Exponentially Decaying Quantum Simulation Error with Noisy Devices

TL;DR

The paper analyzes noisy quantum Trotter simulations and shows that, when state information is accounted for, both physical and algorithmic errors decay exponentially with the number of Trotter steps. It provides state-dependent upper bounds, phase diagrams, and formulas for the optimal Trotter number and noise thresholds, demonstrating substantial resource savings for fault-tolerant implementations. Through both analytic proofs and extensive numerics across models and noise channels, the work delineates parameter regimes where practical quantum advantage via simulation remains feasible. These results offer a concrete, quantitative path toward robust quantum simulation on NISQ-era and fault-tolerant devices, with implications for scalable quantum dynamics and chemistry simulations.

Abstract

Quantum simulation is a promising way toward practical quantum advantage, but noise in current quantum hardware poses a significant obstacle. We prove that not only the physical error but also the algorithmic error in a single Trotter step decreases exponentially with the circuit depth. This theoretical finding is validated by our numerical results over various Hamiltonians, initial states, and noise channels. Furthermore, we derive the optimal number of Trotter steps and the noise requirement to guarantee total simulation precision. To explicitly show the requirements for robust quantum simulation, we plot a phase diagram of the accumulated error in terms of circuit depth and noise rate. At last, we demonstrate that our improved error analysis leads to significant resourcesaving for fault-tolerant Trotter circuits. By addressing these aspects, this work provides fresh and systematic insight on the practical quantum advantage through quantum simulation.

Figures (14)

• Figure 1: The regime of potential quantum advantage by quantum simulation. For low-depth quantum circuits or strong noise circuits, the quantum advantage would be diminished by classical algorithms.
• Figure 2: Our model of noisy Trotter simulation circuits. The noisy Trotter circuit is the repeated Trotter steps interspersed by a one-qubit depolarizing noise channel marked by solid red dots. The gray box covers a Trotter step repeated $r$ times. The depolarizing noise renders states close to the maximally mixed state, which results in error decay.
• Figure 3: The one-step physical and algorithmic errors exponentially decay with Trotter steps. The physical error (left panel), algorithmic error (right panel) at every Trotter step with different noise rates $\gamma\in[0.003,0.008]$. The $y$-axis is in the log scale. The color of the line indicates the strength of the noise. We take the TFI Hamiltonian $H_{\mathrm{TFI}}=J \sum_{j=1}^n X_jX_{j+1}+ h\sum_{j=1}^n Z_j$ with parameters ($J$=2, $h$=1, $n$=10), the periodic boundary condition. The Hamiltonian can be grouped into two commuting parts as $H_{\mathrm{TFI}}=H_{X}+H_{Z}$ where $H_X = J \sum_{j=1}^n X_jX_{j+1}$ and $H_Z = h\sum_{j=1}^n Z_j$. We use the second-order product formula PF2 and take the initial state as $\ket{0}^{\otimes n}$, evolution time $t=n$, and Trotter number $r=100$. The dashed lines are the worst-case theoretical upper bounds with the corresponding noise rates. We adopt the upper bound $2n\gamma$ by the diamond norm for the worst-case physical error, while the commutator bound childsTheoryTrotterError2021 is used for the worst-case algorithmic (Trotter) error. See supplementary material for details on the worst-case analysis.
• Figure 4: The impact of initial states on the error decay. We use the standard setup described in \ref{['fig:decay']} but with the fixed noise rate $\gamma=0.005$. The different initial states include the product state $\ket{0}^{\otimes n}$, the worst-case state of one-step Trotter (the state that maximizes the one-step algorithmic error), the ground state of the Hamiltonian, and an ensemble of 20 Haar random states with error bars indicating standard deviations.
• Figure 5: Fitting and extrapolating the one-step physical error $\epsilon_{2,\gamma}^{\textup{phy}}=C\gamma e^{-c \gamma d}$ and algorithmic error $\epsilon_{2,\gamma}^{\textup{alg}}= B \frac{t^{3}}{r^{3}} e^{-b \gamma d}$ of the noisy PF2 with $r=100$ Trotter steps. We adopt the standard Trotter setup described in \ref{['fig:decay']} with varying system sizes (number of qubits $n$). (a) Both physical prefactor and decay coefficients scale linearly with noise rate $\gamma$. (b) Both algorithmic prefactor and decay coefficients scale linearly with noise rate $\gamma$. (c) The fitted prefactor coefficients $C$ and $B$ scale linearly with number of qubits $n$. We extrapolate them to large $n$ cases by the dashed lines. (d) The fitted decaying coefficients $c$ and $b$ do not strongly depend on number of qubits $n$.

Theorems & Definitions (27)

• Definition 1: Noisy Trotter
• Proposition 1: Exponential decay of physical error
• Proposition 2: Exponential decay of algorithmic error
• Theorem 1: Upper bound of noisy Trotter error
• Proposition 3: Optimal Trotter number for noisy Trotter
• Corollary 1: Noise rate requirement for robust noisy Trotter simulation
• Definition 2: Schatten norm
• Definition 3: Diamond norm
• Lemma 1: Diamond distance between Pauli channels magesanCharacterizingQuantumGates2012
• Corollary 2: Diamond distance of 1-qubit depolarizing channel