quantum-riccati-solver

name	quantum-riccati-solver
description	Quantum algorithms for solving algebraic Riccati equations via Riesz projectors and quantum singular value transformations (QSVT). Applications to quantum chemistry RPA and coupled-cluster theory. Activation: quantum riccati, quantum nonlinear matrix, quantum chemistry algorithm, RPA quantum, algebraic riccati equation quantum, quantum singular value transformation chemistry.

Quantum Solvers for Nonlinear Matrix Equations

Quantum algorithm for solving algebraic Riccati equations using Riesz projectors onto invariant subspaces, implemented via contour-integral resolvents and QSVT, with applications to quantum-chemical RPA and coupled-cluster theory. (arXiv:2605.16189)

Metadata

• Source: arXiv:2605.16189
• Authors: Pablo Rodenas-Ruiz, Andrew Zhao, Joonho Lee
• Published: 2026-05-15
• Categories: quant-ph, physics.chem-ph

Core Methodology

Key Innovation

First quantum algorithm framework for solving algebraic Riccati equations - a fundamental class of nonlinear matrix equations ubiquitous in control theory, quantum chemistry, and many-body physics. The method achieves linear scaling in system size and polynomial scaling in excitation rank, suggesting exponential advantage over classical local-correlation heuristics.

Technical Framework

1. Block-Encoding via Riesz Projectors

   • Riccati solutions are encoded as Riesz projectors onto invariant subspaces of an associated non-normal matrix
   • The projector P = (1/2πi) ∮_Γ (zI - A)^(-1) dz captures the stabilizing solution
   • Contour Γ encloses eigenvalues corresponding to the stable invariant subspace

2. Contour-Integral Resolvent Implementation

   • The resolvent (zI - A)^(-1) is approximated using quantum linear system algorithms (QLSA)
   • Contour integration discretized via quadrature: sum of weighted resolvent evaluations
   • Each resolvent evaluation implemented as a block-encoded linear system solve

3. Quantum Singular Value Transformation (QSVT)

   • Polynomial transformations of block-encoded matrices extract the projector
   • QSVT provides optimal gate complexity for matrix function approximation
   • Enables efficient estimation of correlation-energy density from the solution

4. Application to RPA (Random-Phase Approximation)

   • m-particle, m-hole RPA reduces to a Riccati equation for cluster amplitudes
   • Under localized-orbital sparsity: O(N) system-size scaling, poly(m) excitation-rank scaling
   • Exponential advantage in m over classical local-correlation methods

Implementation Guide

Prerequisites

• QSVT / quantum signal processing library
• Quantum linear system solver (HHL variant or QSVT-based)
• Block-encoding primitives for the problem matrix A

Step-by-Step

1. Formulate the Riccati Equation

   A^T X + X A - X B X + C = 0
   

   Identify the associated Hamiltonian matrix H = [[A, -B], [-C, -A^T]]

2. Construct Block-Encoding of H

   • Decompose H into efficiently implementable unitary blocks
   • Ensure normalization ||H|| ≤ 1 for QSVT compatibility

3. Define Contour and Quadrature

   • Choose contour Γ enclosing stable eigenvalues
   • Discretize via trapezoidal or Gauss-Legendre quadrature
   • Number of quadrature points O(log(1/ε)) for ε precision

4. Implement QSVT-Based Projector

   • Design polynomial approximation of the indicator function
   • Apply QSVT to block-encoded H
   • Extract projector onto stable subspace

5. Estimate Observables

   • Compute correlation-energy density: E_corr = Tr(X · C)
   • Use amplitude estimation for quadratic speedup in sampling

Code Example (Pseudocode)

# Block-encode the Hamiltonian matrix
H_block = block_encode(H, precision=epsilon)

# Define contour quadrature points
z_points, weights = contour_quadrature(stable_eigenvalues, n_points=O(log(1/eps)))

# Evaluate resolvents via QLSA
projector = sum(w * qlsolve(z_i * I - H_block) for z_i, w in zip(z_points, weights))

# Extract Riccati solution from projector blocks
X = extract_solution(projector)

# Estimate correlation energy
E_corr = amplitude_estimate(lambda X: trace(X @ C))

Applications

• Quantum chemistry: RPA, higher-order RPA, coupled-cluster theory
• Control theory: optimal control, Kalman filtering, LQR
• Many-body physics: Green's function calculations, self-consistent field methods
• Machine learning: discrete-time Riccati equations in filtering/smoothing

Pitfalls

• Non-normal matrices: Standard QSVT assumes normal/Hermitian matrices; non-normal H requires careful block-encoding design
• Sparsity assumptions: Linear scaling relies on localized-orbital sparsity; dense systems revert to polynomial scaling
• Contour selection: Poor contour choice leads to ill-conditioned resolvents; eigenvalue gap must be bounded away from zero
• QSVT degree: Polynomial degree scales with condition number; ill-conditioned problems require high-degree polynomials
• State preparation: Input state quality directly affects solution accuracy

Related Skills

• quantum-block-encoding-linear-algebra
• quantum-linear-algebra-block-encoding
• qsvt-quantum-signal-processing
• quantum-chemistry-algorithms
• quantum-algorithm-framework-designer