Trotterized Time Evolution

Simulated dynamics of a quantum system under time evolution with Trotterization.

Overview

Using IBM Qiskit, I constructed a quantum circuit that approximates the unitary time evolution operator $U(t) = e^{-iHt}$ via First-Order Trotterization.

The Physics & Math

The system consists of a chain of $N$ qubits interacting from an initial state, with the Hamiltonian given by:

$$H = \sum_{j=1}^{N-1} Z_j Z_{j+1} + B \sum_{j=1}^{N} X_j$$

Where $Z_j Z_{j+1}$, $X_j$, and $B$ are Pauli Operators.

Trotter Decomposition

Since the terms $\sum ZZ$ and $\sum X$ do not commute, we cannot simply write $e^{-i(A+B)t} = e^{-iAt}e^{-iBt}$. Instead, we approximate the evolution for total time $t$ by slicing it into $M$ small steps of duration $\delta t = t/M$:

$$|\psi(t)\rangle \approx \prod_{m=1}^{M} \left[ \left( \prod_{j=1}^{N-1} e^{-i \frac{t}{M} Z_j Z_{j+1}} \right) \left( \prod_{j=1}^{N} e^{-i \frac{t}{M} B X_j} \right) \right] |\psi_0\rangle$$

Circuit Implementation

This decomposition maps directly to quantum logic gates:

1. Interaction Terms: $e^{-i \theta ZZ} \rightarrow$ Rzz Gate with angle $2\delta t$.
2. Field Terms: $e^{-i \theta X} \rightarrow$ Rx Gate with angle $2B\delta t$.

The resulting circuit (shown below for $N=6$, $M=5$) consists of alternating layers of single-qubit rotations and entangling gates.

Results Analysis

The simulation was run with the parameters:

• Qubits: $N = 6$
• Parameter: $B = 0.4$
• Time: $t = 0 \to 3.0$

🚀 Usage

1. Prerequisites

pip install qiskit numpy matplotlib