Bosonic devices promise applications in simulation, sensing and quantum error correction, but infinite-dimensional local Hilbert spaces obstruct the locality tools that make qubit dynamics efficiently learnable and simulable. We establish a finite-energy locality principle for geometrically local bosonic open systems satisfying photon-number moment propagation. It compares unbounded generators with Galerkin cutoffs, transferring finite-dimensional Lieb--Robinson, product-formula and circuit techniques to bosons with explicit errors. For this moment-controlled class, we obtain, to our knowledge, the first model-independent weak Lieb--Robinson bounds beyond Bose--Hubbard-type dynamics, together with quantitative Trotter and simulation guarantees for polynomial bosonic GKSL generators. As a central application, coherent-state preparation and local heterodyne detection suffice to learn coefficients of a known bounded-degree polynomial Hamiltonian ansatz local on a bounded-growth interaction graph to accuracy $\varepsilon$ and failure probability $δ$, with sample complexity and total evolution time both $\widetilde{\mathcal{O}}(\varepsilon^{-2}\log(m/δ))$, where $m$ counts on-site and interaction terms. This matches the best known finite-dimensional locality-assisted scaling in accuracy and system size, up to polylogarithmic factors. The assumptions hold for Bose--Hubbard and quadratic dynamics without added dissipation; for more general local polynomial Hamiltonians they can be supplied natively or engineered by known multi-photon loss in stabilized bosonic architectures.
Bosonic devices promise applications in simulation, sensing and quantum error correction, but infinite-dimensional local Hilbert spaces obstruct the locality tools that make qubit dynamics efficiently learnable and simulable. We establish a finite-energy locality principle for geometrically local bosonic open systems satisfying photon-number moment propagation. It compares unbounded generators with Galerkin cutoffs, transferring finite-dimensional Lieb--Robinson, product-formula and circuit techniques to bosons with explicit errors. For this moment-controlled class, we obtain, to our knowledge, the first model-independent weak Lieb--Robinson bounds beyond Bose--Hubbard-type dynamics, together with quantitative Trotter and simulation guarantees for polynomial bosonic GKSL generators. As a central application, coherent-state preparation and local heterodyne detection suffice to learn coefficients of a known bounded-degree polynomial Hamiltonian ansatz local on a bounded-growth interaction graph to accuracy ε and failure probability δ, with sample complexity and total evolution time both O(ε−2log(m/δ)), where m counts on-site and interaction terms. This matches the best known finite-dimensional locality-assisted scaling in accuracy and system size, up to polylogarithmic factors. The assumptions hold for Bose--Hubbard and quadratic dynamics without added dissipation; for more general local polynomial Hamiltonians they can be supplied natively or engineered by known multi-photon loss in stabilized bosonic architectures.
Understanding the dynamics of interacting quantum systems is a central goal of quantum simulation. For geometrically local Hamiltonians, locality suggests that digital simulation should retain much of the parallelism of the underlying dynamics. For a target accuracy ε, we construct a nearest-neighbor quantum circuit that simulates the dynamics of an n-qubit, time-independent, finite-range Hamiltonian with normalized local interaction strength on a fixed-dimensional lattice for time T≥1, with circuit depth O(Tlog(nT/ε)) and gate count O(nTlog(nT/ε)). This reduces the polylogarithmic overhead of Haah, Hastings, Kothari, and Low to a single logarithm. On logarithmic-scale blocks from the Lieb--Robinson decomposition, a shallow but highly accurate implementation is essential for reducing the circuit depth. Our key observation is that spatial localization also controls the cost of high precision. In each block, the error of a constant-order product formula is compensated for by a correction, yielding exponential accuracy. The correction can be implemented with depth proportional to the block diameter, introducing no additional asymptotic depth scale beyond that required by spatial localization.
Furthermore, we prove an unconditional circuit-depth lower bound for uniform XXZ Hamiltonians: at fixed evolution time T in the stated regime and sufficiently small fixed error ε, any nearest-neighbor quantum circuit simulating these dynamics requires depth Ω(logn/loglogn). This matches our O(logn) upper bound up to a doubly logarithmic factor, establishing near-optimal circuit depth for lattice Hamiltonian simulation in this regime. Together, these results clarify the role of locality and precision in determining the circuit complexity of quantum dynamics and suggest broader principles for spatially constrained quantum computation.
We present an efficient protocol for learning an unknown k-local Lindblad generator on n qubits using only product-state preparations, short-time evolution, and single-qubit Pauli measurements, without prior knowledge of the interaction structure. For fixed k and a supplied weighted interaction-strength bound αˉ, the protocol estimates all Hamiltonian and dissipative Pauli--GKSL coefficients with total coefficient error at each site at most ε with probability at least 1−δ using Ok(αˉ2n2k−2ε−2log(n/δ)) samples and polylogarithmically many distinct evolution times. This guarantee requires no degree, sparsity, or tail promise. A convex optimization converts these estimates into a valid k-local Lindblad generator with diamond-norm error at most ε using Ok(αˉ2n2kε−2log(n/δ)) samples and polynomial-time classical postprocessing. With a supplied local effective-sparsity bound r and a sufficiently small sitewise subthreshold tail Ok(ε), coefficient recovery requires only Ok(αˉ2r2ε−2log(n/δ)) samples, without a supplied threshold or coefficient locations and without a coefficient-gap assumption. In particular, exactly sparse bounded-degree models have logarithmic sample dependence on n. We also provide guard-band support recovery and complementary supplied-candidate guarantees, extend the guarantees to model misspecification, and prove complementary sample-complexity lower bounds.
Hamiltonian learning protocols are becoming valuable tools for benchmarking and verifying quantum computers and simulators, yet few rigorous protocols address explicitly time-dependent dynamics. We give a scalable protocol for learning the time-dependent coefficients of a locally interacting n-qubit system on a graph G of effective dimension D. The unknown generator contains geometrically local Hamiltonian terms and weak onsite Pauli-dephasing terms with time-dependent, nonnegative rates. Its coefficient functions and their derivatives admit a certified simultaneous C1 approximation in a known finite-dimensional function class that supports stable interpolation. The only experimental primitives are preparation of product Pauli eigenstates, evolution for chosen times, and product-Pauli measurements. Under an explicit weak-dissipation condition, the Lieb--Robinson estimates proved here, and a certified effective approximation/interpolation degree meff, the protocol reconstructs every coefficient uniformly on a constant interval with error ε and success probability 1−δ using O(ε−2poly(meff)log(nδ−1)) samples and polynomial classical processing. The method estimates derivatives of finite-time channel overlaps by stable interpolation, uses process shadows to learn the required local transfer matrices, and constructs sparse diagonally dominant systems through localized semidefinite programs.
Closed bosonic lattice systems do not generally admit operator norm Lieb-Robinson bounds uniformly in the initial state. The reason is that the information propagation velocity can grow with the local boson occupancy, which can be macroscopically large. Here, we show that local dissipation restores an operator-norm Lieb-Robinson bound. We consider the dissipative Bose-Hubbard model described by a Lindbladian operator with on-site ℓ-photon loss, ℓ>2. For our main result, we establish an almost-ballistic Lieb-Robinson bound. Dissipation rapidly depletes highly occupied sites, thus regularizing the state on the Sobolev-type scale of local particle moments. The resulting moment bounds diverge as time approaches zero but remain integrable near zero at sufficiently low orders, which is crucial for our argument. We extend these ideas to treat cat-code dissipation and, for initial states in the code space, we prove a local adiabatic approximation uniform in the total volume. Further consequences include local channel approximation, a thermodynamic limit, and efficient hybrid bosonic-qubit simulation. The point is that these applications are now available uniformly in the input state, as for quantum spin systems, but in contrast to closed Bose-Hubbard systems.
Quantum many-body systems involving bosonic modes or gauge fields have infinite-dimensional local Hilbert spaces which must be truncated to perform simulations of real-time dynamics on classical or quantum computers. To analyze the truncation error, we develop methods for bounding the rate of growth of local quantum numbers such as the occupation number of a mode at a lattice site, or the electric field at a lattice link. Our approach applies to various models of bosons interacting with spins or fermions, and also to both abelian and non-abelian gauge theories. We show that if states in these models are truncated by imposing an upper limit Λ on each local quantum number, and if the initial state has low local quantum numbers, then an error at most ϵ can be achieved by choosing Λ to scale polylogarithmically with ϵ−1 , an exponential improvement over previous bounds based on energy conservation. For the Hubbard-Holstein model, we numerically compute a bound on Λ that achieves accuracy ϵ , obtaining significantly improved estimates in various parameter regimes. We also establish a criterion for truncating the Hamiltonian with a provable guarantee on the accuracy of time evolution. Building on that result, we formulate quantum algorithms for dynamical simulation of lattice gauge theories and of models with bosonic modes; the gate complexity depends almost linearly on spacetime volume in the former case, and almost quadratically on time in the latter case. We establish a lower bound showing that there are systems involving bosons for which this quadratic scaling with time cannot be improved. By applying our result on the truncation error in time evolution, we also prove that spectrally isolated energy eigenstates can be approximated with accuracy ϵ by truncating local quantum numbers at Λ=polylog(ϵ−1) .
Open-system dynamics govern decoherence, transport, and engineered dissipation in quantum many-body systems. We propose a quantum algorithm for simulating time-independent, finite-range Lindblad dynamics with bounded local strength on a constant-dimensional lattice of n sites, allowing noncommuting jump operators. For time T≥1 and diamond-norm error at most ε, it has depth O(Tpolylog(nT/ε)) and gate count O(nTpolylog(nT/ε)). These results extend the HHKL spatial decomposition to open systems. The main obstacle is that the reversible overlap corrections used for unitary dynamics have no direct analogue for dissipative channels. We overcome this by retaining the environment in a shared unitary dilation, keeping information lost from the system coherently available and making the spatial corrections reversible. We prove exponential occupation bounds for the environmental states generated by the decomposition, allowing compact representation and coherent simulation of local factors on the inputs that arise. The construction reduces each short-time evolution to a constant number of parallel layers on regions of logarithmic diameter, using nearest-neighbour gates and polylogarithmic auxiliary density.
We also prove unconditional worst-case lower bounds using time-independent, purely dissipative processes that transport quantum information. For circuits with local quantum and classical communication, the lower bounds are Ω(nT) for the expected gate count and Ω(T) for depth on time scales up to the lattice diameter. They match the upper bounds' leading time and volume dependence up to polylogarithmic factors. At fixed time and sufficiently small fixed error, we further prove a depth lower bound of Ω(logn/loglogn), showing that global accuracy can require increasing depth as the lattice grows.