Iterated Function Systems (IFS) generate invariant measures with fractal structures through the random iteration of a family of contraction mappings, servingas a standard framework for describing scale generation phenomena in classicalnonlinear dynamics. This paper generalizes classical IFS to the quantum domain,establishing a rigorous mathematical formulation of Quantum Iterated FunctionSystems (QIFS): classical contraction mappings are replaced by families of completely positive maps acting on the space of density operators, and the quantumattractor state is defined as the fixed point satisfying a self-similarity equation. Onthis basis, we prove that the rescaled position distribution of discrete-time quantum random walks with decoherence converges to a convex combination of normaldistributions, with the convergence rate controlled by the spectral gap of the superoperator. Furthermore, we establish a quantum-classical correspondence principle:in the limit of vanishing decoherence rate, the support of the Wigner function ofthe quantum attractor state converges in distribution to the classical IFS attractor,and the convergence exponents satisfy a scaling relation with the classical contraction rates. This framework unifies fractal geometry, quantum open systems, anddecoherence theory into a coherent generative language, providing a rigorous mathematical foundation for understanding scaling behavior in quantum informationprocessing.
Iterated Function Systems (IFS) generate invariant measures with fractal structures through the random iteration of a family of contraction mappings, servingas a standard framework for describing scale generation phenomena in classicalnonlinear dynamics. This paper generalizes classical IFS to the quantum domain,establishing a rigorous mathematical formulation of Quantum Iterated FunctionSystems (QIFS): classical contraction mappings are replaced by families of completely positive maps acting on the space of density operators, and the quantumattractor state is defined as the fixed point satisfying a self-similarity equation. Onthis basis, we prove that the rescaled position distribution of discrete-time quantum random walks with decoherence converges to a convex combination of normaldistributions, with the convergence rate controlled by the spectral gap of the superoperator. Furthermore, we establish a quantum-classical correspondence principle:in the limit of vanishing decoherence rate, the support of the Wigner function ofthe quantum attractor state converges in distribution to the classical IFS attractor,and the convergence exponents satisfy a scaling relation with the classical contraction rates. This framework unifies fractal geometry, quantum open systems, anddecoherence theory into a coherent generative language, providing a rigorous mathematical foundation for understanding scaling behavior in quantum informationprocessing.
An iterated function system (IFS) is defined by specifying a set of functions in a classical phase space, which act randomly on an initial point. In an analogous way, we define a quantum IFS (QIFS), where functions act randomly with prescribed probabilities in the Hilbert space. In a more general setting, a QIFS consists of completely positive maps acting in the space of density operators. This formalism is designed to describe certain problems of nonunitary quantum dynamics. We present exemplary classical IFSs, the invariant measure of which exhibits fractal structure, and study properties of the corresponding QIFSs and their invariant states.
The mathematical structure of quantum groups, as q-deformations of Lie algebras, has been extensively studied, yet their deep connection with the dynamicsof open quantum systems remains to be elucidated. Based on the framework ofQuantum Iterated Function Systems (QIFS), this paper establishes an emergencemechanism from quantum groups to classical Lie groups. We first construct a concrete QIFS whose generators are a family of completely positive maps acting onthe space of quantum states, and prove that the Lie algebra structure of its lowenergy effective theory corresponds to the Drinfeld–Jimbo quantum group Uq(g).Next, we derive the relation between the q-parameter and the decoherence rate Γ,characteristic coherence time τc as q = e−Γτc, revealing the physical origin of theq-deformation. Based on this, we prove that in the limit of vanishing decoherencerate Γ → 0, the quantum group degenerates to the classical Lie group g. Finally, weestablish a correspondence between the representation theory of quantum groupsand the spectrum of QIFS, providing a unified mathematical framework for understanding the emergence mechanism of quantum symmetries. This result links theconcept of decoherence in quantum information theory with the algebraic structureof quantum groups, revealing a deeper picture where classical Lie group symmetriesemerge as the semiclassical limit of open quantum system dynamics.
The computational advantage of quantum algorithms over classical algorithmshas been partially proven, but its deep connection with the dynamics of open quantum systems has yet to be elucidated. This paper establishes a formal correspondence between quantum algorithm complexity and dissipative dynamics based onthe framework of Quantum Iterated Function Systems (QIFS). We first prove thatthe convergence rate of a QIFS—governed by the spectral gap ∆—directly determines the query complexity and time complexity of the corresponding quantumalgorithm. Second, we establish an exponential relationship between the quantumgroup parameter q and the success probability of the algorithm, revealing the dualrole played by decoherence in quantum search algorithms: moderate decoherenceaccelerates convergence, while excessive decoherence causes the algorithm to fail.Building on this, we propose the “dissipative speedup” theorem: there exists an optimal decoherence rate Γopt such that the quantum algorithm described by the QIFSachieves minimal time complexity, and this optimal value satisfies Γopt = −lnq/tc.Finally, we explore the connections between this framework and adiabatic quantum computation as well as quantum walk algorithms, and propose experimentalschemes to test the theoretical predictions on superconducting quantum processors.This work unifies the theory of open quantum systems, quantum group algebra,and quantum computational complexity into a common generative framework, providing a new perspective for understanding the persistence of quantum advantagein noisy environments.
This paper establishes a rigorous correspondence between quantum generativedynamics and quantum algorithmic complexity, reducing the query complexity andtime complexity of algorithms to the spectral properties of the generative process.Byembedding quantum algorithms within the framework of quantum iterated function systems, it is proven that the convergence rate of an algorithm is uniquely determined by the spectral gap ∆ of the Markov operator: the required number of iterations T ∼ (1/∆)ln(1/ε). Furthermore, the dual role of decoherence in quantumsearch is revealed-the competition between coherent and dissipative contributionsleads to the existence of an optimal decoherence rate, given by Γopt ∼ −(1/4)lnN,which corresponds to the optimal complexity of Grover’s algorithm. By relatingthe spectral gap to the quantum group parameter q = e−Γtc, we derive ∆ = 1−q2,establishing a direct correspondence between algebraic parameters and computational resources. This framework unifies open quantum system theory, quantumgroup algebra, and quantum computational complexity under the language of generative dynamics, providing a new theoretical foundation for understanding thebehavior of quantum algorithms in noisy environments.
The classical Schrödinger bridge seeks the most likely probability law for a diffusion process, in path space, that matches marginals at two end points in time; the likelihood is quantified by the relative entropy between the sought law and a prior. Jamison proved that the new law is obtained through a multiplicative functional transformation of the prior. This transformation is characterised by an automorphism on the space of endpoints probability measures, which has been studied by Fortet, Beurling, and others. A similar question can be raised for processes evolving in a discrete time and space as well as for processes defined over non-commutative probability spaces. The present paper builds on earlier work by Pavon and Ticozzi and begins by establishing solutions to Schrödinger systems for Markov chains. Our approach is based on the Hilbert metric and shows that the solution to the Schrödinger bridge is provided by the fixed point of a contractive map. We approach, in a similar manner, the steering of a quantum system across a quantum channel. We are able to establish existence of quantum transitions that are multiplicative functional transformations of a given Kraus map for the cases where the marginals are either uniform or pure states. As in the Markov chain case, and for uniform density matrices, the solution of the quantum bridge can be constructed from the fixed point of a certain contractive map. For arbitrary marginal densities, extensive numerical simulations indicate that iteration of a similar map leads to fixed points from which we can construct a quantum bridge. For this general case, however, a proof of convergence remains elusive.
The classical Schrödinger bridge seeks the most likely probability law for a diffusion process, in path space, that matches marginals at two end points in time; the likelihood is quantified by the relative entropy between the sought law and a prior. Jamison proved that the new law is obtained through a multiplicative functional transformation of the prior. This transformation is characterised by an automorphism on the space of endpoints probability measures, which has been studied by Fortet, Beurling, and others. A similar question can be raised for processes evolving in a discrete time and space as well as for processes defined over non-commutative probability spaces. The present paper builds on earlier work by Pavon and Ticozzi and begins by establishing solutions to Schrödinger systems for Markov chains. Our approach is based on the Hilbert metric and shows that the solution to the Schrödinger bridge is provided by the fixed point of a contractive map. We approach, in a similar manner, the steering of a quantum system across a quantum channel. We are able to establish existence of quantum transitions that are multiplicative functional transformations of a given Kraus map for the cases where the marginals are either uniform or pure states. As in the Markov chain case, and for uniform density matrices, the solution of the quantum bridge can be constructed from the fixed point of a certain contractive map. For arbitrary marginal densities, extensive numerical simulations indicate that iteration of a similar map leads to fixed points from which we can construct a quantum bridge. For this general case, however, a proof of convergence remains elusive.