QArray+: A physics-informed GPU-accelerated simulator for quantum dot arrays

Pranav Vaidhyanathan; Barnaby van Straaten; Alice Petrillo; Rahul Marchand; Edwin De Nicolo; Menno Veldhorst; Brucek Khailany; Taylor L. Patti; Natalia Ares

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QArray+: A physics-informed GPU-accelerated simulator for quantum dot arrays

Pranav Vaidhyanathan, Barnaby van Straaten, Alice Petrillo, Rahul Marchand, Edwin De Nicolo, Menno Veldhorst, Brucek Khailany, Taylor L. Patti, Natalia Ares

arXiv:2609.02736v1Sep 2, 2026cond-mat.mes-hallcs.DCquant-ph

TL;DR

Large quantum-dot arrays need realistic, scalable simulators because existing steady-state models miss non-equilibrium charge dynamics and coherent hybridization. QArray+ combines gate-dependent tunnel coupling with stochastic and Lindblad-based modeling, achieving scalable simulation while representing both latching and hybridization. Its accelerator execution supports high-throughput synthetic datasets for automated tuning.

Problem
Existing steady-state simulators assume rapid relaxation, limiting their ability to represent experimentally relevant non-equilibrium latching and coherent interdot hybridization.
Method
QArray+ combines gate-dependent capacitance and tunnel parameters with stochastic, Hubbard, and Lindblad open-system simulation models.
Results
0.17 s computes a 100×100 charge-stability scan over 64 dots on 8×H100 GPUs.
Takeaways & Limitations
QArray+ supports large-scale synthetic datasets and automated tuning pipelines by combining scalable computation with realistic charge-dynamics features.

Abstract

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Semiconductor quantum-dot arrays are a compelling platform for scalable quantum technologies, yet their practical operation is hindered by the complexity of tuning large-scale devices. Existing automation tools rely on simplified physical models---such as constant-capacitance approximations and equilibrium Hubbard models---which assume instantaneous relaxation to a steady state. These frameworks fail in experimentally critical regimes where measurement rates exceed tunneling dynamics, necessitating more sophisticated non-equilibrium control strategies. To bridge this gap, we introduce QArray+, an extension of the QArray framework that incorporates gate-dependent tunnel coupling and a quantum open-system description of dissipative processes. This approach enables the unified simulation of coherent interdot charge-state hybridization and the non-equilibrium latching dynamics essential for training robust machine-learning models for automated device operation. Implemented in JAX with GPU acceleration, QArray+ scales across GPUs and multi-node systems. For example, a charge stability diagram for a 100X100 grid of gate voltages over 64 dots can be computed in $\sim0.17\,\mathrm{s}$ on multiple GPUs. Since interdot interactions are short-ranged and the corresponding tuning corrections are local, simulations at these scales capture the physics relevant to even larger devices. These capabilities support high-throughput dataset generation for automated device tuning.

I. INTRODUCTION

QArray+ addresses the limitations of steady-state simulators by modeling non-equilibrium dynamics and coherent interdot effects while supporting scalable, accelerator-based simulation. Its models provide complementary trade-offs, with the open-system approach capturing latching and hybridization together.

I. INTRODUCTION: Machine-learning charge-tuning strategies require realistic simulators and large labeled datasets that are impractical to obtain experimentally.Fast physics-based simulation is therefore used to create training data for automated tuning pipelines.
I. INTRODUCTION: Existing steady-state models assume charge configurations relax faster than measurement times, missing slow dynamics, hysteresis, latching, and coherent hybridization.Reservoir coupling can be exponentially suppressed with distance, making relaxation times experimentally relevant.
I. INTRODUCTION: QArray+ extends QArray with interdot tunnel coupling, finite reservoir tunneling rates, gate-dependent parameters, and JAX-based GPU and multi-node execution.Its polynomially reformulated electrostatic core targets accelerated, dataset-scale charge-stability-diagram synthesis.
I. INTRODUCTION: The stochastic model captures latching, the spinless Hubbard model captures hybridization, and the open quantum system model captures both phenomena.The open-system model combines coherent and incoherent processes through a Lindblad master equation and stochastic quantum-jump trajectories.

II. MODELS FOR SIMULATING CHARGE STABILITY DIAGRAMS

This section introduces the dynamical models used to compute charge stability diagrams in QArray+.

II. MODELS FOR SIMULATING CHARGE STABILITY DIAGRAMS: The section describes the dynamical models used by QArray+.
II. MODELS FOR SIMULATING CHARGE STABILITY DIAGRAMS: These models compute charge stability diagrams.
II. MODELS FOR SIMULATING CHARGE STABILITY DIAGRAMS: QArray+ uses three dynamical models for charge-stability-diagram computation.

A. Stochastic Capacitance Model

The stochastic capacitance model simulates non-equilibrium charge evolution as discrete stochastic transitions across measurement pixels. It captures latching with polynomial scaling, while its approximations constrain accuracy outside the slow-tunneling regime.

A. Stochastic Capacitance Model: Each measurement integration window is treated as one diagram pixel, with at most one charge transition initially allowed per window.The assumption targets regimes where tunnel rates are low relative to the sampling timescale.
A. Stochastic Capacitance Model: The model enumerates reservoir loading/unloading and interdot tunneling transitions, then updates the integer charge state using stochastic transition trials.Interdot transitions conserve total electron number, while loading and unloading change a dot occupation by one.
A. Stochastic Capacitance Model: Gate-dependent capacitance matrices define the electrostatic energy landscape, while user-specified tunneling rates are modulated by Fermi factors using energy changes and inverse temperature.The capacitance model does not explicitly include quantized orbital level spacing.
A. Stochastic Capacitance Model: The stochastic model captures latching but not coherent interdot tunneling, which is reserved for the Lindblad open-system simulator.
A. Stochastic Capacitance Model: The single-transition update is accurate when Γkτ ≪1, but random traversal biases competing-channel selection relative to exact branching ratios.Subdividing each integration window into Nr sub-intervals suppresses this bias when multiple transitions must be resolved.
A. Stochastic Capacitance Model: Polynomial scaling follows from evaluating only Nrndot(ndot + 1) transitions, unlike steady-state solvers that examine exponentially many charge states.This scaling improves tractability as the number of dots increases.

B. Tunnel coupling

QArray+ extends the capacitance model with coherent, gate-dependent interdot tunneling and computes tunnel-coupled ground states in a truncated charge basis. This truncation keeps calculations controlled by a small low-energy manifold rather than the exponentially growing full basis.

B. Tunnel coupling: Finite interdot tunnel coupling hybridizes near-degenerate charge configurations, producing avoided crossings and fractional charge expectation values.The spinless Hubbard Hamiltonian represents coherent hopping directly in the occupation-number basis.
B. Tunnel coupling: QArray+ uses the capacitance-model electrostatic energy as the diagonal Hamiltonian contribution and adds optional gate-dependent tunnel matrix elements.Gate-dependent capacitances define a per-pixel energy landscape, while relative energies determine the ground state.
B. Tunnel coupling: For each gate-voltage point, the simulator selects the lowest-energy eigenstate of the tunnel-coupled Hamiltonian and reports expected dot occupations.Occupation expectations are obtained from the ground-state expansion in the charge basis.
B. Tunnel coupling: QArray+ constructs a fixed-size truncated basis near the relaxed continuous-charge solution, then evaluates candidate energies and retains the ntruncate lowest-energy configurations.The candidate neighborhoods are heuristic and may vary with gate coupling.
B. Tunnel coupling: The truncated Hamiltonian is solved with dense eigensolvers for small bases or sparse Lanczos iterations for larger ones, avoiding full-basis exponential scaling.The full scan is controlled by candidate scoring, tunnel-graph sparsity, and the chosen eigensolver.

C. Open quantum system

QArray+ models the quantum-dot array as an open system whose coherent Hamiltonian dynamics coexist with phonon- and lead-induced dissipation. Instantaneous-eigenbasis stochastic wavefunction propagation preserves hybridized states while representing experimentally relevant transitions and decoherence as quantum jumps.

C. Open quantum system: The open-system model combines a tunnel-coupled Hamiltonian with phonon and fermionic-lead dissipators in a Lindblad master equation.Phonons induce relaxation and dephasing within fixed-total-charge sectors, while leads mediate charge-changing transitions.
C. Open quantum system: The instantaneous-eigenbasis stochastic Schrödinger equation allows coherent superpositions to survive at avoided crossings between dissipative events.This differs from Fock-basis jump models that project onto unhybridized charge configurations at each jump.
C. Open quantum system: The simulated pixel signal is the expected occupation vector after propagation, combining fractional hybridized charge with stochastic latching dynamics.The initial state is the instantaneous Hamiltonian ground state, then the state is propagated across pixels.
C. Open quantum system: A single quantum-jump trajectory models an individual experimental scan, while averaging many trajectories reproduces the master equation’s predictions.The simulator therefore uses single trajectories rather than ensemble averages for typical few-shot charge-stability measurements.
C. Open quantum system: Within each sensor timestep, non-Hermitian evolution produces coherent phase accumulation and dissipative decay, followed probabilistically by either no jump or a stochastic collapse.The eigenbasis remains fixed during a pixel because gate voltages are constant over that timestep.

A. Stochastic Capacitance Model

The stochastic capacitance model treats charge configurations as classical Markov states and simulates non-equilibrium transitions on the sensor-integration timescale. Its charge-stability diagrams reproduce finite-rate effects, measurement averaging, blockade-induced latching, and gate-dependent device behavior.

A. Stochastic Capacitance Model: Figure 4 compares original-QArray steady-state diagrams with QArray+ stochastic diagrams spanning infinite-rate, finite-rate latched, and fully isolated regimes.The diagrams use a 200 × 200 gate-voltage grid and overlay the charge state.
A. Stochastic Capacitance Model: Increasing recursion depth Nr enables multiple jumps within one timestep, yielding a time-averaged double-step signal characteristic of finite-bandwidth measurements.The raw stochastic events are averaged because the measurement setup does not resolve every transition.
A. Stochastic Capacitance Model: The model simulates Pauli spin blockade as latching at the interdot boundary and supports gate-voltage dependence in capacitance matrices and tunnel rates.These features extend the simulated behavior beyond a fixed-parameter capacitance model.

B. Spinless Hubbard model

The spinless Hubbard model isolates coherent interdot hybridization in equilibrium charge-stability diagrams. Increasing tunnel coupling rounds interdot transitions through avoided crossings and fractional occupations, while preserving sharp boundaries between total-charge sectors.

B. Spinless Hubbard model: The classical stochastic model captures latching but not hybridization, motivating equilibrium analysis with the spinless Hubbard model.The Hubbard treatment addresses the complementary coherent effect.
B. Spinless Hubbard model: At t12 = 0, the zero-temperature diagram has piecewise-constant integer occupations and sharp boundaries forming the familiar honeycomb pattern.This is the classical ground-state limit.
B. Spinless Hubbard model: For finite t12, interdot degeneracies become avoided crossings with gap ∼2t12, and occupations interpolate continuously across transitions.The sensor response progressively broadens and rounds interdot lines as t12 increases.
B. Spinless Hubbard model: Because the coherent tunnel Hamiltonian conserves total charge, interdot hybridization occurs within fixed-occupation manifolds while lead-transition boundaries remain sharp.This separates coherent charge delocalization from dissipative effects in the equilibrium solver.
B. Spinless Hubbard model: Figure 4’s fully isolated case contains only interdot transitions because the dots are decoupled from charge reservoirs.The initial lower-left charge state is propagated stochastically, preventing parallelization in that regime.

C. Open quantum system

QArray+ combines coherent interdot hybridization with finite-rate dissipative tunneling to model the crossover between equilibrium behavior and scan-history-dependent latching. The open-system solver captures how tunnel coupling, reservoir rates, and pixel dwell time shape charge-stability diagrams, while remaining computationally more expensive than equilibrium simulation.

Model and regime: The open-system model combines coherent interdot hybridization with finite-rate tunneling to leads and evolves the system through stochastic quantum jumps.Its behavior depends on the coherent scale t12, dissipative lead rates Γlead,i, and the effective pixel dwell time τint.
Model and regime: When Γrelaxτint ≫ 1, dissipative relaxation is fast within a pixel and the charge-stability diagram approaches equilibrium.The crossover is controlled by the competition between relaxation, coherent tunneling, reservoir exchange, and scan dwell time.
Reservoir asymmetry: With only dot 1 reservoir-coupled, coherent hybridization provides an effective relaxation pathway and produces comparatively smooth, near-equilibrated transitions.Coupling only dot 2 creates a different bottleneck relative to the scan axes, allowing charge trapping and non-equilibrium features.
Experimental realism: The simulations include finite-temperature stochastic switching, Pauli spin blockade, and gate-dependent interdot capacitive coupling and lead tunnel rates.For Nr = 10, resolving multiple stochastic transitions per pixel captures the experimentally observed doubled transition, whereas Nr = 1 produces artifacts.
Reservoir asymmetry: Increasing hybridization suppresses non-equilibrium artifacts by improving connectivity between relevant charge configurations and accelerating relaxation through the lead-coupled dot.This pushes the diagram toward the equilibrium limit.
Computational trade-off: The Lindblad solver is more expensive than the equilibrium ground-state solver at the same grid resolution, although JAX keeps large-scale simulations tractable on accelerators.Its higher cost reflects propagating the quantum state across the scan and sampling stochastic quantum jumps.

IV. BENCHMARKING

QArray+ benchmarking shows accelerator-enabled scalability across stochastic latching, tunnel-coupled ground-state, and open-system workloads. Multi-GPU execution extends tractable simulation to larger quantum-dot arrays and substantially reduces end-to-end runtime.

Stochastic latching: QArray+ latching scales polynomially with dot count and remains tractable to 64 dots, unlike the baseline, which becomes impractical beyond approximately 16 dots.The baseline's rapid runtime growth is consistent with exponential state-space growth in steady-state formulations.
Stochastic latching: At 64 dots, a 100×100 scan takes approximately 2.8 s on CPU, 0.4 s on one H100, and 0.17 s on 8×H100.These timings demonstrate substantial end-to-end accelerator speedups for stochastic latching.
Tunnel-coupled ground state: QArray+ remains in the few-second to few-tens-of-seconds regime up to 20 dots, while QDarts reaches hundreds of seconds per scan beyond approximately 5 dots.The tunnel-coupled solver uses a fixed-size truncated charge basis and sparse linear algebra to avoid full-basis enumeration and diagonalization.
Open-system solver: At 10 dots, the open-system solver decreases from approximately 1.3×10^3 s on CPU to 110 s on one H100 and 30 s on 8×H100.Accelerator execution substantially extends the accessible regime of this fully physics-informed workload.
Overall scaling: Across the tested workloads, JAX accelerator execution—especially multi-GPU sharding—delivers order-of-magnitude reductions in end-to-end runtime.The benchmarks support high-throughput dataset generation across multiple physical regimes.

V. CONCLUSION

QArray+ unifies coherent interdot hybridization with dissipative, non-equilibrium charge dynamics while supporting scalable simulation and automated-tuning datasets. Its models address simulator limitations involving latching, tunnel coupling, computational complexity, and realistic device behavior.

Conclusion: QArray+ uses Lindblad master-equation evolution to jointly model coherent dynamics, dissipative tunneling, latching, and interdot tunnel coupling.The framework can generate charge-stability diagrams containing both latching and interdot tunnel-coupling effects.
Conclusion: Polynomial scaling in the stochastic model enables large-array simulations that would be computationally prohibitive with conventional steady-state methods.Steady-state solvers typically evaluate at least O(2^ndot) charge states, whereas the stochastic approach has more favorable dot-count scaling.
Conclusion: QArray+ incorporates gate-voltage-dependent capacitances, spin blockade, finite-temperature effects, sensor readout, and noise models.These features support integration into automated tuning pipelines and synthetic machine-learning datasets.
Conclusion: Multi-GPU execution maintains sub-second runtimes across dot counts up to 64 for 100 × 100 charge-stability-diagram benchmarks.The benchmark compares QArray+ with the original QArray implementation on CPU, single-GPU, and 8-GPU configurations.
Conclusion: QArray+ is intended to support offline testing, verification, and rapid data generation for autonomous tuning of scalable semiconductor quantum devices.The authors describe the framework as a prospective digital twin for large-scale devices.
Conclusion: Dense tunnel graphs increase runtime because more nonzero couplings and associated rates must be evaluated at each voltage point.The benchmark identifies JAX/XLA fusion and multi-GPU or multi-node execution as important for high-throughput generation under strong interdot connectivity.

Appendix D: Experimentally measured CSD

Figure 13 presents an experimentally measured charge-stability diagram with a double transition line at a dot–reservoir charge transition.

Appendix D: Experimentally measured CSD: Figure 13 shows an experimentally measured charge-stability diagram exhibiting a double transition line at a dot–reservoir charge transition.The passage identifies the figure as an experimental observation.