We introduce the symmetric-folding principle, a geometric quantum-control framework that lifts quantum state steering to exact propagator synthesis. The required structure is a folding symmetry: a unitary transformation that reverses the drift Hamiltonian while preserving the admissible control space. The method recasts propagator synthesis on the unitary group as steering of a chosen quantum state variable between fibers of an equivariant quotient map, followed by a prescribed folded continuation of the controls. We prove that control inputs satisfying the resulting fiber-transfer condition, when extended by the prescribed folded continuation, generate the desired unitary propagator in the associated symmetry-selected target family. This gives two key structural advantages. The fiber-transfer condition constrains the midpoint propagator only to a coset of the folding symmetry's centralizer. The prescribed fold cancels this residual freedom exactly. Thus a full group-level endpoint constraint is replaced by a lower-codimension fiber/coset condition, giving additional freedom in control design. For quantum ensembles, different members may reach different midpoint propagators within the same coset, while the folded sequence still yields a common final propagator for all ensemble members. This parameter-dependent freedom can be especially significant for robust ensemble-propagator design. We demonstrate the construction on an SO(3) Bloch ensemble and an SU (4) scalar-coupled two-spin system.
We introduce the symmetric-folding principle, a geometric quantum-control framework that lifts quantum state steering to exact propagator synthesis. The required structure is a folding symmetry: a unitary transformation that reverses the drift Hamiltonian while preserving the admissible control space. The method recasts propagator synthesis on the unitary group as steering of a chosen quantum state variable between fibers of an equivariant quotient map, followed by a prescribed folded continuation of the controls. We prove that control inputs satisfying the resulting fiber-transfer condition, when extended by the prescribed folded continuation, generate the desired unitary propagator in the associated symmetry-selected target family. This gives two key structural advantages. The fiber-transfer condition constrains the midpoint propagator only to a coset of the folding symmetry's centralizer. The prescribed fold cancels this residual freedom exactly. Thus a full group-level endpoint constraint is replaced by a lower-codimension fiber/coset condition, giving additional freedom in control design. For quantum ensembles, different members may reach different midpoint propagators within the same coset, while the folded sequence still yields a common final propagator for all ensemble members. This parameter-dependent freedom can be especially significant for robust ensemble-propagator design. We demonstrate the construction on an SO(3) Bloch ensemble and an SU (4) scalar-coupled two-spin system.
We develop a constructive control framework for distance-and configuration-selective operations in neutral-atom arrays. The starting point is the symmetric folding principle, which reformulates propagator synthesis as a structured state-or fiber-steering problem followed by a symmetry-prescribed folded continuation of the control sequence. We specialize this principle to spectrally programmed (SU(2)) spin propagators whose action is prescribed as a function of the spectral detuning. For a driven target atom coupled to an undriven controller, the Rydberg interaction decomposes the two-atom Hamiltonian into invariant target subspaces with effective detunings (∆) and (∆ -V (ρ)). Interatomic distance is thereby converted into a spectral control parameter. Distance-selective gates can consequently be formulated as passband/stopband designs in which a chosen target rotation is produced over a prescribed distance range and the identity is produced over a separated stopband, with a finite transition region. The framework extends to one driven target and multiple conserved controller configurations, provided that the corresponding effective spectral detunings remain distinguishable.
In the current framework of geometric quantum machine learning, the canonical method for constructing a variational ansatz that respects the symmetry of some group action is by forcing the circuit to be equivariant, i.e., to commute with the action of the group. This can, however, be an overzealous constraint that greatly limits the expressivity of the circuit, especially in the case of continuous symmetries. We propose an alternative paradigm for the symmetry-informed construction of variational quantum circuits, based on homogeneous spaces, relaxing the overly stringent requirement of equivariance. We achieve this by introducing horizontal quantum gates, which only transform the state with respect to the directions orthogonal to those of the symmetry. We show that horizontal quantum gates are much more expressive than equivariant gates, and thus can solve problems that equivariant circuits cannot. For instance, a circuit comprised of horizontal gates can find the ground state of an SU ( 2 ) -symmetric model where the ground state spin sector is unknown—a task where equivariant circuits fall short. Moreover, for a particular subclass of horizontal gates based on symmetric spaces, we can obtain efficient circuit decompositions for our gates through the KAK theorem. Finally, we highlight a particular class of horizontal quantum gates that behave similarly to general SU ( 4 ) gates, while achieving a quadratic reduction in the number of parameters for a generic problem.
This thesis is concerned with the control and filtering problems for quantum systems. Two kinds of control problems are considered for a class of quantum systems: optimal control and trajectory tracking control. Filtering of a class of linear quantum systems is also investigated. \n \nFor the optimal control of quantum systems, the system in question evolves on a manifold inR^3 and is modeled as a bilinear control form whose states are represented as coherence vectors. An associated matrix Lie group system with state space S O(3) is introduced in order to facilitate solving the given problem. The controllability as well as the reachable set of the system is analyzed in detail. The maximum principle is applied to the optimal control for systems evolving on the Lie group of special orthogonal matrices of dimension 3, with cost that is quadratic in the control input. Controlling the evolutions of coherence vector is equivalent to controlling the states of an associated matrix Lie group system. Thus, the optimal control problem for single spin-1/2 systems can be transformed to steer state of the matrix Lie group system from the identity matrix to a final matrix corresponding to a target state of single spin-1/2 systems in an optimal fashion. The obtained result is employed to perform a reversible logic quantum operation NOT on single spin-1/2 systems. Explicit expressions for the optimal control are given which are linked to the initial state of the system. The main contribution of the result is to extend the existing result to a larger class of systems, generalizing existing wave-function-based results to single spin-1/2 systems whose states are represented by density matrices such that it allows the inclusion of mixed states. \n \nFor the tracking control problem of closed single spin-1/2 quantum ensembles, based on the Liouville-von Neumann equation, mathematical model of dynamics for single spin-1/2 quantum ensembles are formulated as a bilinear form with states evolving on Bloch sphere. For a typical setting in NMR, one in general would not cancel the applied electromagnetic field along a specified direction, thus leading to the dynamics in a plane perpendicular to that direction on the Bloch sphere. One natural question arises that how can one design a feedback controller such that given an initial condition for quantum ensembles, the system evolutions can be steered to the pre-determined trajectory. An associated matrix Lie group differential equation describing transfers from the initial state being an identity matrix is naturally introduced. The tracking problem for the system of interest is transformed to a stabilization problem such that the control techniques are able to be used. A distance function as well as a distance vector defined on the Lie group SO(3) are introduced, which can be used to construct a Lyapunov function to deal with the tracking problem. In order to deal with the tracking problem, the Lyapunov-based feedback design method is employed to obtain the feedback control such that the dynamics of system can converge to the reference trajectory. \n \nFor filtering of quantum systems, the filter problem of a class of linear quantum stochastic systems with quantum noise is investigated. The quantum system model is first analyzed, then the necessary conditions of nondemolition are obtained by applying the observation principle. The relations between the input and output noise are necessary for the nondemolition measurement of the quantum output channels. Given prescribed quantum system, the filter of the system is derived using those necessary conditions.
Quantum Geometry, a novel framework based on the manipulation of quantum states as geometric shapes, offers a promising pathway to stabilizing and controlling quantum states. This paper explores the potential of utilizing this approach to design stable and controllable quantum states, providing a new tool for quantum computing. We detail the core mechanism – the transformation of quantum states into geometric representations – and present preliminary results demonstrating the feasibility of manipulating these shapes to achieve desired states. The paper concludes with a discussion of future research directions and the potential impact of this methodology on the field of quantum state design.
We establish a global, worst-case non-identifiability theorem for periodically driven finite-dimensional quantum systems. On the unrestricted smooth periodic Hamiltonian class, we show that the period-averaged Fubini-Study metric component of a fixed initial state cannot, in general, be reconstructed from exact one-period propagators indexed by every starting time and external parameter. Thus, even complete starting-time-resolved endpoint data are insufficient to determine this intra-period geometric quantity. The obstruction is characterized exactly. The starting-time-indexed endpoint data determine the conjugation path of the monodromy, but not its particular unitary lift. On each fixed-monodromy slice, the observational fibres are precisely the right orbits generated by smooth parameter-dependent based loops taking values in the pointwise centralizer of the monodromy. The Fubini-Study functional is not invariant under this fibre action and therefore does not factor through the endpoint observation map. An explicit real-analytic two-level witness demonstrates the obstruction within a commuting, one-generator Hamiltonian family, so neither non-Abelian time ordering nor Floquet-logarithm ambiguity is required. A continuous family of Hamiltonians produces identical starting-time-indexed one-period endpoint data while yielding different, and on the unrestricted class arbitrarily separated, period-averaged Fubini-Study geometry. Non-identifiability persists under any prescribed uniform bound on the parameter derivative of the Hamiltonian. The result is a deterministic global statement, not a claim of generic non-identifiability or experimental impossibility. It identifies a precise information gap between complete one-period endpoint data and full intra-period dynamics: the missing information is the unitary lift of the observed monodromy path rather than ordinary scalar phase freedom.