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arXiv:0709.1268v1 [quant-ph] 10 Sep 2007 rr ubr fbt.Mlietr r eerepresented here are arbi- for working bits. visualization of in- the numbers of to trary method of was new paper one present a the Therefore, troduce writing think- for of motivations terms. habit first three-dimensional our to in due bits ing three than more involving ready. sically etse,dn n[] a Acntuto fall a of for construction needed of ingredients GA reformulation formal GA gates. a essential quantum was three-bit the and Therefore, [5], two-, The one-, in [4]. elementary in the done [3] step, problem was Simon [1] next the name from to the construction generalized terms quickly (hence The visualization geometric computation’). cartoon in ‘cartoon for interpretable allowed was and step Each Ap- quantum the [1]. a in of of given known. version was widely GA algorithm [2]) a be (Deutsch-Jozsa of example to quantum first of seem the version parently, not GA a does for computation allow and states entangled be ways. different can several and in multivectors interpretations visualized and geometric Blades numerous multivectors. possess called are blades a of resentation eidxdb euneof sequence a by indexed be e ocp 1 n sroe ntefc htteset the 2 contains that ) fact real Euclidean a the of in vectors rooted sis is and of [1] concept new interpretations. very common-sense with quantum resist and, exceptions, of counterintuitive few structures highly The are states. computation products entangled of called sequence Nonfac- simple are a of qubits. and of superpositions qubit, products linear tensor torizable a by by represented represented is here bits is bit computation. quantum A and theory information quantum hr eesm rbeswt iulzn situations visualizing with problems some were There h atta utvcoscnpa oeaaoosto analogous role a play can multivectors that fact The oigbsddrcl ngoercagba(A sa is (GA) geometric on directly based Coding from known well is products tensor on based Coding blades goercGasanCiodpout fba- of products Grassmann-Clifford (geometric ASnmes 36.x 03.65.Ud b 03.67.Lx, visualized numbers: are PACS states Hadamard and geom GHZ, sim explicit Bell-, by detail formulas the in the of discuss a illustrate We and is approaches . theory re two quantum resulting are involve The products not tensor does multivectors. where by framework states quantum entangled the to native unu optto sbsdo esrpout n entangl and products tensor on based is computation Quantum .INTRODUCTION I. n btnme.Lna obntosof combinations Linear number. -bit 1 n all dmninlEcieno pseudo- or Euclidean -dimensional etu e pse CE)adFudtoso h xc Scie Exact the of Foundations and (CLEA) Apostel Leo Centrum fqatmcmuainaeba- are computation quantum of esrpoutv.goercpoutcoding geometric- vs. Tensor-product n lmns ahbaecan Each elements. n 2 isadtu sarep- a is thus and bits aer iyiToeyze nomtk Kwantowej Informatyki i Teoretycznej Fizyki Katedra rj nvrietBusl 00Busl,Belgium Brussels, 1050 Brussel, Universiteit Vrije oiehiaGank,8-5 d´s,Poland Gda´nsk, 80-952 Gda´nska, Politechnika 3 idrkAerts Diederik STSD ahleeUiesti Leuven, Universiteit Katholieke ESAT-SCD, 01Lue,Belgium Leuven, 3001 1 n ae Czachor Marek and n edo ope ubr.Orrpeetto fthe of representation Our number’ numbers. ‘imaginary without complex factors) structure of phase complex need number. and the any gates a with elementary deal by for to (needed blade how a explain also mul- of We geometric multiplication geometrically over and We addition tiplication polylines. of colored distributivity oriented interpret of sets by ain sdi A h esni htteuulrepre- usual the that is reason The where sentation, GA. in used tations fsm motn nage ttsocrigi quantum theory. in we occurring information states IV, entangled important Section some analogues in of Finally, of interpretation quantum geometric blades). discuss interpretation of of deterministic nature superpositions probabilistic of vs. the principle compu- superposition quantum (eg. to occur some respect where tation with formalism concen- GA differences we the of important Instead, elements elsewhere. those on found trate be can do We , these curiosity. and since a gates elementary just introduce are explicitly stage generaliza- not present further the some at in but mixed tions, useful and prove product may that states) (eg. constructions out- we and certain differences, III line and Section similarities coding. important In geometric-product We and object. tensor-product concepts geometric the compare of appropriate each illustrate an and by coding, of way GA the this from free complex is are formalism difficulty. Our coefficients multivectors. whose unique states into entangled map to pc,addnt t rhnra ai etr by vectors basis orthonormal its denote and an space, consider Indeed, ural. 1 where ≤ ebgn nScinI,wt ealdepaainof explanation detailed a with II, Section in begin, We h rcdr s nfc,eteeysml n nat- and simple extremely fact, in is, procedure The k I EMTI-RDC CODING GEOMETRIC-PRODUCT II. k ≤ ti bet hr utvco versions multivector where objects etric aoost unu optto but computation quantum to nalogous 1 en fclrdoine polylines. oriented colored of means y k < n lrte n ieecsbtenthe between differences and ilarities A . lcdb emti rdcsand products geometric by placed 2 dsae.W ics nalter- an discuss We states. ed < blade 2 i · · · , 3 sa prpit ld,de o allow not does blade, appropriate an is i k < iesfo h tnadrepresen- standard the from differs sdfie by defined is cs(FUND) nces j h ai etr (one-blades) vectors basis The . n dmninlra Euclidean real -dimensional b k 1 ...k j = b k 1 b . . . b k k j , , 2

Combs to be multiplied: satisfy the Clifford algebra

bkbl + blbk =2δkl.

The binary number associated with bk1...kj can be read Multiplication: out by the following recipe: Take a basis vector bk and check if it occurs in bk1...kj . If it does — the kth bit is 1st sign change Ak = 1, otherwise Ak = 0. Check in this way all the basis vectors. For notational reasons it is useful to denote the blades in binary parametrization by a character different from 2nd sign change b, say, c. The blades parametrized in a binary way will be termed the combs [5]. Blades and combs are related by the rule 3rd sign change

A1 An cA1...An = b1 ...bn (1) 0 1 where it is understood that bk = . Combs and blades Teeth located at the same position annihilate: are, by definition, normalized if they are constructed by taking geometric products of mutually orthonormal vec- tors.

Sometimes it is useful to be able to speak of real and imaginary blades and combs. This can be achieved by FIG. 1: Mechanical interpretation of the comb multiplication. an additional bit, represented by a vector b0. If b0 occurs (a) Take two combs and flip one of them. Move one comb in the direction of the other. Each time the teeth located in in bk1...kj , the blade is imaginary — otherwise it is real. Denoting logical negation by the prime, 0′ = 1, 1′ = 0, different places meet — the comb changes its sign. (b) Teeth we define the ‘imaginary unit’ by the complex structure located at the same position annihilate each another. map

A0 ′ of objects. To understand why it is so, consider the icA0A1...An = ( 1) cA0A1...An . (2) − algebra of a plane with basis vectors b1, b2. The ori- A general ‘complex’ element of GA can be written in a ented plane segment b12 = b1b2 is unaffected by rota- ′ ′ form analogous to the usual representation of complex tions b1 = b1 cos α b2 sin α, b2 = b1 sin α + b2 cos α, ′ − ′ −1 numbers or rescalings b1 = λb1, b2 = λ b2. Fig. 2 shows an alternative way of visualizing blades in a 6-dimensional

zA1...An = x00A1...An + iy00A1...An (3) Euclidean space (i.e. 6-bit combs), whose is sufficiently counterintuitive. The idea is adapted from = x00A1...An + y10A1...An . (4) a configuration space of three 2-dimensional particles.

Here x00A1...An = xc00A1...An and y00A1...An = We distinguish particles by color and tilting of the basis yc00A1...An denote elements of GA that are proportional vectors. Oriented segments are represented by oriented to a real comb c00A1...An , and the proportionality factors multi-color polylines. x, y are also real. Blades can be added and multiplied by numbers. Fig. 3 There exists a ‘mechanical’ justification of the ‘comb’ illustrates in what sense one can speak of distributiv- terminology. In order to understand it, let us recall the ity of addition over geometric multiplication. Addition formula for multiplication of two normalized combs [1] of two identical blades results in a blade one of whose segments is twice bigger. The three polylines shown Pk

tation chosen in [1], where higher than three led to obvious difficulties. 1 b1 b2 b3 b4 b5 b6

III. SIMILARITIES AND DIFFERENCES b12 b13 b14 b15 b16 BETWEEN TENSOR AND GEOMETRIC CODINGS

Let us now list the basic similarities and differences b23 b24 b25 b26 . . . between coding based on tensor and geometric products.

A. Partial separability

b123 b1234 b1324 = – b1234 . . . Two (k + l)-bit kets that share the same part of bits,

FIG. 2: (Color online) Colored polyline interpretation of say, A1 ...AkB1 ...Bl and A1 ...AkC1 ...Cl , possess blades. the following| partial separbilityi | property i

α A ...A B ...B + β A ...A C ...C = A ...A α B ...B + β C ...C . (7) | 1 k 1 li | 1 k 1 li | 1 ki | 1 li | 1 li The property is essential for teleportation protocols. In geometric-product coding we have an analogous rule,

αc + βc = c αc + βc , (8) A1...Ak B1...Bl A1...AkC1...Cl A1...Ak 01...0l 01...0kB1...Bl 01...0k C1...Cl  and thus teleportation protocols can be formulated in purely geometric ways. A1 An Since the link between blades and combs can be written as cA1...An = b1 ...bn the above rule means simply that

αbA1 ...bAk bB1 ...bBl + βbA1 ...bAk bC1 ...bCl = bA1 ...bAk αbB1 ...bBl + βbC1 ...bCl . (9) 1 k k+1 k+l 1 k k+1 k+l 1 k k+1 k+l k+1 k+l Hence, yet another notation is possible

αc + βc = c αc + βc . (10) A1...AkBk+1...Bk+l A1...AkCk+1...Ck+l A1...Ak Bk+1...Bk+l Ck+1...Ck+l 

Here we are making use of the fact that the number of Indeed, GA of a plane consists of 1, b1, b2, and b1b2. The 2 zeros occuring in combs such as cA1...Ak 01...0l is a matter latter satisfies (b1b2) = 1, and thus ‘complex numbers’ of convention: It reflects the freedom of looking at an are often represented in a− GA context by multivectors of n-dimensional Euclidean space from the perspective of the form x 1+ yb1b2, where x, y are real. This type of higher dimensions, and treating it as an n-dimensional complex structure is employed in [6]. subspace of something bigger. The notions of product and entangled states can be However, there is a simple reason why such a type of introduced in the GA formalism in exact analogy to the ‘i’ is not applicable in our formalism. For assume that quantum case. i = b1b2. Then ic00 = b1b2 1 = c11 whose quantum counterpart should read i 00 = 11 , making 00 and 11 linearly dependent. We| havei to| proceedi differently.| i B. Phase factors | i A way out of the difficulty was proposed in [5] and is Complex phase factors play a crucial role in quantum based on i defined by (2). Intuitively, this i is equivalent computation and are responsible for interference effects. to a π/2 rotation in a real plane (the convention used in An analogous structure occurs also in our geometric for- [5] differed by a sign from (2), but we prefer the latter malism, but we first have to comment on the meaning of choice). The additional bit A0 introduces the doubling of i. the dimension analogous to the one associated with real In it is usual to treat i as a . and imaginary parts of a single . 4

(a) Our definition also implies that i2 = 1 so that − iφ e c00A1...An = cos φc00A1...An + sin φic00A1...An(11) = – = = = cos φc00A1...An + sin φc10A1...An (12)

has all the required properties of, simultaneously, a com- b1b2b5b6b2 = – b1b2b5b2b6 = b1b2b2b5 b 6 = b1b5 b 6 plex number multiplied by a complex phase factor, and a 2-dimensional real vector rotated by an angle φ. (b) Let us illustrate these considerations by a GA repre- sentation of the state

1 + = = = ψ = 10 + eiφ 01 . (13) | i √2| i | i

Now we have to use three bits and a three dimensional b1b2b4b6 + b1b2b5b6 = b1b2(b4b6+b 5b6) = (b1b2b4+ b1b2b5)b6 space spanned by b0, b1, and b2. The GA analogue reads = b1b2(b4+b5)b6

1 iφ (c) ψ = c010 + e c001 , (14) √  2 1 = c010 + cos φc001 + sin φic001 (15) + = 2 = = √2  1 = c010 + cos φc001 + sin φc101 (16) √2  = b4b5b6 + b4b5b6 = 2 b4b5b6 = (2 b4)b5b6 1 = b1 + cos φb2 + sin φb02 . (17) √2 

= = = = Obviously, the multivector (17) can be easily visualized in various ways.

= b4b5(2 b6) = b4(2 b5)b6 C. Scalar product

FIG. 3: (Color online) Arithmetic of blades in 6-dimensional space. (a) Two subsequent segments can be interchanged but Scalar product does not seem important for GA com- orientation (i.e. overall sign) is then changed. Two iden- putation (we do not really need ‘bras’). But just for the tical (here unit) segments annihilate when placed one after sake of completeness let us mention the following con- another. (b) Distributivity of addition of two blades. (c) struction. Multiplication of a blade by a number is derived from dis- A1 An Consider a comb cA1...An = b1 ...bn . Its re- tributivity. The blade 2b4b5b6 is illustrated by means of three ∗ An A1 verse is cA ...A = bn ...b1 . The geometric product different polylines belonging to the same equivalence class. ∗ 1 n cA1...An cB1...Bn equals 1 if and only if (A1,...,An) = (B1,...,Bn). If the two sequences of bits are not identi- ∗ cal, the product cA1...An cB1...Bn is a blade different from 1. Let now Π denote the projection of a multivector on (a) 0 the scalar part 1 = c0...0. Then

∗ Π0 cA ...A cB1...Bn = δA1B1 ...δAnBn . (18) (b) 1 n

5 The latter formula might be used to define a GA scalar product, if needed. FIG. 4: (Color online) Two equivalent bag-of-shapes represen- tations of the multivector 5+1.5b1 −b2 +b1b4 +3b5b6 +2b1b4b5. The black vector represents 1.5b1 − b2. Representation of D. Elementary gates scalars by numbers, as in (b), is perhaps more convenient than in terms of ‘charged points’ — represented by five circles GA analogues of elementary gates (Pauli, Hadamard, in (a). Yet another representation involves three-dimensional phase, π/8, cnot, and Toffoli) were described in [5]. Here visualization where the scalar part, here 5, is the height of we want to shed some light on the issue of how many suspension of the plane containing the collection of polylines. elementary operations are associated with networks of gates. 5

We have to begin with yet another additional dimen- If we do not know how to treat multivectors as sin- sion, represented by the basis vector bn+1. This addi- gle geometric objects, then computing H1 ...Hnψ will tional dimension will not be used for coding, but for defin- involve an exponential number of steps. So the key in- ing certain bivector operations. We do it as follows [5]. gredient of efficient geometric-algebra computation is to Let a = b b , 0 j n, and consider any complex implement all the needed gates in a geometric manner. j j n+1 ≤ ≤ — in the sense of (4) — vector zA1...Ak...An . Negation of In Section IV C we give a concrete example of realization a kth bit, 1 k n, of H ...H ψ in 2n steps, if ψ = c . ≤ ≤ 1 n 01...0n

k−1 ∗ k−1 n z = b a z a E. Reading superposed information k A1...Ak...An k Y j  A1...Ak...An Y j j=0 j=0 = z ′ , (19) A1...Ak...An An important difference between quantum and geo- metric coding is in the ways one gets information from is defined in purely algebraic terms. The same with an- superpositions of states. In quantum coding a measure- other important , multiplication by ( 1)Ak , ment projects a superposition on a randomly selected ba- − sis vector. Such measurements destroy the original state. a∗z a = ( 1)Ak z . (20) k A1...Ak...An k − A1...Ak...An In GA coding the ontological status of superpositions is different. Here one has a collection of geometric objects All the elementary one-, two-, and three-bit gates can be and thus can perform many measurements on the same defined in terms of n , ( 1)Ak , and i [5]. Of particular k − system without destroying its state. As a consequence importance is the certain standard ingredients of quantum algorithms are not needed in GA-based computations. 1 1 ∗ AkzA1...Ak ...An = zA1...Ak ...An akzA1...Ak...An ak Shor’s algorithm [7] for factoring 15 into 3 5 pro- 2 − 2 vides a simple illustration. The entangled state· ψ = = A z . (21) 15 | i k A1...Ak...An x ax mod15 , for a = 2, reads Px=0 | i| i Now let X be any map of GA into itself satisfying ψ = 0 + 4 + 8 + 12 1 X(0) = 0. Then Xk = 1 Ak + X Ak is a control- | i | i | i | i | i| i − ◦ + 1 + 5 + 9 + 13 2 X, controlled by the kth bit. Let us note that Ak is a | i | i | i | i| i projector on the subspace spanned by those blades that + 2 + 6 + 10 + 14 4 contain the vector b . In particular, in Fig. 5 the selection | i | i | i | i| i k + 3 + 7 + 11 + 15 8 . of blades containing the red is performed, algebraically | i | i | i | i| i ր speaking, by means of A8. The goal is to find the period of the function x In order to understand how to count the number of 2x mod 15, but the problem is that the sequences of num-7→ algorithmic steps let us take a concrete example of, say, bers correlated with the values 20 = 1, 21 = 2, 22 = 4, n Hadamard gates acting on different bits. Let ψ be a 23 = 8, are periodic, but shifted by, respectively, 0, 1, multivector, not necessarily a single blade, but a general 2, and 3. To get rid of this shift one performs quantum combination of 2n possible blades. The Hadamard gate Fourier transformation on the first register. simultaneously affecting the kth bits of all the 2n blades We claim that in a GA version of the algorithm the of ψ is [5] Fourier transform step will not be needed. What we have to do is to localize the vectors x 1 and the smallest 1 ∗ x> 0 is the solution. The GA analog| i| i of the calculation Hkψ = nkψ + a ψak . (22) √2 k  is given by the multivector

Similarly to ψ, H ψ is a single multivector. c01020304 + c01120304 + c11020304 + c11120304 c05060718 k  The latter observation is trivial, perhaps, but crucial + for the problem. A simple illustration of a 2-bit multivec- c0 0 0 1 + c0 1 0 1 + c1 0 0 1 + c1 1 0 1 c0 0 1 0 tor ψ = ψ 1+ψ b +ψ b +ψ b b is a 2-dimensional ori- 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4  5 6 7 8 0 1 1 2 2 12 1 2 + ented plane segment (represented by ψ12b1b2), suspended at the height ψ0, and whose center of is above the c01021304 + c01121304 + c11021304 + c11121304 c05160708 ′  point ψ1b1+ψ2b2. A gatemaps ψ into some new ψ which + has a similar geometric interpretation. c0 0 1 1 + c0 1 1 1 + c1 0 1 1 + c1 1 1 1 c1 0 0 0 . So when it comes to the question of how many oper- 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4  5 6 7 8 ations are performed while computing H1 ...Hnψ, the The cartoon version of this computation is shown in answer is this: Two for computing Hnψ, another two Fig. 5. Selecting an appropriate subset of shapes we find for computing Hn−1Hnψ, yet another two for computing that the period is x = 4. The factorization is given by x/2 Hn−2Hn−1Hnψ, and so on. Finally, we need 2n opera- 2 1. We have not needed the Fourier transform. An tions. The same argument applies to all the other ele- analogous± observation was made in the context of the mentary gates. Simon algorithm in [4]. 6

Basis: quantum correlation functions.

b1 b2 b3 b4 b5 b6 b7 b8

Resulting superposition: G. GA versions of quantum algorithms

We will not give here explicit GA versions of quan-

tum algorithms, since separate papers [1, 4, 11] were de- voted to this subject. The general conclusion is that any

quantum algorithm has a GA analogue, a fact follow- ing from the GA construction of the elementary quan-

tum gates [5]. The main formal difference between GA

and quantum computation is that the GA formalism is

not bound to use unitary gates. Indeed, the unitarity of quantum gates follows from the Schr¨odinger equation for- mula U = exp( iHt), which is irrelevant for GA compu- t − tation. What is relevant in the GA framework are those

operations that have a geometric meaning. In particular, one of the most important non-unitary geometric opera- Selection of : tions is a projection on a subspace. This projection has

nothing to do with the projection postulate of quantum

mechanics. Indeed, in the projec- tion ‘collapses’ a superposition on a basis vector. The

The smallest x>0: geometric projection projects a multivector on another multivector, but in general not on a single comb. = b2 b 8 = c01000001 A situation where geometric projections play a simpli- fying role in GA analogues of quantum algorithms is the

FIG. 5: (Color online) Shor-type algorithm. Euclidean space problem of deleting intermediate ‘carry bits’ in quantum is 8-dimensional. Oracle first produces the multivector play- adder networks [12, 13, 14, 15]. A glimpse at the network ing a role of the entangled state. Then a selection of blades proposed in [12] shows that the number of gates could be containing the red ր is performed. The blade c01000001 corre- reduced by almost a half if one did not insist on perform- sponding to the smallest number x > 0 is selected. The first ing this task in a reversible way. This is especially clear half of bits represents x = 4. if one compares alternative adder networks discussed in [15]. In terms of GA computation this concrete part of the F. Mixed states network will be replaced by an appropriate projection which, in spite of being irreversible, is geometrically al- The example of the Shor algorithm shows clearly that lowed. For example, in a 3-bit case the operation of re- setting the third bit, c c , corresponds to the multivector coefficients, as opposed to wave functions, ABC → AB0 do not have a probabilistic interpretation. For this rea- following set of projections son the GA analogues of quantum algorithms are not probabilistic. Still, probabilistic algorithms will occur if 1= c000 c000 =1, (23) one replaces multivectors by multivector-valued random → b1 = c100 c100 = b1, (24) variables. The resulting states will be mixed in the usual → b = c c = b , (25) meaning of this term (probability measures defined on 2 010 → 010 2 the set of pure states) but nevertheless will not, in gen- b3 = c001 c000 =1, (26) → eral, lead to a density formalism (the latter oc- b1b2 = c110 c110 = b1b2, (27) curs only in theories where pure-state averages of random → b1b3 = c101 c100 = b1, (28) variables are bilinear functionals of pure states). → b b = c c = b , (29) In this context we should mention the paper [8] where 2 3 011 → 010 2 b b b = c c = b b . (30) multivector-valued hidden variables were used to violate 1 2 3 111 → 110 1 2 an analogue of the Bell inequality. The construction em- ploys a hidden-variable state that is mixed in our sense. Each of them has a geometric interpretation: b b b 1 2 3 → Although the problem posed in [8] is not exactly equiva- b1b2 squeezes a cube into its x y wall; b2b3 b2 squeezes lent to the one addressed in the original Bell construction a square lying in the y z plane− into its side→ parallel to [9], it is nevertheless interesting from our point of view, the y axis, and so on. An− interpretation in terms of the and shows a way of generating certain GA analogues of polylines is left to the readers. 7

(a) Bell basis B. GHZ state

Ψ+ = , Ψ– = The 3-bit GHZ state reads 1 Φ+ = , Φ– = ΨGHZ = 000 + 111 . (35) | i √2| i | i

(b) GHZ state The GA representation 1 1 ΨGHZ = ΨGHZ = c000 + c111 = 1+ b123 (36) √2  √2  (c) 3-bit Hadamard state is shown in a cartoon form in Fig. 6b.

= C. n-fold Hadamard state

An n-fold Hadamard state is obtained if one acts with FIG. 6: (Color online) Cartoon versions of entangled states: an nth tensor power of a Hadamard gate on a ‘vacuum’ (a) The Bell basis, and (b) the 3-bit GHZ state. (c) Action 0 ... 0 . As a result one gets a superposition of all the n- of a 3-bit Hadamard gate on c000. The combination b1 + b2 is bit| numbersi A ...A . Such a state is the usual starting shown as a single black vector. All the blades are assumed to | 1 ni be appropriately normalized. point for quantum computation. In the GA formalism the corresponding multivector reads [5]

2−n/2(1 + b ) ... (1 + b )=2−n/2 c . (37) IV. MULTIVECTOR ANALOGUES OF 1 n X A1...An IMPORTANT PURE STATES A1...An Fig. 6c shows its 3-bit illustration. n Let us finally give explicit multivector counterparts of Let us note that the superposition of 2 combs cA1...An , some important entangled states occurring in quantum representing all the n-bit numbers, is here obtained by information problems. means of n aditions and n multiplications. This step is as efficient as its quantum version and agrees with our previous analysis of the n-fold Hadamard gate in Section A. Bell basis III D.

The Bell basis consists of four mutually orthogonal 2- V. CONCLUSIONS qubit entangled states:

1 It seems fair to say that quantum computation looks Ψ± = 01 10 , (31) | i √2| i ± | i from the GA perspective as a particular implementation 1 of a more general way of computing. The implementation Φ± = 00 11 . (32) based on tensor products of qubits and quantum super- | i √2| i ± | i position principle is of the quantum world. There are two bits and thus a 2-dimensional Euclidean However, the formalism of quantum computation loses its space will suffice as long as we do not need complex num- micro-world flavor when viewed from the GA standpoint. bers. Let the basis be b1, b2. The corresponding multi- Actually, there is no reason to believe that quantum com- vectors then read putation has to be associated with systems described by quantum mechanics. GA occurs whenever some geome- 1 1 try comes into play. It is enough to thumb the mono- Ψ± = c01 c10 = b2 b1 , (33) √2 ±  √2 ±  graphs of the subject [16, 17, 18, 19, 20, 21, 22, 23] to 1 1 understand its ubiquity, interdisciplinary character, and Φ± = c00 c11 = 1 b12 . (34) √2 ±  √2 ±  vast scope of applications. The question of concrete practical implementation of Fig. 6a shows the corresponding sets of blades. The GA coding is an open one and is certainly worth of fur- blades Ψ± are represented simply by two unit vectors ro- ther studies. tated by π/4 with respect to the axis spanned by b1. It is interesting± that an analogous simple representation of an entangled state occurs in quantum optics formulated Acknowledgments in the so-called -representation of canonical commuta- tion relations [10].∞ The two basic blades correspond there This work was supported by the Flemish Fund for Sci- to two modes of light behind a beam splitter. entific Research (FWO), project G.0452.04. MC thanks 8

J. Rembieli´nski for support and encouragement, and Z. Oziewicz for comments.

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