  Effective quantum computation relies upon making good use of the exponential
information capacity of a quantum machine. A large barrier to designing quantum
algorithms for execution on real quantum machines is that, in general, it is
intractably difficult to construct an arbitrary quantum state to high
precision. Many quantum algorithms rely instead upon initializing the machine
in a simple state, and evolving the state through an efficient (i.e. at most
polynomial-depth) quantum algorithm. In this work, we show that there exist
families of quantum states that can be prepared to high precision with circuits
of linear size and depth. We focus on real-valued, smooth, differentiable
functions with bounded derivatives on a domain of interest, exemplified by
commonly used probability distributions. We further develop an algorithm that
requires only linear classical computation time to generate accurate
linear-depth circuits to prepare these states, and apply this to well-known and
heavily-utilized functions including Gaussian and lognormal distributions. Our
procedure rests upon the quantum state representation tool known as the matrix
product state (MPS). By efficiently and scalably encoding an explicit amplitude
function into an MPS, a high fidelity, linear-depth circuit can directly be
generated. These results enable the execution of many quantum algorithms that,
aside from initialization, are otherwise depth-efficient.
