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Canadian Journal of Physics

Combinatorial Solution of the Free Fermion Model

F.W. Wiegel

Published Jun 15, 1975• 9 citations
Publisher

Abstract

It is shown that the free fermion model can be solved by means of a combinatorial method which is tantamount to counting free random walks.
DOI
10.1139/p75-145

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We give a self-contained exposition of the combinatorial solution of quantum mechanical systems of coupled spins on a one-dimensional lattice. Using Trotter formula, we write the partition function as a generating function of a spanning subgraph of a two-dimensional lattice and solve the combinatorial problem by the method of Pfaffians provided the weights satisfy the so-called free fermion condition. The free energy and the ground state energy as a function of the inverse temperature, couplings J and magnetic fields h, for the XY model in a transverse field with period p=1 and 2, is then obtained.
Search algorithms based on quantum walks have emerged as a promising approach to solve computational problems across various domains, including combinatorial optimization, and cryptography. Stating a generic search problem in terms of a (quantum) search over a graph makes the efficiency of the algorithmic method depend on the structure of the graph itself. In this work, we propose a complete implementation of a quantum walk search on Johnson graphs, speeding up the solution of the subset-sum problem. We provide a detailed design of each sub-circuit, quantifying their cost in terms of gate number, depth, and width. We compare our solution against a Grover quantum search, showing a reduction of the T-count and T-depth for practically solvable problems. The proposed design provides a building block for the construction of efficient quantum search algorithms that can be modelled on Johnson graphs, filling the gap with the existing theoretical complexity analyses.
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