Tensor network methods, particularly those based on Matrix Product States (MPS), provide a powerful framework for simulating quantum many-body systems. A persistent computational challenge in these methods is the selection of the bond dimension chi, which controls the trade-off between accuracy and computational cost. Fixed bond dimension strategies either waste resources in low-entanglement regions or lose fidelity in high-entanglement regions. This work introduces an adaptive bond dimension management framework that uses von Neumann entropy feedback coupled with a Proportional-Integral-Derivative (PID) controller to dynamically adjust chi at each bond during simulation. An Exponential Moving Average (EMA) filter stabilizes entropy measurements against transient fluctuations, and a predictive scheduling module anticipates future bond dimension requirements from entropy trends. The per-bond granularity of the allocation ensures that computational resources concentrate where entanglement is largest. The framework integrates GPU-accelerated Singular Value Decomposition (SVD) via CuPy and the cuSOLVER backend, achieving individual SVD speedups of 4.1x at chi=256 and 7.1x at chi=2048 relative to CPU-based NumPy for isolated matrix factorisations (measured on an NVIDIA A100-SXM4-40GB GPU with CuPy 13.4.1 and CUDA 12.8). At the system level, benchmarks on the spin-1/2 antiferromagnetic Heisenberg chain demonstrate a 2.7x reduction in total DMRG wall time compared to fixed-chi simulations, with energy accuracy within 0.1% of the Bethe ansatz solution. Integration with the Density Matrix Renormalization Group (DMRG) algorithm yields ground-state energies per site converging to E/N = -0.4432 for the isotropic Heisenberg model at chi = 128. Validation against Amazon Web Services (AWS) Braket SV1 statevector simulator confirms agreement within 2-5% for small systems.
Tensor network methods, particularly those based on Matrix Product States (MPS), provide a powerful framework for simulating quantum many-body systems. A persistent computational challenge in these methods is the selection of the bond dimension chi, which controls the trade-off between accuracy and computational cost. Fixed bond dimension strategies either waste resources in low-entanglement regions or lose fidelity in high-entanglement regions. This work introduces an adaptive bond dimension management framework that uses von Neumann entropy feedback coupled with a Proportional-Integral-Derivative (PID) controller to dynamically adjust chi at each bond during simulation. An Exponential Moving Average (EMA) filter stabilizes entropy measurements against transient fluctuations, and a predictive scheduling module anticipates future bond dimension requirements from entropy trends. The per-bond granularity of the allocation ensures that computational resources concentrate where entanglement is largest. The framework integrates GPU-accelerated Singular Value Decomposition (SVD) via CuPy and the cuSOLVER backend, achieving individual SVD speedups of 4.1x at chi=256 and 7.1x at chi=2048 relative to CPU-based NumPy for isolated matrix factorisations (measured on an NVIDIA A100-SXM4-40GB GPU with CuPy 13.4.1 and CUDA 12.8). At the system level, benchmarks on the spin-1/2 antiferromagnetic Heisenberg chain demonstrate a 2.7x reduction in total DMRG wall time compared to fixed-chi simulations, with energy accuracy within 0.1% of the Bethe ansatz solution. Integration with the Density Matrix Renormalization Group (DMRG) algorithm yields ground-state energies per site converging to E/N = -0.4432 for the isotropic Heisenberg model at chi = 128. Validation against Amazon Web Services (AWS) Braket SV1 statevector simulator confirms agreement within 2-5% for small systems.
Out-of-equilibrium dynamics of non-integrable Hamiltonian many-body quantum systems are characterized by highly entangled wave functions. Near-maximal entanglement arises in systems exhibiting thermalization or pre-thermalization, where the system converges to a steady state with a fixed energy density. Classical simulation of the time dependence of such wave functions requires exponential resources. However, typical computations aim to estimate expectation values of local operators and correlation functions to some expected precision. For thermalizing systems at sufficiently high energy densities, such computations can be done without storing the full wave function by instead simulating the evolution of the local operator, which requires significantly fewer resources. Nonetheless, constructing such resource-efficient classical algorithms remains a challenge for intermediate energy densities, where simulating both the wave function and operator evolution is costly. In this paper, we propose a heuristic approach to accelerate the convergence of Matrix Product State (MPS) simulations of expectation values, applicable across a broad range of energy densities. We estimate the desired observables by rescaling the MPS results at low bond dimensions with a factor that depends on the fidelity of the MPS wave function. Using this technique, we simulated the dynamics of the two-dimensional Transverse-Field Ising Model (TFIM) on a 7×8 grid with periodic boundary conditions, using a maximum bond dimension of χ=4096 on a single A100 GPU, as well as the dynamics of the two-dimensional XY model on grids of size up to 9×9. We compare our TFIM results to similar simulations on a digital quantum processor [R. Haghshenas et al., Nature 653, 56 (2026)], demonstrating excellent agreement and confirming the predictive power of our method.
We present a method for allocating bond dimensions in matrix product state (MPS) simulations based on the commensurateness structure of the system’s frequency spectrum. For each pair of frequencies (ωᵢ, ωⱼ), the quantity sin(Cᵢⱼ/2) — where Cᵢⱼ measures the distance of the ratio ωᵢ/ωⱼ from the nearest low-order rational — predicts the entanglement contribution of the corresponding bond. When ωᵢ/ωⱼ is close to a simple rational, the underlying mode dynamics is confined to KAM tori: the effective coupling is suppressed and the bond carries little entanglement. When the ratio is far from all simple rationals, the dynamics is chaotic, the coupling is active, and the bond requires significant χ. We prove an upper bound on the entanglement entropy per bond in terms of sin(C/2) and the coupling angles, under the assumption that the commensurateness filter acts on gate angles in the same way it acts on the kick parameter in the quantum kicked rotor. Empirically, the method is ineffective for nearest-neighbor circuits with uniform frequencies — correctly, as no commensurateness structure is present — but yields substantial gains for molecular Hamiltonians: on H₂O UCCSD (8 qubits) we observe +1.6 percentage points of fidelity at χ=8; on LiH full (12 qubits, 6× coupling variation) we observe +27 percentage points at χ=12, and exact fidelity at χ=16 where uniform allocation requires χ=32 — a factor-of-two reduction in required bond dimension. The channel analysis scales as O(N·r²) with a sparse approximation at radius r=5, enabling 100,000 qubits in under one second on a single CPU core.
Classical simulation of quantum circuits remains indispensable for algorithm development, hardware validation, and error analysis in the noisy intermediate-scale quantum (NISQ) era. However, state-vector simulation faces exponential memory scaling, with an n-qubit system requiring O(2^n) complex amplitudes, and existing simulators often lack the flexibility to exploit heterogeneous computing resources at runtime. This paper presents a GPU-accelerated quantum circuit simulation framework that introduces three contributions: (1) an empirical backend selection algorithm that benchmarks CuPy, PyTorch-CUDA, and NumPy-CPU backends at runtime and selects the optimal execution path based on measured throughput; (2) a directed acyclic graph (DAG) based gate fusion engine that reduces circuit depth through automated identification of fusible gate sequences, coupled with adaptive precision switching between complex64 and complex128 representations; and (3) a memory-aware fallback mechanism that monitors GPU memory consumption and gracefully degrades to CPU execution when resources are exhausted. The framework integrates with Qiskit, Cirq, PennyLane, and Amazon Braket through a unified adapter layer. Benchmarks on an NVIDIA A100-SXM4 (40 GiB) GPU demonstrate speedups of 64x to 146x over NumPy CPU execution for state-vector simulation of circuits with 20 to 28 qubits, with speedups exceeding 5x from 16 qubits onward. Hardware validation on an IBM quantum processing unit (QPU) confirms Bell state fidelity of 0.939, a five-qubit Greenberger-Horne-Zeilinger (GHZ) state fidelity of 0.853, and circuit depth reduction from 42 to 14 gates through the fusion pipeline. The system is designed for portability across NVIDIA consumer and data-center GPUs, requiring no vendor-specific compilation steps.
Tensor network algorithms can efficiently simulate complex quantum many-body systems by utilizing knowledge of their structure and entanglement. These methodologies have been adapted recently for solving the Navier-Stokes equations, which describe a spectrum of fluid phenomena, from the aerodynamics of vehicles to weather patterns. Within this quantum-inspired paradigm, velocity is encoded as matrix product states (MPS), effectively harnessing the analogy between interscale correlations of fluid dynamics and entanglement in quantum many-body physics. This particular tensor structure is also called quantics tensor train (QTT). By utilizing NVIDIA's cuQuantum library to perform parallel tensor computations on GPUs, our adaptation speeds up simulations by up to 12.1 times. This allows us to study the algorithm in terms of its applicability, scalability, and performance. By simulating two qualitatively different but commonly encountered 2D flow problems at high Reynolds numbers up to 1 × 10 7 using a fourth-order time stepping scheme, we find that the algorithm has a potential advantage over direct numerical simulations in the turbulent regime as the requirements for grid resolution increase drastically. In addition, we derive the scaling χ = O ( poly ( 1 / ε ) ) for the maximum bond dimension χ of MPS representing turbulent flow fields, with an error ε , based on the spectral distribution of turbulent kinetic energy. Our findings motivate further exploration of related quantum algorithms and other tensor network methods.
Quantum computer simulators are an indispensable tool for prototyping quantum algorithms and verifying the functioning of existing quantum computer hardware. The current largest quantum computers feature more than one thousand qubits, challenging their classical simulators. Statevector quantum simulators are challenged by the exponential increase of representable quantum states with respect to the number of qubits, making more than fifty qubits practically unfeasible. A more appealing approach for simulating quantum computers is adopting the tensor network approach, whose memory requirements fundamentally depend on the level of entanglement in the quantum circuit, and allows simulating the current largest quantum computers. This work investigates and evaluates the CUDA-Q tensor network simulators on an Nvidia Grace Hopper system, particularly the Matrix Product State (MPS) formulation. We compare the performance of the CUDA-Q state vector implementation and validate the correctness of MPS simulations. Our results highlight that tensor network-based methods provide a significant opportunity to simulate large-qubit circuits, albeit approximately. We also show that current GPUaccelerated computation cannot fully utilize GPU efficiently in the case of MPS simulations.
This work presents a matrix product state (MPS) simulation framework for adaptive non-local observable variational quantum circuits (ANO-VQCs) in image super-resolution (SR) beyond the practical limits of statevector simulation. Runtime benchmarks on a single NVIDIA RTX 4070 GPU show that, under the tested shallow-circuit setting, MPS completes ANO-VQC forward feature extraction for individual inputs up to 16 x 16 pixels (256 qubits), whereas statevector simulation encounters a memory bottleneck at 6 x 6 inputs (36 qubits) and exact tensor-network (Exact TN) contraction becomes computationally impractical beyond 12 x 12 inputs (144 qubits). For a fixed 7 x 7 input (49 qubits), a bond-dimension sweep over depths L = 1 to L = 4 shows that the required MPS bond dimension increases with circuit depth. Using Exact TN contraction as the reference, the bond dimension required for near-exact agreement increases from chi = 2 at L = 1 to chi = 16 at L = 4. Finally, 7 x 7 to 28 x 28 Fashion-MNIST SR training with chi = 16 shows that the shallow L = 1 model achieves the lowest loss, lowest LPIPS, and highest PSNR and SSIM among the tested depths. These results highlight MPS as a scalable and controllable simulation backend for ANO-VQC image SR and as a practical tool for studying large-scale quantum algorithms.