H. Zurek, Physics Today 44: No. 10,36 (1991); J.R. Anglin, J.Paz, and W. H. Zurek, Phys. Rev A 55:4041 (1997); W. G. Unruh,Phys. Rev. A 51:992 (1995).., .The Representation oj Numbers by States in Quantum Mechanics 9(12] R. Laflamme, C. Miquel, J. P. Paz, and W. H. Zurek, Phys. 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