Efficient quantum state preparation with Walsh series

An approximate quantum state preparation method is introduced, called the Walsh series loader (WSL). The WSL approximates quantum states defined by real-value 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 O(1/root epsilon), where epsilon is the precision of the approximation in terms of infidelity. The circuit depth is also O(1/root epsilon), the size is O(n + 1/root epsilon), and only one ancilla qubit is needed. The second method accurately represents 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 epsilon with a depth O(sk), a size O(n + sk), and one ancilla qubit. In both cases, the protocol is a repeat-until-success procedure with a probability of success P = Theta(epsilon), giving an averaged total time of O(1/epsilon(3/2)) for the WSL and O(sk/root epsilon) for the sparse WSL. Amplitude amplification can be used to reach a probability of success P = Theta(1), modifying the quantum circuit size to (O) over tilde((n + 1/root epsilon)/root epsilon) and (O) over tilde((n + sk)/root epsilon) and the depth to O([log(n)(3) + 1/root epsilon]root epsilon) and O([log(n)(3) + sk]/root epsilon), respectively. Amplitude amplification reduces by a factor O(1/root epsilon) the total time dependence on epsilon 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-value, multivariate, almost-everywhere-differentiable function. The repeat-until-success Walsh series loader is so far the only method that prepares a quantum state with a circuit depth and an averaged total time independent of the number of qubits.