Other derivation of (4.68) which could be useful in quantum computing is as follows.

Let $H^p$ the Hilbert space with orthonormal basis $\left\{ |\omega_p> \right\} $. We define a unitary transformation

$$ U : H^p \rightarrow H^p$$

by the formula·

$$ U = (-1)^F b {e}^{-\beta\,H} $$

with

$$ U |\omega_p>= (-1)^p {e}^{-\beta\,E_p} b |\omega_p> $$

Let a quantum state given by

$$|\Psi>= \sum _p|\omega_p>.$$

Will all these ingredients we make the following computation.

$$Tr((-1)^F b {e}^{-\beta\,H} ) = Tr(U)= < \Psi |U|\Psi>= \sum _q\sum _p<\omega_q|U|\omega_p>,$$

$$Tr((-1)^F b {e}^{-\beta\,H} ) = Tr(U)= < \Psi |U|\Psi>=\sum _q\sum _p<\omega_q|(-1)^p {e}^{-\beta\,E_p} b |\omega_p>,$$

$$Tr((-1)^F b {e}^{-\beta\,H} ) = Tr(U)= < \Psi |U|\Psi>=\sum _q\sum _p(-1)^p {e}^{-\beta\,E_p} <\omega_q|b |\omega_p>,$$

$$Tr((-1)^F b {e}^{-\beta\,H} ) = Tr(U)= < \Psi |U|\Psi>=\sum _q\sum _p(-1)^p {e}^{-\beta\,E_p} <\omega_p|b |\omega_p>\delta_{{q,p}},$$

$$Tr((-1)^F b {e}^{-\beta\,H} ) = Tr(U)= < \Psi |U|\Psi>=\sum _p(-1)^p {e}^{-\beta\,E_p} <\omega_p|b |\omega_p>,$$

$$Tr((-1)^F b {e}^{-\beta\,H} ) = Tr(U)= < \Psi |U|\Psi>= \sum _p(-1)^p \chi_{{p}} \left( b \right) = \chi \left( M_{{b}} \right) .$$

Reference : Louis Kauffman, arXiv:1001.0354v3 [math.GT] 31 Jan 2010 Topological Quantum Information, Khovanov Homology and the Jones Polynomial (http://lanl.arxiv.org/pdf/1001.0354)