We introduce a tensor-network channel framework for compressed quantum dynamics on operator subsystems associated with variational many-body manifolds. Starting from a tensor-network manifold, we construct manifold-induced Heisenberg channels acting on observables and study the resulting sequential dynamics via their Schr\"odinger duals. The central mathematical result is a compression-stability statement establishing controlled approximation of sequential manifold-induced evolution, yielding a well-defined operator-system description of compressed dynamics with quantitatively bounded leakage. To connect this operator-algebraic structure with tensor-network geometry, we formulate a representability assumption asserting that intrinsic states on the compressed operator subsystem admit tensor-network representatives of bounded bond dimension. Under this assumption, tensor-network geometric structure emerges as a consequence of compressed operator dynamics rather than being imposed a priori as a variational ansatz. All geometric statements are therefore conditional on this representability assumption and do not follow from the operator-algebraic framework alone. In this setting, we establish a graph-cut mutual-information bound for representable states and introduce a geometric correlation-capacity functional determined by bond dimension and network architecture. These results provide quantitative constraints on correlation propagation within the compressed subsystem and relate information flow directly to tensor-network geometry. The framework separates a rigorous operator-algebraic layer—governing compressed observable dynamics and stability under sequential projection—from a conditional geometric layer arising under representability. This separation isolates a mathematically precise core independent of tensor-network structure while identifying the representability problem as the key bridge to geometry. From a physical perspective, tensor-network geometry becomes a dynamical constraint on information propagation and correlation capacity in compressed quantum systems. Illustrative examples involving matrix-product-state manifolds and comparisons across tensor-network architectures demonstrate how graph geometry controls representable correlation scaling. The resulting picture establishes a unified connection between compressed quantum dynamics, operator systems, tensor-network geometry, and quantum information theory, with geometry emerging from operator-algebraic compression under representability.
We introduce a tensor-network channel framework for compressed quantum dynamics on operator subsystems associated with variational many-body manifolds. Starting from a tensor-network manifold, we construct manifold-induced Heisenberg channels acting on observables and study the resulting sequential dynamics via their Schr"odinger duals. The central mathematical result is a compression-stability statement establishing controlled approximation of sequential manifold-induced evolution, yielding a well-defined operator-system description of compressed dynamics with quantitatively bounded leakage. To connect this operator-algebraic structure with tensor-network geometry, we formulate a representability assumption asserting that intrinsic states on the compressed operator subsystem admit tensor-network representatives of bounded bond dimension. Under this assumption, tensor-network geometric structure emerges as a consequence of compressed operator dynamics rather than being imposed a priori as a variational ansatz. All geometric statements are therefore conditional on this representability assumption and do not follow from the operator-algebraic framework alone. In this setting, we establish a graph-cut mutual-information bound for representable states and introduce a geometric correlation-capacity functional determined by bond dimension and network architecture. These results provide quantitative constraints on correlation propagation within the compressed subsystem and relate information flow directly to tensor-network geometry. The framework separates a rigorous operator-algebraic layer—governing compressed observable dynamics and stability under sequential projection—from a conditional geometric layer arising under representability. This separation isolates a mathematically precise core independent of tensor-network structure while identifying the representability problem as the key bridge to geometry. From a physical perspective, tensor-network geometry becomes a dynamical constraint on information propagation and correlation capacity in compressed quantum systems. Illustrative examples involving matrix-product-state manifolds and comparisons across tensor-network architectures demonstrate how graph geometry controls representable correlation scaling. The resulting picture establishes a unified connection between compressed quantum dynamics, operator systems, tensor-network geometry, and quantum information theory, with geometry emerging from operator-algebraic compression under representability.
We propose a unified geometric framework in which space and time emerge as observer-dependent projections of quantum correlations embedded in the universal wavefunction. In this approach, we formulate six axioms (including a duality between entanglement and time) that establish a correspondence between correlation structures and effective spacetime geometry. We introduce the "correlationhedron", a convex geometric object representing the set of all permissible correlation vectors of a given state, and show how different observer "slicings" (modeled by projection maps) yield distinct emergent spacetimes from this object. An emergent metric is defined on each observer's spacetime via the second derivative of a correlation density (analogous to a Fisher information metric), linking gradients in quantum entanglement to curvature. This paper establishes the conceptual foundations of this approach as the first in a planned series of investigations. We illustrate the framework with conceptual and quantitative examples, including an explicit calculation of the emergent geometry for a two- qubit system in both separable and entangled states. Connections to holography (AdS/CFT) (Maldacena1998, Ryu2006), tensor networks (Swingle2012), and relational quantum approaches (Rovelli1996) are discussed, along with experimental signatures (such as curvature variations with entanglement) that could support the theory. Finally, we outline the limitations of the framework, particularly regarding dynamics and observer-dependent causality, to guide future investigations. This kinematic framework establishes the conceptual skeleton of emergent spacetime; detailed constructions, proofs of smoothness, and dynamical laws will appear in forthcoming work.
Many diagnostics of quantum dynamics focus on operator growth, scrambling, complexity generation, or information spreading. Examples include out-of-time-order correlators, Krylov complexity, operator entanglement, and related algebraic formulations of scrambling. These diagnostics provide complementary perspectives on how operators evolve under unitary dynamics, but they generally emphasize expansion into progressively larger regions of operator space. In this work we develop a filtration-relative viewpoint. Given a nested sequence of operator algebrasA0⊂A1⊂⋯⊂An=B(H),the Hilbert--Schmidt structure of operator space induces an orthogonal decomposition into filtration sectors. Hamiltonian evolution generates transport among these sectors. We define transport coefficients that quantify the strength of sector-to-sector coupling and show that they induce a weighted operator-flow network. Recoverability is then described geometrically relative to a chosen coarse-graining. Recoverable regions, boundary transport, persistence, leakage pathways, and bottlenecks arise naturally from the network structure generated by the filtration. The resulting framework does not attempt to replace existing scrambling diagnostics. Rather, it provides a complementary coarse-grained description adapted to physically meaningful operator-algebra structures. A central structural result is a reciprocity relation for transport coefficients generated by Hermitian Hamiltonians,diTji=djTij,where di denotes the dimension of the i-th filtration sector. Consequently, transport support graphs induced by Hamiltonian evolution are intrinsically undirected, although normalized transport weights may remain asymmetric. This reciprocity theorem clarifies the geometric structure underlying filtration-induced transport. Several finite-dimensional examples are developed explicitly, including diagonal and block-algebra filtrations. We further analyze an associated effective leakage model illustrating how network bottlenecks can generate long persistence times. This model is presented as an illustrative coarse-grained approximation rather than as a consequence of the underlying Hamiltonian framework. The principal contribution of the paper is therefore not a new theory of recoverability, but a filtration-based geometry that organizes persistence, leakage, and recoverability properties relative to nested operator-algebra structures.
We develop a block-encoding compiler that treats matrix product operators as compressed, virtual-path linear combination of unitaries programs. The compiler constructs conditional PREP and local SELECT stages directly from a parent matrix product operator, establishing tensor networks as structured quantum intermediate representations that can be efficiently compiled to block-encoded circuits. We apply the construction to real-time evolution in the Heisenberg chain and two perturbed Heisenberg-family models. Across the regimes studied, the compressed, approximately unitary propagator MPOs retain mild bond dimension and LCU normalization. Relative to an LCU that explicitly lists the O(NK) Pauli-product branches of an order-K truncated Taylor polynomial, our virtual-transition implementation replaces combinatorial branch enumeration by a circuit complexity scaling as O(αMPONχ2), approaching O(Nχ2) when αMPO remains mild. We numerically characterize how truncation order, bond-dimension budget, and system size affect approximation error, normalization, and compiler cost. These results demonstrate how classically compressed tensor-network representations can serve as quantum compiler intermediate representations for block encoding and opens new avenues to accelerate quantum algorithms.
We study entanglement reachability in quantum many-body systems evolving under physically admissible local dynamics. While Lieb--Robinson bounds constrain the propagation of information, and complexity-based approaches characterize the resources required to represent or prepare quantum states, these frameworks do not by themselves determine which entanglement structures are dynamically attainable. The central question addressed in this work is therefore: which entanglement structures can arise under constrained quantum evolution, and which remain inaccessible despite being kinematically allowed? We introduce a unified axiomatic framework for entanglement reachability based on the notion of \emph{structural confinement}. The framework incorporates finite-range locality, finite-dimensional memory degrees of freedom, and abstract representability constraints expressed through operator subsystems. Rather than characterizing accessibility directly in Hilbert space, we formulate the problem on an induced structural space of entanglement configurations and quantum information patterns. We establish a structural confinement theorem showing that admissible dynamics generate only a restricted reachable region in structural space. The theorem provides a general criterion for identifying forbidden structures whenever their defining information cannot be preserved within the admissible representational framework. Thus, ocality, memory, and representability constraints do not merely limit the rate of information propagation; they impose additional dynamical restrictions on the set of quantum structures that can actually be realized. The generality of the framework is demonstrated through three explicit realizations involving multipartite coherence, tensor-network representability, and protected logical nformation in quantum error-correcting codes. Although these realizations involve distinct physical structures, they are governed by the same underlying confinement mechanism: dynamical accessibility is determined by the compatibility between physical evolution and the structural information retained by the chosen representation. These results establish structural confinement as a general organizing principle for entanglement reachability under constrained quantum dynamics. The framework provides a unified language for deriving dynamical accessibility limits beyond propagation bounds alone and for classifying which quantum structures are reachable, forbidden, or conditionally accessible in locally constrained many-body systems.
Quantum kernels admit Fourier representations whose frequencies are determined by the data-encoding gates of the underlying feature map. We show that entangling tensor kernels are matrix product operator factorizations of the corresponding Fourier coefficient tensors, thereby identifying quantum kernels as spectral tensor networks. By grouping gate-level frequency configurations that yield the same feature-wise frequency, we obtain a grouped Fourier form that induces a more compact spectral tensor network representation of the kernel. We further show that kernel target alignment serves as a bridge between the Fourier and tensor network views. On a grid that resolves the accessible Fourier modes, it becomes the Frobenius cosine similarity between Fourier coefficient tensors. Our numerical experiments show that layered quantum kernels admit accurate representations with small bond dimension, revealing a compressibility governed by correlations between Fourier modes. This compressibility provides a diagnostic of classical representability and of whether kernel evaluation is likely to remain classically tractable.
We develop an operator-based framework in which quantum dynamics, interaction structure, and geometric features arise from the modular properties of a density operator constructed from a non-commuting pair of Hermitian operators. A positive state is defined via exponential coarse-graining, and time evolution is derived from the associated modular Hamiltonian rather than postulated. Using a systematic Baker–Campbell–Hausdorff expansion, interaction terms are shown to be organized by nested commutators, yielding a natural algebraic hierarchy analogous to effective field theory. Interpreting the coarse-graining parameter as a renormalization scale, we derive flow equations for operator-valued couplings, establishing a renormalization group structure in operator space. A continuum field-theoretic extension and a lattice regularization suitable for numerical investigation are constructed. Local automorphisms of the operator algebra give rise to gauge connections, while the quantum Fisher information metric defines an emergent notion of geometry. Under additional assumptions of locality and approximate equilibrium, the framework reproduces the structure of entropic approaches to gravitational dynamics. These results provide a unified algebraic perspective in which dynamics, interactions, and geometry are encoded in the modular structure of quantum states.