Not all functions on a quantum memory register preserve the
superposition of the state vector. For example, measurement destroys
the superposition in the register. Operations that collapse the state
vector are called *measurements*, and any complex linear
transformation of the state vector is called an
*operator*. We can represent any operator on an *N*-bit
quantum memory register in
as a matrix

Quantum mechanics imposes conditions on which linear transformations
are legal operators. In particular, the operation must be reversible,
and it must preserve the length of the state vector
[9]. If we impose the condition that the sum of
the kinetic and potential energy (called the Hamiltonian) of our
quantum memory register is constant, then all legal operators have
*unitary* matrix representations. A matrix *T* is unitary if the
transpose of its complex conjugate is *T*^{-1}
[9]. Systems with time-dependent Hamiltonians
are not required to perform either Grover's or Shor's algorithm, and
are not within the scope of this thesis.