clifford-algebraic random walks on the hypercube - Springer Link

CLIFFORD-ALGEBRAIC RANDOM WALKS ON THE HYPERCUBE G. Stacey Staples Department of Mathematics and Statistics Southern Illinois University at Edwardsville Edwardsville, IL 62026-1653 [email protected] (Received: March 29, 2005;

Accepted: May 30, 2005)

Abstract. The n-dimensional hypercube is a simple graph on 2n vertices labeled by binary strings, or words, of length n. Pairs of vertices are adjacent if and only if they differ in exactly one position as binary words; i.e., the Hamming distance between the words is one. A discrete-time random walk is easily defined on the hypercube by “flipping” a randomly selected digit from 0 to 1 or vice-versa at each time step. By associating the words as blades in a Clifford algebra of particular signature, combinatorial properties of the geometric product can be used to represent this random walk as a sequence within the algebra. A closed-form formula is revealed which yields probability distributions on the vertices of the hypercube at any time k ≥ 0 by a formal power series expansion of elements in the algebra. Furthermore, by inducing a walk on a larger Clifford algebra, probabilities of self-avoiding walks and expected first hitting times of specific vertices are recovered. Moreover, because the Clifford algebras used in the current work are canonically isomorphic to fermion algebras, everything appearing here can be rewritten using fermion creation/annihilation operators, making the discussion relevant to quantum mechanics and/or quantum computing. MR Subject Classifications: 15A66, 05C65, 60C05, 16W60 keywords: combinatorial probability, algebraic probability, self-avoiding walks

1. Introduction The n-dimensional cube or hypercube Qn is the simple graph whose vertices are the n-tuples with entries in {0, 1} and whose edges are the pairs of ntuples that differ in exactly one position. This graph has natural applications in computer science, symbolic dynamics, and coding theory [6]. The structure Advances in Applied Clifford Algebras 15 No. 2, 213-232 (2005)

214

Clifford-Algebraic Random . . . G. Stacey Staples

of the hypercube allows one to construct a random walk on the hypercube by “flipping” a randomly selected digit from 0 to 1 or vice-versa. Combinatorial properties of Clifford algebras are employed to represent such a random walk as sequences within a particular Clifford algebra. A closed-form formula is revealed which yields probability distributions on the vertices of the hypercube at any time k ≥ 0 from a formal power series expansion of an element in the algebra. By construction of a larger auxiliary Clifford algebra, probabilities of self-avoiding walks and expected first hitting times of specific vertices are recovered. Let b be a block, or word, of length n; that is, let b be a sequence of n zeros and ones. The weight of b is defined as the number of 1’s in the sequence. The binary sum of two such words is the sequence resulting from addition modulotwo of the two sequences. The Hamming distance between two binary words is defined as the weight of their binary sum. Example 1.1. Let b1 = 01101100 and let b2 = 11001111. Then the weights are w(b1 ) = 4 and w(b2 ) = 6. The Hamming distance between b1 and b2 is w(10100011) = 4. A simple graph G = (V, E) is a set V of vertices and a set E of unordered pairs of vertices, or edges, such that no unordered pair occurs more than once in E, and there is no unordered pair of the form {v, v}. Another way of saying this is that a simple graph contains no multiple edges and no loops. Two vertices v1 , v2 ∈ V are said to be adjacent if {v1 , v2 } ∈ E. Definition 1.2. The n-dimensional hypercube Qn is the simple graph whose vertices are the 2n n-tuples from {0, 1} and whose edges are defined by the rule {v1 , v2 } ∈ E(Q) iff w(v1 + v2 ) = 1. (1.1) Here v1 + v2 is bitwise addition modulo-two, and w is the weight. In other words, two vertices of the hypercube are adjacent if and only if their Hamming distance is 1. What follows in the remainder of this introductory subsection is a more-orless standard introduction to Clifford algebras and their properties. The reader is referred to works such as [3] and [4] for essential background information on Clifford algebras.

Advances in Applied Clifford Algebras 15, No. 2 (2005)

Fig. 1.

215

The three-dimensional hypercube.

Definition 1.3. For fixed n ≥ 0, let V be an n-dimensional vector space having orthonormal basis e1 , . . . , en . The 2n -dimensional Clifford algebra of signature (p, q), where p + q = n, is defined as the associative algebra generated by the collection {ei } along with the scalar e∅ = 1 ∈ R, subject to the following multiplication rules:

ei 2

ei ej + ej ei = 0 for i 6= j, and ( 1, if 1 ≤ i ≤ p = −1, if p + 1 ≤ i ≤ p + q = n.

(1.2) (1.3)

We denote the Clifford algebra of signature (p, q) by C`p,q . Generally the vectors generating the algebra do not have to be orthogonal. When they are orthogonal, as in the definition above, the resulting multivectors are called blades. Let [n] = {1, , 2, . . . , n} and denote arbitrary, canonically ordered subsets of [n] by underlined Roman characters. The basis elements of C`p,q can then be indexed by these finite subsets if we write Y ek . (1.4) ei = k∈i

Arbitrary elements of C`p,q have the form X u= u i ei , i∈2[n]

where ui ∈ R for each i ∈ 2[n] .

(1.5)

216


Definition 1.4. By the degree of a monomial in C`p,q we shall mean the cardinality of its index. For example, deg e1 3 4 = |{1, 3, 4}| = 3. Definition 1.5. For 0 ≤ k ≤ n, we define the degree-k part of u ∈ C`p,q as the sum of degree-k monomials in the expansion of u. In other words, X huik = u i ei . (1.6) i∈2[n] |i|=k

Notation We use the notation hhuiik to denote the sum of the coefficients in the degree-k part of u. That is, X ui . (1.7) hhuiik = i∈2[n] |i|=k

Definition 1.6. We define the Berezin integral as the linear functional R : C`p,q → R such that B

Z X

ui ei = u12...n .

(1.8)

[n] B i∈2

In other words, the Berezin integral is the “top-form” coefficient in the expansion of u. Remark 1.7. This use of the Berezin integral is not standard but follows naturally from Berezin’s original construction on the Grassmann algebra [1]. X X Definition 1.8. Given arbitrary u = ui ei and v = vi ei the Clifford i∈2[n]

i∈2[n]

inner product of u and v is defined by hu, vi =

X

ui vi .

(1.9)

i∈2[n]

Consequently, the expansion of u ∈ C`p,q can be written X u= hu, ei i ei .

(1.10)

i∈2[n]

This inner-product defines a norm on C`p,q by 1

kuk = hu, ui 2 . This norm is referred to as the Clifford inner-product norm.

(1.11)


217

It is not difficult to see that C`0,1 ∼ = C via the correspondence e0 ' 1, e1 ' ı, and that C`0,2 ∼ = H (the algebra of quaternions) via e0 ' 1, e1 ' i, e2 ' j, and e12 ' k. It is also easy to verify that C`n,n is isomorphic to the n-particle fermion algebra (cf. [2]) via the correspondence 1 (ei − en+i ) , for 1 ≤ i ≤ n 2 1 ' (ei + en+i ) , for 1 ≤ i ≤ n. 2

fi ' fi +

(1.12) (1.13)

Here fi , fi + are thought of as the fermion annihilation and creation operators, respectively. This concludes the introductory material. The remainder of the current work is original with the author. 1.1. The algebra C`n sym Definition 1.9. For fixed n > 0, the algebra C`n sym is defined as the 2n dimensional associative algebra generated by the elements ςi = ei en+i ∈ C`n,n for 1 ≤ i ≤ n along with the scalar ς∅ = 1 ∈ R. It is easy to see that C`n sym is a commutative algebra whose generators satisfy ςi 2 = 1 for 1 ≤ i ≤ n; i.e., the generators are unipotent. Basis elements of C`n sym can again be indexed by canonically-ordered subsets of [n] so that arbitrary elements have the form u=

X

ui ςi .

(1.14)

i∈2[n]

By the properties of Clifford multiplication, we see that for arbitrary i, j ∈ 2[n] we have ςi ςj = ςi4j , (1.15) where 4 = (i ∪ j) \ (i ∩ j) is the symmetric difference of i and j. It is evident that the generators of ςi of C`n sym (bivectors in C`n,n ) generate a multiplicative group Σn isomorphic to the group generated by reflections across orthogonal hyperplanes in the real vector space Rn , for these also satisfy Ri Rj = Rj Ri and Ri 2 = id. It is equally evident that Σn ∼ = (2[n] , 4), the group consisting of the power set of [n] = {1, 2, . . . , n} with the set symmetric difference operator. These groups are also isomorphic to the additive

218


abelian group Z2 + · · · + Z2 . Because of these isomorphisms, we see that the | {z } n-times

Cayley graph of Σn is the n-dimensional hypercube, a structure having natural connections with information and coding theory. Lemma 1.10. Let ςi be an arbitrary blade in C`n sym and let ai ∈ R. Then eai ςi = cosh(ai ) + sinh(ai ) ςi .

(1.16)

Proof. eai ςi =

∞ X X 1 X 1 1 k k ai ςi = ai k + ai k ςi = cosh(ai ) + sinh(ai ) ςi . k! k! k! k even k odd k=0 (1.17)

Lemma 1.11. Let ςi , ςj be two blades in C`sym n , and let a, b ∈ R. Then ea ςi +b ςj = ea ςi eb ςj .

(1.18)

Proof. ea ςi +b ςj =

∞ ∞ m X X 1 1 X m k m−k k m−k (a ςi + b ςj )m = a b ςi ςj . (1.19) m! m! k m=0 m=0 k=0

Then by inspection, one sees that (1.19) is a sum of the following four parts: X 1 X m ak bm−k (scalar part) (1.20) m! k m even k even X 1 X m ak bm−k ςi (1.21) m! k m odd k odd X 1 X m ak bm−k ςj (1.22) m! k m odd k even X 1 X m ak bm−k ςi4j . (1.23) m! k m even k odd

By Lemma 1.10, one has ea ςi eb ςj = cosh(a) + sinh(a) ςi

cosh(b) + sinh(b) ςj

= cosh(a) cosh(b) + cosh(a) sinh(b) ςj + cosh(b) sinh(a) ςi + sinh(a) sinh(b) ςi4j . (1.24)


219

The scalar part of this equation is 1 b 1 a cosh(a) cosh(b) = e + e−a e + e−b 2 2 1 a+b = e + e−(a+b) + ea−b + e−(a−b) 4 ∞ m ∞ m 1 X 1 X m k m−k 1 X 1 X m m = a b + (−1) ak bm−k 4 m=0 m! k 4 m=0 m! k k=0 k=0 ∞ m 1 X 1 X m k (1.25) m−k m−k + a (−1) b 4 m=0 m! k k=0 ∞ m 1 X 1 X m + (−1)m ak (−1)m−k bm−k 4 m=0 m! k k=0 ∞ m 1 X 1 X m = 4 m=0 m! k k=0 ak bm−k + (−1)m−k ak bm−k + (−1)m (ak bm−k + (−1)m−k ak bm−k ) . It is clear that if k and m are both even, this term is X X 1 m ak bm−k , m! k m even

(1.26)

k even

while in all other cases this term is zero, in agreement with (1.20). Similarly, 1 b 1 a e + e−a e − e−b 2 2 1 a+b = e − e−(a+b) − ea−b + e−(a−b) 4 ∞ m ∞ m 1 X 1 X m k m−k 1 X 1 X m m = a b − (−1) ak bm−k 4 m=0 m! k 4 m=0 m! k k=0 k=0 ∞ m 1 X 1 X m k m−k m−k a (−1) b − 4 m=0 m! k k=0 ∞ m 1 X 1 X m + (−1)m ak (−1)m−k bm−k 4 m=0 m! k k=0 ∞ m 1 X 1 X m = 4 m=0 m! k cosh(a) sinh(b) =

k=0 ak bm−k − (−1)m−k ak bm−k − (−1)m (ak bm−k − (−1)m−k ak bm−k ) .

(1.27)

220


Here, the only nonzero terms correspond to odd values of m and even values of k, in agreement with (1.22). Similar calculations yield ∞ m 1 X 1 X m cosh(b) sinh(a) = 4 m=0 m! k k=0

ak bm−k + (−1)m−k ak bm−k − (−1)m (ak bm−k + (−1)m−k ak bm−k ) , (1.28) ∞ m 1 X 1 X m 4 m=0 m! k

sinh(a) sinh(b) =

k=0

k m−k

a b

m−k k m−k

− (−1)

a b

+ (−1)m (ak bm−k − (−1)m−k ak bm−k ) , (1.29)

which agree with equations (1.21) and (1.23), and the proof is complete. Because C`n sym is commutative and associative, the following corollaries are obtained. Corollary 1.12. 

 exp 

X

ai ςi  =

i∈2[n]

Y

cosh(ai ) + sinh(ai ) ςi .

(1.30)

i∈2[n]

The following Corollary gives a formal expression for the real coefficient of ςi in ea . X ai ςi . Then Corollary 1.13. Let i ∈ 2[n] , i 6= ∅ be fixed and let a = i∈2[n] n

a

he , ςi i =

2 X m=1

X

Y

sinh(aj1 ) · · · sinh(ajm )

cosh(ak ). (1.31)

k∈J / m

Jm ={j1 ,...,jm }⊂2[n] j1 4···4jm =i

The coefficient of ς∅ , i.e., the scalar part of ea is n

a

he , ς∅ i =

Y

cosh(ai ) +

i∈2[n]

sinh(aj1 ) · · · sinh(ajm )

2 X m=1

Q

X Jm ={j1 ,...,jm }⊂2[n] j1 4···4jm =∅

k∈J / m

cosh(ak ).

(1.32)


221

2. Random Walks on the n-Dimensional Hypercube The n-dimensional hypercube n is the Cayley graph of the group Σn , yielding LQ n aLgroup isomorphism Σn ∼ = i=1 Z2 . An algebra isomorphism C`n sym → R ⊗ n i=1 Z2 , is defined by linear extension of the mapping a ⊗ ςi 7→ a ⊗ zi ,

(2.1)

where zi can be thought of as a binary n-vector with 1’s only in the positions specified by the multiindex i. To ensure this mapping is an algebra isomorphism, one requires (2.2) a ⊗ ςi b ⊗ ςj 7→ ab ⊗ zi + zj and

( (a + b) ⊗ zi a ⊗ ςi + b ⊗ ςj 7→ a ⊗ zi + b ⊗ zj

if i = j otherwise.

(2.3)

Three bijective mappings become necessary for the remainder of the current work: φ : {0, 1}n → 2[n] η : {0, 1}n → [2n ]

(2.4) (2.5)

ψ : 2[n] → [2n ].

(2.6)

The mapping φ takes binary strings of length n to elements of the power set 2[n] . The mapping η takes binary strings of length n, considered as base-two integers contained in the interval [0, 2n − 1], to their corresponding base-10 representations. The mapping ψ is defined as the composition ψ = η ◦ φ−1 . One uniquely associates each vertex to an element of 2[n] by using the bijection φ : {0, 1}n → 2[n] defined by [ φ(v) = {i}. (2.7) 1≤i≤n vi =1

Now each vertex v ∈ V (Qn ) is uniquely associated with a unique in C`n sym via the bijection v 7→ ςφ(v) . Example 2.1. Let v = 100110 ∈ V (Qn ), then φ(v) = {1, 4, 5} , and η(v) = ψ({1, 4, 5}) = 38.

(2.8) (2.9)

222


To construct a random walk on the n-dimensional hypercube, one can think of a binary string of length n and consider the effect of “flipping” a single digit, either from 0 to 1 or vice-versa, at each discrete time step. This can be accomplished within C`n sym simply by multiplying the Clifford representation of a vertex by a Clifford representation of the digit being flipped. Any probability distribution on the vertices of Qn represented as a vector n in R2 can be uniquely represented as an element of C`n sym via the bijection φ. To simplify the notation, let us write vi to mean the vertex of Qn associated with ςi . The initial distribution is then defined by X

ξ0 =

pi (0) ςi

(2.10)

i∈2[n]

where pi (0) is the probability that the initial vertex of the random walk is vi . An equivalent formulation is ξ0 =

X

pv (0) ςφ(v) .

(2.11)

v∈Qn

Proposition 2.2. Let Y be a random variable taking values in [n] ∪ {0} with probabilities pi = Pr{Y = i} for each 1 ≤ i ≤ n, and let {Yk }k>0 be the sequence of independent random variables obtained from repeated observations of Y . Let ξ0 ∈ C`n sym represent any initial probability distribution on the n X vertices of Qn . Let τ = pi ςi . Then for k > 0, the distribution on Qn at i=0

time k is given by ξk = ξ0 τ k .

(2.12)

Proof. When k = 0, this is true by hypothesis. Let pi (k) be defined as the probability of being at vertex ςi at time step k. We see that the distribution at time k = 1 is given by

ξ1 =

X i∈2[n]

pi (1) ςi =

n X X

pi4{j} (1) ςi4{j} =

i∈2[n] j=1

=

n X X

pi (0)pj ςi ςj

i∈2[n] j=1

X i∈2[n]

pi (0) ςi

n X j=1

 pj ςj = ξ0 

n X j=1

 pj ςi  , (2.13)


223

where 4 denotes the symmetric difference of sets. Assuming true for k > 0 and proceeding by induction, we find X

ξk+1 =

=

pi (k) ςi

i∈2[n]

pi4{j} (k + 1) ςi4{j}

i∈2[n] j=1

i∈2[n]

X

n X X

pi (k + 1) ςi = n X

   k   n n n X X X pj ςj = ξk  pj ςj  = ξ0  pj ςj   pj ςj 

j=1

j=1

j=1

j=1

 k+1 n X = ξ0  pj ςj  . j=1

(2.14) Remark 2.3. Because Qn is a simple graph, i.e. contains no loops, random walks would not ordinarily allow consecutively repeated vertices. Real-world applications of such random walks, on the other hand, may require this possibility. Here multiplying by ς0 = 1 represents the event that no digit is “flipped,” i.e., no step is taken. Hence, the value of p0 determines which type of walk is being considered. The sequence in (2.12) is referred to as the vertex distribution sequence associated with the random walk on the hypercube. By construction, the random walk described corresponds to the more traditional construction x~k = x~0 T k , where T is a stochastic matrix giving the transition probabilities of the walk. To immediately see the benefits of the Clifford-algebraic approach, consider the following three-dimensional case. Example 2.4. In considering random walks on the three-dimensional hypercube Q3 , one traditionally considers powers of the transition probability matrix T given below.

 p0 p3  p2  p1 T = 0  0  0 0

p3 p0 0 0 p2 p1 0 0

p2 0 p0 0 p3 0 p1 0

p1 0 0 p0 0 p3 p2 0

0 p2 p3 0 p0 0 0 p1

0 p1 0 p3 0 p0 0 p2

0 0 p1 p2 0 0 p0 p3

 0 0  0  0  p1   p2   p3  p0

(2.15)

224


Fig. 2.

The three-dimensional hypercube labeled with generators of C`3 sym .

By contrast, in the Clifford-algebraic formalism one considers powers of τ ∈ C`n sym defined by τ = p0 ς0 + p1 ς1 + p2 ς2 + p3 ς3 .

(2.16)

Theorem 2.5. Let Y be a random variable taking values in {0, 1, 2, . . . , n} with probabilities pi = Pr{Y = i} for each 0 ≤ i ≤ n, and let {Yk }k>0 be the sequence of independent random variables obtained from repeated observations of Y . Let ξk ∈ C`n sym represent the distribution on the vertices of Qn at time step k ≥ 0 corresponding to the random walk induced by the sequence {Yk }, n X and let τ = pi ςi . Then for k > 0 and real parameter t 6= 0, i=0 t p0

ξk = k! · ξ0 e

n Y

(cosh(t pi ) + sinh(t pi ) ςi ) .

(2.17)

tk

i=1

Here the notation ]tk on the right-hand side represents the Clifford-valued coefficient of tk in the formal power series expansion of the product. Proof. Let τ ∈ C`n sym be of the form τ =

n X

pi ςi , and let t be any real nonzero

i=1

parameter. Applying Corollary 1.12, one has et τ =

n Y

(cosh(t pi ) + sinh(t pi ) ςi ) .

i=0

The result follows immediately.

(2.18)


Corollary 2.6. If there exists an element ξ =

225

X

xi ςi such that

i∈2[n]

t p0

lim k k! · ξ0 e

k→∞

n Y

! (cosh(t pi ) + sinh(t pi ) ςi ) − ξk = 0

i=1

(2.19)

tk

then ξ represents a stationary distribution on the vertices of Qn corresponding to the random walk in the proposition. The norm used here is the Clifford inner-product norm. Example 2.7. An example computed with Mathematica: 30 steps of a random walk on the 5-dimensional hypercube. Set the number of generators (dimension of the hypercube)... n=5; (*p[[1]] represents probability of "no jump" remaining values are probabilities of "flipping" (i + 1)^th bit *) p0 = Table[Random[\ ], {n+1}]; p = p0 Sum[p0[[i]], {i,1,n+1}] {0.120874, 0.107259, 0.248378, 0.250487, 0.0817183, 0.191283} Xi0 = 1; (*Initial distribution the "vacuum state"*) maxk=30; (*Max time steps*) starttime=AbsoluteTime[]; For[k = 1, k 1},t^ k]] Print["Elapsed time of calculation: ",AbsoluteTime[]-starttime," seconds."]; Elapsed time of calculation: 11.4218750 seconds. The distribution at time step k = 10, i.e. ξ10 , is found to be

226


0.0421027 +0.0316885ς1 +0.0376143ς2 +0.0357539ς1 ς2 +0.0376197ς3 +0.0357593ς1 ς3 +0.0419747ς2 ς3 +0.0315618ς1 ς2 ς3 +0.0269712ς4 +0.0255019ς1 ς4 +0.030628ς2 ς4 +0.0219918ς1 ς2 ς4 +0.0306332ς3 ς4 +0.0219969ς1 ς3 ς4 +0.0268484ς2 ς3 ς4 +0.0253796ς1 ς2 ς3 ς4 +0.0371508ς5 +0.0352973ς1 ς5 +0.0414989ς2 ς5 +0.0311066ς1 ς2 ς5 +0.0415043ς3 ς5 +0.0311119ς1 ς3 ς5 +0.0370231ς2 ς3 ς5 +0.0351702ς1 ς2 ς3 ς5 +0.0302017ς4 ς5 +0.0215858ς1 ς4 ς5 +0.0264237ς2 ς4 ς5 +0.0249617ς1 ς2 ς4 ς5 +0.0264288ς3 ς4 ς5 +0.0249668ς1 ς3 ς4 ς5 +0.0300785ς2 ς3 ς4 ς5 +0.0214639ς1 ς2 ς3 ς4 ς5


227

2.1. Self-Avoiding Walks on the Hypercube Combinatorial properties of the algebra C`n sym can also be used to “sieve out” walks which revisit any vertex. If vertices are labeled with generators of the algebra, and transitions between vertices correspond to multiplying these labels, it is clear that a walk revisiting a vertex will result in a product of reduced degree. For example, let the vertices of Q3 be labeled with the generators of C`8 sym and consider the 3-walk {v1 v2 v6 v2 }. As an element of C`8 sym this corresponds to the blade ς1 2 6 2 = ς1 6 which has degree 4 because vertex 2 was revisited. On the other hand, a self-avoiding 3-walk such as {v1 v2 v3 v4 } corresponds to the degree-8 blade ς1 2 3 4 . While representing random walks on the hypercube as random walks on C`n sym is appealing because of the notational convenience, studying self-avoiding random walks on Qn is not straightforward. Let V be a real 2n -dimensional vector space with orthonormal basis {~vi }0≤i≤2n −1 , and let V ∗ denote its dual. For ease of notation, we assume all multi-indices are lexicographically ordered and index the vectors ~vi by multiindices i ∈ 2[n] . Let L(V ) denote the space of linear operators on V . In accordance with Dirac notation, denote elements of V by |vi and elements of V ∗ by hv|. Sequences in C`2n sym ⊗ L(V ) can now be defined which correspond to the random walk on Qn . Begin with the initial distribution on vertices of Qn , written as pi (0) ςi , where i ranges over all subsets of [n]. Let η : {0, 1}n → [2n ] be defined as the bijection taking vertices of Qn , which are base-two integers in the range 0 to 2n − 1, inclusive, to their base-10 representations plus one. That is, η(v) = φ−1 (v)10 + 1.

(2.20)

Here φ−1 (v) is the base-two representation of a multi-index from (2.7) and φ−1 (v)10 denotes the conversion to base-ten. We require the minimum value to be 1 because ς0 is the unit scalar, which is not useful as a vertex label, because it undergoes no reduction in degree when squared. Each blade ςi ∈ C`n sym is now associated with a Clifford bivector ςη(φ−1 (i)) ∈ C`2n sym , and the bit representation of the vertices of Qn is now associated with an integer-valued (base-ten) labeling. Walks on the hypercube written as sequences of vertices are represented as products of generators from C`2n sym , and the degree of any such product can be used to determine whether the represented walk revisited a vertex.

228


Example 2.8. The three-dimensional hypercube and vertex labellings for the random walk. v ∈ V (Q3 ) 000 001 010 011 100 101 110 111

Fig. 3.

ςφ(v) ∈ C`3 sym ς∅ ς3 ς2 ς23 ς1 ς13 ς12 ς123

ςη(v) ∈ C`8 sym ς1 ς2 ς3 ς4 ς5 ς6 ς7 ς8

The three-dimensional hypercube relabeled with generators of C`8 sym .

Define ψ = η ◦ φ−1 : 2[n] → [2n ]. Using this, define X pi (0) ςψ(i) ⊗ |vψ(i) ihvψ(i) | ∈ C`2n sym ⊗ L(V ), Ξ0 =

(2.21)

i∈2[n]

and Ψ1 =

n X X i∈2[n]

j=1

pj ςψ(i4{j}) ⊗ |vψ(i) ihvψ(i4{j}) | ∈ C`2n sym ⊗ L(V ).

(2.22)


229

Definition 2.9. For k > 1, let the step distribution sequence be defined by k

Ψk = (Ψ1 ) .

(2.23)

Let the walk distribution sequence be defined by Ξk = Ξ0 Ψk .

(2.24)

Theorem 2.10. For fixed vertices v0 and vt , let W be a random walk on Qn described above. The probability that an m-step walk with initial vertex v0 terminates at vertex vt and forms a path, i.e. revisits no vertices, is given by 1 ⊗ hvη(v0 ) | hhΨm ii2(m+1) 1 ⊗ |vη(vt ) i . (2.25) Proof. Let W be the random walk on the vertices of Qn corresponding to independently “flipping” digits in the binary representation of the vertices. Letting the conditional probabilities Pr{Wk+1 = vk+1 |Wi = vi } be denoted by Pr{vi+1 |vi }, the probability of the random walk Wk = {v0 , v1 , v2 , . . . , vk } is equal to Pr{v0 }Pr{v1 |v0 }Pr{v2 |v1 } · · · Pr{vk |vk−1 }. Observing that because pj denotes the probability of “flipping” bit j in the representation of a vertex, one finds that pj = Pr{Wk+1 = φ−1 (i4{j})|Wk = φ−1 (i)}. (2.26) The theorem is proved by showing that 1 ⊗ hvη(v0 ) | Ξm 1 ⊗ |vη(vt ) i is a sum of terms corresponding to all random m-walks v0 → vt ∈ Qn with respective probabilities as coefficients. Let v0 ∈ V (Qn ) be fixed. It is claimed that 1 ⊗ hvη(v0 ) | Ξm = X (2.27) Pr{W } ς{ψ(φ(v0 ))}4···4{ψ(φ(vm ))} ⊗ hvη(vm ) |. m-walks W beginning at v0

By construction of Ξ0 and Ψ1 , Ξ1 = Ξ0 Ψ1 = X

pi (0) ςψ(i) ⊗ |vψ(i) ihvψ(i) |

=

i∈2[n]

pj ςψ(`4{j}) ⊗ |vψ(`) ihvψ(`4{j}) |

`∈2[n] j=1

i∈2[n]

X

n X X

pi (0) ςψ(i)

n X

pj ςψ(i4{j}) ⊗ |vψ(i) i|hvψ(i4{j}) |.

j=1

(2.28)

230


Thus, when m = 1,

n X 1 ⊗ hvψ(i0 ) | Ξ1 = pi0 (0) ςψ(i0 ) pj ςψ(i0 4{j}) ⊗ hvψ(i0 4{j}) | j=1

=

n X

Pr{v0 = φ−1 (i0 )}Pr{v1 = φ−1 (i0 4{j})|v0 = φ−1 (i0 )}⊗hvη(φ−1 (i0 4{j})) |

j=1

X

=

Pr{W } ς{ψ(i0 )}4{ψ(k)} ⊗ hvψ(i0 4{j}) | (2.29)

1-walks W :φ−1 (i0 )→φ−1 (k)

Assuming true for m ≥ 1 and proceeding by induction, 1 ⊗ hvψ(i0 ) | Ξm+1 = 1 ⊗ hvψ(i0 ) | Ξm Ψ1 X = Pr{W } ς{ψ(i0 )}4···4{ψ(im )} m-walks W beginning at φ−1 (i0 )

⊗hvψ(im ) |

n X X

pj ςψ(i4{j}) ⊗ |vψ(i) ihvψ(i4{j}) |

i∈2[n] j=1

X

=

Pr{W } ς{ψ(i0 )}4···4{ψ(im )}

m-walks W beginning at φ−1 (i0 ) n X

pj ςψ(im 4{j}) ⊗ hvψ(im 4{j}) |

j=1

=

X

Pr{W } ς{ψ(i0 )}4···4{ψ(im+1 )} ⊗ hvψ(im+1 ) |

m + 1-walks W beginning at φ−1 (i0 )

(2.30) −1 that φ (i ) = η(v ) proves the claim. Letting i0 be the multi-index such 0 0 Now multiplying on the right by 1 ⊗ |vψ(it ) i = 1 ⊗ |vη(vt ) i eliminates all terms from the sum except those corresponding to m + 1-walks from v0 to vt . Since Ξ0 contributes 2 to the degree of each term, walks of length-m are selfavoiding if and only if they correspond to degree 2(m+1) Clifford multivectors. Summing coefficients over terms of degree 2(m + 1) then gives the probability of W . The Clifford-algebraic method also allows one to determine the expected “hitting time” of a fixed vertex of Qn . Given the n-dimensional hypercube, define


the Clifford algebra C`2n ,2n ,1   1 2 ei = −1   0

231

with squaring rules if 1 ≤ i ≤ 2n , i 6= β if 2n + 1 ≤ i ≤ 2n+1 , i 6= n + β if i = 2n+1 + 1.

(2.31)

Choosing an arbitrary vertex vΩ of the hypercube, construct C`2n sym over C`2n ,2n ,1 exactly as before, but with one exception: the bivector ςη(vΩ ) corresponding to vertex vΩ should be defined by ςη(vΩ ) = eψ(vΩ ) e2n+1 +1 . Now for any ςi ∈ C`2n ,2n −1,1 it is evident that ( 1 if 0 ≤ i ≤ 2n , i 6= Ω 2 ςi = (2.32) 0 if i = η(vΩ ). Proposition 2.11 (Expected hitting time). Given the n-dimensional hypercube Qn , let one vertex vΩ be set aside and use C`2n sym obeying (2.32) to label the vertices of Qn , with vΩ labeled by ςη(vΩ ) . Let {Ξk } be the associated walk-distribution sequence of the random walk and let H denote a random variable taking nonnegative integer values such that H is the first time the walk with initial vertex v0 “hits” vertex vΩ . Then EH ⊗ I =

∞ X

k ⊗ hvη(v0 ) | hhΞk ii 1 ⊗ |vη(vΩ ) i .

(2.33)

k=1

Proof. We observe that k ⊗ hvη(v0 ) | hhΞk ii 1 ⊗ |vη(vΩ ) i X = khh Pr{k-walk W exists} ς{η(v0 )}4···4{η(vΩ )} ii ⊗ I

(2.34)

k-walks W :v0 →vΩ

where the sum is over k-walks from v0 to vΩ visiting vΩ for the first time, for if a walk W revisits vertex vΩ , ςη(vΩ ) 2 = 0 removes that walk from the sum. Summing over k, the result follows immediately. In the finite-dimensional case, Ψ1 is the standard transition probability matrix associated with a time-homogeneous Markov chain. The use of Dirac notation makes convenient an extension of the above results to infinite dimensions. One can generate infinite-dimensional Clifford algebras using the orthonormal basis of an infinite-dimensional Hilbert space. Some additional considerations would need to be made concerning the maximum “weights” of vertices reached by the walk in finite time, but the possibilities here are vast.

232


3. Conclusion The methods employed here allow one to model combinatorial structures with elements of Clifford algebras. The specific model of the hypercube has applications in computer science and coding theory. While such objects have been studied successfully without Clifford algebras, it is hoped that this approach will promote new ways of thinking about existing combinatorial structures and related problems. In particular, since the algebra C`n sym is constructed within the algebra C`n,n , which is isomorphic to the n-particle fermion algebra, everything appearing here could be rewritten using fermions. The author believes that this in itself makes the methods interesting in the context of quantum computing. References [1] [2] [3] [4] [5] [6]

Berezin F. A., “The Method of Second Quantization”, Academic Press, New York, 1966. Feinsilver P., R. Schott, “Algebraic Structures and Operator Calculus” Vol. III, Kluwer, Dordrecht 1996. Lounesto P., “Clifford Algebras and Spinors”, Cambridge University Press, Cambridge, 2001. Porteous I., Clifford Algebras and the Classical Groups, Cambridge Studies in Advanced Mathematics 50, Cambridge University Press, Cambridge, (1995). R´enyi A., “Foundations of Probability”, Holden-Day, San Francisco, 1970. West D., “Introduction to Graph Theory”, Second Ed., Prentice Hall, Upper Saddle River, 2001.