We demonstrate that time-evolved operators can construct a Krylov space to compute operator complexity and introduce Krylov observability as a measure of effective phase space dimension in quantum systems. We test Krylov observability in the framework of quantum reservoir computing and show that it closely mirrors information processing capacity, a data-driven expressivity metric, while achieving computation times that are orders of magnitude faster. Our results validate operator complexity and give the interpretation that data in a quantum reservoir are mapped onto the Krylov space.
We demonstrate that time-evolved operators can construct a Krylov space to compute operator complexity and introduce Krylov observability as a measure of effective phase space dimension in quantum systems. We test Krylov observability in the framework of quantum reservoir computing and show that it closely mirrors information processing capacity, a data-driven expressivity metric, while achieving computation times that are orders of magnitude faster. Our results validate operator complexity and give the interpretation that data in a quantum reservoir are mapped onto the Krylov space.
This study employs Krylov-based information measures to understand task performance in quantum reservoir computing, a subfield of quantum machine learning. In our study, we show that fidelity and spread complexity can only explain the task performance for short time evolutions of the quantum systems. We then discuss two measures, Krylov expressivity and Krylov observability, and compare them to task performance and the information processing capacity. Our results show that Krylov observability exhibits almost identical behavior to information processing capacity, while being three orders of magnitudes faster to compute. In the case when the system is undersampled, Krylov observability best captures the behavior of the task performance.
A bstract In this work, we propose a quantum-mechanically measurable basis for the computation of spread complexity. Current literature focuses on computing different powers of the Hamiltonian to construct a basis for the Krylov state space and the computation of the spread complexity. We show, through a series of proofs, that time-evolved states with different evolution times can be used to construct an equivalent space to the Krylov state space used in the computation of the spread complexity. Afterwards, we introduce the effective dimension, which is upper-bounded by the number of pairwise distinct eigenvalues of the Hamiltonian. The computation of the spread complexity requires knowledge of the Hamiltonian and a classical computation of the different powers of the Hamiltonian. The computation of large powers of the Hamiltonian becomes increasingly difficult for large systems. The first part of our work addresses these issues by defining an equivalent space, where the original basis consists of quantum-mechanically measurable states. We demonstrate that a set of different time-evolved states can be used to construct a basis. We subsequently verify the results through numerical analysis, demonstrating that every time-evolved state can be reconstructed using the defined vector space. Based on this new space, we define an upper-bounded effective dimension and analyze its influence on finite-dimensional systems. We further show that the Krylov space dimension is equal to the number of pairwise distinct eigenvalues of the Hamiltonian, enabling a method to determine the number of eigenenergies the system has experimentally. Lastly, we compute the spread complexities of both basis representations and observe almost identical behavior, thus enabling the computation of spread complexities through measurements.
In the Wigner-Weyl phase space formulation of quantum mechanics, we analyse the problem of the spreading of an initial state or an initial operator under time evolution when described in terms of the Krylov basis. After constructing the phase space functions corresponding to the Krylov basis states generated by a Hamiltonian from a given initial state by using the Weyl transformation, we subsequently use them to cast the Krylov state complexity as an integral over the phase space in terms of the Wigner function of the time-evolved initial state, so that the contribution of the classical Liouville equation and higher-order quantum corrections to the Wigner function time evolution equation towards the Krylov state complexity can be identified. Next, we construct the double phase space functions associated with the Krylov basis for operators by using a suitable generalisation of the Weyl transformation applicable for superoperators, and use them to rewrite the Krylov operator complexity as an integral over the double phase space in terms of a generalisation of the usual Wigner function. These results, in particular, show that the complexity measures based on the expansion of a time-evolved state (or an operator) in the Krylov basis can be thought to belong to a general class of complexity measures constructed from the expansion coefficients of the time-dependent Wigner function in an orthonormal basis in the phase space, and help us to connect these complexity measures with measures of complexity of time evolved state based on harmonic expansion of the time-dependent Wigner function.
Diese Dissertation untersucht das Zusammenspiel von Quantendynamik, Quanteninformationstheorie und Quantum Reservoir Computing (QRC). Im Zentrum steht die Frage, wie sich Quantensysteme zeitlich entwickeln und inwieweit das Verständnis dieser Zeitentwicklung Einblicke in QRC geben kann. Die Arbeit führt zunächst in Grundlagen zu Quantenmechanik, dynamischen Systemen und Reservoir Computing ein. Anschließend wird die Krylov Complexity untersucht, wobei gezeigt wird, dass Krylov-Räume auch aus zeitentwickelten Zuständen oder Operatoren gebildet werden können, was ein Vorteil für QRC ohne bekannten Hamiltonian ist. Zudem wird festgestellt, dass diese Räume oft eine geringere Komplexität besitzen, was bisherige Annahmen infrage stellt. Zur Charakterisierung werden die Konzepte Krylov Expressivity und Krylov Observability eingeführt. In Kapitel IV wird gezeigt, dass Krylov Observability die data expressivity im Quantum Reservoir Computing (QRC) besser beschreibt als die Krylov Spread Complexity. Dies unterstützt die Annahme, dass QRC Daten in den zugehörigen Krylov-Raum abbildet. Darüber hinaus wird ein modellhafter Ansatz mit beschränktem Gedächtnis vorgestellt, der das Problem des Zustandskollapses bei Messungen umgeht und die Effizienz sowie die Leistung durch Ausnutzung der natürlichen fading memory-Eigenschaft verbessert. Kapitel V untersucht Phasenübergänge in ungeordneten Quantensystemen mithilfe quanteninformativer Maße. Dabei zeigt sich, dass die Krylov Complexity nicht nur zwischen ergodischem und lokalisiertem Verhalten, sondern auch zwischen verschiedenen lokalisierten Phasen unterscheiden kann. Eine daraus abgeleitete charakteristische Zeitgröße korreliert am stärksten mit der data expressivity im QRC. Die Arbeit verbindet Methoden aus Quanteninformation, Quantenstatistik, Krylov-Complexity, maschinellem Lernen, Phasenübergangstheorie und Dynamik. Sie liefert neue Einsichten in das Verhalten und die Steuerung quantenmechanischer Systeme und eröffnet Perspektiven für weitere Forschung in der Quanteninformation.
Quantum simulation of complex many-body systems beyond classical computational capabilities provides a promising route toward understanding novel quantum phases and their transitions. In particular, analog quantum simulators with global control fields have attracted considerable attention due to their potential to simulate arbitrary Hamiltonians and perform quantum computing tasks. However, a clear, quantitative measure for the complexity of implementing specific quantum operations in such systems is still lacking. In this Letter, we address this challenge by introducing generalized Krylov complexity, a concept originating from operator growth dynamics, as a direct diagnosis for this synthesis complexity. We construct the block Krylov basis generated by a set of Hamiltonians, which naturally organizes the operator space achievable through the simulator's native interactions and their nested commutators. By analyzing representative systems including Rydberg atom arrays, we demonstrate that the generalized Krylov complexity of a target operation serves as a strong predictor of the minimum time required for its realization. Our results establish Krylov complexity as an intuitive and predictive tool for designing efficient control protocols in analog quantum simulators.
We study the quantum dynamics in Krylov space of a spatially flat, homogeneous, and isotropic universe sourced with a massless scalar field within Wheeler-DeWitt (WDW) quantum cosmology and loop quantum cosmology (LQC) frameworks. The availability of a physical Hilbert space and physical Hamiltonian and the presence of an internal clock enable us to construct the Krylov basis analytically by applying the Lanczos algorithm. We then evaluate both the Krylov state and operator complexity for WDW quantum cosmology and LQC on this basis. In regimes where the wave function of the universe is sharply peaked, our results indicate that the Krylov complexity grows quadratically with the scalar field clock for the state and operator complexities in both the WDW quantum cosmology and LQC. We further show that the operator complexity is exactly twice the state complexity in these regimes. We discuss the interpretation of the global behavior of these systems by calculating the Krylov entropy for both quantum cosmological frameworks. We observe that in LQC, the Krylov complexity and entropy remain finite at the bounce, whereas in the WDW quantum cosmology, they diverge at the big bang/crunch singularity. Our work provides the first example of computing Krylov complexity for a system with a totally constrained Hamiltonian and no external time, a framework to calculate a purely quantum-mechanical entropy in quantum cosmology, and, to our knowledge, the first direct bridge between Krylov complexity and canonical quantum cosmology, as a first step toward understanding how polymerized quantum geometry modifies complexity and entropy.