Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2601.10876 · IEEE Signal Processing Letters

Efficient Quantum Circuits for the Hilbert Transform

Henry Zhang, Joseph Li

quant-pheess.SPSubmitted Jan 15, 2026• Announced Jan 19, 2026• 2 citations
arXivHTMLPDF

Abstract

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 $O(d\log N)$. Simulations demonstrate effectiveness for tasks such as power systems control and image processing, with exact agreement with classical results.

DOI
10.1109/LSP.2026.3654893
Comments
6 pages, 5 figures, accepted to IEEE Signal Processing Letters

Related papers

Efficient quantum algorithm for all quantum wavelet transforms

Mohsen Bagherimehrab, Alán Aspuru‐Guzik

Abstract Wavelet transforms are widely used in various fields of science and engineering as a mathematical tool with features that reveal information ignored by the Fourier transform. Unlike the Fourier transform, which is unique, a wavelet transform is specified by a sequence of numbers associated with the type of wavelet used and an order parameter specifying the length of the sequence. While the quantum Fourier transform, a quantum analog of the classical Fourier transform, has been pivotal in quantum computing, prior works on quantum wavelet transforms (QWTs) were limited to the second and fourth order of a particular wavelet, the Daubechies wavelet. Here we develop a simple yet efficient quantum algorithm for executing any wavelet transform on a quantum computer. Our approach is to decompose the kernel matrix of a wavelet transform as a linear combination of unitaries (LCU) that are compilable by easy-to-implement modular quantum arithmetic operations and use the LCU technique to construct a probabilistic procedure to implement a QWT with a known success probability. We then use properties of wavelets to make this approach deterministic by a few executions of the amplitude amplification strategy. We extend our approach to a multilevel wavelet transform and a generalized version, the packet wavelet transform, establishing computational complexities in terms of three parameters: the wavelet order M , the dimension N of the transformation matrix, and the transformation level d . We show the cost is logarithmic in N , linear in d and superlinear in M . Moreover, we show the cost is independent of M for practical applications. Our proposed QWTs could be used in quantum computing algorithms in a similar manner to their well-established counterpart, the quantum Fourier transform.

arXiv:2309.09350•Quantum Science and Technology•Apr 11, 2024•14 citations

Fast Laplace transforms on quantum computers

Julien Zylberman

While many classical algorithms rely on Laplace transforms, it has remained an open question whether these operations could be implemented efficiently on quantum computers. In this work, we introduce the Quantum Laplace Transform (QLT), which enables the implementation of $N\times N$ discrete Laplace transforms on quantum states encoded in $\lceil \log_2(N)\rceil$-qubits. In many cases, the associated quantum circuits have a depth that scales with $N$ as $O(\log(\log(N)))$ and a size that scales as $O(\log(N))$, requiring exponentially fewer operations and double-exponentially less computational time than their classical counterparts. These efficient scalings open the possibility of developing a new class of quantum algorithms based on Laplace transforms, with potential applications in physics, engineering, chemistry, machine learning, and finance.

arXiv:2412.05173•Dec 6, 2024•0 citations

On the need for large Quantum depth

Nai-Hui Chia, Kai-Min Chung, Ching–Yi Lai

Near-term quantum computers are likely to have small depths due to short coherence time and noisy gates. A natural approach to leverage these quantum computers is interleaving them with classical computers. Understanding the capabilities and limits of this hybrid approach is an essential topic in quantum computation. Most notably, the quantum Fourier transform can be implemented by a hybrid of logarithmic-depth quantum circuits and a classical polynomial-time algorithm. Therefore, it seems possible that quantum polylogarithmic depth is as powerful as quantum polynomial depth in the presence of classical computation.

ACM Symposium on Theory of Computing (STOC)•Jun 7, 2020•20 citations

Quantum Fourier transform toolbox

Carli Bruinsma, Pietro M. Posta, Joppe Stokvis +2 more

Quantum Fourier transforms (QFTs) are essential primitives in quantum algorithms. While abelian groups admit efficient QFT circuits, with circuit size polynomial in the logarithm of the group order, efficient constructions are known for relatively few non-abelian families. We develop two new approaches to QFT circuit construction, based on Mackey theory and Clifford theory, respectively, and use them to show exponential improvement in circuit cost for specific group families. Using the Mackey-theoretic approach, we obtain explicit quantum circuits for the QFT over $\mathrm{GL}_2(F_q)$ that scale polynomially in $\log q$, rather than polynomially in $q$. Using the Clifford-theoretic approach, we obtain QFT circuits for wreath products $F\wr S_n$, whose cost depends on the cost of a QFT over $F$ and the size of its representation registers. This removes the restriction $|F|=\operatorname{poly}(n)$ required by previous generic constructions and can yield exponential improvements when $F$ itself has an efficient QFT. Together, these methods provide new systematic tools to construct QFTs for broad classes of finite groups.

quant-pharXiv:2608.28573•Aug 31, 2026•0 citations

Decoherence-Optimized Circuits for Multidimensional and Multilevel-Decomposable Quantum Wavelet Transform

Naveed Mahmud, Andrew MacGillivray, Manu Chaudhary +1 more

Algorithms such as quantum wavelet transform (QWT) can be utilized on quantum processors to gain significant speedup compared to their classical counterparts. Domains such as high-energy-physics and remote-sensing hyperspectral imagery require high compute capabilities and can benefit from applying QWT for dimension reduction of multidimensional data. However, quantum circuits for QWT include permutations that contribute to large overall circuit depth. Deep circuits pose a critical challenge for state-of-the-art quantum computers because of quantum decoherence. In this work, we propose QWT circuits, which are optimized in terms of circuit depth, to account for the effects of decoherence, and resulting in high fidelity and efficient implementation on quantum processors. We present the circuits in generalized forms and show that they can be used for multilevel decomposable, multidimensional wavelet operations. Experimental evaluation of the proposed circuits is performed through simulation using MATLAB and IBM-Q qasm, and implemented on a real 15-qubit quantum processor from IBM.

IEEE Internet Computing•Dec 9, 2021•8 citations

A Polylogarithmic-Time Quantum Algorithm for the Laplace Transform

Akash Kumar Singh, Ashish Kumar Patra, Anurag K. S. V. +3 more

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 $N \times N$ discrete Laplace transform in gate complexity that grows as $O((log\,N)^3)$, ignoring the state preparation cost, where $N=2^n$ and $n$ is the number of qubits, which is a superpolynomial speedup in number of gates over the best classical counterpart that has complexity $O(N\cdot log\,N)$ for the same cases. Also, the circuit width grows as $O(log\,N)$. 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.

quant-pharXiv:2512.17980•Dec 22, 2025•0 citations
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 NNN, exponentially fewer operations than classical algorithms for the same mapping. We generalize this algorithm to create any ddd-dimensional Hilbert transform in depth O(dlog⁡N)O(d\log N)O(dlogN). Simulations demonstrate effectiveness for tasks such as power systems control and image processing, with exact agreement with classical results.
Abstract Wavelet transforms are widely used in various fields of science and engineering as a mathematical tool with features that reveal information ignored by the Fourier transform. Unlike the Fourier transform, which is unique, a wavelet transform is specified by a sequence of numbers associated with the type of wavelet used and an order parameter specifying the length of the sequence. While the quantum Fourier transform, a quantum analog of the classical Fourier transform, has been pivotal in quantum computing, prior works on quantum wavelet transforms (QWTs) were limited to the second and fourth order of a particular wavelet, the Daubechies wavelet. Here we develop a simple yet efficient quantum algorithm for executing any wavelet transform on a quantum computer. Our approach is to decompose the kernel matrix of a wavelet transform as a linear combination of unitaries (LCU) that are compilable by easy-to-implement modular quantum arithmetic operations and use the LCU technique to construct a probabilistic procedure to implement a QWT with a known success probability. We then use properties of wavelets to make this approach deterministic by a few executions of the amplitude amplification strategy. We extend our approach to a multilevel wavelet transform and a generalized version, the packet wavelet transform, establishing computational complexities in terms of three parameters: the wavelet order M , the dimension N of the transformation matrix, and the transformation level d . We show the cost is logarithmic in N , linear in d and superlinear in M . Moreover, we show the cost is independent of M for practical applications. Our proposed QWTs could be used in quantum computing algorithms in a similar manner to their well-established counterpart, the quantum Fourier transform.
While many classical algorithms rely on Laplace transforms, it has remained an open question whether these operations could be implemented efficiently on quantum computers. In this work, we introduce the Quantum Laplace Transform (QLT), which enables the implementation of N×NN\times NN×N discrete Laplace transforms on quantum states encoded in ⌈log⁡2(N)⌉\lceil \log_2(N)\rceil⌈log2​(N)⌉-qubits. In many cases, the associated quantum circuits have a depth that scales with NNN as O(log⁡(log⁡(N)))O(\log(\log(N)))O(log(log(N))) and a size that scales as O(log⁡(N))O(\log(N))O(log(N)), requiring exponentially fewer operations and double-exponentially less computational time than their classical counterparts. These efficient scalings open the possibility of developing a new class of quantum algorithms based on Laplace transforms, with potential applications in physics, engineering, chemistry, machine learning, and finance.
Near-term quantum computers are likely to have small depths due to short coherence time and noisy gates. A natural approach to leverage these quantum computers is interleaving them with classical computers. Understanding the capabilities and limits of this hybrid approach is an essential topic in quantum computation. Most notably, the quantum Fourier transform can be implemented by a hybrid of logarithmic-depth quantum circuits and a classical polynomial-time algorithm. Therefore, it seems possible that quantum polylogarithmic depth is as powerful as quantum polynomial depth in the presence of classical computation.
Quantum Fourier transforms (QFTs) are essential primitives in quantum algorithms. While abelian groups admit efficient QFT circuits, with circuit size polynomial in the logarithm of the group order, efficient constructions are known for relatively few non-abelian families. We develop two new approaches to QFT circuit construction, based on Mackey theory and Clifford theory, respectively, and use them to show exponential improvement in circuit cost for specific group families. Using the Mackey-theoretic approach, we obtain explicit quantum circuits for the QFT over GL2(Fq)\mathrm{GL}_2(F_q)GL2​(Fq​) that scale polynomially in log⁡q\log qlogq, rather than polynomially in qqq. Using the Clifford-theoretic approach, we obtain QFT circuits for wreath products F≀SnF\wr S_nF≀Sn​, whose cost depends on the cost of a QFT over FFF and the size of its representation registers. This removes the restriction ∣F∣=poly⁡(n)|F|=\operatorname{poly}(n)∣F∣=poly(n) required by previous generic constructions and can yield exponential improvements when FFF itself has an efficient QFT. Together, these methods provide new systematic tools to construct QFTs for broad classes of finite groups.
Algorithms such as quantum wavelet transform (QWT) can be utilized on quantum processors to gain significant speedup compared to their classical counterparts. Domains such as high-energy-physics and remote-sensing hyperspectral imagery require high compute capabilities and can benefit from applying QWT for dimension reduction of multidimensional data. However, quantum circuits for QWT include permutations that contribute to large overall circuit depth. Deep circuits pose a critical challenge for state-of-the-art quantum computers because of quantum decoherence. In this work, we propose QWT circuits, which are optimized in terms of circuit depth, to account for the effects of decoherence, and resulting in high fidelity and efficient implementation on quantum processors. We present the circuits in generalized forms and show that they can be used for multilevel decomposable, multidimensional wavelet operations. Experimental evaluation of the proposed circuits is performed through simulation using MATLAB and IBM-Q qasm, and implemented on a real 15-qubit quantum processor from IBM.
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×NN \times NN×N discrete Laplace transform in gate complexity that grows as O((log N)3)O((log\,N)^3)O((logN)3), ignoring the state preparation cost, where N=2nN=2^nN=2n and nnn is the number of qubits, which is a superpolynomial speedup in number of gates over the best classical counterpart that has complexity O(N⋅log N)O(N\cdot log\,N)O(N⋅logN) for the same cases. Also, the circuit width grows as O(log N)O(log\,N)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.