Noisy quantum circuit simulation with the tensor jump method

Abstract: 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.

• The paper introduces cTJM, a trajectory-based tensor-network framework that combines TDVP-based MPS gate evolution with Monte Carlo wave-function sampling for sparse Pauli-Lindblad noise, including long-range correlated channels without SWAP overhead.
• The paper develops analog and projector unravelings with exact generator matching and variance bounds, showing that projector sampling can maintain variance near 0.4 in 127-qubit kicked-Ising benchmarks while analog methods perform well under weak noise.
• The paper demonstrates that unraveling choice jointly controls sampling variance and MPS bond-dimension growth, while also identifying limits from non-Clifford entanglement, multi-jump events, and TDVP projection errors.

The paper introduces the circuit Tensor Jump Method (cTJM), a trajectory-based tensor network framework for classical simulation of noisy quantum circuits that combines local time-dependent variational principle (TDVP) gate application on the matrix product state (MPS) manifold with Monte Carlo wave function (MCWF) sampling in the spirit of the Tensor Jump Method (TJM) (2607.01323). The framework is specialized to sparse Pauli-Lindblad models (SPLM) of hardware noise and introduces two variance-aware unravelings—an analog unitary-mixture scheme and a projector-jump scheme—each with distinct variance and entanglement properties. The method is validated on a 25-qubit noisy XY quench and IBM's 127-qubit kicked-Ising benchmark, demonstrating reduced Monte Carlo variance and controlled bond dimension growth relative to standard Kraus-insertion baselines.

Motivation and positioning

Exact noisy simulation via density matrices scales as $\mathcal{O}(4^n)$ in memory; MPO/MPDO compression softens but does not remove this barrier for generic circuits with correlated noise. Stabilizer and Pauli-propagation methods scale well only under structural restrictions (Clifford dominance, sparse Pauli expansions). Trajectory methods such as MCWF retain $\mathcal{O}(2^n)$ per-trajectory scaling and compose naturally with MPS compression, but suffer from trajectory entanglement growth and Monte Carlo variance, which inflates the required number of samples $N$ through the standard error $\mathrm{SE}=\sqrt{\operatorname{Var}(X)/N}$. cTJM addresses precisely these two bottlenecks: it applies gates variationally to limit bond dimension growth, and it exploits unraveling freedom to reduce estimator variance.

Method: circuit TJM with SPLM noise

Gates are realized as short variational evolutions on the MPS manifold via 1-TDVP/2-TDVP, which accommodates long-range gates without SWAP insertion and permits dynamic bond dimension adjustment. Noise is applied stochastically once per gate window using a local Lindblad generator restricted to jump operators whose support intersects the gate support $Q_g$.

The SPLM choice is algorithmically decisive. Because Pauli jumps satisfy $P_m^\dagger P_m = 1$, the jump hazard becomes state-independent ($\delta p_m = \delta t\,\gamma_m$), so jump probabilities can be precomputed per layer as $p_m = \gamma_m/\sum_j \gamma_j$, eliminating the auxiliary stochastic MPS used in prior TJM work. Moreover, the dissipative contraction collapses to a global scalar $$e^{-\frac{1}{2}\delta t\,\Gamma_{\mathrm{tot}}}\mathbb{1}$$ that cancels upon renormalization, reducing the dissipative step to trivial rescaling. The authors note this restriction is pragmatic rather than fundamental, since coherent errors can be Pauli-tailored into stochastic Pauli noise via randomized compiling.

A notable capability is support for arbitrary Lindblad correlated multi-qubit channels consistent with hardware connectivity, including two-qubit channels on non-adjacent qubits. In both unravelings introduced here, every collapse operator reduces up to a scalar to $a\,\mathbb{1} + b\,P$, which admits an exact bond-dimension-2 MPO independent of operator separation. Long-range crosstalk can therefore be applied without SWAP overhead or operator bond dimension inflation—a direct advantage over simulators requiring locality reductions.

Variance-aware unravelings

Analog sampling unravels each Pauli-Lindblad term $\mathcal{O}(2^n)$0 via rate-free unitary kicks $\mathcal{O}(2^n)$1 drawn from a symmetric angle law $\mathcal{O}(2^n)$2, with algorithmic intensity $\mathcal{O}(2^n)$3 fixed by exact generator matching $\mathcal{O}(2^n)$4, where $\mathcal{O}(2^n)$5. Two-point and Gaussian angle laws are analyzed explicitly. Small angles yield frequent near-identity kicks, reducing single-trajectory fluctuations while preserving the ensemble mean exactly. The dissipative contraction again reduces to a global scalar, with total hazard $\mathcal{O}(2^n)$6.

Projector sampling uses collapse operators $\mathcal{O}(2^n)$7, shown exactly equivalent to the original generator. Its central result is a closed-form Bernoulli variance law (Theorem 1): within an absorbing window—a gate-aligned noise window with no unitary dynamics where the readout observable $\mathcal{O}(2^n)$8 anticommutes with all active channels—the estimator is binary, with mean $\mathcal{O}(2^n)$9 and variance $N$0. For general windows mixing commuting and anticommuting channels, the paper derives sharp sandwich bounds on the per-step variance, decomposed into between-event spread $N$1 plus a bounded within-commuting contribution. Both bounds are attainable. This is a strong quantitative claim, though its exact form holds only when the back-propagated observable remains a single Pauli string—true for Clifford segments such as CNOT/CZ/SWAP/H layers and Ising layers at Clifford angles, but not for generic non-Clifford layers, where the formulas serve only as bounds or approximations.

Both unravelings are unbiased and inherit $N$2 convergence with reduced constants.

Numerical validation

Two-qubit bit-flip check: with equal rates on $N$3, all unravelings reproduce the density-matrix mean decay $N$4 over 2000 trajectories. Variances match theory: the projector unraveling follows $N$5, the standard unraveling approaches unit variance as $N$6, and both analog schemes settle at the predicted plateau $N$7. The appendix derives these analog variances via a two-copy Liouvillian closed on a three-dimensional operator subspace, yielding closed-form transients for both angle laws.

25-qubit XY quench: 20 Trotter steps with correlated nearest-neighbor $N$8 noise at $N$9, compared against Qiskit Aer's MPS reference with 200 trajectories and $\mathrm{SE}=\sqrt{\operatorname{Var}(X)/N}$0. All unravelings remain unbiased. A clear regime separation emerges: at weak noise, analog schemes reduce variance while preserving coherent transport dynamics; at moderate-to-strong noise, the projector unraveling simultaneously minimizes variance and suppresses entanglement growth—staying below the bond dimension cap through depth 20 at intermediate noise where other schemes saturate near $\mathrm{SE}=\sqrt{\operatorname{Var}(X)/N}$1 around step 14, and maintaining average bond dimension $\mathrm{SE}=\sqrt{\operatorname{Var}(X)/N}$2 at strong noise.

127-qubit kicked Ising: five Trotter steps of the IBM Eagle benchmark with two-qubit depolarizing noise after each entangling gate, 100 trajectories, $\mathrm{SE}=\sqrt{\operatorname{Var}(X)/N}$3. For $\mathrm{SE}=\sqrt{\operatorname{Var}(X)/N}$4, the projector unraveling keeps the variance of $\mathrm{SE}=\sqrt{\operatorname{Var}(X)/N}$5 uniformly bounded around $\mathrm{SE}=\sqrt{\operatorname{Var}(X)/N}$6—including over longer runs of up to 20 Trotter steps—whereas the standard unraveling approaches order-one variance at strong noise in the identity-equivalent baseline. In the strongly scrambling Clifford case $\mathrm{SE}=\sqrt{\operatorname{Var}(X)/N}$7, the projector variance can exceed the standard one, an effect the paper attributes correctly to the within-event contribution $\mathrm{SE}=\sqrt{\operatorname{Var}(X)/N}$8 from commuting projector jumps, since commuting unitary Pauli flips leave $\mathrm{SE}=\sqrt{\operatorname{Var}(X)/N}$9 unchanged. The non-Clifford case $Q_g$0 is entanglement-limited, reaching $Q_g$1 at low noise for both unravelings—an honest indication that variance reduction does not eliminate the fundamental entanglement barrier.

Limitations and open questions

The paper concedes several constraints. The absorbing-window variance law holds exactly only under restrictive conditions (no Hamiltonian action, full anticommutation with the readout); outside Clifford-dominated segments the closed forms fail, although empirical variance reduction typically persists. Only one jump per two-qubit gate per trajectory is allowed, leaving multi-jump events unaddressed despite their relevance at larger hazards. The TDVP projection error onto finite-bond-dimension manifolds admits no general-purpose estimator beyond special cases such as strictly nearest-neighbor Hamiltonians under 2-TDVP. Finally, the regime separation between analog and projector sampling is established empirically rather than derived from first principles; the paper proposes hybrid, window-adaptive unraveling selection driven by a combined variance–bond-dimension cost proxy as an open direction, along with extensions beyond Pauli-Lindblad noise via structured approximations.

Conclusion

This paper consolidates trajectory-based noisy circuit simulation into a unified framework in which SPLM structure renders jump hazards state-independent and dissipative contractions trivial, while unraveling freedom is exploited quantitatively rather than heuristically. The combination of exact generator matching for analog mixtures, closed-form Bernoulli variance laws for projector jumps, and bond-2 MPO representations of long-range correlated noise yields demonstrable reductions in both Monte Carlo variance and bond dimension growth at hardware-relevant scale. The main unresolved question is whether adaptive, window-level unraveling selection can provably minimize the combined cost of variance and trajectory entanglement across generic non-Clifford circuits.