We present a new primitive for quantum algorithms that implements a discrete Hermite transform efficiently, in time that is polylogarithmic in the dimension and the inverse of the allowable error. This transform, which maps basis states to states whose amplitudes are proportional to the Hermite functions, can be interpreted as the Gaussian analogue of the Fourier transform. Our algorithm is based on a method to exponentially fast-forward the evolution of the quantum harmonic oscillator, giving a simulation algorithm with nearly optimal circuit complexity for a fundamental Hamiltonian more than four decades after Feynman posed the simulation of quantum physics as an application of quantum computers.
We present a new primitive for quantum algorithms that implements a discrete Hermite transform efficiently, in time that is polylogarithmic in the dimension and the inverse of the allowable error. This transform, which maps basis states to states whose amplitudes are proportional to the Hermite functions, can be interpreted as the Gaussian analogue of the Fourier transform. Our algorithm is based on a method to exponentially fast-forward the evolution of the quantum harmonic oscillator, giving a simulation algorithm with nearly optimal circuit complexity for a fundamental Hamiltonian more than four decades after Feynman posed the simulation of quantum physics as an application of quantum computers.
We introduce a quantum algorithm to perform the Laplace transform on quantum computers. Already, the quantum Fourier transform (QFT) is the cornerstone of many quantum algorithms, but the Laplace transform or its discrete version has not seen any efficient implementation on quantum computers due to its dissipative nature and hence non-unitary dynamics. However, a recent work has shown an efficient implementation for certain cases on quantum computers using the Taylor series. Unlike previous work, our work provides a completely different algorithm for doing Laplace Transform using Quantum Eigenvalue Transformation and Lap-LCHS, very efficiently at points which form an arithmetic progression. Our algorithm can implement N×N discrete Laplace transform in gate complexity that grows as O((logN)3), ignoring the state preparation cost, where N=2n and n is the number of qubits, which is a superpolynomial speedup in number of gates over the best classical counterpart that has complexity O(N⋅logN) for the same cases. Also, the circuit width grows as O(logN). Quantum Laplace Transform (QLT) may enable new Quantum algorithms for cases like solving differential equations in the Laplace domain, developing an inverse Laplace transform algorithm on quantum computers, imaginary time evolution in the resolvent domain for calculating ground state energy, and spectral estimation of non-Hermitian matrices.
Gaussian states hold a fundamental place in quantum mechanics, quantum information, and quantum computing. Many subfields, including quantum simulation of continuousvariable 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 continuousvariable systems, the practical interest in digital, gate-based quantum computers demands discrete approximations of Gaussian distributions over a computational basis of size2n. 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 singlequbit 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 inO(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.
A quantum orthogonal polynomial transform (QOPT) is an algorithmic primitive that maps a superposition of standard basis states ∑kαk∣k⟩ coherently to a basis of normalized univariate polynomials orthogonal with respect to a probability measure μ(x). We provide new discrete and continuous QOPTs for polynomial families in the Askey scheme extending the results for the quantum Hermite transform (Jain et al., STOC'26). Our efficient QOPT algorithms require time O(polylog(N,1/ε)), where N is the grid size and ε is the error, for the discrete Charlier, Meixner and Krawtchouk transforms and for the integer-order Laguerre transform in the continuous setting. The efficient QOPTs are obtained by uncovering the links between Askey scheme polynomials and Gaussian quantum optical gates and developing a compilation framework for SU(2) and SU(1,1) optical gates on Cartesian grids extending the framework developed by Iyer et al. (arXiv:2602.15180). Further, we reduce the continuous Jacobi transform to the discrete Hahn transform and provide an O(Npolylog((N+α+β+1)/ε))-time Hahn transform, a quadratic speedup over the naive implementation. This is based on a more efficient compilation of the Clebsch--Gordan transform for coupling SU(2) representations with spins (j1,j2). Finally, we develop a new framework for Laguerre transforms for all orders ν>0 by fast-forwarding the corresponding radial oscillator using a 3-term chirp decomposition and an efficient algorithm for the Quantum Hankel Transform on a logarithmic grid.
The quantum Fourier transform and quantum wavelet transform have been cornerstones of quantum information processing. However, for non-stationary signals and anomaly detection, the Hilbert transform can be a more powerful tool, yet no prior work has provided efficient quantum implementations for the discrete Hilbert transform. This letter presents a novel construction for a quantum Hilbert transform in polylogarithmic size and logarithmic depth for a signal of length N, exponentially fewer operations than classical algorithms for the same mapping. We generalize this algorithm to create any d-dimensional Hilbert transform in depth O(dlogN). Simulations demonstrate effectiveness for tasks such as power systems control and image processing, with exact agreement with classical results.
Quantum Kravchuk Transform using su(2) fast-forwarding
We present a quantum algorithm for the Kravchuk transform that scales logarithmically in both the dimension and the inverse of the error parameter. The quantum Kravchuk transform maps computational basis states to states with amplitudes proportional to Kravchuk functions. We achieve this by combining two key techniques: the structural relationship between the Kravchuk transform and the Lie algebras su(2), and a recent fast-forwarding simulation method for su(2) operators in the oscillator representation. More precisely, we first establish the map from Kravchuk transform in computational basis to su(2) in Fock basis. Then built on this connection, we apply the fast-forwarding to achieve an efficient quantum Kravchuk transform.
The quantum Fourier transform (QFT) is a fundamental primitive in quantum computation and quantum information. In this work, we extend the framework for the QFT from finite groups to finite-dimensional semisimple algebras, and give efficient QFTs for the partition algebra Pn(d), Brauer algebra Bn(d), and walled Brauer algebra Br,s(d). These algebras play important roles in generalized Schur-Weyl duality, statistical physics and many-body systems, and have recently found several applications in quantum algorithms.
Unlike the group case, the Fourier transform over a semisimple algebra can be non-unitary. Nevertheless, we show that when the parameter d is sufficiently large, the Fourier transform is well approximated by a unitary operator. Furthermore, we show that for each of the algebras A from above, such an approximate Fourier transform can be implemented efficiently: we give a quantum algorithm with gate complexity poly(n,logd,log(1/ε)) for approximating the Fourier transform to error (d−1/2+ε)⋅poly(∣A∣). Along the way, we establish several properties of the Fourier basis of semisimple algebras that may be of independent interest.