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Classical Simulation of Noiseless Quantum Dynamics without Randomness

Jue Xu, Chu Zhao, Xiangran Zhang, Shuchen Zhu, Qi Zhao

Abstract

Simulating noiseless quantum dynamics classically faces a fundamental dilemma: tensor-network methods become inefficient as entanglement saturates, while Pauli-truncation approaches typically rely on noise or randomness. To close the gap, we propose the Low-weight Pauli Dynamics (LPD) algorithm that efficiently approximates local observables for short-time dynamics in the absence of noise. We prove that the truncation error admits an average-case bound without assuming randomness, provided that the state is sufficiently entangled. Counterintuitively, entanglement--usually an obstacle for classical simulation--alleviates classical simulation error. We further show that such entangled states can be generated either by tensor-network classical simulation or near-term quantum devices. Our results establish a rigorous synergy between existing classical simulation methods and provide a complementary route to quantum simulation that reduces circuit depth for long-time dynamics, thereby extending the accessible regime of quantum dynamics.

Classical Simulation of Noiseless Quantum Dynamics without Randomness

Abstract

Simulating noiseless quantum dynamics classically faces a fundamental dilemma: tensor-network methods become inefficient as entanglement saturates, while Pauli-truncation approaches typically rely on noise or randomness. To close the gap, we propose the Low-weight Pauli Dynamics (LPD) algorithm that efficiently approximates local observables for short-time dynamics in the absence of noise. We prove that the truncation error admits an average-case bound without assuming randomness, provided that the state is sufficiently entangled. Counterintuitively, entanglement--usually an obstacle for classical simulation--alleviates classical simulation error. We further show that such entangled states can be generated either by tensor-network classical simulation or near-term quantum devices. Our results establish a rigorous synergy between existing classical simulation methods and provide a complementary route to quantum simulation that reduces circuit depth for long-time dynamics, thereby extending the accessible regime of quantum dynamics.
Paper Structure (6 sections, 21 theorems, 54 equations, 5 figures, 1 table, 1 algorithm)

This paper contains 6 sections, 21 theorems, 54 equations, 5 figures, 1 table, 1 algorithm.

Table of Contents

  1. Preliminaries
  2. Proofs of lemmas and theorems
  3. Proof of Lemma 1
  4. Trotter formula error analysis
  5. Pauli truncation error analysis
  6. Runtime and memory analysis

Key Result

Lemma 1

Given a local observable $O=\sum_j O_j$, if the input state $\ket{\psi_{S}}$ has the subsystem entanglement entropy $S(\rho_{j,j'})\ge\abs{\textup{supp}(O^\dagger_j O_{j'})}-\frac{1}{2}\norm{O}_{\bar{2}}^4/(\sum_j\norm{O_j})^4$ for all subsystems $\textup{supp}(O^\dagger_j O_{j'})$, then the squared

Figures (5)

  • Figure 1: Regimes of efficient classical simulations for Hamiltonian dynamics. Tensor network methods efficiently represent quantum states with low entanglement, whereas existing theoretical guarantees for Pauli-truncation methods rely on noise or randomness. Our work shows the state entanglement actually suppresses the Pauli truncation error in the noiseless case, thereby broadening the regime of classical simulation.
  • Figure 2: The illustration of the $\textup{LPD}$ (Low-weight Pauli Dynamics) algorithm. (a) The Trotterized backward evolution of local observable with high-weight truncation and light-cone argument. The initial product state is evolved to a sufficiently entangled state by the MPS method. The expectation value is evaluated by the low-weight observable and the entangled state. (b) The local operator flow under local Pauli rotations. The norm of Paulis with certain weight flowing to higher weights is damped by small angle $\textup{d} t$. The darker color represents a larger norm contribution of Paulis with certain weight to the norm of the evolved observable. (c) Truncation of high-weight Pauli operators in one step. The colors and the sizes of the bubbles represent the magnitudes of the Pauli coefficients. In this example, the Pauli operators with weight larger than $w^*=2$ are truncated.
  • Figure 3: The numerical results of Alg. \ref{['alg:noiseless']} for the QMFI model with $n=10$ qubits. We set evolution time $t=5$ and the second-order Trotter formula with $r=50$ steps. (a) The expectation values at Trotter every step. The solid line represents the ideal expectation value. The purple dotted line is the Trotter expectation without truncation. The dashed line is the one with the truncation. (b) The dashed line is Trotter error with the product input state, while the line with error bars is the average Trotter error over 100 Haar random states that are mostly entangled states. (c) In the short-time regime, the low-weight Paulis (dark green) dominate the norm distribution over weights, while the high-weight Paulis (light green) gradually accumulate in longer time. The empty regime is the norm lost by the Pauli truncation. (d) The dashed line is the Pauli truncation error in expectation values with the product input state, while the line with error bars is the average Pauli truncation error over Haar random (entangled) states.
  • Figure 4: $\textup{LPD}$ complements $\textup{MPS}$ for the dynamics of QMFI. (a) The entanglement entropy $S_2$ ($2$ qubits subsystem) of the forward-simulated state by $\textup{MPS}$. The dashed line indicates the entanglement entropy of the ideally evolved state, while the dots are the entanglement entropy of the state simulated by $\textup{MPS}$ with bond dimension $\chi=32$. (b) The entropy of Pauli coefficients of the backward-evolved observable $O(t)$ (also known as operator magic). The dashed line indicates the operator magic of the ideally evolved observable, while the dots are the operator magic of the observable simulated by $\textup{LPD}$ with truncation threshold $w^*=5$. (c) The hybrid simulation by $\textup{LPD}$ (blue) and $\textup{MPS}$ (red) with a total evolution time $t=10$ matches the exact expectation value (black) well.
  • Figure 5: The Pauli branch and weight change in one Trotter step. (a) An illustration of constant layers of local Pauli rotations acting on a local (low-weight) Pauli operator. For simplicity, assume the initial Pauli operator is XZIIIIIXIIII with weight 3, the two-layer brickwork Pauli rotations are $e^{-\textup{i} XX \textup{d} t}$ colored red. In the first layer, only two Pauli rotations (red) has support on the initial Pauli operator. One commutes with the initial Pauli operator and thus does not change it; the other anti-commutes and thus generates a new Pauli operator XYIIIIIXIIII with weight 3, the coefficient damped by $\sin(\textup{d} t)$ (with the lighter color). Similarly, in the second layer, one Pauli rotation commutes and the other anti-commutes, generating another two new Paulis. At the end of one Trotter step, all Pauli operators are low-weight (below the threshold) due to truncation. Since high-weight Paulis flowed from low-weight Paulis experienced many Pauli rotations, their contribution must be damped by small angle $\textup{d} t$ many times. (b) The 2-norm distribution over Pauli weights with(above)/without(below) truncation. Without truncation, the top figure shows that in the short-time regime, the low-weight Paulis (dark green) dominate the 2-norm distribution, while the high-weight Paulis (light green) gradually accumulate 2-norm in longer time. So, it is feasible to truncate high-weight Paulis in the short-time regime and Pauli truncation is unlikely to work in long-time (i.e., $t\sim n$) regime. The bottom figure shows that with the high-weight truncation, the 2-norm distribution is strictly limited to low-weight Paulis, and the 2-norm gradually loses as time evolves.

Theorems & Definitions (45)

  • Lemma 1: Pauli 2-norm upper bound on local observable expectation value with entangled states
  • Lemma 2: Damped local norm flow
  • Theorem 1: Runtime of Alg. \ref{['alg:noiseless']}, informal
  • Lemma 3: Upper bound on local observable expectation value
  • proof : Proof of \ref{['thm:entangle_ob_bound']}
  • proof : Sketch proof of \ref{['thm:weight_decay']}
  • Proposition 1: Truncation threshold of $\textup{LPD}$
  • proof : Sketch proof of \ref{['thm:truncation_threshold']}
  • Definition A.1: Normalized Schatten 2-norm, or Pauli 2-norm
  • Definition A.2: State $k$-design meleIntroductionHaarMeasure2024
  • ...and 35 more