Use this Skill for quantum computing experiments with Qiskit: qubit circuits, gates, measurement, VQE/QAOA variational algorithms, noise models, and statevector vs shot simulation.
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Use this Skill for quantum computing experiments with Qiskit: qubit circuits, gates, measurement, VQE/QAOA variational algorithms, noise models, and statevector vs shot simulation.
TL;DR — Build and simulate quantum circuits with Qiskit. Use statevector
or shot-based simulation, add realistic noise models, implement VQE for ground
state energy estimation, and QAOA for combinatorial optimization (MaxCut).
When to Use
Use this Skill when you need to:
Design and test quantum gate circuits (H, CNOT, Ry, Rz, T, S, SWAP)
Simulate quantum circuits on classical hardware (Aer statevector / QASM)
"""
Build a Bell state circuit, simulate with statevector and shot-based simulators,
and plot the measurement histogram.
"""import numpy as np
import matplotlib.pyplot as plt
from qiskit import QuantumCircuit, transpile
from qiskit_aer import AerSimulator
from qiskit.visualization import plot_histogram
defcreate_bell_circuit() -> QuantumCircuit:
"""
Create a 2-qubit Bell state circuit |Φ+⟩ = (|00⟩ + |11⟩)/√2.
Circuit:
q0: ──H──●──
q1: ─────X──
"""
qc = QuantumCircuit(2, 2)
qc.h(0) # Hadamard on qubit 0
qc.cx(0, 1) # CNOT: control=0, target=1
qc.measure([0, 1], [0, 1])
return qc
defsimulate_statevector(qc_no_measure: QuantumCircuit) -> np.ndarray:
"""
Get exact statevector amplitudes before measurement.
Args:
qc_no_measure: Circuit without measurement instructions.
Returns:
Complex statevector array of length 2^n.
"""
sim = AerSimulator(method="statevector")
qc_sv = qc_no_measure.copy()
qc_sv.save_statevector()
job = sim.run(transpile(qc_sv, sim))
result = job.result()
sv = np.array(result.get_statevector())
return sv
defsimulate_shots(qc: QuantumCircuit, n_shots: int = 4096) -> dict:
"""
Run shot-based simulation and return measurement counts.
Args:
qc: Circuit with measurement gates.
n_shots: Number of repetitions.
Returns:
Dictionary of bitstring: count.
"""
sim = AerSimulator(method="qasm")
job = sim.run(transpile(qc, sim), shots=n_shots)
return job.result().get_counts()
defplot_counts(counts: dict, title: str = "Bell State", output: str = "bell.png") -> None:
"""Plot measurement histogram."""
fig, ax = plt.subplots(figsize=(5, 4))
states = sorted(counts.keys())
vals = [counts[s] / sum(counts.values()) for s in states]
ax.bar(states, vals, color="#4C72B0")
ax.set_xlabel("Measurement outcome")
ax.set_ylabel("Probability")
ax.set_title(title)
fig.tight_layout()
fig.savefig(output, dpi=150)
print(f"Histogram saved to {output}")
if __name__ == "__main__":
# Statevector
qc_bare = QuantumCircuit(2)
qc_bare.h(0)
qc_bare.cx(0, 1)
sv = simulate_statevector(qc_bare)
print("Statevector amplitudes:", sv)
print(f"|00⟩ probability: {abs(sv[0])**2:.4f}")
print(f"|11⟩ probability: {abs(sv[3])**2:.4f}")
# Shot-based
qc = create_bell_circuit()
counts = simulate_shots(qc, n_shots=8192)
print("Shot counts:", counts)
plot_counts(counts, title="Bell State |Φ+⟩", output="bell_histogram.png")
Step 2 — VQE for H₂ Ground State Energy
"""
VQE: Variational Quantum Eigensolver for estimating the ground state energy
of a simple 2-qubit Hamiltonian representing H₂ at fixed bond length.
H = -1.0523 * II + 0.3979 * ZI - 0.3979 * IZ - 0.0112 * ZZ + 0.1809 * XX
(Jordan-Wigner encoding of H₂ minimal basis Hamiltonian)
"""import numpy as np
from qiskit.circuit.library import EfficientSU2
from qiskit.quantum_info import SparsePauliOp
from qiskit_aer.primitives import Estimator
from qiskit_algorithms import VQE
from qiskit_algorithms.optimizers import COBYLA, SPSA
import matplotlib.pyplot as plt
defbuild_h2_hamiltonian() -> SparsePauliOp:
"""
Construct the H₂ Hamiltonian in the minimal (STO-3G) basis at R=0.735 Å.
Returns:
SparsePauliOp representing the qubit Hamiltonian (energy in Hartree).
"""# Pauli coefficients from Jordan-Wigner transformation# Strings are in reverse qubit order (Qiskit convention)
H = SparsePauliOp.from_list([
("II", -1.0523732),
("IZ", 0.3979374),
("ZI", -0.3979374),
("ZZ", -0.0112801),
("XX", 0.1809312),
])
return H
defrun_vqe(
hamiltonian: SparsePauliOp,
reps: int = 2,
max_iter: int = 200,
seed: int = 42,
) -> dict:
"""
Run VQE with EfficientSU2 ansatz to find ground state energy.
Args:
hamiltonian: Target Hamiltonian as SparsePauliOp.
reps: Number of repetition layers in EfficientSU2.
max_iter: Maximum optimizer iterations.
seed: Random seed for reproducibility.
Returns:
Dictionary with keys: energy, optimal_params, num_evals.
"""
n_qubits = hamiltonian.num_qubits
# Ansatz: hardware-efficient EfficientSU2
ansatz = EfficientSU2(n_qubits, reps=reps, entanglement="linear")
print(f"Ansatz parameters: {ansatz.num_parameters}")
# Estimator primitive (Aer)
estimator = Estimator(approximation=True)
# COBYLA optimizer (gradient-free)
optimizer = COBYLA(maxiter=max_iter)
np.random.seed(seed)
initial_params = np.random.uniform(-np.pi, np.pi, ansatz.num_parameters)
# Track convergence
energies = []
defcallback(nfev, params, energy, stddev):
energies.append(energy)
iflen(energies) % 50 == 0:
print(f" Iter {len(energies):4d}: E = {energy:.6f} Ha")
vqe = VQE(estimator=estimator, ansatz=ansatz, optimizer=optimizer,
callback=callback, initial_point=initial_params)
result = vqe.compute_minimum_eigenvalue(hamiltonian)
print(f"\nVQE ground state energy: {result.eigenvalue.real:.6f} Hartree")
print(f" FCI (exact) reference: -1.137270 Hartree")
print(f" Error: {abs(result.eigenvalue.real - (-1.137270)):.2e} Ha")
# Plot convergence
fig, ax = plt.subplots(figsize=(7, 4))
ax.plot(energies, color="#2196F3", linewidth=1.2)
ax.axhline(-1.137270, color="red", linestyle="--", linewidth=0.8, label="FCI energy")
ax.set_xlabel("Optimizer iteration")
ax.set_ylabel("Energy (Hartree)")
ax.set_title("VQE Convergence for H₂")
ax.legend()
fig.tight_layout()
fig.savefig("vqe_convergence.png", dpi=150)
return {
"energy": result.eigenvalue.real,
"optimal_params": result.optimal_parameters,
"num_evals": result.cost_function_evals,
"convergence": energies,
}
if __name__ == "__main__":
H = build_h2_hamiltonian()
result = run_vqe(H, reps=2, max_iter=300)
print(f"Converged in {result['num_evals']} function evaluations")
Step 3 — QAOA for MaxCut on a 4-Node Graph
"""
QAOA: Quantum Approximate Optimization Algorithm for the MaxCut problem.
MaxCut: partition graph nodes into two sets S and V\S to maximize the number
of edges between them.
Cost Hamiltonian: H_C = Σ_{(i,j)∈E} (I - Z_i Z_j) / 2
"""import numpy as np
import matplotlib.pyplot as plt
import networkx as nx
from qiskit import QuantumCircuit
from qiskit.quantum_info import SparsePauliOp
from qiskit_aer.primitives import Sampler
from qiskit_algorithms import QAOA
from qiskit_algorithms.optimizers import COBYLA
defbuild_maxcut_hamiltonian(graph: nx.Graph) -> SparsePauliOp:
"""
Build the MaxCut cost Hamiltonian from a NetworkX graph.
H_C = Σ_{(i,j)∈E} (I - Z_i Z_j) / 2
Args:
graph: NetworkX graph with integer node labels 0..n-1.
Returns:
SparsePauliOp for the MaxCut Hamiltonian.
"""
n = graph.number_of_nodes()
pauli_list = []
for u, v in graph.edges():
# Z_u Z_v term: Pauli string of length n with Z at positions u and v
zz_str = ["I"] * n
zz_str[u] = "Z"
zz_str[v] = "Z"# Qiskit uses reversed qubit ordering
pauli_list.append(("".join(reversed(zz_str)), -0.5))
pauli_list.append(("I" * n, 0.5))
return SparsePauliOp.from_list(pauli_list)
defrun_qaoa_maxcut(
graph: nx.Graph,
p: int = 2,
max_iter: int = 200,
) -> dict:
"""
Run QAOA to solve MaxCut on the given graph.
Args:
graph: NetworkX graph.
p: QAOA depth (number of cost+mixer layers).
max_iter: Maximum optimizer iterations.
Returns:
Dictionary with keys: best_cut, best_bitstring, counts, energy.
"""
hamiltonian = build_maxcut_hamiltonian(graph)
n = graph.number_of_nodes()
sampler = Sampler()
optimizer = COBYLA(maxiter=max_iter)
qaoa = QAOA(sampler=sampler, optimizer=optimizer, reps=p)
result = qaoa.compute_minimum_eigenvalue(hamiltonian)
# Get best measurement outcome
counts = result.best_measurement
best_bitstring = result.best_measurement["bitstring"]
partition = [int(b) for b in best_bitstring]
# Count cut edges
cut_value = sum(
1for u, v in graph.edges() if partition[u] != partition[v]
)
max_possible = graph.number_of_edges()
print(f"QAOA (p={p}) MaxCut result:")
print(f" Best bitstring: {best_bitstring}")
print(f" Cut edges: {cut_value} / {max_possible}")
print(f" Approximation ratio: {cut_value / max_possible:.3f}")
return {
"best_bitstring": best_bitstring,
"cut_value": cut_value,
"energy": result.eigenvalue.real,
}
defplot_maxcut_result(graph: nx.Graph, bitstring: str, output: str = "maxcut.png") -> None:
"""Visualize the MaxCut partition on the graph."""
partition = [int(b) for b in bitstring]
colors = ["#E91E63"if p == 0else"#2196F3"for p in partition]
fig, ax = plt.subplots(figsize=(5, 4))
pos = nx.spring_layout(graph, seed=42)
nx.draw(graph, pos, ax=ax, node_color=colors, with_labels=True,
node_size=600, font_color="white", font_weight="bold")
cut_edges = [(u, v) for u, v in graph.edges() if partition[u] != partition[v]]
nx.draw_networkx_edges(graph, pos, edgelist=cut_edges, ax=ax,
edge_color="green", width=2.5, style="dashed")
ax.set_title(f"MaxCut: {len(cut_edges)} edges cut\n(cut edges in dashed green)")
fig.tight_layout()
fig.savefig(output, dpi=150)
print(f"MaxCut visualization saved to {output}")
if __name__ == "__main__":
# 4-node cycle graph (max cut = 4)
G = nx.cycle_graph(4)
G.add_edge(0, 2) # 5-edge graph with max cut = 4
result = run_qaoa_maxcut(G, p=2)
plot_maxcut_result(G, result["best_bitstring"], "maxcut_result.png")
Advanced Usage
Noise Model Simulation
"""
Add a realistic noise model to a circuit and compare noisy vs ideal results.
"""import numpy as np
from qiskit import QuantumCircuit, transpile
from qiskit_aer import AerSimulator
from qiskit_aer.noise import NoiseModel, depolarizing_error, thermal_relaxation_error
defbuild_noise_model(
p_depol_1q: float = 0.001,
p_depol_2q: float = 0.01,
T1: float = 50e3, # ns
T2: float = 70e3, # ns
gate_time_1q: float = 50, # ns
gate_time_2q: float = 300, # ns) -> NoiseModel:
"""
Build a noise model with depolarizing + thermal relaxation errors.
Args:
p_depol_1q: Depolarizing probability for single-qubit gates.
p_depol_2q: Depolarizing probability for two-qubit gates.
T1: Longitudinal relaxation time (ns).
T2: Transverse relaxation time (ns).
gate_time_1q: Duration of single-qubit gates (ns).
gate_time_2q: Duration of two-qubit gates (ns).
Returns:
Qiskit NoiseModel.
"""
noise_model = NoiseModel()
# Thermal relaxation (T1, T2 decoherence)
error_1q = thermal_relaxation_error(T1, T2, gate_time_1q)
error_2q = thermal_relaxation_error(T1, T2, gate_time_2q).expand(
thermal_relaxation_error(T1, T2, gate_time_2q))
# Depolarizing noise
dep_1q = depolarizing_error(p_depol_1q, 1)
dep_2q = depolarizing_error(p_depol_2q, 2)
# Combine: apply both thermal + depolarizing
noise_model.add_all_qubit_quantum_error(error_1q.compose(dep_1q), ["h", "ry", "rz"])
noise_model.add_all_qubit_quantum_error(error_2q.compose(dep_2q), ["cx"])
return noise_model
defcompare_ideal_vs_noisy(n_qubits: int = 4, n_shots: int = 8192) -> None:
"""Build a GHZ state circuit and compare ideal and noisy measurements."""
qc = QuantumCircuit(n_qubits, n_qubits)
qc.h(0)
for i inrange(n_qubits - 1):
qc.cx(i, i + 1)
qc.measure_all()
# Ideal simulation
ideal_sim = AerSimulator()
ideal_counts = ideal_sim.run(transpile(qc, ideal_sim), shots=n_shots).result().get_counts()
# Noisy simulation
noise_model = build_noise_model()
noisy_sim = AerSimulator(noise_model=noise_model)
noisy_counts = noisy_sim.run(transpile(qc, noisy_sim), shots=n_shots).result().get_counts()
# Compare fidelity
all_states = set(ideal_counts) | set(noisy_counts)
ideal_probs = {s: ideal_counts.get(s, 0) / n_shots for s in all_states}
noisy_probs = {s: noisy_counts.get(s, 0) / n_shots for s in all_states}
fidelity = sum(np.sqrt(ideal_probs[s] * noisy_probs[s]) for s in all_states) ** 2print(f"GHZ state ({n_qubits} qubits) fidelity (noisy vs ideal): {fidelity:.4f}")
Circuit Transpilation and Optimization
"""
Transpile a circuit to a specific basis gate set with optimization.
"""from qiskit import QuantumCircuit, transpile
from qiskit.transpiler import PassManager
from qiskit_aer import AerSimulator
deftranspile_and_compare(qc: QuantumCircuit) -> None:
"""Show circuit depth before and after transpilation."""
backend = AerSimulator()
print(f"Original circuit depth: {qc.depth()}")
print(f"Original gate count: {qc.count_ops()}")
for opt_level in [0, 1, 2, 3]:
qc_t = transpile(qc, backend=backend, optimization_level=opt_level)
print(f"Opt level {opt_level}: depth={qc_t.depth()}, "f"cx gates={qc_t.count_ops().get('cx', 0)}")
Troubleshooting
Error
Cause
Fix
ProviderError: No Backend found
Old IBM provider API
Use qiskit-ibm-runtime for real hardware
ValueError: The instruction save_statevector is not supported
Wrong simulator method
Use AerSimulator(method="statevector")
VQE not converging
Poor initial parameters or ansatz
Try SPSA optimizer; increase reps; use parameter shift gradient
