clifford algebras, random graphs, and quantum random variables george stacey staples southern...
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Clifford Algebras, Random Graphs, and Quantum Random VariablesGeorge Stacey StaplesSouthern Illinois University at Edwardsville
René SchottUniversité Henri Poincaré Nancy 1
Clifford Algebras of signature (p,q)
1
12ie
pi 1
qpip 1
Clp,q is the associative algebra of dimension 2p+q
generated by the collection satisfying qpiie 1
ijji eeee ji
These algebras are have inherent connections with quantum probability and graph theory.
Clifford Algebras of signature (p,q)
.][2
ni
iieuu
A typical element of Clp,q can be written in the following form:
I.e., elements are linear combinations of scalar multiples of multi-vectors (or blades) indexed by subsets of the n-set [n]={1, 2, … , n}.
The geometric product is closely related to the logical XOR operator.
Pauli spin matrices
01
10x
0
0
i
iy
10
01z
10
010
Quantum coin flips
Satisfy σ*2 = σ0 = Identity
Anticommute
Generate the “Algebra of Physical Space” Cl3,0
Examples of Clifford Algebras
Quaternions
0,nCl
0,3Cl
2,0Cl
3,1Cl
nnCl ,
1,0Cl Complex number field
Algebra of Physical Space (APS)
SpaceTime Algebra (STA)
Fermion Fock space
Fermion algebra
Motivation: QP & CA
Algebraic, geometric, and combinatorial properties of Clifford algebras make them interesting objects of study in a variety of contexts.
● Stochastic processes on Clifford algebras of arbitrary signature
Accardi, Applebaum, Hudson, von Waldenfels, and others have contributed to the study of processes on the fermion field.
Processes on the finite-dimensional fermion algebra and fermion “toy Fock” space are special cases in the study of stochastic processes on Clifford algebras of arbitrary signature.
Motivation: Random Graphs, QP, & CA
● Using combinatorial properties of Clifford algebras to study random graphs and Markov chains
Hashimoto, Hora, and Obata have contributed significant work applying quantum probabilistic methods to the study of random graphs. The adjacency matrix of any simple graph is an observable. Defining the quantum decomposition of the adjacency matrix makes application of quantum probabilistic methods natural.
In particular, comb graphs, star graphs, Cayley graphs, Hamming graphs, and distance-regular graphs have been considered with this approach. These graphs are related to notions of independence in QP.
Motivation
● Using combinatorial properties of Clifford algebras to study stochastic processes on Clifford algebras.
The next logical step: make use of the combinatorial properties to compute expected numbers of cycles, cycle lengths, etc. of random walks on Clifford algebras.
The Problem
How do we count the cycles in a simple graph on 6 vertices?
Terms along the diagonal of the kth power of the adjacency matrix reveal the closed k-walks based at each vertex.
The Approach
Use the combinatorial properties of Clifford algebras to “sieve out” those walks that revisit a vertex.
Vertices should “cancel” when revisited.
The adjacency matrix will be replaced with a matrix having entries in a Clifford algebra.
Vertices will be labeled with nilpotent or unipotent generators of an abelian subalgebra of a Clifford algebra. Some additional tools are needed for this.
Tools: Abelian unipotent-generated subalgebras
12 i
inx
ii
01
0
Clnsym is the associative algebra of dimension 2n
generated by the collection satisfying nii 1
ijji
Clnsym is realized within Cln,n by writing
In terms of quantum random variables:
inii ee
Clnnil is realized within Cl2n,2n by writing
where fi+ denotes the ith fermion creation operator in
the 2n-particle fermion algebra.
Tools: Abelian nilpotent-generated subalgebras
02 i
inyx
ii i 0
10 )(
Clnnil is the associative algebra of dimension 2n
generated by the collection satisfying nii 1
ijji
Using Pauli matrices:
inii ff
and
The Problem
How do we count the cycles in a simple graph on 6 vertices?
The Solution
Replace the adjacency matrix with a Clifford adjacency matrix.
The matrix entries are elements of Cl6
nil. The generators commute and square to zero. Note: These generators were denoted by on the previous slide!
Elements along the diagonal of Ak now reveal k-cycles based at each vertex!
7 Vertices
Q. How many 5-cycles does the following graph contain?
The adjacency matrix reveals 10 closed 5-walksbased at the pendant vertex.
7 Vertices
Q. How many 5-cycles does the following graph contain?
The Clifford adjacency matrixreveals that the graph contains 6 5-cycles (ignoring orientationand dividing out the 5 choices of base point for each cycle).
Obtaining the number of 7-cycles in a random graph on 12 vertices requires computing the 7th power of a 12-by-12 matrix having entries in a 4096-dimensional algebra!
Note that within the Clifford algebra context, this requires only 6(123)=10,368 multiplications! A strong argument for a computer based on Clifford architecture.
Another Example
An algebraic probability space
Let A denote the algebra generated by the n by n Clifford adjacency matrices with involution * defined by a*=(a). Here, denotes the dual defined by ei=e[n]\i , and denotes the matrix transpose.
Let ‹‹u›› denote the sum of the real scalar coefficients in the canonical expansion of uCln
nil.
Define the norm of aA by
.)( 1
*2dedeaatra n
Remark
Allowing the entries of the matrix a to be in an arbitrary Clifford algebra with complex coefficients, the dual a is defined by linear extension of
(βei) (βei)=|β|2e[n]
applied to each entry of a.
The norm of a is given by the same expression as before.
Define
One finds φ(a*a) 0 for all a in A, and φ(1A) = 1; i.e., φ is a state and (A, φ) is an algebraic probability space.
Each adjacency matrix aA is a quantum random variable whose mth moment corresponds to the number of m-cycles in a graph on n vertices. Letting Xm denote the number of m-cycles in the graph associated with a,
An algebraic probability space
.)(
n
atra
.2
mm X
n
ma
Remarks:
1. The Clifford adjacency matrix can be formed for any finite graph (directed, undirected, having multiple edges, loops, etc.). Graphs containing multiple edges require labeling edges as well as vertices with Clifford elements.
2. The method can also be applied to edge-existence matrices associated with random graphs.
3. The method can be applied to stochastic matrices & Markov chains.
A notion of quantum decomposition
Let Un , Ln , and Dn denote, respectively, the algebras of n by n upper triangular matrices, lower triangular matrices, and diagonal matrices with entries in Cln
nil and involution * defined by a*=(a).
The state φ and the norm ||a|| are defined on each algebra as before. Hence, the three algebraic probability spaces (Un,φ), (Ln, φ), and (Dn,φ) are obtained. Now the Clifford adjacency matrix A for any directed graph on n vertices is the sum of three algebraic random variables…
Algebraic probability spaces
A = a+ + a- + ao (Un, φ)(Ln, φ)(Dn, φ).
Any Clifford matrix A can be decomposed in this manner. When A is the Clifford adjacency matrix of any finite graph,
φ(A*A) 0, and φ(Am) = φ((a+ + a- + ao)m)
coincides with the number of m-cycles in the graph.
Note:
Given the edge-probability Clifford adjacency matrix of a random graph on n vertices, φ(am) corresponds to the expected number of m-cycles in a random graph.
More Combinatorial Applications
The number of Hamiltonian cycles in a graph on n vertices is recovered from the Berezin integral of the trace of An. In the Clifford algebra context this has complexity O(n4). Hence, the NP-complete Hamiltonian-cycle problem is of polynomial time complexity in the Clifford algebra context.
Expected hitting times of specific states in Markov chains can be computed.
The number of Euler circuits in a finite graph can be obtained by labeling edges in place of vertices.
Applying these methods to partitions, Stirling numbers of the second kind, Bell numbers, and Bessel numbers are recovered.
A combinatorial approach to iterated stochastic integrals (in classical probability) is obtained in which the iterated stochastic integral of a process appears as the limit in mean of a sequence of Berezin integrals in an ascending chain of Clifford algebras.
Edge-disjoint cycle decompositions
Edge-disjoint cycle decompositions
is canonical projection.
is an evaluation. It can also be considered canonical projection.
Notation:
Edge-disjoint cycle decompositions
Edge-disjoint cycle decompositions
Let n be fixed and consider the abelian nilpotent-generated algebra Cln
nil. Define
Partitions and Counting Numbers
niln
jj ClA
n
][2
),(!
11 knSddA
k nk
Then for k between 1 and n inclusive,
and
nnA Bdde 1
Stirling numbers of the 2nd kind
nth Bell number
Partitions and Counting Numbers
Non-overlapping Partitions and Bessel Numbers
Bessel Numbers
Canonical raising operator on Cl3,3
Canonical lowering operator on Cl3,3
Operator Calculus and Appell Systems
Clifford Appell Systems
In the Clifford algebra context, the operator considered is the canonical lowering operator, . This operator has the property 2=0, so a sequence of related lowering operators is considered to obtain an Appell system with more than two nonzero elements. This sequence can be depicted graphically.
Clifford Appell Systems
Clifford Appell Systems
Clifford Appell Systems
Pictured here is the graph associated with the multiplicative random walk on the basis blades of the Clifford algebra Cl0,2 .
Idea: Use properties of random walks on hypercubes to establish limit theorems for random walks on Clifford algebras of arbitrary signature
Clifford algebras as directed hypercubes
Preprints, links, etc. can be found at my web page: www.siue.edu/~sstaple
Email: [email protected]