[Paper Review] Quantum structures from association schemes

Radhakrishnan Balu|arXiv (Cornell University)|Feb 22, 2019

Quantum Computing Algorithms and Architecture10 references4 citations

TL;DR

This paper constructs quantum Markov chains (QMCs) and interacting Fock spaces (IFS) from association schemes derived from finite groups, using Bose-Mesner algebras and intersection numbers. It introduces entangled QMCs and establishes a quantum central limit theorem for Grassmann schemes, showing convergence to product measures with explicit asymptotic parameters.

ABSTRACT

Starting from an association scheme induced by a finite group and the corresponding Bose-Mesner algebra we construct quantum Markov chains (QMC), their entangled versions, and interacting Fock spaces (IFS) using the quantum probabilistic approach. Our constructions are based on the intersection numbers and their duals Krien parameters of the schemes with examples focused on regular (distance-regular and distance-transitive) graphs.

Motivation & Objective

• To develop quantum Markov chains (QMCs) on the hypergroups dual to association schemes derived from finite groups.
• To extend QMCs to entangled versions that canonically embed classical Markov chains via creation, annihilation, and preservation operators (CAPs).
• To construct an interacting Fock space (IFS) based on the Bose-Mesner algebra of distance-regular graphs, using generalized Jacobi parameters.
• To establish asymptotic convergence to product measures via the quantum central limit theorem (QCLT) for Grassmann schemes.
• To enable quantum walk models on growing graphs with multi-mode, strata-based degrees of freedom for quantum information and simulation applications.

Proposed method

• Uses the Bose-Mesner algebra of an association scheme as a *-algebra closed under matrix multiplication, with basis matrices corresponding to adjacency relations.
• Defines creation, annihilation, and preservation operators (CAPs) on the algebra using intersection numbers $p^n_{j;1,n}$ and dual Krien parameters.
• Constructs the IFS as a pre-Hilbert space spanned by product vectors of mode states, with vacuum state $\Phi_0 = 1_{\mathcal{P}}$ and recursive state generation via $\Phi_{j,n} = a^{+}_{jn} \cdots a^{+}_{j1} \Phi_0$.
• Applies the quantum central limit theorem (QCLT) to show convergence of individual mode measures in the Grassmann scheme $J_q(n,d)$ to a product measure in the limit.
• Derives explicit asymptotic parameters: $p^{n-1}_{j;1,n} = (2-n)(v-n)$, $p^n_{j;1,n} = n(v-2)$, $p^n_{j;1,n} = n(v-2n)$, and $p^0_{1,1} = d(n-d)$ for $n \leq \min\{d, v-d\}$.
• Establishes the Jacobi relation $B_j P_n = P_{n+1} B_j P_n + P_n B_j P_n + P_{n-1} B_j P_n$ to define the recursive structure of the IFS.

Experimental results

Research questions

• RQ1 How can quantum Markov chains be constructed from the hypergroups dual to association schemes induced by finite groups?
• RQ2 What is the role of intersection numbers and Krien parameters in defining creation, annihilation, and preservation operators in the quantum probabilistic framework?
• RQ3 How can entangled versions of QMCs be systematically constructed to embed classical Markov chains while preserving quantum correlations?
• RQ4 In what way does the interacting Fock space (IFS) generalize the Fock space of a quantum harmonic oscillator for multi-mode quantum walks on graphs?
• RQ5 What asymptotic behavior emerges in the limit of large graphs, particularly for Grassmann schemes, and how does the quantum central limit theorem apply?

Key findings

• The construction of QMCs on the dual hypergroups of association schemes yields a quantum probabilistic framework where classical Markov chains emerge as restrictions to commutative subalgebras.
• Entangled QMCs are realized through canonical embeddings of classical chains into a non-commutative algebraic structure using CAP operators derived from intersection numbers.
• The interacting Fock space (IFS) is built recursively from the Bose-Mesner algebra, with state vectors generated via creation operators acting on a vacuum state $\Phi_0 = 1_{\mathcal{P}}$.
• For the Grassmann scheme $J_q(n,d)$, the asymptotic parameters of the IFS are explicitly derived: $p^{n-1}_{j;1,n} = (2-n)(v-n)$, $p^n_{j;1,n} = n(v-2)$, and $p^n_{j;1,n} = n(v-2n)$ for $n \leq \min\{d, v-d\}$.
• The quantum central limit theorem (QCLT) ensures convergence of the product measure of individual modes to a limiting state, with $p^0_{1,1} = d(n-d)$ as the ground state parameter.
• The framework supports the construction of non-classical quantum states such as Schrödinger cat states and squeezed states on the IFS, analogous to quantum harmonic oscillator Fock states.