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.
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.
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.
Quantum computing holds promise for computational chemistry, but near-term quantum hardware remains limited by noise and scale, motivating classical bridging technologies such as quantum emulators. We present an application-specific systems-level benchmark of matrix product state (MPS) tensor-network emulation for real-time Hamiltonian evolution of a density matrix embedding theory (DMET)-embedded sulfonyl-fluoride pharmaceutical fragment, using state-vector simulation as reference. We evaluate runtime, accuracy, resource requirements, and entanglement growth across active spaces from 4 to 24 qubits for a one-body temporal observable used as a quantum fingerprint for reactivity prediction. The results identify a practical boundary for this workflow. At fixed bond dimension, MPS emulation retains favorable scaling, but the bond dimension required to estimate the observable within a 1.6 mHa chemical-accuracy threshold grows rapidly with active-space size. At 20-24 qubits, it approaches the maximum available MPS representation, eliminating the runtime advantage over state-vector simulation. Entanglement entropy analysis shows that increasing bipartite entanglement in the time-evolved state drives this cost growth, consistent with a mismatch between a one-dimensional MPS ansatz and the non-local correlations generated by molecular electronic dynamics. We do not claim a universal crossover across molecules, observables, mappings, or tensor-network geometries. Rather, this study measures where MPS emulation ceases to be an efficient classical surrogate for this chemically motivated Hamiltonian-simulation workflow. The results motivate entanglement-aware algorithm design, orbital-ordering and mapping optimization, alternative tensor-network geometries, and ultimately fault-tolerant quantum hardware for regimes where accurate molecular dynamics generate non-compressible entanglement.
The advent of quantum computers promises exponential speed ups in the execution of various computational tasks. While their capabilities are hindered by quantum decoherence, they can be exactly simulated on classical hardware at the cost of an exponential scaling in terms of number of qubits. To circumvent this, quantum states can be represented as matrix product states (MPS), a product of tensors separated by so-called bond dimensions. Limiting bond dimensions growth approximates the state, but also limits its ability to represent entanglement. Methods based on this representation have been the most popular tool at simulating large quantum systems. But how to trust resulting approximate quantum states for such intractable systems sizes ? I propose here a method for inferring the fidelity of an approximate quantum state without direct comparison to its exact counterpart, and use it to design an ``entanglement-aware'' (EA) algorithm for both pure and mixed states. As opposed to state of the art methods which limit bond dimensions up to an arbitrary maximum value, this algorithm receives as input a fidelity, and adapts dynamically its bond dimensions to both local entanglement and noise such that the final quantum state fidelity at least reaches the input fidelity. I show that this algorithm far surpasses standard fixed bond dimension truncation schemes. In particular, a noiseless random circuit of 300 qubits and depth 75 simulated using MPS methods takes one week of computation time, while EA-MPS only needs 2 hours to reach similar quantum state fidelity.
We aim to rigorously address how local noise affects classical simulability of quantum dynamics benchmarked by tensor-network methods. Using operator entanglement entropy (OEE), we prove the following: (1) For single-qubit depolarizing noise on arbitrary circuits, tensor networks with poly(n) bond dimension suffice for fixed absolute Hilbert-Schmidt error after $\order{1}$ depth, while relative error demands $\order{\log n}$ depth; and this bound is optimal. (2) For single-qubit depolarizing noise on 1D local circuits, the existence of whole-trajectory error-bounded matrix product operator (MPO) of poly(n) bond dimension at all depths. (3) For general single-qubit noise in 1D brickwall circuits, random two-design gates with contraction coefficient c<1/3 yield an $\order{1}$ OEE plateau with probability 1−Te−Ω(n), while arbitrary gates with c<1/48 give $\order{\log n}$ OEE in the worst case. (4) In higher dimensions, these bounds yield uniform-in-depth poly(n) average boundary-bond dimensions for projected entangled pair operators~(PEPO) across every cut -- under depolarizing noise at either absolute or relative accuracy, and under general noise with strong contraction at absolute accuracy. Our results establish a rigorous connection between certain noise models, circuit types, and their classical simulability.
The trading of fidelity for scale enables approximate classical simulators such as matrix product states (MPSs) to run quantum circuits beyond exact methods.A control parameter, the so-called bond dimension χ for MPSs, governs the allocated computational resources and the output fidelity.Here, we characterize the fidelity for the quantum approximate optimization algorithm by the expectation value of the cost function that it seeks to minimize and find that it follows a scaling law F ln χ N , where N is the number of qubits.With ln χ amounting to the entanglement that a MPS can encode, we show that the relevant variable for investigating the fidelity is the entanglement per qubit.Importantly, our results calibrate the classical computational power required to achieve the desired fidelity and benchmark the performance of quantum hardware in a realistic setup.For instance, we quantify the hardness of performing better classically than a noisy superconducting quantum processor by readily matching its output to the scaling function.Moreover, we relate the global fidelity to that of individual operations and establish its relationship with χ and N .We sharpen the requirements for noisy quantum computers to outperform classical techniques at running a quantum optimization algorithm in speed, size, and fidelity.
Abstract The exact treatment of Markovian models of complex systems requires knowledge of probability distributions exponentially large in the number of components n. Mean-field approximations provide an effective reduction in complexity of the models, requiring only a number of phase space variables polynomial in system size. However, this comes at the cost of losing accuracy close to critical points in the systems dynamics and an inability to capture correlations in the system. In this work, we introduce a tunable approximation scheme for Markovian spreading models on networks based on matrix product states (MPSs). By controlling the bond dimensions of the MPS, we can investigate the effective dimensionality needed to accurately represent the exact 2 n dimensional steady-state distribution. We introduce the entanglement entropy as a measure of the compressibility of the system and find that it peaks just after the phase transition on the disordered side, in line with the intuition that more complex states are at the ‘edge of chaos’. We compare the accuracy of the MPS with exact methods on different types of small random networks and with Markov chain Monte Carlo methods for a simplified version of the railway network of the Netherlands with 55 nodes. The MPS provides a systematic way to tune the accuracy of the approximation by reducing the dimensionality of the systems state vector, leading to an improvement over second-order mean-field approximations for sufficiently large bond dimensions.