Complex systems in quantum technologies

124 p.En esta Tesis, se propone una serie de protocolos de información cuántica, analizando la viabilidad con la tecnología actual, en plataformas de iones atrapados y circuitos superconductores. Encontramos que los protocolos propuestos tienen que adaptarse a las ventajas e inconvenientes de cada plataforma. Se prueba que un qubit protegido, basado en una representación dual de una cadena fermiónica topológica, puede ser codificado en un sistema de trampa de iones, debido a sus propiedades específicas. Se analiza la simulación cuántica de fermiones, encontrando una mayor eficiencia debido a puertas colectivas que son realizables con la tecnología de iones atrapados. Dentro de este espíritu, estimamos las posibilidades de los circuitos superconductores de simular modelos de espines, sistemas fermiónicos y bosónicos. Extendemos estos conceptos a la simulación cuántica de sistemas dinámicos clásicos, encontrando que una simulación de la dinámica de Boltzmann discreta puede ser codificada en sistemas acoplados de qubits con bosones. Estos son los primeros pasos para explorar las simulaciones de dinámica de fluidos en un ordenador cuántico

Second-quantized fermionic operators with polylogarithmic qubit and gate complexity

We present a method for encoding second-quantized fermionic systems in qubits when the number of fermions is conserved, as in the electronic structure problem. When the number $F$ of fermions is much smaller than the number $M$ of modes, this symmetry reduces the number of information-theoretically required qubits from $\Theta(M)$ to $O(F\log M)$. In this limit, our encoding requires $O(F^2\log^4 M)$ qubits, while encoded fermionic creation and annihilation operators have cost $O(F^2\log^5 M)$ in two-qubit gates. When incorporated into randomized simulation methods, this permits simulating time-evolution with only polylogarithmic explicit dependence on $M$. This is the first second-quantized encoding of fermions in qubits whose costs in qubits and gates are both polylogarithmic in $M$, which permits studying fermionic systems in the high-accuracy regime of many modes.Comment: up to date with published version; 19 pages, 4 figure

An analytic theory for the dynamics of wide quantum neural networks

Parametrized quantum circuits can be used as quantum neural networks and have the potential to outperform their classical counterparts when trained for addressing learning problems. To date, much of the results on their performance on practical problems are heuristic in nature. In particular, the convergence rate for the training of quantum neural networks is not fully understood. Here, we analyze the dynamics of gradient descent for the training error of a class of variational quantum machine learning models. We define wide quantum neural networks as parameterized quantum circuits in the limit of a large number of qubits and variational parameters. We then find a simple analytic formula that captures the average behavior of their loss function and discuss the consequences of our findings. For example, for random quantum circuits, we predict and characterize an exponential decay of the residual training error as a function of the parameters of the system. We finally validate our analytic results with numerical experiments.Comment: 26 pages, 5 figures. Comments welcom

Hierarchical Clifford transformations to reduce entanglement in quantum chemistry wavefunctions

The performance of computational methods for many-body physics and chemistry is strongly dependent on the choice of basis used to cast the problem; hence, the search for better bases and similarity transformations is important for progress in the field. So far, tools from theoretical quantum information have been not thoroughly explored for this task. Here we take a step in this direction by presenting efficiently computable Clifford similarity transformations for quantum chemistry Hamiltonians, which expose bases with reduced entanglement in the corresponding molecular ground states. These transformations are constructed via block diagonalization of a hierarchy of truncated molecular Hamiltonians, preserving the full spectrum of the original problem. We show that the bases introduced here allow for more efficient classical and quantum computation of ground state properties. First, we find a systematic reduction of bipartite entanglement in molecular ground states as compared to standard problem representations. This entanglement reduction has implications in classical numerical methods such as ones based on the density matrix renormalization group. Then, we develop variational quantum algorithms that exploit the structure in the new bases, showing again improved results when the hierarchical Clifford transformations are used.Comment: 14 pages, 11 figure