Measurement-Driven Quantum Algorithms: Efficient Eigensolvers and Evaluation of Matrix Functions
Daan Camps
Lawrence Berkeley National Laboratory
NERSC
A fundamental task in quantum computing is extracting information from time evolution: computing eigenvalues, estimating thermal properties, or evaluating functions of operators. This talk explores how measurement-driven approaches which construct classical representations from quantum data, provide a versatile framework for a range of problems across the NISQ-to-FTQC transition. We present two complementary perspectives on leveraging real-time evolution through classical post-processing.
First, we introduce multi-observable dynamic mode decomposition (MODMD) [1], which combines signal processing approaches with classical shadow tomography to extract multiple low-lying eigenvalues. By replacing Hadamard-test circuits with protocols for predicting low-rank observables, MODMD exploits random scrambling to construct signal subspaces encoding spectral information with exponentially reduced resource requirements. We establish theoretical guarantees showing that spectral error decays exponentially in simulation time, providing justification for the rapid convergence observed in our numerical simulations.
Second, we explore quantum algorithms for evaluating matrix elements ?ψ1|f(U)|ψ0? through Szegö quadrature [2]. This approach collects measurement data using single-ancilla quantum circuits to construct quadrature rules that achieve optimal scaling between polynomial degree and circuit count when f is a Laurent polynomial. Crucially, the quadrature rules can be applied to arbitrary functions f after quantum execution, without requiring function approximations during the quantum computation. This flexibility makes the method broadly applicable to spectral characterization tasks, from analyzing Hamiltonian spectra through Green's functions, to estimating Gibbs state properties.
Together, these methods illustrate that classical post-processing transforms quantum measurement data into powerful tools for spectral analysis, enabling algorithms that are becoming practical on current hardware while scaling naturally toward fault-tolerant implementations.
References
[1] Efficient measurement-driven eigenenergy estimation with classical shadows, Y. Shen, Buzali A., Hu H.-Y., Klymko K., Camps D., Yelin S. F., Van Beeumen R., arXiv:2409.13691, Accepted in PRX Quantum.
[2] Quantum Krylov Algorithm for Szegö Quadrature, W. Kirby, Y. Shen, D. Camps, A. Chowdhury, Klymko K., Van Beeumen R., arXiv:2509.19195.
