Efficient Gaussian State Preparation in Quantum Circuits

Gaussian states hold a fundamental place in quantum mechanics, quantum information, and quantum computing. Many subfields, including quantum simulation of continuous-variable systems, quantum chemistry, and quantum machine learning, rely on the ability to accurately and efficiently prepare states that reflect a Gaussian profile in their probability amplitudes. Although Gaussian states are natural in continuous-variable systems, the practical interest in digital, gate-based quantum computers demands discrete approximations of Gaussian distributions over a computational basis of size \(2^n\). Because of the exponential scaling of naive amplitude-encoding approaches and the cost of certain block-encoding or Hamiltonian simulation techniques, a resource-efficient preparation of approximate Gaussian states is required. In this work, we propose and analyze a circuit-based approach that starts with single-qubit rotations to form an exponential amplitude profile and then applies the quantum Fourier transform to map those amplitudes into an approximate Gaussian distribution. We demonstrate that this procedure achieves high fidelity with the target Gaussian state while allowing optional pruning of small controlled-phase angles in the quantum Fourier transform, thus reducing gate complexity to near-linear in \(\mathcal{O}(n)\). We conclude that the proposed technique is a promising route to make Gaussian states accessible on noisy quantum hardware and to pave the way for scalable implementations on future devices. The implementation of this algorithm is available at the Classiq library: https://github.com/classiq/classiq-library.