Controllability-Based Input Allocation in Quantum System

AI summary

Authors: Riku Yaegashi, Kazuhiro Sato (Department of Mathematical Informatics, Graduate School of Information Science and Technology, The University of Tokyo), Kentaro Ohki (Department of Applied Computer Engineering, School of Information Science and Technology, Tokai University) arXiv: 2610.38415v1 [quant-ph], 28 Oct 2026

Overview

Research area: Quantum control theory — specifically output controllability analysis of Schrödinger input–output dynamics, combined with tensor-network numerics for quantum many-body systems. It sits at the intersection of mathematical systems/control theory, quantum information, and computational many-body physics.

Technical level: Advanced. The paper relies on differential geometry of complex projective space, controllability Gramians, log-determinant optimization over the simplex, and matrix product state (MPS) / time-evolving block decimation (TEBD) methods.

Scope (one sentence): The paper defines a controllability-based, trajectory-dependent scoring rule (V-CAS) that allocates virtual input intensities among local Hamiltonian perturbation directions in a quantum system, proves when that allocation is unique, and develops an MPS/TEBD implementation with an error bound for many-body use.

What This Paper Is About

Existing quantum "centrality" measures — the correlation- and entanglement-based measures cited in the paper as [19, 11] — are computed from static properties of a state or Hamiltonian and therefore do not directly describe how a local perturbation propagates dynamically under unitary evolution. The authors instead ask: given a nominal (uncontrolled) Schrödinger trajectory, a set of measured observables, and a set of candidate local Hamiltonian perturbation directions, how should a fixed budget of virtual input intensity be split among those directions so that the first-order set of reachable terminal outputs is as large as possible? The resulting optimal split is the score they call the volumetric controllability allocation score (V-CAS).

Key Contributions

1. A quantum formulation of controllability allocation scoring. Building on the multiobservable quantum-control Gramian [5] and the controllability allocation score (CAS) framework for linear systems [15], the authors introduce virtual local Hamiltonian perturbations along a nominal Schrödinger trajectory and resolve the terminal-output response into channel-wise Gramian contributions $W_j(T)$. These are combined through a simplex-constrained log-determinant optimization, giving an allocation that is both output-dependent and trajectory-dependent and is defined through physically phase-invariant observables (expectation values of Hermitian observables $O_1,\dots,O_r$).

2. A drift-independent sufficient condition for uniqueness. Because the allocation weights are the score, non-uniqueness would produce ambiguous rankings of local Hamiltonian directions. The authors derive a sufficient condition — depending only on the initial state, the local perturbation Hamiltonians, and the differential of the output map at the initial state — that guarantees restricted injectivity, hence affine independence of the finite-time Gramian contributions and strict convexity of the scoring problem, for almost every terminal time at which the problem is feasible. Notably, the condition does not involve the drift Hamiltonian $H_0$.

3. An MPS/TEBD-compatible implementation. For many-body systems, the required sensitivities are evaluated by forward propagation of the nominal state, backward propagation of observable ("adjoint") states $z_\ell(T,t) = \Phi(T,t)^{*} O_\ell \bar\psi(T)$, local operator applications, and MPS inner-product contractions — avoiding explicit construction of exponentially large state-transition matrices. The authors describe this as generalizing the product-state ansatz used in mean-field theory.

4. An error bound for the discretized Gramian. The paper derives a bound that propagates both Trotter error and MPS-truncation error to the resulting discrete output Gramian.

The paper also characterizes the range and minimum-energy properties of the output Gramian (Propositions 1 and 2).

Main Findings

• Reachable outputs equal the Gramian's image. Proposition 1 shows $\operatorname{Im} W(T) = \mathcal{R}_y(T)$, the reachable subspace of terminal output variations, and that positive definiteness of $W(T)$, surjectivity of the input-to-output map $J_y$ onto $\mathbb{R}^r$, and output controllability of the linearized system at time $T$ are equivalent.

• Minimum input energy is given by the inverse Gramian. Proposition 2 states that for a reachable terminal output variation $z$, the unique minimum-norm input variation is $\delta u^\star = J_y^{*} W(T)^{+} z$ and the minimum energy is $E^y_{\min}(z) = z^{\top} W(T)^{+} z$ (Moore–Penrose pseudoinverse in the general case).

• Feasibility has a simple test. The allocation problem is feasible if and only if $\sum_{j=1}^{m} W_j(T) \succ 0$; when this holds, the uniform allocation $p^{\mathrm{uni}} = \frac{1}{m}\mathbf{1}$ is feasible and an optimal solution exists (using results quoted from [16]).

• Uniqueness hinges on affine independence. Restricted injectivity $\ker \mathcal{W}T \cap \mathcal{L}\Delta = {0}$ is equivalent to affine independence of $W_1(T),\dots,W_m(T)$, equivalently linear independence of $W_1(T)-W_m(T),\dots,W_{m-1}(T)-W_m(T)$; the drift-independent condition guarantees this for almost every terminal time.

• The score measures cooperative, not isolated, influence. A larger optimal weight $\hat p_j$ indicates that direction $H_j$ carries a large share of the jointly optimized allocation, not that it is influential on its own. The authors note in Remark 1 that a channel generating a stronger Hamiltonian under the same input-energy budget may receive a larger weight — this dependence is stated to be intentional.

• Log-determinant rather than trace criteria. The negative log-determinant objective is chosen because it rewards generating variation across multiple output directions, and because, unlike trace-based criteria such as $\operatorname{tr}(W^{-1})$, the optimal weights are invariant under invertible linear transformations of the output coordinates [17].

• Numerical results. The paper states that experiments on quantum many-body models show the score captures finite-time influence patterns that can differ from static correlation- and entanglement-based measures, and can inform target-independent control-channel preselection. The truncated content available here does not include the specific models, system sizes, parameter values, or quantitative outcomes of Section 5.

Methodology in Plain English

1. Set up the quantum input–output system. The state obeys the Schrödinger equation with $\hbar = 1$ and a Hamiltonian split into a drift term $H_0$ plus control terms $\sum_j u_j(t) H_j$. Because physical pure states are defined only up to a global phase, states live in the complex projective space $\mathbb{CP}^{d-1}$, and the measurable outputs are expectation values of chosen Hermitian observables — quantities that are unchanged by the phase.

2. Linearize around a nominal trajectory. The authors fix a reference control (in the scoring problem, the uncontrolled evolution $\bar u = 0$) and compute how small input variations produce small terminal-output variations. This gives a linear map $J_y$ from input variations to output changes, and its Gramian $W = J_y J_y^{*}$.

3. Treat local Hamiltonian directions as virtual input channels. Each candidate perturbation direction $H_j$ is given a nonnegative weight $p_j$, entering the dynamics as $\sqrt{p_j}$. Because the Gramian is linear in these weights, $W(p,T) = \sum_j p_j W_j(T)$ — the total is a weighted sum of per-channel contributions, each computed as an integral over $[0,T]$ of an outer product of sensitivity vectors.

4. Optimize the allocation. The weights are constrained to the probability simplex (nonnegative, summing to one), and the objective is $-\log\det W(p,T)$, maximizing the volume of the first-order reachable output set. The optimum $\hat p$ is the V-CAS.

5. Make it computable at scale. Rather than building the exponentially large propagator, the sensitivities are rewritten using adjoint states $z_\ell(T,t) = \Phi(T,t)^{*} O_\ell \bar\psi(T)$, so each component becomes $2,\mathrm{Re}\langle z_\ell(T,t), (-iH_j)\bar\psi(t)\rangle$. This is evaluated with MPS representations, TEBD time evolution, and local operator contractions, with a bound tracking how Trotter and truncation errors reach the discrete Gramian.

Why This Matters

Impact on research. The work moves quantum centrality/influence analysis from static quantities (correlations, entanglement) to a dynamical, controllability-theoretic notion: which local Hamiltonian directions most effectively enlarge the reachable output set over a finite time window. It also extends the classical CAS/log-determinant framework [14, 15, 16] into the quantum setting, handling two issues that are specific to it — global-phase invariance and non-uniqueness of the allocation — and provides a tensor-network route to the required sensitivities, with quantified error control.

Real-world applications (as motivated by the paper's framing):

• Control-channel preselection for quantum devices — the authors state the score "can provide useful information for target-independent control-channel preselection," i.e., deciding which actuators to use before a specific control target is fixed.
• Quantum simulation of many-body systems — the paper situates local Hamiltonian models and locally accessible operations as standard settings in quantum simulation and quantum control [6, 8].
• Quantum network controllability — the paper cites manipulation of a local Hamiltonian on a small subsystem as a standard local-control setting [3], which the score can rank.
• Observable-driven experiment design — because the score depends on the selected observables and terminal time, it speaks to which local perturbations matter for the specific quantities an experiment measures.

Industry relevance. Quantum hardware platforms with limited, local actuation and readout stand to benefit from a principled, quantitative way to prioritize which control channels to build, calibrate, or keep, rather than relying on static correlation heuristics.

Future Directions

• Non-local and structured perturbation directions. Assumption 1 restricts virtual Hamiltonians to single-site operators; extending the score to two-site, multi-site, or symmetry-constrained directions is a natural next step that the paper's own remarks leave open.
• Tighter and more scalable error control. The derived bound propagates Trotter and MPS-truncation errors to the discrete Gramian; how this bound behaves for long terminal times and large bond dimensions — and whether it can guide adaptive bond-dimension or Trotter-step selection — is not settled in the material available here.
• Relation to actual achievable control. The virtual inputs exist only to probe sensitivity; connecting V-CAS rankings to performance of real optimized control pulses (rather than preselection heuristics) is an open question.
• Choice of terminal time and observables. Since the score depends on $T$, on the initial state, and on the observable set, understanding how rankings vary with these choices — and whether a time-averaged or worst-case variant is preferable — remains open.

Target Audience

Graduate students and researchers in quantum control, mathematical systems theory, and quantum information; practitioners working on controllability Gramians and actuator/channel selection; and computational many-body physicists interested in tensor-network (MPS/TEBD) algorithms for sensitivity and controllability analysis. Readers need comfort with linear algebra, differential geometry of state spaces, convex optimization, and tensor networks — the paper is not beginner-friendly.

Authors’ abstract

Understanding how local perturbations influence global quantum dynamics is important for analyzing and designing large-scale quantum systems. Existing quantum centrality measures often rely on static correlations or entanglement and do not directly capture perturbation propagation under unitary evolution. We formulate controllability allocation scoring for quantum input-output dynamics governed by the Schrodinger equation. Using the multiobservable quantum-control Gramian, we introduce virtual local Hamiltonian perturbations along a nominal trajectory and decompose the terminal-output response into channel-wise Gramian contributions. A simplex-constrained log-determinant optimization combines these contributions to define a volumetric controllability allocation score (V-CAS), specifying an optimal allocation of virtual input intensities among local Hamiltonian perturbation directions. The score measures their cooperative dynamical influence and depends on the output and trajectory. We characterize the output Gramian's range and minimum-energy properties and derive a drift-independent sufficient condition ensuring score uniqueness for almost every terminal time at which the problem is feasible. For quantum many-body systems, we develop a matrix-product-state and time-evolving-block-decimation implementation that avoids explicitly constructing exponentially large state-transition matrices. We also derive an error bound propagating Trotter and MPS-truncation errors to the discrete output Gramian. Numerical experiments show that the score captures finite-time influence patterns distinct from those identified by static correlation- and entanglement-based measures and can inform target-independent control-channel preselection.