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.
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