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.
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 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.
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.
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.
We propose the Generative Reality Hypothesis (GRH), a novel interpretation of quantum mechanics that reconceptualizes physical reality as an information-processing system analogous to generative artificial intelligence models. Under this framework, quantum superposition represents the latent space of possible states, wave function collapse corresponds to sampling from probability distributions during information queries (measurements), and the Born rule emerges as the fundamental sampling algorithm of reality. We present rigorous mathematical foundations establishing formal equivalences between quantum dynamics and specific generative models. Our framework recasts the measurement problem as a question of computational process, explains non-local correlations as the enforcement of consistency within a shared generative model, and provides a mechanistic explanation for wave-particle duality as a transition between two computational modes: continuous latent evolution (wave) and discrete sampling (particle). We propose five specific, falsifiable experimental tests using current quantum technologies that can distinguish GRH from standard quantum mechanics through quantitative predictions about complexity-dependent decoherence, measurement dynamics, and information flow. Statistical criteria for falsification are established with required experimental precision specified. The framework connects quantum mechanics to information theory and computational complexity, offering a computationally grounded foundation for quantum phenomena while making testable predictions that differ measurably from existing interpretations.
This dissertation is concerned with a variety of problems in quantum computing and the spectral theory of quantum-mechanical channels and Hamiltonians. We begin with the problem of quantum gate synthesis. The challenge is to find a fixed, finite set S of quantum gates such that, given a quantum circuit component U acting on n qubits and target precision ε, there is a short, efficiently computable composition of gates from S that approximates U to precision ε. This problem has already led to a rich literature in the case n=1 with deep connections to modern number theory. In this dissertation we present the state-of-the-art algorithm for a single qubit, achieving length 7 log (1/ε). The techniques used include the lattice structure of certain dense subgroups of PU(2) and Lenstra's algorithm for integer programming in convex domains. The next topic considered is scientific computing algorithms intended to run on a quantum computer. Specifically, we study a framework for computing functions of normal matrices using a quantum computer, with the objective of controlling the ancilla overhead, probability of success, and end precision. This arises as a special case of the multivariate quantum eigenvalue transform, for computing function of multiple commuting Hermitian matrices. The techniques applied include approximation of regular functions by low-degree polynomials and modern quantum primitives like quantum signal processing and linear combinations of unitaries. The third component is a spectral gap comparison result for Davies generators, a class of Lindbladians used for Gibbs state preparation and which are a well-studied example of quantum Markov dynamics, and which contain a canonical classical Markov chain. The main result shows that if the spectrum of the Hamiltonian driving the dynamics is free of long arithmetic progressions, then the spectral gaps of the quantum and classical Markov chains are comparable. Thus, classical techniques may be leveraged to control the behavior of the quantum dynamics. The proof is based on transferring any "off-diagonal" eigenvector of the Davies generator to an observable which commutes with the Hamiltonian and has a comparable Lindbladian Rayleigh quotient. Finally, we study the infimum of the spectrum, or ground state energy, of a discrete Schrödinger operator on the scaled integer grid in n dimensions, parameterized by a nonnegative potential V and a small frequency parameter θ. We relate this ground state energy to that of a corresponding continuous semiclassical Schrödinger operator on the full n-dimensional space with parameter θ, arising from the same choice of potential, and show that these quantities are equal up to vanishing factors depending on the regularity of V. The proofs are elementary and rely on sampling and interpolation to map low-energy functions for the discrete operator on the scaled grid to low-energy functions for the continuous operator on the full space, and vice versa. Coauthors on these works were J. Basso, T. R. Blackman, Y. Borns-Weil, I. Detherage, S. Ganguly, N. Srivastava, T. Saffat, A. Sinclair, and T.-D. Vuong.
Quantum computing is the offspring of quantum mechanics and computer science, two great scientific fields founded in the 20th century. Quantum computing is a relatively young field and is recognized as having the potential to revolutionize science and technology in the coming century. The primary question in this field is essentially to ask which problems are feasible with potential quantum computers and which are not. In this dissertation, we study this question with a physical bent of mind. We apply tools from computer science and mathematical physics to study the complexity of simulating quantum systems. In general, our goal is to identify parameter regimes under which simulating quantum systems is easy (efficiently solvable) or hard (not efficiently solvable). This study leads to an understanding of the features that make certain problems easy or hard to solve. We also get physical insight into the behavior of the system being simulated. In the first part of this dissertation, we study the classical complexity of simulating quantum dynamics. In general, the systems we study transition from being easy to simulate at short times to being harder to simulate at later times. We argue that the transition timescale is a useful measure for various Hamiltonians and is indicative of the physics behind the change in complexity. We illustrate this idea for a specific bosonic system, obtaining a complexity phase diagram that delineates the system into easy or hard for simulation. We also prove that the phase diagram is robust, supporting our statement that the phase diagram is indicative of the underlying physics. In the next part, we study open quantum systems from the point of view of their potential to encode hard computational problems. We study a class of fermionic Hamiltonians subject to Markovian noise described by Lindblad jump operators and illustrate how, sometimes, certain Lindblad operators can induce computational complexity into the problem. Specifically, we show that these operators can implement entangling gates, which can be used for universal quantum computation. We also study a system of bosons with Gaussian initial states subject to photon loss and detected using photon-number-resolving measurements. We show that such systems can remain hard to simulate exactly and retain a relic of the "quantumness" present in the lossless system. Finally, in the last part of this dissertation, we study the complexity of simulating a class of equilibrium states, namely ground states. We give complexity-theoretic evidence to identify two structural properties that can make ground states easier to simulate. These are the existence of a spectral gap and the existence of a classical description of the ground state. Our findings complement and guide efforts in the search for efficient algorithms.