introduce second order formula

src/boostvqe/compiling_XXZ.py

@@ -81,3 +81,57 @@ def nqubit_XXZ_decomposition(nqubits,t,delta=0.5,steps=1):
     for _step in range(steps):
         multi_layer += circuit
     return multi_layer
+
+
+def nqubit_XXZ_decomposition_second_order(nqubits, t, delta=0.5, steps=1):
+    """
+    Constructs an XXZ model circuit for n qubits, given by:
+    .. math::
+        H = \\sum_{i=0}^{N-1} \\left( X_i X_{i+1} + Y_i Y_{i+1} + \\delta Z_i Z_{i+1} \\right)
+
+    This function decomposes the time evolution operator of the XXZ model
+    into a sequence of quantum gates applied to a quantum circuit using second order formula
+
+    Args:
+    nqubits (int): Number of qubits.
+    t (float): Total evolution time.
+    delta (float): Coefficient for the Z component (default 0.5).
+    steps (int): Number of time steps (default 1).
+
+    Returns:
+    Circuit: The final multi-layer circuit.
+
+    Example:
+        .. testcode::
+
+            from boostvqe.compiling_XXZ_utils import *
+            # Create circuit to decompose 6 qubits XXZ with 3 decomposition steps
+            circ = nqubit_XXZ_decomposition_second_order(nqubits=6,t=0.01,delta=0.5,steps=3)
+            print(circ.draw())
+    """
+    circuit = Circuit(nqubits=nqubits)
+
+    # Create lists of even and odd qubit indices
+    even_numbers = [num for num in range(nqubits) if num % 2 == 0]
+    odd_numbers = [num for num in range(nqubits) if num % 2 == 1]
+
+    if nqubits % 2 == 0:
+        # Handle even number of qubits
+        even_numbers_end_0 = [num for num in range(1, nqubits) if num % 2 == 0]
+        even_numbers_end_0.append(0)
+        circuit = _add_gates(circuit, odd_numbers, even_numbers_end_0, t/2, delta, steps)
+        circuit = _add_gates(circuit, even_numbers, odd_numbers, t, delta, steps)
+        circuit = _add_gates(circuit, odd_numbers, even_numbers_end_0, t/2, delta, steps)
+    elif nqubits % 2 == 1:
+        # Handle odd number of qubits (since XXZ model has periodic boundary conditions)
+        circuit = _add_gates(circuit, odd_numbers, even_numbers[1:], t/2, delta, steps)
+        circuit = _add_gates(circuit, [nqubits - 1], [0], t/2, delta, steps)
+        circuit = _add_gates(circuit, even_numbers[:-1], odd_numbers, t, delta, steps)
+        circuit = _add_gates(circuit, odd_numbers, even_numbers[1:], t/2, delta, steps)
+        circuit = _add_gates(circuit, [nqubits - 1], [0], t/2, delta, steps)
+
+    # Create a multi-layer circuit with the time evolution steps
+    multi_layer = Circuit(nqubits=nqubits)
+    for _step in range(steps):
+        multi_layer += circuit
+    return multi_layer