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:2502.08962 · npj Quantum Information

Quantum circuit for non-unitary linear transformation of basis sets

Guorui Zhu, Joel Bierman, Jianfeng Lu, Yingzhou Li

Published Nov 25, 2025• 1 citation
arXivHTMLPDF

Abstract

This paper presents a novel approach for implementing non-unitary basis transformations on quantum computational platforms, extending beyond conventional unitary methods. By integrating Singular Value Decomposition (SVD), the method achieves operational depth O ( n ) with about n ancilla qubits, enhancing the analysis of fermionic systems. The non-unitary transformation enables mapping a wavefunction between different bases, allowing the calculation of overlaps of states residing in distinct Hilbert subspaces. This capability supports the use of state-specific ansatzes to compute multiple energy eigenstates under orbital-optimized settings, potentially improving accuracy in variational quantum eigensolver (VQE) and related frameworks. Overall, the method expands the range of accessible quantum states and phenomena, offering new opportunities for applying quantum computing in physics and chemistry.

DOI
10.1038/s41534-025-01145-3

Related papers

Block Encoding of Non-Unitary Matrices Into Quantum Framework

Varun Puram, K. M. George, Johnson P. Thomas

Block encoding of non-unitary matrices in a quantum framework is a transformative technique for broadening the capabilities of quantum algorithms, allowing them to address a wider range of computational challenges. This paper introduces a novel embedding method to construct unitary matrices from non-unitary matrices, enabling their efficient encoding into quantum circuits. Utilizing the Gram-Schmidt process, the proposed approach embeds an$n \times n$non-unitary matrix$A$into a$2 n \times 2 n$unitary matrix$U_{A}$. When Compared to conventional methods such as Singular Value Decomposition (SVD), our approach achieves a 20 % faster execution time while maintaining high accuracy. Block encoding method has the potential to significantly advance quantum computing applicability by building unitary matrices from non-unitary ones in areas like optimization, quantum machine learning, and large-scale quantum simulations.

IEEE International Conference on Quantum Control, Computing and Learning (qCCL)•Jun 25, 2025•0 citations

Non-unitary matrices embedding in quantum framework

Varun Puram

In quantum computing, every operation must be unitary to ensure reversibility and preserve probability amplitudes, which is a fundamental requirement for any quantum process. However, many real-world computations in areas such as machine learning, signal processing, and dynamical modeling inherently involve non-unitary matrices. Since these cannot be directly implemented on quantum hardware, a practical workaround is required. One promising solution is block-encoding, a framework that embeds a non-unitary matrix A in C^{n x n} into a larger unitary matrix U_A in C^{2n x 2n}, enabling the simulation of non-unitary actions within quantum circuits through ancilla preparation and post-selection. This thesis presents a novel and efficient method for block-encoding based on Gram-Schmidt orthogonalization. The algorithm constructs a unitary U_A with A embedded as the top-left block and remaining components filled to ensure global unitarity. Compared to traditional Singular Value Decomposition (SVD)-based approaches, the proposed method achieves at least 20% improvement in execution time, without compromising output fidelity or numerical stability. A corresponding quantum circuit is developed to extract A|psi> (that is, the transformed version of the input quantum state |psi> after applying the linear operator A) using post-selection on ancilla qubits, with the success probability determined by

•Dec 1, 2025•0 citations

Improving the Accuracy of Variational Quantum Eigensolvers with Fewer Qubits Using Orbital Optimization

Joel Bierman, Yingzhou Li, Jianfeng Lu

Near-term quantum computers will be limited in the number of qubits on which they can process information as well as the depth of the circuits that they can coherently carry out. To date, experimental demonstrations of algorithms such as the Variational Quantum Eigensolver (VQE) have been limited to small molecules using minimal basis sets for this reason. In this work we propose incorporating an orbital optimization scheme into quantum eigensolvers wherein a parametrized partial unitary transformation is applied to the basis functions set in order to reduce the number of qubits required for a given problem. The optimal transformation is found by minimizing the ground state energy with respect to this partial unitary matrix. Through numerical simulations of small molecules up to 16 spin orbitals, we demonstrate that this method has the ability to greatly extend the capabilities of near-term quantum computers with regard to the electronic structure problem. We find that VQE paired with orbital optimization consistently achieves lower ground state energies than traditional VQE when using the same number of qubits and even frequently achieves lower ground state energies than VQE methods using more qubits.

arXiv:2208.14431•Journal of Chemical Theory and Computation•Jan 25, 2023•24 citations

Efficient Application of the Factorized form of the Unitary Coupled-Cluster Ansatz for the Variational Quantum Eigensolver Algorithm by Using Linear Combination of Unitaries

Luogen Xu, James K. Freericks

The variational quantum eigensolver is one of the most promising algorithms for near-term quantum computers. It has the potential to solve quantum chemistry problems involving strongly correlated electrons with relatively low-depth circuits, which are otherwise difficult to solve on classical computers. The variational eigenstate is constructed from a number of factorized unitary coupled-cluster terms applied onto an initial (single-reference) state. Current algorithms for applying one of these operators to a quantum state require a number of operations that scale exponentially with the rank of the operator. We exploit a hidden SU(2) symmetry to allow us to employ the linear combination of unitaries approach, Our Prepare subroutine uses n+2 ancilla qubits for a rank-n operator. Our Select(U^) scheme uses O(n)Cnot gates. This results in a full algorithm that scales like the cube of the rank of the operator n3, a significant reduction in complexity for rank five or higher operators. This approach, when combined with other algorithms for lower-rank operators (when compared to the standard implementation), will make the factorized form of the unitary coupled-cluster approach much more efficient to implement on all types of quantum computers.

arXiv:2302.08679•Symmetry•Jul 16, 2023•3 citations

A circuit-generated quantum subspace algorithm for the variational quantum eigensolver

Mark R. Hirsbrunner, J. Wayne Mullinax, Yizhi Shen +4 more

Recent research has shown that wavefunction evolution in real and imaginary time can generate quantum subspaces with significant utility for obtaining accurate ground state energies. Inspired by these methods, we propose combining quantum subspace techniques with the variational quantum eigensolver (VQE). In our approach, the parameterized quantum circuit is divided into a series of smaller subcircuits. The sequential application of these subcircuits to an initial state generates a set of wavefunctions that we use as a quantum subspace to obtain high-accuracy groundstate energies. We call this technique the circuit subspace variational quantum eigensolver (CSVQE) algorithm. By benchmarking CSVQE on a range of quantum chemistry problems, we show that it can achieve significant error reduction in the best case compared to conventional VQE, particularly for poorly optimized circuits, greatly improving convergence rates. Furthermore, we demonstrate that when applied to circuits trapped at local minima, CSVQE can produce energies close to the global minimum of the energy landscape, making it a potentially powerful tool for diagnosing local minima.

arXiv:2404.06534•The Journal of Chemical Physics•Oct 22, 2024•3 citations

Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics

András Gilyén, Yuan Su, Guang Hao Low +1 more

An n-qubit quantum circuit performs a unitary operation on an exponentially large, 2n-dimensional, Hilbert space, which is a major source of quantum speed-ups. We develop a new “Quantum singular value transformation” algorithm that can directly harness the advantages of exponential dimensionality by applying polynomial transformations to the singular values of a block of a unitary operator. The transformations are realized by quantum circuits with a very simple structure - typically using only a constant number of ancilla qubits - leading to optimal algorithms with appealing constant factors. We show that our framework allows describing many quantum algorithms on a high level, and enables remarkably concise proofs for many prominent quantum algorithms, ranging from optimal Hamiltonian simulation to various quantum machine learning applications. We also devise a new singular vector transformation algorithm, describe how to exponentially improve the complexity of implementing fractional queries to unitaries with a gapped spectrum, and show how to efficiently implement principal component regression. Finally, we also prove a quantum lower bound on spectral transformations.

arXiv:1806.01838•ACM Symposium on Theory of Computing (STOC)•Jun 20, 2019•645 citations
This paper presents a novel approach for implementing non-unitary basis transformations on quantum computational platforms, extending beyond conventional unitary methods. By integrating Singular Value Decomposition (SVD), the method achieves operational depth O ( n ) with about n ancilla qubits, enhancing the analysis of fermionic systems. The non-unitary transformation enables mapping a wavefunction between different bases, allowing the calculation of overlaps of states residing in distinct Hilbert subspaces. This capability supports the use of state-specific ansatzes to compute multiple energy eigenstates under orbital-optimized settings, potentially improving accuracy in variational quantum eigensolver (VQE) and related frameworks. Overall, the method expands the range of accessible quantum states and phenomena, offering new opportunities for applying quantum computing in physics and chemistry.
Block encoding of non-unitary matrices in a quantum framework is a transformative technique for broadening the capabilities of quantum algorithms, allowing them to address a wider range of computational challenges. This paper introduces a novel embedding method to construct unitary matrices from non-unitary matrices, enabling their efficient encoding into quantum circuits. Utilizing the Gram-Schmidt process, the proposed approach embeds ann×nn \times nn×nnon-unitary matrixAAAinto a2n×2n2 n \times 2 n2n×2nunitary matrixUAU_{A}UA​. When Compared to conventional methods such as Singular Value Decomposition (SVD), our approach achieves a 20 % faster execution time while maintaining high accuracy. Block encoding method has the potential to significantly advance quantum computing applicability by building unitary matrices from non-unitary ones in areas like optimization, quantum machine learning, and large-scale quantum simulations.
In quantum computing, every operation must be unitary to ensure reversibility and preserve probability amplitudes, which is a fundamental requirement for any quantum process. However, many real-world computations in areas such as machine learning, signal processing, and dynamical modeling inherently involve non-unitary matrices. Since these cannot be directly implemented on quantum hardware, a practical workaround is required. One promising solution is block-encoding, a framework that embeds a non-unitary matrix A in C^{n x n} into a larger unitary matrix U_A in C^{2n x 2n}, enabling the simulation of non-unitary actions within quantum circuits through ancilla preparation and post-selection. This thesis presents a novel and efficient method for block-encoding based on Gram-Schmidt orthogonalization. The algorithm constructs a unitary U_A with A embedded as the top-left block and remaining components filled to ensure global unitarity. Compared to traditional Singular Value Decomposition (SVD)-based approaches, the proposed method achieves at least 20% improvement in execution time, without compromising output fidelity or numerical stability. A corresponding quantum circuit is developed to extract A|psi> (that is, the transformed version of the input quantum state |psi> after applying the linear operator A) using post-selection on ancilla qubits, with the success probability determined by
Near-term quantum computers will be limited in the number of qubits on which they can process information as well as the depth of the circuits that they can coherently carry out. To date, experimental demonstrations of algorithms such as the Variational Quantum Eigensolver (VQE) have been limited to small molecules using minimal basis sets for this reason. In this work we propose incorporating an orbital optimization scheme into quantum eigensolvers wherein a parametrized partial unitary transformation is applied to the basis functions set in order to reduce the number of qubits required for a given problem. The optimal transformation is found by minimizing the ground state energy with respect to this partial unitary matrix. Through numerical simulations of small molecules up to 16 spin orbitals, we demonstrate that this method has the ability to greatly extend the capabilities of near-term quantum computers with regard to the electronic structure problem. We find that VQE paired with orbital optimization consistently achieves lower ground state energies than traditional VQE when using the same number of qubits and even frequently achieves lower ground state energies than VQE methods using more qubits.
The variational quantum eigensolver is one of the most promising algorithms for near-term quantum computers. It has the potential to solve quantum chemistry problems involving strongly correlated electrons with relatively low-depth circuits, which are otherwise difficult to solve on classical computers. The variational eigenstate is constructed from a number of factorized unitary coupled-cluster terms applied onto an initial (single-reference) state. Current algorithms for applying one of these operators to a quantum state require a number of operations that scale exponentially with the rank of the operator. We exploit a hidden SU(2) symmetry to allow us to employ the linear combination of unitaries approach, Our Prepare subroutine uses n+2 ancilla qubits for a rank-n operator. Our Select(U^) scheme uses O(n)Cnot gates. This results in a full algorithm that scales like the cube of the rank of the operator n3, a significant reduction in complexity for rank five or higher operators. This approach, when combined with other algorithms for lower-rank operators (when compared to the standard implementation), will make the factorized form of the unitary coupled-cluster approach much more efficient to implement on all types of quantum computers.
Recent research has shown that wavefunction evolution in real and imaginary time can generate quantum subspaces with significant utility for obtaining accurate ground state energies. Inspired by these methods, we propose combining quantum subspace techniques with the variational quantum eigensolver (VQE). In our approach, the parameterized quantum circuit is divided into a series of smaller subcircuits. The sequential application of these subcircuits to an initial state generates a set of wavefunctions that we use as a quantum subspace to obtain high-accuracy groundstate energies. We call this technique the circuit subspace variational quantum eigensolver (CSVQE) algorithm. By benchmarking CSVQE on a range of quantum chemistry problems, we show that it can achieve significant error reduction in the best case compared to conventional VQE, particularly for poorly optimized circuits, greatly improving convergence rates. Furthermore, we demonstrate that when applied to circuits trapped at local minima, CSVQE can produce energies close to the global minimum of the energy landscape, making it a potentially powerful tool for diagnosing local minima.
An n-qubit quantum circuit performs a unitary operation on an exponentially large, 2n-dimensional, Hilbert space, which is a major source of quantum speed-ups. We develop a new “Quantum singular value transformation” algorithm that can directly harness the advantages of exponential dimensionality by applying polynomial transformations to the singular values of a block of a unitary operator. The transformations are realized by quantum circuits with a very simple structure - typically using only a constant number of ancilla qubits - leading to optimal algorithms with appealing constant factors. We show that our framework allows describing many quantum algorithms on a high level, and enables remarkably concise proofs for many prominent quantum algorithms, ranging from optimal Hamiltonian simulation to various quantum machine learning applications. We also devise a new singular vector transformation algorithm, describe how to exponentially improve the complexity of implementing fractional queries to unitaries with a gapped spectrum, and show how to efficiently implement principal component regression. Finally, we also prove a quantum lower bound on spectral transformations.