This thesis describes aspects of dynamics of solid state systems which are relevant to quantum metrology and computation. It may be divided into three research directions (parts). For the first part, a new method to enhance precision measurements that makes use of a sensor’s environment to amplify its response to weak external perturbations is described. In this method a “central” spin is used to sense the dynamics of surrounding spins, which are affected by the external perturbations that are being measured. The enhancement in precision is determined by the number of spins that are coupled strongly to the central spin and is resilient to various forms of decoherence. For polarized environments, nearly Heisenberg-limited precision measurements can be achieved. The second part of the thesis focuses on the decoherence of Majorana fermions. Specializing to the experimentally relevant case where each mode interacts with its own bath we present a method to study the effect of external perturbations on these modes. We analyze a generic gapped fermionic environment (bath) interacting via tunneling with individual Majorana modes - components of a qubit. We present examples with both static and dynamic perturbations (noise), and derive a rate of information loss for Majorana memories, that depends on the spectral density of both the noise and the fermionic bath. For the third part of the thesis we discuss vortices in topological superconductors which we model as closed finite systems, each with an odd number of real fermionic modes. We show that even in the presence of many-body interactions, there are always at least two fermionic operators that commute with the Hamiltonian. There is a zero mode corresponding to the total Majorana operator [1] as well as additional linearly independent zero modes, one of which is continuously connected to the Majorana mode in the non-interacting limit. We also show that in the situation where there are two or more well separated vortices their zero modes have non-Abelian Ising statistics under braiding.
This thesis describes aspects of dynamics of solid state systems which are relevant to quantum metrology and computation. It may be divided into three research directions (parts). For the first part, a new method to enhance precision measurements that makes use of a sensor’s environment to amplify its response to weak external perturbations is described. In this method a “central” spin is used to sense the dynamics of surrounding spins, which are affected by the external perturbations that are being measured. The enhancement in precision is determined by the number of spins that are coupled strongly to the central spin and is resilient to various forms of decoherence. For polarized environments, nearly Heisenberg-limited precision measurements can be achieved. The second part of the thesis focuses on the decoherence of Majorana fermions. Specializing to the experimentally relevant case where each mode interacts with its own bath we present a method to study the effect of external perturbations on these modes. We analyze a generic gapped fermionic environment (bath) interacting via tunneling with individual Majorana modes - components of a qubit. We present examples with both static and dynamic perturbations (noise), and derive a rate of information loss for Majorana memories, that depends on the spectral density of both the noise and the fermionic bath. For the third part of the thesis we discuss vortices in topological superconductors which we model as closed finite systems, each with an odd number of real fermionic modes. We show that even in the presence of many-body interactions, there are always at least two fermionic operators that commute with the Hamiltonian. There is a zero mode corresponding to the total Majorana operator [1] as well as additional linearly independent zero modes, one of which is continuously connected to the Majorana mode in the non-interacting limit. We also show that in the situation where there are two or more well separated vortices their zero modes have non-Abelian Ising statistics under braiding.
The work presented in this thesis is divided into two main parts: strong zero modes in quantum spin chains with open boundaries and large deviations in classical spin systems. The results concerning the strong zero modes are presented in chapters 2 and 3, while chapter 4 contains the results about large deviations in systems of many, but independent, classical degrees of freedom. These two distinct classes of problems in spin chains are both about aspects of slow relaxation to equilibrium in many-body systems. Nowadays slow dynamics and non-ergodicity in many-body quantum systems are receiving much attention due to their notable technological potential for storing and processing, for very long times, the quantum information encoded in boundary degrees of freedom. In the first part of the work presented in this thesis we mainly focus on quantum spin chains with open boundaries, which can display coherence times for edge spins, that diverge with the system size, as a consequence of the emergence, in the thermodynamic limit, of conserved local quantities, called strong zero modes. First, we discuss the fate of these coherence times when dynamics becomes dissipative. Understanding the behavior of these long correlation times in open quantum systems, undergoing Markovian (memoryless) dynamics, would allow for the possibility of exploiting the protected information encoded in the boundary spins for applications in quantum devices in realistic non-equilibrium settings. Second, we study one possible simple generalization of the transverse field Ising model to a class of ladder spin systems with plaquette interactions in the presence of a transverse field. Also for these systems one can explicitly construct the strong zero mode operators and understand the behavior of the consequential long coherence times for the edge degrees of freedom under isolated (unitary) dynamics. This is possible because the models we consider, while appearing complex superficially, can be brought to simple forms by exploiting an extensive number of conserved quantities they possess. The results presented in chapters 2 and 3 are closely related, even if they focus on different models and investigate different aspects of their dynamics. Nevertheless, the underlying concept they are based on is the same one, that is, an eigenstate phase transition for the strong zero modes that has interesting consequences for the full energy spectrum. In the second part of the thesis we focus on a different type of phase transitions. Here we address the issue of dynamical large deviation phase transitions from a disordered phase to an ordered one, in classical systems which consist in a collection of independent Ising spins evolving stochastically. Surprisingly, by reweighing trajectories accordingly to a certain type of time-integrated observable, even a system of independently evolving spins can display some critical behavior, at the fluctuation level, in trajectory space in the large size and long time limit. By making use of the tilted generator techniques we study the statistics of two particular time-integrated observables. Their associated tilted generators have an exact Kramers-Wannier duality and they look like, up to a sign and an additive constant, some specific well known quantum Hamiltonians. In this sense, the dynamical large deviation phase transitions we observe in these classical stochastic systems translate in ground state quantum phase transitions taking place at the self dual point of the tilted generators we study.
This thesis presents a collection of projects spanning different sub-fields of quantum science, undertaken with the goal of understanding and advancing quantum technologies. Broadly, it can be divided into two thematic parts — parameter estimation and open system dynamics. Each project addresses inverse and ill-posed problems in Physics, which lie at the heart of developing robust and scalable quantum devices.A central theme in quantum technology is the challenge of inference of microscopic system parameters from limited, indirect observations. The first part of the thesis investigates this through the problem of Hamiltonian reconstruction via local measurements. Indirect estimation of couplings in quantum systems relies on the measurement of the spectrum and the overlap of eigenvectors with some reference state. For a one-dimensional chain, this data can be obtained by local measurement on one end of the chain, thereby reducing the need for full Hamiltonian tomography. Information about couplings is extracted from the temporal data obtained by measuring the expectation values of the time evolution 𝑒−𝑖𝐻𝑡 for a reference state. In Chapter 3, we analyze the robustness of this estimation scheme under perturbations arising from beyond nearest neighbour interactions. We show that the error scales linearly with the strength of such interactions and on average, the error increases with the distance of interaction between sites.Chapter 4 examines whether coupling information can be extracted from local thermal measurements, using expectation values of the Gibbs state at varying temperatures rather than temporal data. We develop a procedure to obtain the coefficients of the Taylor expansion of these expectation values and relate them to the Hamiltonian parameters. For very small chains, this mapping can be carried out analytically, while for larger systems it reveals the algebraic structure that links local measurements to the underlying couplings and highlights the trade-offs between measurement resources, numerical stability, and identifiability. The analysis also clarifies how additional local fields or self-interaction terms enhance the information content of thermal data, suggesting a role for local thermal probes as practical tools for Hamiltonian verification and parameter estimation.Building on the ideas of controlling and inferring information about a quantum system, we now turn to active intervention through adaptive measurement. In Chapter 5, we develop a process tensor driven procedure for cooling the spin bath that is coupled to a nitrogen vacancy (NV) center in diamond. By modeling the sequence of intermediate quantum instruments acting on a correlated system-environment process, we optimize a pair of completely positive (CP) maps to maximize the population in a lower-energy state. Unlike existing approaches that rely solely on positive operator valued measures (POVMs), our method encompasses a broader class of CP maps, including adaptive feed-forward and mixed operations. We explicitly account for initial correlation between system and environment and distinguish between the controllable and uncontrollable parts of the dynamics. Our framework supports non-Markovian operations and benchmarks them against optimal Markovian strategies. The optimization is performed via Semi-definite Programming (SDP), ensuring global optimality. We also assess robustness to Hamiltonian noise and provide a proposal for implementation using NV center in diamonds.Finally, Chapter 6 explore the relaxation dynamics of topological qubits, focusing on a tetron qubit formed from two uncoupled Kitaev chains coupled to a finite fermionic bath. Using the correlation matrix formalism for purely quadratic couplings, we show that the relaxation is fundamentally independent of temperature when either the bath is large, or the zero-mode energy of the system is close to zero. This challenges a key assumption in the field that cooling enhances coherence. Interestingly, we find that introducing glassy disorder into the system slows down relaxation, offering a potential route to prolong qubit lifetimes.Collectively, these studies form a coherent narrative around extraction, control and utilization of quantum information under fundamental constraints. By tackling such inverse and often ill-posed problems, this thesis contributes new tools and insights essential for the realization of future quantum technologies.
Recently there have been numerous proposals to create Majorana zero modes in solid state heterojunctions, superconducting wires, and optical lattices. Putatively the information stored in qubits constructed from these modes is protected from various forms of decoherence. Here we present a generic method to study the effect of external perturbations on these modes. We focus on the case where there are no interactions between different Majorana modes either directly or through intermediary fermions. To quantify the rate of loss of the information stored in the Majorana modes we study the two-time correlators for qubits built from them. We analyze a generic gapped fermionic environment (bath) interacting via tunneling with different components of the qubit (different Majorana modes). We present examples with both static and dynamic perturbations (noise), and using our formalism derive a rate of information loss, for Majorana memories, that depends on the spectral density of both the noise and the fermionic bath.
Entanglement as a fundamental aspect of quantum mechanics plays a crucial role in many-body systems. Quantum dynamics, due to the intricate interplay between the locality of the interaction and the non-local nature of entanglement, greatly enriches the phenomena and brings the subject to the next complexity. In this dissertation, I will investigate entanglement and dynamics in low-dimensional systems from various perspectives. Chapter 1 examines the relationship between entanglement and topology in gapped quantum systems. We introduce a new framework called entanglement response, which offers a new perspective on the previously conjectured modular commutator formula that extracts the chiral central charge. Moreover, by incorporating global symmetries, we derive a new formula for the quantum Hall conductance. Chapter 2 extends our analysis of entanglement to mixed states. In particular, we inquire how entanglement can be used to define topological order in mixed states that are obtained by applying local decoherence to gapped ground states. This problem is naturally related to the active error correction, as we will address. Chapter 3 focuses on the dynamics of one-dimensional conformal field theories under specially engineered Floquet driving. We show that the Floquet dynamics support a heating phase that exhibits robust superuniversal features that rely only on conformal invariance. This heating phase is also shown to be useful in efficiently cooling. Chapter 4 delves into the entanglement dynamics of one-dimensional random unitary circuit doped with projective measurement. We focus on the limit where the measurement happens with a small probability. We will offer an explanation of the stability of the steady state by invoking the idea of quantum error correction.
We show that for closed finite-sized systems with an odd number of real fermionic modes, even in the presence of many-body interactions, there are always at least two fermionic operators that commute with the Hamiltonian. There is a zero mode corresponding to the total Majorana operator, as shown by Akhmerov [Phys. Rev. B 82, 020509 (2010)], as well as additional linearly independent zero modes, one of which is (1) continuously connected to the Majorana mode solution in the noninteracting limit, and (2) less prone to decoherence when the system is opened to contact with an infinite bath. We also show that in the idealized situation where there are two or more well-separated zero modes each associated with a finite number of interacting fermions at a localized vortex, these modes have non-Abelian Ising statistics under braiding. Furthermore the algebra of the zero mode operators makes them useful for fermionic quantum computation [S. B. Bravii and A. Y. Kitaev, Ann. Phys. 298, 210 (2002)].
Quantum simulation platforms such as cold atoms, superconducting circuits, and trapped ions enable the experimental investigation of quantum many-body systems, with important implications for quantum technologies. In this thesis, we investigate how entanglement and correlations develop in such systems and how they can be harnessed both for the theoretical study of models relevant to condensed matter physics and for practical applications in quantum metrology. We begin by focusing on the adiabatic preparation of quantum many-body ground states. In particular, we show that quantum simulators can probe the link between spectral degeneracy and Spontaneous Symmetry Breaking, as pioneered by P.W. Anderson. We prove that odd-sized spin-½ systems exhibiting time-reversal symmetry sustain a net magnetization even away from the thermodynamic limit, thanks to an exact spectral degeneracy. Moreover, the symmetry-broken state exhibits scalable spin squeezing, a key resource for quantum metrology. We then turn to a different paradigm of evolution: quench dynamics. In particular, we show that monitoring the real-time evolution of correlations allows for the experimental measurement of the dispersion relation of fundamental excitations. In the case of two-dimensional long-range ordered systems, such as the dipolar XY model, these excitations are spin waves, leading to sharp features in the evolution of two-point correlation functions. Conversely, in one-dimensional systems, fundamental excitations have a fermionic nature, and their detection requires the measurement of non-local correlation functions. The latter, while inaccessible in condensed matter, are instead well suited to quantum simulators with individually addressable degrees of freedom.