Publications of Paula García-Molina

This work discusses the solution of partial differential equations (PDEs) using matrix product states (MPS). The study focuses on the search for the lowest eigenstates of a Hamiltonian equation, for which five algorithms are introduced: imaginary-time evolution, steepest gradient descent, an improved gradient descent, an implicitly restarted Arnoldi method, and density matrix renormalization group (DMRG) optimization. The first four methods are engineered using a framework of limited-precision linear algebra, where operations between MPS and matrix product operators (MPOs) are implemented with finite resources. All methods are benchmarked using the PDE for a quantum harmonic oscillator in up to two dimensions, over a regular grid with up to $2^{28}$ points. Our study reveals that all MPS-based techniques outperform exact diagonalization techniques based on vectors, with respect to memory usage. Imaginary-time algorithms are shown to underperform any type of gradient descent, both in terms of calibration needs and costs. Finally, Arnoldi like methods and DMRG asymptotically outperform all other methods, including exact diagonalization, as problem size increases, with an exponential advantage in memory and time usage.

Noisy Intermediate-Scale Quantum (NISQ) devices lack error correction, limiting scalability for quantum algorithms. In this context, digital-analog quantum computing (DAQC) offers a more resilient alternative quantum computing paradigm that outperforms digital quantum computation by combining the flexibility of single-qubit gates with the robustness of analog simulations. This work explores the impact of noise on both digital and DAQC paradigms and demonstrates DAQC’s effectiveness in error mitigation. We compare the quantum Fourier transform and quantum phase estimation algorithms under a wide range of single and two-qubit noise sources in superconducting processors. DAQC consistently surpasses digital approaches in fidelity, particularly as processor size increases. Moreover, zero-noise extrapolation further enhances DAQC by mitigating decoherence and intrinsic errors, achieving fidelities above 0.95 for 8 qubits, and reducing computation errors to the order of $10^{-3}$. These results establish DAQC as a viable alternative for quantum computing in the NISQ era.

In this work, we develop a highly efficient representation of functions and differential operators based on Fourier analysis. Using this representation, we create a variational hybrid quantum algorithm to solve static, Schr\”odinger-type, Hamiltonian partial differential equations (PDEs), using space-efficient variational circuits, including the symmetries of the problem, and global and gradient-based optimizers. We use this algorithm to benchmark the performance of the representation techniques by means of the computation of the ground state in three PDEs, i.e., the one-dimensional quantum harmonic oscillator, and the transmon and flux qubits, studying how they would perform in ideal and near-term quantum computers. With the Fourier methods developed here, we obtain low infidelities of order $10^{-4}-10^{-5}$ using only three to four qubits, demonstrating the high compression of information in a quantum computer. Practical fidelities are limited by the noise and the errors of the evaluation of the cost function in real computers, but they can also be improved through error mitigation techniques.