Quantum algorithms take advantage of quantum phenomena such as entanglement and superposition to solve problems unreachable by classical computers, or to outperform them in the performance of similar tasks. While the applications of quantum computing are wide reaching, we are still in the NISQ era, where noise and small numbers of qubits dominate our quantum capabilities. Guided by these constraints, our group is dedicated to the research of novel quantum algorithms capable of working with short and medium term quantum computers, such as: partial differential equations (PDE’s) solvers, exploring quantum algorithms to solve hard optimization problems, and developing quantum-inspired algorithms that run in classical computers.

### Quantum optimization

A wide range of optimization problems can be mapped into a QUBO problem or an Ising Hamiltonian, from finance and the more academic MaxCut problem to finding the ground state of spin glass Hamiltonians. Quantum variational optimization has emerged as a promising framework to solve these problems faster and at a larger scale than what classical methods allow. In this regard, we study the performance of state-of-art algorithms such as the Variational Quantum Eigensolver [1]. Furthermore, we explore practical applications in finance, where we have developed a hybrid quantum-classical algorithm to choose a series of assets that track a particular financial index [2].

### Quantum PDE solvers

The solution of partial differential equations is a ubiquitous problem, with a broad impact in many areas of science and technology, from physics and chemistry to finance and engineering. Existing numerical methods are limited in the size and complexity of the problems they can solve. It is therefore of great interest to find alternative techniques that can provide size or time speedups in the manipulation of PDE and their solutions.

In our group we are exploring the use of a variational hybrid quantum-classical algorithm to tackle this problem. The idea is to encode a complex multidimensional function as the superposition state of a quantum computer, manipulating this state to find the best approximation to the solution of a PDE.

In our first work in this topic [3], we have studied the performance of existing and new variational ansätze and optimization techniques, gauging their application in current and near-term NISQ hardware. This work illustrates the power of quantum Fourier interpolation as an acceleration technique to reduce the dimensionality of the quantum states, obtaining accurate representations with exponentially reduced numbers of qubits. It also confirms the limitations of existing hardware and qubit quality in the application of these ideas.

### Tensor networks

Tensor networks are a mathematical description to represent quantum-many body states based on the entanglement structure. This formalism allows to reduce the number of parameters that represent a state, and hence is efficient for the simulation of many body systems. This can be applied to quantum computing, by using tensor networks to represent qubits, allowing for the use of a greater number of qubits than current quantum computers, both real and simulators. By using tensor networks we aim at developing quantum-inspired algorithms for a variety of problems, such as multivariate analysis [4].

#### Group members working on this field

#### Publications

[1] Quantum variational optimization: the role of entanglement and problem hardness, *Pablo Díez-Valle, Diego Porras, Juan José García-Ripoll, *Phys. Rev. A 104, 062426 (2021).

[2] Hybrid quantum-classical optimization for financial index tracking, *Samuel Fernández-Lorenzo, Diego Porras, Juan José García-Ripoll*, Quantum Sci. Technol. 6 034010 (2021).

[3] Quantum Fourier analysis for multivariate functions and applications to a class of Schrödinger-type partial differential equations, *Paula García-Molina, Javier Rodríguez-Mediavilla, Juan José García-Ripoll*, Phys. Rev. A 105, 012433 (2022).

[4] Quantum-inspired algorithms for multivariate analysis: from interpolation to partial differential equations, *Juan José García-Ripoll*, Quantum 5, 431 (2021).