Efficient simulation of open quantum systems is central to modeling noisy quantum hardware and many-body dynamics. In trajectory-based tensor network methods, cost is often associated with trajectory-level quantities such as entanglement growth or bond dimension. However, the total cost of a fixed-accuracy simulation also depends on statistical sampling, and the interplay between per-trajectory complexity and sampling effort remains poorly understood. Here we introduce a cost-resolved framework for matrix product state (MPS)-based quantum trajectory simulations that decomposes total cost into memory per trajectory, runtime per trajectory, and sampling effort. We show that physically equivalent stochastic unravelings of the same Lindblad dynamics do not necessarily reduce total cost, but instead redistribute cost between trajectory complexity and statistical convergence. This trade-off is quantified by two dimensionless inflation factors: a bond dimension inflation $α$ and a sampling inflation $κ$, which together determine the preferred unraveling under hardware-dependent memory and parallelism constraints. We provide a practical protocol for extracting $(α,κ)$ from modest pilot simulations and demonstrate it using benchmarks across multiple noise channels. The resulting decision maps show that the computationally favorable unraveling can change with noise strength, time-step resolution, system size, and available parallelism. These results establish unraveling choice as a hardware-aware simulation design problem rather than an intrinsic optimization of trajectory entanglement alone.
Efficient simulation of open quantum systems is central to modeling noisy quantum hardware and many-body dynamics. In trajectory-based tensor network methods, cost is often associated with trajectory-level quantities such as entanglement growth or bond dimension. However, the total cost of a fixed-accuracy simulation also depends on statistical sampling, and the interplay between per-trajectory complexity and sampling effort remains poorly understood. Here we introduce a cost-resolved framework for matrix product state (MPS)-based quantum trajectory simulations that decomposes total cost into memory per trajectory, runtime per trajectory, and sampling effort. We show that physically equivalent stochastic unravelings of the same Lindblad dynamics do not necessarily reduce total cost, but instead redistribute cost between trajectory complexity and statistical convergence. This trade-off is quantified by two dimensionless inflation factors: a bond dimension inflation α and a sampling inflation κ, which together determine the preferred unraveling under hardware-dependent memory and parallelism constraints. We provide a practical protocol for extracting (α,κ) from modest pilot simulations and demonstrate it using benchmarks across multiple noise channels. The resulting decision maps show that the computationally favorable unraveling can change with noise strength, time-step resolution, system size, and available parallelism. These results establish unraveling choice as a hardware-aware simulation design problem rather than an intrinsic optimization of trajectory entanglement alone.
Classical simulation of noisy quantum circuits is essential for validating algorithms, benchmarking hardware, and assessing error-mitigation strategies, but remains limited by the exponential cost of density-matrix methods and the high variance of standard trajectory sampling. We introduce a variance-aware tensor network framework that combines the tensor jump method with local TDVP gate evolution on matrix product states and sparse Pauli-Lindblad hardware noise models. Gates are applied as short variational evolutions on the MPS manifold, while noise is sampled per circuit window from Pauli-Lindblad jump sets with state-independent hazards and dissipative contractions that reduce to irrelevant global factors after renormalization. The method supports correlated multi-qubit Lindblad noise consistent with hardware connectivity, including long-range operators on non-adjacent qubits, enabling direct simulation of crosstalk and other connectivity-induced errors beyond local noise models. We develop two unbiased variance-aware unravelings. An analog unitary-mixture unraveling matches the Lindblad generator exactly under symmetric Gaussian or two-point angle laws, while a projector-jump unraveling yields state-independent hazards and closed-form variance laws. Both retain the standard 1/sqrt(N) Monte Carlo convergence but with reduced prefactors. Empirically, projector sampling strongly reduces trajectory variance and bond-dimension growth across many circuit architectures, whereas analog sampling is most effective at weak noise. We demonstrate accurate, scalable noisy-circuit simulation on a 25-qubit noisy XY quench and IBM's 127-qubit kicked-Ising benchmark with long-range depolarizing noise, achieving reduced Monte Carlo variance and favorable MPS bond-dimension growth compared with standard Kraus-insertion baselines.
Classical simulation of noisy quantum circuits is essential for validating algorithms, bench-marking hardware, and guiding error-mitigation strategies. Yet it remains challenging due to either the exponential memory footprint of density matrix methods or the high variance in trajectories of standard Kraus-insertion approaches. We introduce a variance aware tensor network framework that unifies the tensor jump method (TJM) with a local time-dependent variational principle (TDVP) and a sparse Pauli-Lindblad model (SPLM) of hardware noise. Gates are realized as short variational evolutions on the matrix product state (MPS) manifold, while noise is applied stochastically per circuit window using Pauli-Lindblad jump sets that render the sampling hazard state-independent and the dissipative contraction trivial after renormalization. Strikingly, our method supports arbitrary Lindblad correlated multi-qubit noise channels consistent with hardware connectivity, including long-range noise operators on non-adjacent subsets, enabling direct simulation of crosstalk and other connectivity-induced errors beyond local noise models. Building on prior work that introduced analog unitary unravelings, we adapt this scheme to a framework based on the Lindblad master equation with exact generator matching under discretized angle laws and a parameter choice. This exactly matches the Lindblad generator, while selecting Gaussian or two-point angle laws. In parallel, we introduce an MPS-projector-jump unraveling that yields state-independent hazards and closed-form variance laws. Both schemes are unbiased and inherit the standard 1/√N Monte Carlo convergence, but with substantially smaller constants due to variance reduction. Empirically, projector sampling markedly reduces the required bond dimension per trajectory across many circuit architectures, while analog sampling excels at low noise. We demonstrate accurate, scalable noisy-circuit simulation on a 25-qubit noisy XY quench and IBM’s 127-qubit kicked Ising benchmark with long-range depolarizing noise, achieving substantially reduced Monte Carlo variance and favorable bond dimension growth compared to standard Kraus insertion baselines.
Classical simulations of noisy quantum circuits are instrumental to our understanding of the behavior of real-world quantum systems and the identification of regimes where one expects quantum advantage. In this work, we present a highly parallelizable tensor-network-based classical algorithm, equipped with rigorous accuracy guarantees, for simulating n -qubit quantum circuits with arbitrary single-qubit noise. Our algorithm represents the state of a noisy quantum system by a particular ensemble of matrix product states from which we stochastically sample. Each pure state evolved under a single-qubit noise process is then represented by the ensemble of states that achieves the minimal average entanglement (the entanglement of formation) between the noisy qubit and the remainder. This approach lets us use a more compact representation of the quantum state for a given accuracy requirement and noise level. For a given maximum bond dimension χ and circuit, our algorithm comes with an upper bound on the simulation error, runs in poly ( n , χ ) -time, and improves upon related prior work (1) in scope by extending analytic methods from the three commonly considered noise models to general single-qubit noise, (2) in performance by deriving an exact, closed-form solution to the local entanglement minimization problem, previously approached via variational heuristics, thereby guaranteeing local optimality without numerical optimization overhead, and (3) in conceptual contribution by showing that the fixed unraveling used in prior work becomes equivalent to our choice of unraveling in the special case of depolarizing, dephasing, and amplitude damping noise acting on a maximally entangled state.
Monte-Carlo trajectory (quantum-jump) methods are the practical route to simulating noisy quantum circuits once the exact density-matrix method is precluded by its 4n memory cost. Their bottleneck is estimator variance: resolving one expectation value can demand thousands of trajectories. Recent tensor-network work shows that \emph{variance-reduced unravelings} -- projector and analog sampling -- sharply cut this variance, but only on CPU matrix-product-state backends, with no path into production tooling. We implement both unravelings on a \emph{GPU dense-statevector} trajectory engine and validate them against the exact density matrix (ideal-circuit fidelity 1−2.2×10−16; 1/N convergence; all unravelings unbiased to trace distance <0.01). On a single consumer GPU, projector unraveling reaches a target standard error with 20.8× fewer trajectories than Qiskit-Aer's \texttt{batched\_shots\_gpu} at n=10, a factor that holds at 19--26× across n=8--20. A regime map places analog sampling optimal at weak noise and projector at strong noise, crossing near γt≈0.35. We further report a systems finding: Qiskit-Aer applies noise at the \emph{channel} level and reconstructs a canonical Kraus decomposition at apply time, discarding any user-supplied unraveling, so variance-reduced unravelings cannot be delivered through its public API. Because Aer's Born-rule collapse machinery already exists, we specify a minimal change that would unlock the technique in production.
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 dynamics of open quantum systems can be simulated by unraveling it into an ensemble of pure state trajectories undergoing nonunitary monitored evolution, which has recently been shown to undergo measurement-induced entanglement phase transition. Here, we show that, for an arbitrary decoherence channel, one can optimize the unraveling scheme to lower the threshold for entanglement phase transition, thereby enabling efficient classical simulation of the open dynamics for a broader range of decoherence rates. Taking noisy random unitary circuits as a paradigmatic example, we analytically derive the optimum unraveling basis that on average minimizes the threshold. Moreover, we present a heuristic algorithm that adaptively optimizes the unraveling basis for given noise channels, also significantly extending the simulatable regime. When applied to noisy Hamiltonian dynamics, the heuristic approach indeed extends the regime of efficient classical simulation based on matrix product states beyond conventional quantum trajectory methods. Finally, we assess the possibility of using a quasi-local unraveling, which involves multiple qubits and time steps, to efficiently simulate open systems with an arbitrarily small but finite decoherence rate.