  We propose a formalism that captures the algebraic structure of many-body
two-level quantum systems, and directly motivates an efficient numerical
method. This formalism is based on the binary representation of the
enumeration-indices of the elements of the corresponding Lie algebra. The
action of arbitrarily large elements of that algebra reduces to a few bit-wise
exclusive-or operations. This formalism naturally produces sparse
representations of many-body density operators, the size of which we control
through a dynamic truncation method. We demonstrate how this formalism applies
to real-time evolution, dissipative Lindblad action, imaginary-time evolution,
and projective measurement processes. We find that this approach to calculating
quantum dynamics scales close to linearly with the number of non-zero
components in the density operator. We refer to this exclusive-or represented
quantum algebra as ORQA. As a proof of concept, we provide a numerical
demonstration of this formalism by simulating quantum annealing processes for
the maximum independent set problem for up to 22 two-level systems.
