  In this paper we give a quantum statistical interpretation for the bracket
polynomial state sum <K> and for the Jones polynomial. We use this quantum
mechanical interpretation to give a new quantum algorithm for computing the
Jones polynomial. This algorithm is useful for its conceptual simplicity, and
it applies to all values of the polynomial variable that lie on the unit circle
in the complex plane. Letting C(K) denote the Hilbert space for this model,
there is a natural unitary transformation U from C(K) to itself such that <K> =
<F|U|F> where |F> is a sum over basis states for C(K). The quantum algorithm
arises directly from this formula via the Hadamard Test. We then show that the
framework for our quantum model for the bracket polynomial is a natural setting
for Khovanov homology. The Hilbert space C(K) of our model has basis in
one-to-one correspondence with the enhanced states of the bracket state
summmation and is isomorphic with the chain complex for Khovanov homology with
coefficients in the complex numbers. We show that for the Khovanov boundary
operator d defined on C(K) we have the relationship dU + Ud = 0. Consequently,
the unitary operator U acts on the Khovanov homology, and we therefore obtain a
direct relationship between Khovanov homology and this quantum algorithm for
the Jones polynomial. The formula for the Jones polynomial as a graded Euler
characteristic is now expressed in terms of the eigenvalues of U and the Euler
characteristics of the eigenspaces of U in the homology. The quantum algorithm
given here is inefficient, and so it remains an open problem to determine
better quantum algorithms that involve both the Jones polynomial and the
Khovanov homology.
