A new approximate Quantum State Preparation (QSP) method is introduced,
called the Walsh Series Loader (WSL). The WSL approximates quantum states
defined by real-valued functions of single real variables with a depth
independent of the number n of qubits. Two approaches are presented: the
first one approximates the target quantum state by a Walsh Series truncated at
order O(1/ϵ​), where ϵ is the precision of the
approximation in terms of infidelity. The circuit depth is also
O(1/ϵ​), the size is O(n+1/ϵ​) and only one
ancilla qubit is needed. The second method represents accurately quantum states
with sparse Walsh series. The WSL loads s-sparse Walsh Series into n-qubits
with a depth doubly-sparse in s and k, the maximum number of bits with
value 1 in the binary decomposition of the Walsh function indices. The
associated quantum circuit approximates the sparse Walsh Series up to an error
ϵ with a depth O(sk), a size O(n+sk) and one ancilla qubit. In
both cases, the protocol is a Repeat-Until-Success (RUS) procedure with a
probability of success P=Θ(ϵ), giving an averaged total time of
O(1/ϵ3/2) for the WSL (resp. O(sk/ϵ) for the sparse WSL).
Amplitude amplification can be used to reduce by a factor
O(1/ϵ​) the total time dependency with ϵ but increases
the size and depth of the associated quantum circuits, making them linearly
dependent on n. These protocols give overall efficient algorithms with no
exponential scaling in any parameter. They can be generalized to any
complex-valued, multi-variate, almost-everywhere-differentiable function. The
Repeat-Until-Success Walsh Series Loader is so far the only method which
prepares a quantum state with a circuit depth and an averaged total time
independent of the number of qubits