In this paper the stability of specific quantum control protocols is analyzed through a simulation. Two-level quantum systems driven by resonant external fields are studied. The focus of attention are those protocols requiring long sequences of external pulses. It is proven that these techniques are prone to cumulative error. Inspired by the methodology of the treatment of noise in signal processing, noise effects of practical implementations of single qubit quantum algorithms are also considered. The numerical results are consistent when compared to experimental data previously reported.
In this paper the stability of specific quantum control protocols is analyzed through a simulation. Two-level quantum systems driven by resonant external fields are studied. The focus of attention are those protocols requiring long sequences of external pulses. It is proven that these techniques are prone to cumulative error. Inspired by the methodology of the treatment of noise in signal processing, noise effects of practical implementations of single qubit quantum algorithms are also considered. The numerical results are consistent when compared to experimental data previously reported.
The main obstacles to implementing ideal quantum operations are unwanted interactions of quantum systems with the environment and noise in control fields. This problem can be tackled by methods of quantum control. Among these methods are \ncomposite pulse (CP) sequences which have long been employed in nuclear magnetic resonance (NMR) to mitigate the effects of systematic errors in the control. CP sequences have been initially developed to correct for static but otherwise unknown \nerrors in the amplitude or frequency of the driving field. One of the challenges to the systematic incorporation of these control protocols into practical quantum information systems remains the limited understanding of CP performance in the presence \nof time-dependent noise. Treating the influence of time-dependent noise processes on quantum control operations has been facilitated by recent advances in dynamical \nerror suppression based on open-loop Hamiltonian engineering. These approaches provide a general framework for understanding and mitigating non-Markovian time- \ndependent noise in a finite-dimensional open quantum system. Particularly, arbitrary single-qubit control characteristics may be captured quantitatively in filter-transfer \nfunctions (FF) using methods of spectral overlap in the frequency domain. In this thesis work, we present a systematic study of control pulse sequences in the presence of time-dependent noise. We use a combination of analytic formulations based on FFs and numerical simulations to demonstrate that CPs are able to effectively suppress control errors caused by time-dependent processes possessing realistic noise \npower spectra. We provide a geometric interpretation of CP performance under time-dependent amplitude noise, further linking the FF formalism with known techniques in CP construction. We also develop new optimized pulse sequences that act as notch filters for time-dependent noise. These high-fidelity control protocols effectively sup- \npress errors from the noise sources with sharp features in spectral densities and can be used practically on various quantum architectures. We also present our work \non simulation of randomized benchmarking protocols and CPs that have been used experimentally by our collaborators to measure gate errors.
We present novel techniques for modelling and mitigating the effects of time-dependent noise created by interactions between a quantum system and its environment, and by errors in applied controls. Our scope encompasses quantum storage and nontrivial QIP tasks in both single- and multi-qubit systems, with a particular focus on control operations applied to trapped-ion qubits. The work begins with the introduction of a computationally efficient approach to characterizing the impact of time-dependent noise in single-qubit systems, in terms of the spectral characteristics of the noise and a set of generalized control filter functions. In conjunction with brute-force numerical simulation, we show that this novel analytical technique can used to model the evolution of a noisy single-qubit system subject to an arbitrary piecewise-constant control sequence, and to accurately predict the effects of experimentally engineered noise in single-qubit trapped ion systems. We also obtain a number of new and important results that extend the range of utility of state-preserving dynamical decoupling control sequences. These device-independent results provide a means to create a practically useful high-fidelity, long-time quantum memory in both single- and multi-qubit systems. We go on to explain how, given the freedom to arbitrarily and quasi-instantaneously change the phase of the interaction between a system of qubits and a finite collection of oscillators, frequency-specific phase modulation protocols can be implemented that decouple the qubits from targeted oscillator modes at pre-determined times. This technique can be used to generate fast, high-fidelity multi-qubit entangling operations in, for example, ion trap QIP devices.
[SPA] El objetivo principal es el diseño, simulación y análisis de secuencias de detección magnética para sensores cuánticos basados en centros NV de diamante. Estos sensores se están estableciendo entre los sensores cuánticos como candidatos preferentes para su integración en chips. Para su control se utilizan técnicas de resonancia magnética nuclear que caracterizan la respuesta del sistema frente a campos magnéticos. En particular, se implementa y estudia una versión modificada de la secuencia de pulsos Hahn Echo, adaptada al Hamiltoniano dependiente del tiempo de un sistema de espín 1. Para ello, se ha desarrollado e implementado un modelo numérico en Python que resuelve la ecuación de Schrödinger dependiente del tiempo para el sistema, utilizando el método de Runge–Kutta de cuarto orden (RK4) para garantizar precisión y estabilidad en la simulación de la evolución temporal del estado cuántico. La secuencia implementada consiste en un pulso inicial π/2, seguido de un número variable de bloques compuestos por un pulso π y un intervalo de evolución libre de duración τ , finalizando con un último pulso π/2. Los resultados de la simulación demuestran que la señal de detección, cuantificada como la probabilidad final de encontrar el sistema en el estado inicial |0i, depende críticamente de tres parámetros experimentales: la intensidad de los pulsos de microondas (Ω), el número de intervalos de evolución libre (N) y la fase inicial (φ) del campo magnético oscilante externo. Se observa que el uso de pulsos de alta intensidad y un mayor número de intervalos de espera permite una despoblación más efectiva del estado inicial |0i, mejorando la sensibilidad. Además, la fase inicial φ modula la forma de la señal, transformándola de un perfil con un mínimo único (para φ = 0) a una estructura bimodal (para φ = π/2), lo que conlleva un compromiso entre una mayor sensibilidad y una potencial pérdida de robustez frente a la decoherencia. En conjunto, este trabajo sienta las bases para el diseño optimizado de sensores cuánticos de alta precisión. Los resultados subrayan la utilidad de la simulación numérica como herramienta indispensable para explorar y validar protocolos cuánticos complejos antes de su implementación experimental. [ENG] The main objective is the design, simulation, and analysis of magnetic detection sequences for quantum sensors based on diamond NV centers. These sensors are becoming established among quantum sensors as preferred candidates for integration into chips. For their control, nuclear magnetic resonance techniques are used to characterize the systems response to magnetic fields. In particular, a modified version of the Hahn Echo pulse sequence, adapted to the time-dependent Hamiltonian of a spin-1 system, is implemented and studied. To this end, a numerical model has been developed and implemented in Python that solves the time-dependent Schrödinger equation for the system, using the fourth-order Runge–Kutta (RK4) method to ensure precision and stability in the simulation of the time evolution of the quantum state. The implemented sequence consists of an initial π/2 pulse, followed by a variable number of blocks composed of a π pulse and a free evolution interval of duration τ , ending with a final π/2 pulse. The simulation results show that the detection signal, quantified as the final probability of finding the system in the initial state |0i, critically depends on three experimental parameters: the intensity of the microwave pulses (Ω), the number of free evolution intervals (N), and the initial phase (φ) of the external oscillating magnetic field. It is observed that the use of high-intensity pulses and a greater number of waiting intervals allows a more effective depopulation of the initial state |0i, improving sensitivity. Furthermore, the initial phase φ modulates the shape of the signal, transforming it from a profile with a single minimum (for φ = 0) to a bimodal structure (for φ = π/2), which entails a trade-off between higher sensitivity and a potential loss of robustness against decoherence. Altogether, this work lays the foundations for the optimized design of high-precision quantum sensors. The results highlight the usefulness of numerical simulation as an indispensable tool for exploring and validating complex quantum protocols prior to their experimental implementation.
Esta tesis trata del desarrollo de herramientas que faciliten el análisis de la eficiencia y deterioro de los algoritmos cuánticos en sistemas abiertos. Una de las características de los sistemas cuánticos, la correlación, es considerada como una de las principales causas de la mejora de desempeño de los algoritmos cuánticos en comparación con sus homólogos clásicos. Una de las grandes dificultades en el estudio de su influencia es la diversidad de propuestas de medidas existentes, no siempre equivalentes, y la imposibilidad de comparación entre sistemas de diferentes dimensiones utilizando estas medidas. Otro factor, la presencia de errores provenientes de diversas causales, es de fundamental análisis para la determinación de las limitaciones existentes en las propuestas de algoritmos cuánticos. En esta tesis se propone una medida de correlación multiqubit para estados cuánticos mezcla. Esta medida es definida en forma recursiva, acumulando las correlaciones de los subespacios, simplificando el cálculo sin la necesidad del uso de regresión no lineal. A diferencia de las propuestas existentes, esta medida es continua, aditiva y refleja la dimensión del espacio, permitiendo la comparación entre estados de diferentes dimensiones. Con la finalidad de caracterizar, y analizar, la propagación de errores en los algoritmos cuánticos se presentan índices de isotropía. En particular, se presenta un nuevo índice que separa un estado mezcla en dos componentes. El primero, o componente isótropo, cuantifica la falta de información existente en el estado. El segundo, la alineación, representa la desviación del estado respecto a un estado puro de referencia. Este doble índice se representa mediante una gráfica triangular. Utilizando las técnicas propuestas se analizan dos ejemplos de algoritmos cuánticos de búsquedas, considerando sistemas abiertos. En el primer estudio se analiza el cambio de desempeño del algoritmo de Grover con error modelado como un canal de despolarización del estado completo y como un error de despolarización local en cada qubit. En objetivo no es la corrección del error, sino analizar cómo se caracteriza la degradación del desempeño debido a cada tipo de error. En el segundo se analiza un algoritmo de Tulsi modificado mediante mediciones parciales en el qubit de control. Se investiga con diferentes valores del intervalo de tiempo entre mediciones, determinando relaciones existentes entre la probabilidad de éxito y las correlaciones existentes en el estado cuántico del algoritmo. Se estima el orden en cada caso, resultando que para determinada elección de parámetros el algoritmo con mediciones parciales resulta con mejor desempeño que en un sistema cerrado. Como parte del trabajo de tesis se ha desarrollado un simulador para computación cuántica, QuantumLab V2.1, utilizado en todos los trabajos presentados. ABSTRACT This thesis deals with the development of tools that facilitate the analysis of the efficiency and deterioration of quantum algorithms in open systems. One of the characteristics of quantum systems, correlation, is considered as the main cause of performance improvement of quantum algorithms, compared to their classical counterparts. A great difficult in the study of its influence is the diversity of existing correlation measures, not always equivalent, and the impossibility of comparison between systems with different dimensions using these measures. Furthermore, the presence of errors from various causes, must be analyzed in order to determine the limitations in quantum algorithms. At first a measure of correlation for multiqubit mixed states is presented. The measure is defined recursively, accumulating the correlation of the subspaces, making it simple to calculate without the use of regression. Unlike usual measures, the proposed measure is continuous, additive and reflects the dimensionality of the state space, allowing to compare states with different dimensions. Isotropic indexes are proposed in order to characterize, and analyze, the propagation of errors in the quantum algorithms. In particular, a novel index that separates the resultant mixed error state in two components is presented. The first component, or isotropic component, quantifies the lack of information. The second one, the dis-alignment component, represents the shift between the current state and the original pure quantum state. The Isotropic Triangle, a graphical representation that fits naturally with the proposed index, is also introduced. Using the proposed techniques, two examples of quantum search algorithms are analyzed, considering open systems. In the first example the effect of noise on Grover's algorithm is analyzed, modeled as a total depolarizing channel and as a local depolarizing channel in each qubit. The focus was not in error correction (e.g. by the fault-tolerant method), but to provide an insight to the kind of error, or degradation, that needs to be corrected. In the second example a modification of Tulsi's quantum search algorithm with intermediate measurements of the control qubit is presented. In order to analyze the effect of measurements in quantum searches, a different choice of the angular parameter is used. The study is performed for several values of time lapses between measurements, finding close relationships between probabilities and correlations. The order of this modified algorithm is estimated, showing that for some time lapses the performance is improved, and becomes of order Q(N) (classical brute-force search) when measurements are taken in every step. As part of this thesis, a simulator for quantum computing has been developed, QuantumLab V2.1, and it has been used in all the former examples.
Quantum computing holds immense promise for revolutionizing computation, yet its practical realization is severely hampered by the pervasive issue of quantum noise. This noise, arising from various sources including environmental interactions and imperfect control, introduces errors into quantum computations, degrading their fidelity. This paper explores novel control algorithms designed to mitigate the detrimental effects of quantum noise. We investigate the underlying mechanisms generating noise and propose a method based on precise qubit control and calibration to minimize its impact. The core claim is that effective noise control is crucial for achieving reliable quantum computation. The proposed algorithms leverage feedback control and adaptive calibration techniques to dynamically compensate for noise fluctuations, aiming to enhance the accuracy and robustness of quantum algorithms. The mathematical framework presented includes noise models, control equations, and performance metrics, demonstrating the feasibility and potential of this approach. Specifically, we introduce a closed-loop control system described by the equation: (Δsn+1 = K(en - sn)) + wn+1, where Δsn+1 represents the change in state, K is the gain, en is the desired state, sn is the actual state, and wn+1 is the noise term. We also examine the effect of decoherence, modeled as: ds = Hdt + Vd, where ds is the time evolution of the quantum state, H is the Hamiltonian describing the system, dt is the time step, and Vd is the damping term due to noise. The research aims to contribute towards the development of more robust and practical quantum computers.