Quantum state preparation without coherent arithmetic
Sam McArdle, András Gilyén, Mario Berta
TL;DR
This work presents a QSVT-based framework for preparing quantum states whose amplitudes follow a known function, without relying on amplitude oracles or handcrafted coherent arithmetic. By block-encoding a sine-based template and applying a degree-d polynomial via QSVT, the method constructs |Psi_f> from a low-cost encoding with only 3 ancilla qubits and resource usage governed by the discretized L2-filling fraction. It delivers concrete, favorable complexity for smooth functions that admit polynomial or Fourier approximations, and demonstrates Gaussian and Kaiser window state preparations with favorable ancilla counts compared to black-box approaches. The approach is versatile, extensible to priors, non-smooth functions, and multivariate cases, and has practical implications for phase estimation, quantum chemistry, and financial modeling.
Abstract
We introduce a versatile method for preparing a quantum state whose amplitudes are given by some known function. Unlike existing approaches, our method does not require handcrafted reversible arithmetic circuits, or quantum table reads, to encode the function values. Instead, we use a template quantum eigenvalue transformation circuit to convert a low cost block encoding of the sine function into the desired function. Our method uses only 4 ancilla qubits (3 if the approximating polynomial has definite parity), providing order-of-magnitude qubit count reductions compared to state-of-the-art approaches, while using a similar number of gates if the function can be well represented by a polynomial or Fourier approximation. Like black-box methods, the complexity of our approach depends on the 'L2-norm filling-fraction' of the function. We demonstrate the algorithmic utility of our method, including preparing Gaussian and Kaiser window states.