arXiv:2605.16189

Quantum Solvers for Nonlinear Matrix Equations in Quantum Chemistry

Pablo Rodenas-Ruiz, Andrew Zhao, Joonho Lee

quant-phphysics.chem-phSubmitted May 15, 2026• Announced May 18, 2026• 0 citations

Abstract

We present a quantum algorithm for solving algebraic Riccati equations, with applications to quantum-chemical random-phase approximation (RPA) and higher-order RPA theories. Our method block-encodes stabilizing Riccati solutions via Riesz projectors onto invariant subspaces of an associated non-normal matrix, implemented using contour-integral resolvents and quantum singular value transformations. Applied to $m$-particle, $m$-hole RPA, our algorithm yields a block-encoding of the amplitude solution and estimates the electronic correlation-energy density with it. Under localized-orbital sparsity assumptions, the end-to-end cost scales linearly with system size and polynomially with excitation rank $m$, suggesting an exponential advantage in $m$ over plausible classical local-correlation heuristics. More broadly, this work provides a framework for quantum algorithms for nonlinear matrix equations in quantum chemistry and opens a possible route toward developing quantum algorithms for coupled-cluster theory.

Comments: 8+30 pages, 1+4 figures

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arXiv:2605.16189

Quantum Solvers for Nonlinear Matrix Equations in Quantum Chemistry

Pablo Rodenas-Ruiz, Andrew Zhao, Joonho Lee

quant-phphysics.chem-phSubmitted May 15, 2026• Announced May 18, 2026• 0 citations

arXiv HTML PDF

Abstract

m

-particle,

m

-hole RPA, our algorithm yields a block-encoding of the amplitude solution and estimates the electronic correlation-energy density with it. Under localized-orbital sparsity assumptions, the end-to-end cost scales linearly with system size and polynomially with excitation rank

m

, suggesting an exponential advantage in

m

over plausible classical local-correlation heuristics. More broadly, this work provides a framework for quantum algorithms for nonlinear matrix equations in quantum chemistry and opens a possible route toward developing quantum algorithms for coupled-cluster theory.

Comments: 8+30 pages, 1+4 figures

Related papers

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We develop quantum algorithms that construct block-encodings of solution matrices for continuous- and discrete-time algebraic Riccati equations (CAREs and DAREs), differential Riccati equations (DREs), and finite Riccati recursions. The Quantum Weighted Riesz Method, unifying four kinds of Riccati problems, constructs projectors onto the solution graphs using weighted Riesz branching operators and recovers the solution matrices. Under the stated access and normalization assumptions, our algorithms for CAREs and DAREs have near-optimal query complexity $Θ(\mathcal Rα)$ up to logarithmic factors, with the generalized singularity factor $\mathcal R$ that reflects spectral separation, and the output normalization $α$ . For DREs and finite-horizon problems, the query complexity has an additional initialization conditioning factor $κ_{\rm init}$ . We establish product query lower bounds for specified input-oracle families, demonstrating necessary joint dependence on spectral separation and initialization or output scale. We also prove that selected-value promise problems are BQP-complete on explicit circuit-generated families of single-control discrete-time algebraic and zero-terminal differential Riccati equations. Applications include classical feedback evaluation for linear-quadratic control, a query-complexity comparison for stable random-phase approximation Riccati equations in quantum chemistry, and heated boundary control with numerical validation.We develop quantum algorithms that construct block-encodings of solution matrices for continuous- and discrete-time algebraic Riccati equations (CAREs and DAREs), differential Riccati equations (DREs), and finite Riccati recursions. The Quantum Weighted Riesz Method, unifying four kinds of Riccati problems, constructs projectors onto the solution graphs using weighted Riesz branching operators and recovers the solution matrices. Under the stated access and normalization assumptions, our algorithms for CAREs and DAREs have near-optimal query complexity $Θ(\mathcal Rα)$ up to logarithmic factors, with the generalized singularity factor $\mathcal R$ that reflects spectral separation, and the output normalization $α$. For DREs and finite-horizon problems, the query complexity has an additional initialization conditioning factor $κ_{\rm init}$. We establish product query lower bounds for specified input-oracle families, demonstrating necessary joint dependence on spectral separation and initialization or output scale. We also prove that selected-value promise problems are BQP-complete on explicit circuit-generated families of single-control discrete-time algebraic and zero-terminal differential Riccati equations. Applications include classical feedback evaluation for linear-quadratic control, a query-complexity comparison for stable random-phase approximation Riccati equations in quantum chemistry, and heated boundary control with numerical validation.

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Annual Symposium on Foundations of Computer ScienceOct 27, 202412 citations

Quantum Eigenvalue Processing

Guang Hao Low, Yuan Su

Abstract. Many problems in linear algebra—such as those arising from non-Hermitian physics, transcorrelated quantum chemistry, and differential equations—can be solved on a quantum computer by processing eigenvalues of the non-normal input matrices. However, the existing Quantum Singular Value Transformation (QSVT) framework is ill-suited for this task because eigenvalues and singular values are different in general. We present a Quantum EigenValue Transformation (QEVT) framework for applying arbitrary polynomial transformations on eigenvalues of block-encoded non-normal operators and a related Quantum EigenValue Estimation (QEVE) algorithm for operators with real spectra. QEVT has query complexity to the block encoding nearly recovering that of the QSVT for a Hermitian input, and QEVE achieves the Heisenberg-limited scaling for diagonalizable input matrices. As applications, we develop a linear differential equation solver with strictly linear time query complexity for average-case diagonalizable operators, as well as a ground state preparation algorithm that upgrades previous nearly optimal results for Hermitian Hamiltonians to diagonalizable matrices with real spectra. Underpinning our algorithms is an efficient method to prepare a quantum superposition of Faber polynomials, which generalize the nearly-best uniform approximation properties of Chebyshev polynomials to the complex plane. Of independent interest, we also develop techniques to generate [Formula: see text] Fourier coefficients with [Formula: see text] gates compared to prior approaches with linear cost.Abstract. Many problems in linear algebra—such as those arising from non-Hermitian physics, transcorrelated quantum chemistry, and differential equations—can be solved on a quantum computer by processing eigenvalues of the non-normal input matrices. However, the existing Quantum Singular Value Transformation (QSVT) framework is ill-suited for this task because eigenvalues and singular values are different in general. We present a Quantum EigenValue Transformation (QEVT) framework for applying arbitrary polynomial transformations on eigenvalues of block-encoded non-normal operators and a related Quantum EigenValue Estimation (QEVE) algorithm for operators with real spectra. QEVT has query complexity to the block encoding nearly recovering that of the QSVT for a Hermitian input, and QEVE achieves the Heisenberg-limited scaling for diagonalizable input matrices. As applications, we develop a linear differential equation solver with strictly linear time query complexity for average-case diagonalizable operators, as well as a ground state preparation algorithm that upgrades previous nearly optimal results for Hermitian Hamiltonians to diagonalizable matrices with real spectra. Underpinning our algorithms is an efficient method to prepare a quantum superposition of Faber polynomials, which generalize the nearly-best uniform approximation properties of Chebyshev polynomials to the complex plane. Of independent interest, we also develop techniques to generate [Formula: see text] Fourier coefficients with [Formula: see text] gates compared to prior approaches with linear cost.

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Accurately predicting the properties of molecules and materials from first principles requires solving the electronic structure problem. However, the computational cost of solving the electronic structure problem scales prohibitively in the number of atoms, and the usual approximate methods designed to reduce the overhead are often inaccurate for strongly correlated systems. For this reason, quantum algorithms have been developed as a promising alternative. In this work, we develop algebraic tools to improve the efficiency of three quantum algorithms for the electronic structure problem, addressing key bottlenecks for both near-term and fault-tolerant quantum computers. The first is the variational quantum eigensolver (VQE), which suffers from a large measurement overhead needed to estimate the Hamiltonian expectation value. To this end, we introduce a general class of measurable fragments of the electronic Hamiltonian, which enable partitioning schemes that significantly reduce the sampling cost for energy estimation. The second are quantum subspace methods (QSMs). We introduce a new QSM, the quantum seniority-based subspace expansion algorithm, which is a scalable alternative to VQE that alleviates VQE's state preparation, measurement, and optimization challenges while reducing qubit requirements. The third is quantum phase estimation (QPE), which requires an encoding of the electronic Hamiltonian directly on the quantum computer, the cost of which scales indirectly with the Hamiltonian spectral range. To address this, we develop linear programming approaches to implement the Block Invariant Symmetry Shift protocol, a method which reduces the spectral range of the Hamiltonian, improving its block-encoding cost for QPE. Collectively, these methods demonstrate that one can exploit the Hamiltonian's algebraic structure to mitigate computational costs, accelerating the path toward practical quantum advantage for electronic structure.Accurately predicting the properties of molecules and materials from first principles requires solving the electronic structure problem. However, the computational cost of solving the electronic structure problem scales prohibitively in the number of atoms, and the usual approximate methods designed to reduce the overhead are often inaccurate for strongly correlated systems. For this reason, quantum algorithms have been developed as a promising alternative. In this work, we develop algebraic tools to improve the efficiency of three quantum algorithms for the electronic structure problem, addressing key bottlenecks for both near-term and fault-tolerant quantum computers. The first is the variational quantum eigensolver (VQE), which suffers from a large measurement overhead needed to estimate the Hamiltonian expectation value. To this end, we introduce a general class of measurable fragments of the electronic Hamiltonian, which enable partitioning schemes that significantly reduce the sampling cost for energy estimation. The second are quantum subspace methods (QSMs). We introduce a new QSM, the quantum seniority-based subspace expansion algorithm, which is a scalable alternative to VQE that alleviates VQE's state preparation, measurement, and optimization challenges while reducing qubit requirements. The third is quantum phase estimation (QPE), which requires an encoding of the electronic Hamiltonian directly on the quantum computer, the cost of which scales indirectly with the Hamiltonian spectral range. To address this, we develop linear programming approaches to implement the Block Invariant Symmetry Shift protocol, a method which reduces the spectral range of the Hamiltonian, improving its block-encoding cost for QPE. Collectively, these methods demonstrate that one can exploit the Hamiltonian's algebraic structure to mitigate computational costs, accelerating the path toward practical quantum advantage for electronic structure.

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Dec 11, 20250 citations

m

-particle,

m

m

, suggesting an exponential advantage in

m

Θ(\mathcal Rα)

up to logarithmic factors, with the generalized singularity factor

\mathcal R

that reflects spectral separation, and the output normalization

α

. For DREs and finite-horizon problems, the query complexity has an additional initialization conditioning factor

κ_{\rm init}

. We establish product query lower bounds for specified input-oracle families, demonstrating necessary joint dependence on spectral separation and initialization or output scale. We also prove that selected-value promise problems are BQP-complete on explicit circuit-generated families of single-control discrete-time algebraic and zero-terminal differential Riccati equations. Applications include classical feedback evaluation for linear-quadratic control, a query-complexity comparison for stable random-phase approximation Riccati equations in quantum chemistry, and heated boundary control with numerical validation.

We develop operator-output quantum algorithms for matrix equations and matrix functions using matrix-sign embeddings. For each problem, an augmented matrix

M

is chosen so that the target operator is a block of

\text{sign}(M)

or is recovered from blocks of

(I-\text{sign}(M))/2

. We approximate the half-plane sign by a logarithmic-sinc formula and implement the shifted inverse families by scaled multiplexing with node-dependent rebalancing. For ordinary Sylvester equations, this yields a block-encoding of the solution with query complexity linear in the relevant conditioning parameters and logarithmic in the inverse error tolerance, under either a field-of-values (FoV) gap or a strip-resolvent bound. The same method extends to generalized Sylvester and Lyapunov equations, principal square and inverse square roots, matrix geometric means, and continuous-time algebraic Riccati equations (CARE), with explicit query-complexity and block-encoding-normalization bounds that cover non-normal and non-diagonalizable cases.

n

Fourier coefficients with 0 (poly log (

n

)) gates compared to prior approaches with linear cost.