A New Graph Polynomial and Generalized Tutte-Grothendieck Invariant from Quantum Circuits

Chaowen Guan, Kenneth W. Regan

Abstract: A new polynomial \(Q_G(x)\) associated to graphs G is defined and studied. The main theorems represent \(Q_G(x)\) as a quasi-specialization of the rank-generating polynomial \(S_G(x,y)\) of Oxley and Whittle, J Comb Theory Ser B 59:210–244, 1993, [10] and show that \(Q_G\) is likewise a generalized Tutte–Grothendieck invariant. The value \(Q_G(1)\) gives the amplitude of acceptance for a class of quantum circuits with associated graphs G. This class, called stabilizer circuits or Clifford circuits, has long been known to have deterministic polynomial time simulation, so \(Q_G(1)\) is polynomial time computable, given G as input. The specialization has \(y = -\sqrt{2}i\), which (along with its complex conjugate) is the only choice that invalidates formulas in a theorem by Noble, Comb. Probab. Comput 15:449–461, 2006, [9] classifying hard and easy real points of \(S_G\), so the complexity of other points \(Q_G(x)\) is open. We reduce the base cases for \(S_G\) by adjoining multiple kinds of isolated nodes and draw possible further implications of the connections between matroid theory and quantum computing developed by this work.