Quantum subsystem structure is typically specified independently of the workload, yet the decomposition that best preserves locality under a Hamiltonian need not be the one that best supports repeated computation. We develop workload–mereology quantum computing, a framework in which generalized tensor-product structure is selected by an explicit competition between dynamical naturalness and computational demand. Physical cost is quantified using the Gaussian scrambling criterion of operational quantum mereology, while task cost measures workload interactions cut by a candidate bipartition. After dimensionless normalization, their competition is captured by Jλ(F) = Ŝ(F) + λĈₜ(F). We prove that a factorization is uniquely optimal for some workload pressure precisely when its cost pair is an exposed point of the lower convex hull, and that a genuine compromise remains optimal over a nonempty interval precisely when it forms a strict intermediate lower-hull vertex. A four-qubit construction yields the transition threshold in closed form. Exhaustive reproducible simulations show that compromise factorizations are not exceptional: at n = 10 and λ = 1, they occur in 61.33% of 300 commuting Pauli-ZZ instances for uniform and lognormal couplings (60.67% for normal couplings) and in 59.00% of generic noncommuting XX+YY+ZZ instances. These results identify a task-dependent regime of quantum subsystem organization in which workload does more than act on a predetermined decomposition: it becomes a variational ingredient in determining which decomposition is preferred.
Quantum subsystem structure is typically specified independently of the workload, yet the decomposition that best preserves locality under a Hamiltonian need not be the one that best supports repeated computation. We develop workload–mereology quantum computing, a framework in which generalized tensor-product structure is selected by an explicit competition between dynamical naturalness and computational demand. Physical cost is quantified using the Gaussian scrambling criterion of operational quantum mereology, while task cost measures workload interactions cut by a candidate bipartition. After dimensionless normalization, their competition is captured by Jλ(F) = Ŝ(F) + λĈₜ(F). We prove that a factorization is uniquely optimal for some workload pressure precisely when its cost pair is an exposed point of the lower convex hull, and that a genuine compromise remains optimal over a nonempty interval precisely when it forms a strict intermediate lower-hull vertex. A four-qubit construction yields the transition threshold in closed form. Exhaustive reproducible simulations show that compromise factorizations are not exceptional: at n = 10 and λ = 1, they occur in 61.33% of 300 commuting Pauli-ZZ instances for uniform and lognormal couplings (60.67% for normal couplings) and in 59.00% of generic noncommuting XX+YY+ZZ instances. These results identify a task-dependent regime of quantum subsystem organization in which workload does more than act on a predetermined decomposition: it becomes a variational ingredient in determining which decomposition is preferred.
We introduce the novel concept of mereological quantum phase transition (m-QPTs). Our framework is based on a variational family of operator algebras defining generalized tensor product structures (g-TPS), a parameter-dependent Hamiltonian, and a quantum scrambling functional. By minimizing the scrambling functional, one selects a g-TPS, enabling a pullback of the natural information-geometric metric on the g-TPS manifold to the parameter space. The singularities of this induced metric -- so-called algebra susceptibility -- in the thermodynamic limit characterize the m-QPTs. We illustrate this framework through analytical examples involving quantum coherence and operator entanglement. Moreover, spin-chains numerical simulations show susceptibility sharp responses at an integrability point and strong growth across disorder-induced localization, suggesting critical reorganizations of emergent subsystem structure aligned with those transitions.
This paper aims to lay some groundwork for a systematic framework for quantum mereology based on the tensor product structures (TPSs) of Hilbert space. While previous work has suggested that such a TPS-based mereology might conform to classical extensional mereology, we demonstrate that the space of all possible tensor product structures for a given Hilbert space lacks the lattice-theoretic structure characteristic of classical partitions. Specifically, we show that this space admits no canonical meet operation, thus violating the global validity of a natural extension of the weak supplementation principle. These structural features suggest that quantum mereology, even when based on TPSs, should exhibit certain non-extensional behavior: what counts as a "part" is decomposition-relative in a stronger sense, with different decompositions allowed to be mutually incompatible. The resulting picture challenges extensional interpretations of quantum composition and underscores the need for a formally richer, genuinely non-classical mereology.
Hybrid quantum-classical workflows are expected to underpin practical quantum computing applications, yet the quantum and HPC communities lack a shared framework for reasoning about where and when their integration requirements matter most. Such a framework must separate two distinct levels of analysis: the application level, where communication overhead affects runtime performance, and the real-time level, where it determines feasibility. To address this, we introduce a runtime model that decomposes workflow execution into quantum compute, classical compute, and communication costs. At the application level, a communication-to-computation ratio from this decomposition quantifies whether a workflow is communication-bound or compute-bound; at the real-time level, a feasibility constraint determines whether timing requirements can be met at all, with the reaction time setting the logical clock speed of fault-tolerant computation once they are. Application of this model to representative workflows demonstrates that co-location of quantum processors with HPC infrastructure offers negligible performance benefit for compute-intensive applications today, while tight integration remains crucial for real-time tasks such as quantum error correction needed for large scale quantum computations. However, we discuss how even at the application level these assessments may shift with hardware evolution, illustrating how the model can identify specific crossover conditions, and how, under fault tolerance, the reaction time can set application-level performance.
A quantum computer is similar to an ordinary computer, except that instead of bits it uses two-level quantum systems (qubits), and the usual logical operations on bits are replaced with noiseless quantum dynamics in the form of unitary gates or Hamiltonian interactions. Many sets of resources that are universal for quantum computation (that is, capable of implementing any quantum computation) have been found, but general conditions for universality are still only partially understood. Improving our understanding of these conditions contributes to two broad goals. On the one hand, understanding what class of physical systems is suitable for quantum computation allows us choose the systems that are most easily controlled or built in the laboratory. On the other hand, identifying the unifying characteristics of such a class provides insight into the fundamental physics responsible for the much greater computational power of quantum computers. The original results presented in this thesis are a contribution to our understanding of the physical requirements for universal quantum computation. These results lead to the identification of a new, large class of universal dynamics. More specifically, the main problem that is addressed is the following: consider a physical system that is made up of a finite number of qubits that are subject to a natural, fixed interaction described by a two-body Hamiltonian. Suppose we have the ability to control the system only by applying fast one-qubit operations. Under what conditions is this system capable of universal quantum computation? The first original result of the thesis is to prove that universal quantum computation is possible under these conditions given the single extra constraint that the Hamiltonian must entangle, or connect, all of the qubits. In other words, since a two-body Hamiltonian is expressible as a sum of coupling terms each of which acts nontrivially on a pair of qubits, consider the graph whose vertices correspond to qubits and whose edges connect qubits that are coupled by a term in the Hamiltonian. Then this Hamiltonian, together with fast one-qubit unitaries, is universal for quantum computation precisely when the corresponding graph is connected. This result suggests many potential generalizations, three of which are explored in this thesis. The first generalization is to show that the same result holds for a two-body Hamiltonian that acts on D-dimensional quantum systems (qudits), given fast one-qudit unitaries. A second extension is to the case of a many-body Hamiltonian, that is, a Hamiltonian that has coupling terms that act on more than two qubits. Again, we find that under mild restrictions, such Hamiltonians are universal provided they are connected and that arbitrary one-qubit unitaries are available. We also discover and explore some surprising structure in many-body Hamiltonians that is associated with the parity of the couplings. A third related problem that is addressed is the question of what two-qubit unitaries are universal for quantum computation. This problem was originally solved by J.-L. and R. Brylinski [BB02], who gave a technical proof of the universality of any two-qudit gate that is not the swap gate or a product of one-qudit unitaries, given one-qudit unitaries as a free resource. We develop an elementary and constructive proof for the case of qubits, and prove that our construction is very close to optimal in the number of uses of the two-qubit gate. Given a qualitative understanding of what dynamical resources are universal for quantum computation, it is natural to consider developing a quantitative measure of the usefulness of quantum dynamics for quantum information processing. This is done in the final chapter of the thesis, in which the beginnings of a quantitative theory of quantum dynamics are developed.
A finite Hilbert space supports many inequivalent tensor-product decompositions, so any Hilbert-space-first account of locality must explain more than why a single candidate performs well in a discrete search; it must also determine whether that preference governs nearby continuous refactorizations. Earlier work in this area established three key ingredients: that decomposition space is empirically non-flat under simple support-based selectors in explicit few-qubit studies; that the physically meaningful object in exact finite searches is a quotient class rather than a uniquely labeled factorization, defined modulo local gauge transformations and subsystem permutations; and that, for exactly graph-local Hamiltonians, the associated quadratic form has a kernel precisely given by the local gauge Lie algebra under explicit visibility and genericity conditions. Building on these results, the present paper proves a local theorem showing that, in a neighborhood of the graph-local class, the static selector is quantitatively equivalent to the square of the quotient distance. As a consequence, the preferred decomposition class is a strictly isolated local minimum: small selector values enforce closeness in the quotient sense, and no sufficiently small nongauge refactorization can replicate the preferred structure. The result is intentionally local in scope. It does not assert global uniqueness, a global spectral gap, or a full reconstruction of geometry, but instead isolates the strongest quantitative statement currently supported by the available analytic framework.
A finite-dimensional Hilbert space admits many inequivalent tensor-product decompositions, so exact uniqueness of subsystem structure must be formulated on a specified class of Hamiltonians and graphs. In the graph-local setting, exact zero-set identifiability means that the static selector vanishes on a single physical quotient class, or equivalently that the Hamiltonian has a unique admissible-support realization modulo local basis changes and subsystem permutation. On the deletion-robust graph class, defined by the requirement that deleting any vertex leave no isolated vertex component, the analysis gives a finite-certificate description of that question. The reference complement-generation locus in exact graph-local coefficient space is nonempty exactly on this graph class, and an explicit family of one-site fields and mixed edge couplings realizes it. Two finite certificates are then introduced on the reference exact slice: a bounded-length omitted-site word-algebra rank test and a quadratic-rigidity determinant for the first forbidden-support quadratic form. On deletion-robust graphs the resulting reference-regular locus is nonempty, Zariski-open, and Euclidean dense. Assuming the exact zero set of a base Hamiltonian is finite, each competitor exact zero determines a local branch whose intersection with the reference exact slice is governed by an explicit Jacobian-rank criterion. Near a regular-transverse base point, exact zero-set identifiability is therefore controlled by finitely many explicit obstruction manifolds. The graph boundary, the reference certificates, and the structural failure modes are all explicit within this nondegenerate exact-uniqueness framework.