Transverse-Field Ising Model with Q-CTRL's Performance Management
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Usage estimate: 2 minutes on a Heron r2 processor. (NOTE: This is an estimate only. Your runtime may vary.)
Background
The Transverse-Field Ising Model (TFIM) is important for studying quantum magnetism and phase transitions. It describes a set of spins arranged on a lattice, where each spin interacts with its neighbors while also being influenced by an external magnetic field that drives quantum fluctuations.
A common approach to simulate this model is to use Trotter decomposition to approximate the time evolution operator, constructing circuits that alternate between single-qubit rotations and entangling two-qubit interactions. However, this simulation on real hardware is challenging due to noise and decoherence, leading to deviations from the true dynamics. To overcome this, we use Q-CTRL's Fire Opal error suppression and performance management tools, offered as a Qiskit Function (see the Fire Opal documentation). Fire Opal automatically optimizes circuit execution by applying dynamical decoupling, advanced layout, routing, and other error suppression techniques, all aimed at reducing noise. With these improvements, the hardware results align more closely with noiseless simulations, and thus we can study TFIM magnetization dynamics with higher fidelity.
In this tutorial we will:
- Build the TFIM Hamiltonian on a graph of connected spin triangles
- Simulate time evolution with Trotterized circuits at different depths
- Compute and visualize single-qubit magnetizations over time
- Compare baseline simulations with results from hardware runs using Q-CTRL's Fire Opal performance management
Overview
The Transverse-field Ising Model (TFIM) is a quantum spin model that captures essential features of quantum phase transitions. The Hamiltonian is defined as:
where and are Pauli operators acting on qubit , is the coupling strength between neighboring spins, and is the strength of the transverse magnetic field. The first term represents classical ferromagnetic interactions, while the second introduces quantum fluctuations through the transverse field. To simulate TFIM dynamics, you use a Trotter decomposition of the unitary evolution operator , implemented through layers of RX and RZZ gates based on a custom graph of connected spin triangles. The simulation explores how magnetization evolves with increasing Trotter steps.
The performance of the proposed TFIM implementation is assessed by comparing noiseless simulations with noisy backends. Fire Opal's enhanced execution and error suppression features are used to mitigate the effect of noise in real hardware, yielding more reliable estimates of spin observables like and correlators .
Requirements
Before starting this tutorial, be sure you have the following installed:
- Qiskit SDK v1.4 or later, with visualization support
- Qiskit Runtime v0.40 or later (
pip install qiskit-ibm-runtime) - Qiskit Functions Catalog v0.9.0 (
pip install qiskit-ibm-catalog) - Fire Opal SDK v9.0.2 or later (
pip install fire-opal) - Q-CTRL Visualizer v8.0.2 or later (
pip install qctrl-visualizer)
Setup
First, authenticate using your IBM Quantum API key. Then, select the Qiskit Function as follows. (This code assumes you've already saved your account to your local environment.)
# Added by doQumentation — required packages for this notebook
!pip install -q matplotlib networkx numpy qctrlvisualizer qiskit qiskit-aer qiskit-ibm-catalog qiskit-ibm-runtime
from qiskit.transpiler.preset_passmanagers import generate_preset_pass_manager
from qiskit import QuantumCircuit
from qiskit_ibm_catalog import QiskitFunctionsCatalog
from qiskit_ibm_runtime import QiskitRuntimeService
from qiskit_ibm_runtime import SamplerV2 as Sampler
from qiskit.quantum_info import SparsePauliOp
from qiskit_aer import AerSimulator
import numpy as np
import networkx as nx
import matplotlib.pyplot as plt
import qctrlvisualizer as qv
catalog = QiskitFunctionsCatalog(channel="ibm_quantum_platform")
# Access Function
perf_mgmt = catalog.load("q-ctrl/performance-management")
Step 1: Map classical inputs to a quantum problem
Generate TFIM graph
We begin by defining the lattice of spins and the couplings between them. In this tutorial, the lattice is constructed from connected triangles arranged in a linear chain. Each triangle consists of three nodes connected in a closed loop, and the chain is formed by linking one node of each triangle to the previous triangle.
The helper function connected_triangles_adj_matrix builds the adjacency matrix for this structure. For a chain of triangles, the resulting graph contains nodes.
def connected_triangles_adj_matrix(n):
"""
Generate the adjacency matrix for 'n' connected triangles in a chain.
"""
num_nodes = 2 * n + 1
adj_matrix = np.zeros((num_nodes, num_nodes), dtype=int)
for i in range(n):
a, b, c = i * 2, i * 2 + 1, i * 2 + 2 # Nodes of the current triangle
# Connect the three nodes in a triangle
adj_matrix[a, b] = adj_matrix[b, a] = 1
adj_matrix[b, c] = adj_matrix[c, b] = 1
adj_matrix[a, c] = adj_matrix[c, a] = 1
# If not the first triangle, connect to the previous triangle
if i > 0:
adj_matrix[a, a - 1] = adj_matrix[a - 1, a] = 1
return adj_matrix
To visualize the lattice we just defined, we can plot the chain of connected triangles and label each node. The function below builds the graph for a chosen number of triangles and displays it.
def plot_triangle_chain(n, side=1.0):
"""
Plot a horizontal chain of n equilateral triangles.
Baseline: even nodes (0,2,4,...,2n) on y=0
Apexes: odd nodes (1,3,5,...,2n-1) above the midpoint.
"""
# Build graph
A = connected_triangles_adj_matrix(n)
G = nx.from_numpy_array(A)
h = np.sqrt(3) / 2 * side
pos = {}
# Place baseline nodes
for k in range(n + 1):
pos[2 * k] = (k * side, 0.0)
# Place apex nodes
for k in range(n):
x_left = pos[2 * k][0]
x_right = pos[2 * k + 2][0]
pos[2 * k + 1] = ((x_left + x_right) / 2, h)
# Draw
fig, ax = plt.subplots(figsize=(1.5 * n, 2.5))
nx.draw(
G,
pos,
ax=ax,
with_labels=True,
font_size=10,
font_color="white",
node_size=600,
node_color=qv.QCTRL_STYLE_COLORS[0],
edge_color="black",
width=2,
)
ax.set_aspect("equal")
ax.margins(0.2)
plt.show()
return G, pos
For this tutorial we will use a chain of 20 triangles.
n_triangles = 20
n_qubits = 2 * n_triangles + 1
plot_triangle_chain(n_triangles, side=1.0)
plt.show()
Coloring graph edges
To implement the spin–spin coupling, it is useful to group edges that do not overlap. This allows us to apply two-qubit gates in parallel. We can do this with a simple edge-coloring procedure [1], which assigns a color to each edge so that edges meeting at the same node are placed in different groups.
def edge_coloring(graph):
"""
Takes a NetworkX graph and returns a list of lists where each inner list contains
the edges assigned the same color.
"""
line_graph = nx.line_graph(graph)
edge_colors = nx.coloring.greedy_color(line_graph)
color_groups = {}
for edge, color in edge_colors.items():
if color not in color_groups:
color_groups[color] = []
color_groups[color].append(edge)
return list(color_groups.values())
Step 2: Optimize problem for quantum hardware execution
Generate Trotterized circuits on spin graphs
To simulate the dynamics of the TFIM, we construct circuits that approximate the time evolution operator.
We use a second-order Trotter decomposition:
where and .
- The term is implemented with layers of
RXrotations. - The term is implemented with layers of
RZZgates along the edges of the interaction graph.
The angles of these gates are determined by the transverse field , the coupling constant , and the time step . By stacking multiple Trotter steps, we generate circuits of increasing depth that approximate the system's dynamics. The functions generate_tfim_circ_custom_graph and trotter_circuits construct a Trotterized quantum circuit from an arbitrary spin interaction graph.
def generate_tfim_circ_custom_graph(
steps, h, J, dt, psi0, graph: nx.graph.Graph, meas_basis="Z", mirror=False
):
"""
Generate a second order trotter of the form e^(a+b) ~ e^(b/2) e^a e^(b/2) for simulating a transverse field ising model:
e^{-i H t} where the Hamiltonian H = -J \\sum_i Z_i Z_{i+1} + h \\sum_i X_i.
steps: Number of trotter steps
theta_x: Angle for layer of X rotations
theta_zz: Angle for layer of ZZ rotations
theta_x: Angle for second layer of X rotations
J: Coupling between nearest neighbor spins
h: The transverse magnetic field strength
dt: t/total_steps
psi0: initial state (assumed to be prepared in the computational basis).
meas_basis: basis to measure all correlators in
This is a second order trotter of the form e^(a+b) ~ e^(b/2) e^a e^(b/2)
"""
theta_x = h * dt
theta_zz = -2 * J * dt
nq = graph.number_of_nodes()
color_edges = edge_coloring(graph)
circ = QuantumCircuit(nq, nq)
# Initial state, for typical cases in the computational basis
for i, b in enumerate(psi0):
if b == "1":
circ.x(i)
# Trotter steps
for step in range(steps):
for i in range(nq):
circ.rx(theta_x, i)
if mirror:
color_edges = [sublist[::-1] for sublist in color_edges[::-1]]
for edge_list in color_edges:
for edge in edge_list:
circ.rzz(theta_zz, edge[0], edge[1])
for i in range(nq):
circ.rx(theta_x, i)
# some typically used basis rotations
if meas_basis == "X":
for b in range(nq):
circ.h(b)
elif meas_basis == "Y":
for b in range(nq):
circ.sdg(b)
circ.h(b)
for i in range(nq):
circ.measure(i, i)
return circ
def trotter_circuits(G, d_ind_tot, J, h, dt, meas_basis, mirror=True):
"""
Generates a sequence of Trotterized circuits, each with increasing depth.
Given a spin interaction graph and Hamiltonian parameters, it constructs
a list of circuits with 1 to d_ind_tot Trotter steps
G: Graph defining spin interactions (edges = ZZ couplings)
d_ind_tot: Number of Trotter steps (maximum depth)
J: Coupling between nearest neighboring spins
h: Transverse magnetic field strength
dt: (t / total_steps
meas_basis: Basis to measure all correlators in
mirror: If True, mirror the Trotter layers
"""
qubit_count = len(G)
circuits = []
psi0 = "0" * qubit_count
for steps in range(1, d_ind_tot + 1):
circuits.append(
generate_tfim_circ_custom_graph(
steps, h, J, dt, psi0, G, meas_basis, mirror
)
)
return circuits
Estimate single-qubit magnetizations
To study the dynamics of the model, we want to measure the magnetization of each qubit, defined by the expectation value .
In simulations, we can compute this directly from the measurement outcomes. The function z_expectation processes the bitstring counts and returns the value of for a chosen qubit index. On real hardware, we evaluate the same quantity by specifying the Pauli operator using the function generate_z_observables, and then the backend computes the expectation value.
def z_expectation(counts, index):
"""
counts: Dict of mitigated bitstrings.
index: Index i in the single operator expectation value < II...Z_i...I > to be calculated.
return: < Z_i >
"""
z_exp = 0
tot = 0
for bitstring, value in counts.items():
bit = int(bitstring[index])
sign = 1
if bit % 2 == 1:
sign = -1
z_exp += sign * value
tot += value
return z_exp / tot
def generate_z_observables(nq):
observables = []
for i in range(nq):
pauli_string = "".join(["Z" if j == i else "I" for j in range(nq)])
observables.append(SparsePauliOp(pauli_string))
return observables
observables = generate_z_observables(n_qubits)
We now define the parameters for generating the Trotterized circuits. In this tutorial, the lattice is a chain of 20 connected triangles, which corresponds to a 41-qubit system.
all_circs_mirror = []
for num_triangles in [n_triangles]:
for meas_basis in ["Z"]:
A = connected_triangles_adj_matrix(num_triangles)
G = nx.from_numpy_array(A)
nq = len(G)
d_ind_tot = 22
dt = 2 * np.pi * 1 / 30 * 0.25
J = 1
h = -7
all_circs_mirror.extend(
trotter_circuits(G, d_ind_tot, J, h, dt, meas_basis, True)
)
circs = all_circs_mirror
Step 3: Execute using Qiskit primitives
Run MPS simulation
The list of Trotterized circuits is executed using the matrix_product_state simulator with an arbitrary choice of shots. The MPS method provides an efficient approximation of the circuit dynamics, with accuracy determined by the chosen bond dimension. For the system sizes considered here, the default bond dimension is sufficient to capture the magnetization dynamics with high fidelity. The raw counts are normalized, and from these we compute the single-qubit expectation values at each Trotter step. Finally, we calculate the average over all qubits to obtain a single curve that shows how the magnetization changes over time.
backend_sim = AerSimulator(method="matrix_product_state")
def normalize_counts(counts_list, shots):
new_counts_list = []
for counts in counts_list:
a = {k: v / shots for k, v in counts.items()}
new_counts_list.append(a)
return new_counts_list
def run_sim(circ_list):
shots = 4096
res = backend_sim.run(circ_list, shots=shots)
normed = normalize_counts(res.result().get_counts(), shots)
return normed
sim_counts = run_sim(circs)
Run on hardware
service = QiskitRuntimeService()
backend = service.backend("ibm_marrakesh")
def run_qiskit(circ_list):
shots = 4096
pm = generate_preset_pass_manager(backend=backend)
isa_circuits = [pm.run(qc) for qc in circ_list]
sampler = Sampler(mode=backend)
res = sampler.run(isa_circuits, shots=shots)
res = [r.data.c.get_counts() for r in res.result()]
normed = normalize_counts(res, shots)
return normed
qiskit_counts = run_qiskit(circs)
Run on hardware with Fire Opal
We evaluate the magnetization dynamics on real quantum hardware. Fire Opal provides a Qiskit Function that extends the standard Qiskit Runtime Estimator primitive with automated error suppression and performance management. We submit the Trotterized circuits directly to an IBM® backend while Fire Opal handles the noise-aware execution.
We prepare a list of pubs, where each item contains a circuit and the corresponding Pauli-Z observables. These are passed to Fire Opal's estimator function, which returns the expectation values for each qubit at each Trotter step. The results can then be averaged over qubits to obtain the magnetization curve from hardware.
backend_name = "ibm_marrakesh"
estimator_pubs = [(qc, observables) for qc in all_circs_mirror[:]]
# Run the circuit using the estimator
qctrl_estimator_job = perf_mgmt.run(
primitive="estimator",
pubs=estimator_pubs,
backend_name=backend_name,
options={"default_shots": 4096},
)
result_qctrl = qctrl_estimator_job.result()
Step 4: Post-process and return result in desired classical format
Finally, we compare the magnetization curve from the simulator with the results obtained on real hardware. Plotting both side by side shows how closely the hardware execution with Fire Opal matches the noiseless baseline across Trotter steps.
def make_correlators(test_counts, nq, d_ind_tot):
mz = np.empty((nq, d_ind_tot))
for d_ind in range(d_ind_tot):
counts = test_counts[d_ind]
for i in range(nq):
mz[i, d_ind] = z_expectation(counts, i)
average_z = np.mean(mz, axis=0)
return np.concatenate((np.array([1]), average_z), axis=0)
sim_exp = make_correlators(sim_counts[0:22], nq=nq, d_ind_tot=22)
qiskit_exp = make_correlators(qiskit_counts[0:22], nq=nq, d_ind_tot=22)
qctrl_exp = [ev.data.evs for ev in result_qctrl[:]]
qctrl_exp_mean = np.concatenate(
(np.array([1]), np.mean(qctrl_exp, axis=1)), axis=0
)
def make_expectations_plot(
sim_z,
depths,
exp_qctrl=None,
exp_qctrl_error=None,
exp_qiskit=None,
exp_qiskit_error=None,
plot_from=0,
plot_upto=23,
):
import numpy as np
import matplotlib.pyplot as plt
depth_ticks = [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22]
d = np.asarray(depths)[plot_from:plot_upto]
sim = np.asarray(sim_z)[plot_from:plot_upto]
qk = (
None
if exp_qiskit is None
else np.asarray(exp_qiskit)[plot_from:plot_upto]
)
qc = (
None
if exp_qctrl is None
else np.asarray(exp_qctrl)[plot_from:plot_upto]
)
qk_err = (
None
if exp_qiskit_error is None
else np.asarray(exp_qiskit_error)[plot_from:plot_upto]
)
qc_err = (
None
if exp_qctrl_error is None
else np.asarray(exp_qctrl_error)[plot_from:plot_upto]
)
# ---- helper(s) ----
def rmse(a, b):
if a is None or b is None:
return None
a = np.asarray(a, dtype=float)
b = np.asarray(b, dtype=float)
mask = np.isfinite(a) & np.isfinite(b)
if not np.any(mask):
return None
diff = a[mask] - b[mask]
return float(np.sqrt(np.mean(diff**2)))
def plot_panel(ax, method_y, method_err, color, label, band_color=None):
# Noiseless reference
ax.plot(d, sim, color="grey", label="Noiseless simulation")
# Method line + band
if method_y is not None:
ax.plot(d, method_y, color=color, label=label)
if method_err is not None:
lo = np.clip(method_y - method_err, -1.05, 1.05)
hi = np.clip(method_y + method_err, -1.05, 1.05)
ax.fill_between(
d,
lo,
hi,
alpha=0.18,
color=band_color if band_color else color,
label=f"{label} ± error",
)
else:
ax.text(
0.5,
0.5,
"No data",
transform=ax.transAxes,
ha="center",
va="center",
fontsize=10,
color="0.4",
)
# RMSE box (vs sim)
r = rmse(method_y, sim)
if r is not None:
ax.text(
0.98,
0.02,
f"RMSE: {r:.4f}",
transform=ax.transAxes,
va="bottom",
ha="right",
fontsize=8,
bbox=dict(
boxstyle="round,pad=0.35", fc="white", ec="0.7", alpha=0.9
),
)
# Axes
ax.set_xticks(depth_ticks)
ax.set_ylim(-1.05, 1.05)
ax.grid(True, which="both", linewidth=0.4, alpha=0.4)
ax.set_axisbelow(True)
ax.legend(prop={"size": 8}, loc="best")
fig, axes = plt.subplots(1, 2, figsize=(10, 4), dpi=300, sharey=True)
axes[0].set_title("Fire Opal (Q-CTRL)", fontsize=10)
plot_panel(
axes[0],
qc,
qc_err,
color="#680CE9",
label="Fire Opal",
band_color="#680CE9",
)
axes[0].set_xlabel("Trotter step")
axes[0].set_ylabel(r"$\langle Z \rangle$")
axes[1].set_title("Qiskit", fontsize=10)
plot_panel(
axes[1], qk, qk_err, color="blue", label="Qiskit", band_color="blue"
)
axes[1].set_xlabel("Trotter step")
plt.tight_layout()
plt.show()
depths = list(range(d_ind_tot + 1))
errors = np.abs(np.array(qctrl_exp_mean) - np.array(sim_exp))
errors_qiskit = np.abs(np.array(qiskit_exp) - np.array(sim_exp))
make_expectations_plot(
sim_exp,
depths,
exp_qctrl=qctrl_exp_mean,
exp_qctrl_error=errors,
exp_qiskit=qiskit_exp,
exp_qiskit_error=errors_qiskit,
)
References
[1] Graph coloring. Wikipedia. Retrieved September 15, 2025, from https://en.wikipedia.org/wiki/Graph_coloring
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