Describing Trotterized Time Evolutions on Noisy Quantum Computers via Static Effective Lindbladians

TL;DR

The paper addresses how decoherence influences Trotterized quantum simulations on noisy devices by formulating a circuit-level Lindblad framework and introducing the noisy algorithm model, where the effective generator is $\mathcal{L}_{\text{eff}}\approx -i[H,\cdot]+\sum_i \gamma_i (L_i\rho L_i^{\dagger}-\tfrac{1}{2}\{L_i^{\dagger}L_i,\rho\})$. It provides a practical method to derive this effective dynamics from a given circuit, showing how noise aggregates as a static Lindblad term whose form depends on the Hamiltonian, the circuit implementation, and gate decompositions, including large gates and SWAPs. Numerical verification on a four-spin transverse-field Ising model demonstrates that the noisy algorithm model accurately reproduces the circuit evolution and outperforms simpler noise models. The framework offers potential use in tailoring circuits for open-system simulations on near-term devices and can be extended to more general Hamiltonians and degrees of freedom, such as fermionic systems via Jordan–Wigner.

Abstract

We consider the extent to which a Trotterized time evolution implemented on a quantum computer is altered by the presence of decoherence. Given a specific set of assumptions regarding the manner in which noise processes acting on such a device can be modeled at the circuit level, we show how the effects of noise can be reinterpreted as a shift to the dynamics of the original system being simulated. In particular, we find that this shift can be described through the use of static Lindblad noise terms, which act in addition to the original unitary dynamics. The form of these noise terms depends not only on the underlying noise processes occurring on the device, but also on the original unitary dynamics, as well as the manner in which these dynamics are simulated on the device, i.e., the choice of quantum algorithm. We call this effectively simulated open quantum system the noisy algorithm model. Our results are confirmed through numerical analysis.

Figures (13)

• Figure 1: Demonstrating the idea of a block decomposition: A ZZ Ising gate can be decomposed into CNOT gates and a single qubit rotation around the $Z$ axis.
• Figure 2: Another example of a gate decomposition: Using Hadamard gates to effectively change the axis of a rotation gate.
• Figure 3: Demonstrating the idea of commuting noise past a gate G: The original noise term N that acted prior to G is replaced by a noise term P acting after G, where both gate sequences are equivalent. How to find P, given G and N, is explained in the main text. Note that these operations may act on one or multiple qubits, depicted through the dash in the horizontal qubit lines, indicating a register.
• Figure 4: An example of a basic SWAP block, where between the swapping operations the qubit indices are effectively scrambled. This is to illustrate that treating SWAP blocks similarly to the decomposition blocks in Section \ref{['sec:large-gates']} would easily lead to very large blocks, potentially causing a substantial computational overhead commuting all noise terms out of the block.
• Figure 5: The two-step noise procedure for SWAP gates. First, noise terms appearing within a LGDB within the circuit need to be commuted out of the block. Here, this is indicated by the orange arrow in the first line: The noise N within the decomposition block between the dashed lines representing a $\texttt{ZZ}$ Ising gate (see Figure \ref{['fig:block']}) is shifted out of the block. After commuting N past the CNOT gate we are left with the (in general two-qubit) noise term P (similar to Figure \ref{['fig:noiseHop']}). Now, the circuit consists only of LGDBs equivalent to small angle gates without noise, separate noise terms, and SWAPs. The second step is to commute P past all SWAPs. When doing this, moving past a LGDB (dashed orange arrows in the second line) does not modify P, moving past a SWAP gate (solid orange arrow in the second line) will change the qubits P is acting on.