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Classical simulation of noisy quantum circuits via locally entanglement-optimal unravelings

Simon Cichy, Paul K. Faehrmann, Lennart Bittel, Jens Eisert, Hakop Pashayan

TL;DR

This work develops a highly parallelizable, tensor-network-based classical algorithm for simulating $n$-qubit quantum circuits under arbitrary single-qubit noise using locally entanglement-optimal unravelings. By reducing the problem to an effective two-qubit state and applying Wootters' entanglement-of-formation decomposition, the authors obtain an exact analytical optimal unraveling that minimizes entanglement and hence the required bond dimension $\\chi$, with rigorous a posteriori error bounds for truncated trajectories. They connect this approach to standard classical tasks (Samp and OEstim), compare it to existing unravelings (Orthogonal, Haar Optimal, Projective), and demonstrate substantial performance gains in Lindbladian and noisy random-circuit settings, including near-optimal fixed unravelings for random circuits and a least-unitary unraveling perspective. The results offer rigorous insights into the classical simulability of noisy quantum systems, providing scalable tools and a principled framework to assess how noise affects quantum advantage in practical, one-dimensional architectures.

Abstract

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 $\mathrm{poly}(n,χ)$-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.

Classical simulation of noisy quantum circuits via locally entanglement-optimal unravelings

TL;DR

This work develops a highly parallelizable, tensor-network-based classical algorithm for simulating -qubit quantum circuits under arbitrary single-qubit noise using locally entanglement-optimal unravelings. By reducing the problem to an effective two-qubit state and applying Wootters' entanglement-of-formation decomposition, the authors obtain an exact analytical optimal unraveling that minimizes entanglement and hence the required bond dimension , with rigorous a posteriori error bounds for truncated trajectories. They connect this approach to standard classical tasks (Samp and OEstim), compare it to existing unravelings (Orthogonal, Haar Optimal, Projective), and demonstrate substantial performance gains in Lindbladian and noisy random-circuit settings, including near-optimal fixed unravelings for random circuits and a least-unitary unraveling perspective. The results offer rigorous insights into the classical simulability of noisy quantum systems, providing scalable tools and a principled framework to assess how noise affects quantum advantage in practical, one-dimensional architectures.

Abstract

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 -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 and circuit, our algorithm comes with an upper bound on the simulation error, runs in -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.

Paper Structure

This paper contains 64 sections, 20 theorems, 305 equations, 15 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Simulation through unraveled quantum trajectory sampling
  4. Truncated Trajectory Sampling Task
  5. Class of Truncated Trajectory Sampling Algorithms
  6. Upper bounds on the simulation errors
  7. Target classical simulation tasks
  8. Our algorithm
  9. Details of MPS-based trajectory sampling
  10. Locally entanglement-optimal unraveling
  11. Problem statement and main result
  12. Reduction to the effective two-qubit problem
  13. Wootters' optimal decomposition
  14. Optimal unraveling unitary
  15. Comparing unraveling strategies
  16. ...and 49 more sections

Key Result

Theorem 1

Given a noisy circuit description $\mathcal{D}$ with a final state $\rho$ and a maximal bond dimension $\chi\in \mathbb{N}$, there exist an $\epsilon\geq 0$ such that with runtime $\mathop{\mathrm{poly}}\nolimits(n, \chi, N)$ the trajectory sampling algorithm returns samples from an $(\epsilon, \chi

Figures (15)

  • Figure 1: Effect of single-qubit dephasing noise on the Bloch sphere (a) and two interpretations of the unraveling of single-qubit dephasing noise into pure-state trajectories. Circuit corresponding to the unraveling of one trajectory in the (b) mixed-unitary picture and (c) projective picture. The trajectories (and their entanglement) have very different behaviors in both.
  • Figure 2: Graphical representation of the unraveling of quantum trajectories based on matrix product states, for noisy circuits Consider the setting of one-dimensional noisy circuits, composed of (two-)local unitary gates, followed by single-qubit noise channels (top left). The unitary gates are dealt with by the standard MPS contraction. The noise results in a statistical mixture of pure states. This mixture is then unraveled to sample one trajectory, and the resulting state is truncated to the allowed bond dimension. The trajectory sampling results from computing the probabilities of each path and sampling one of them, thus applying the corresponding Kraus operator (bottom). Before that, the unraveling is chosen. In our case, we do so based on the computation of the entanglement of formation of the affected effective two-qubit state (top right).
  • Figure 4: Graphical representation of the procedure to find the optimal single-qubit unraveling.
  • Figure 5: Simulation of a $16$ qubit system evolving under a Heisenberg Hamiltonian $H = \sum_i Y_iY_{i+1} + 0.35 X_i + 0.35 Y_i + 0.5 Z_i$ (open boundary conditions) with amplitude damping noise. All results are an average of $500$ independently sampled trajectories and repeated for the Orthogonal and Haar Optimal fixed unravelings (see Table \ref{['tab:unravelings']}) as well as our locally entanglement-optimal unraveling. (a) Average entanglement as a function of circuit depth plotted for various noise rates and unravelings in a simulation with bond dimension $\chi = 50$. (b) Average upper bound on truncation error (Eq. \ref{['eq:MPS_truncation_error']}) as a function of circuit depth, plotted for various bond dimensions in a simulation with noise rate $\gamma^{\mathrm{AD}} = 0.002$. (c) Scaling of the truncation error (at depth $600$) with bond dimension for various noise rates.
  • Figure 6: Simulation of a $12$ qubit open quantum system under random translationally invariant and time-independent Lindbladian evolutions with amplitude damping noise. For each random evolution, we consider a brickwork circuit composed of a fixed gate repeated at each site. The gate itself is obtained by a short-time unitary evolution $U=\mathop{\mathrm{e}}\nolimits^{-\mathop{\mathrm{i}}\nolimits H dt}$ of a random Hamiltonian $H$ (a random Hermitian matrix with normally distributed real and imaginary parts of each entry) and a time step $dt=0.01$. Each trace is the average of 1000 independently computed trajectories truncated to a fixed bond dimension. (a) Average entanglement as a function of circuit depth plotted for various noise rates and unravelings in a simulation with bond dimension $\chi = 20$. (b) to (d) Average upper bound on truncation error (Eq. \ref{['eq:MPS_truncation_error']}) as a function of circuit depth, plotted for various combinations of bond dimensions and noise rates.
  • ...and 10 more figures

Theorems & Definitions (45)

  • Definition 1: Unraveling
  • Definition 2: $(\epsilon, \chi)$-convex MPS
  • Definition 3: Truncated Trajectory Sampling
  • Theorem 1: Truncated Trajectory Sampling error
  • Lemma 1: Output distribution sampling from trajectory sampling
  • Lemma 2: Expectation value estimation from trajectory sampling
  • Corollary 1: Output distribution sampling
  • proof
  • Corollary 2: Expectation value estimation
  • proof
  • ...and 35 more