OTOCs, circuits and noise¶

1. The infinite-temperature OTOC of an SYK realisation, computed exactly and with the interferometric (Hadamard-test) circuit.
2. The T1/T2 noise model: coherence decays as $e^{-t/T_2}$, independent of the number of simulation steps.
3. A noisy Trotter circuit on PennyLane's mixed-state simulator, checked against the NumPy noise model.

In [1]:

import matplotlib.pyplot as plt
import numpy as np

from quantum_chaos.circuits.otoc_calculator import OTOCCalculator
from quantum_chaos.hamiltonians import SYKHamiltonian
from quantum_chaos.noise.nisq_simulator import NISQSimulator
from quantum_chaos.noise.noise_channels import NoiseModel

import matplotlib.pyplot as plt import numpy as np from quantum_chaos.circuits.otoc_calculator import OTOCCalculator from quantum_chaos.hamiltonians import SYKHamiltonian from quantum_chaos.noise.nisq_simulator import NISQSimulator from quantum_chaos.noise.noise_channels import NoiseModel

1. OTOC: exact vs circuit¶

In [2]:

syk = SYKHamiltonian(8, seed=3)
calc = OTOCCalculator(syk.n_qubits, syk.hamiltonian_matrix)
times = np.linspace(0, 4, 21)
W, V = calc.local_pauli("Z", 0), calc.local_pauli("X", 2)
exact = np.real(calc.compute_otoc_exact(W, V, times))
circuit = calc.create_otoc_circuit("Z", 0, "X", 2)  # T = inf via Bell-pair purification
measured = np.array([circuit(t) for t in times[::4]])
print("max |circuit - exact| =", np.max(np.abs(measured - exact[::4])))

fig, ax = plt.subplots(figsize=(6, 3.5))
ax.plot(times, exact, label="exact")
ax.plot(times[::4], measured, "o", label="Hadamard-test circuit")
ax.set_xlabel("t")
ax.set_ylabel(r"Re $F(t)$")
ax.legend()
plt.show()

syk = SYKHamiltonian(8, seed=3) calc = OTOCCalculator(syk.n_qubits, syk.hamiltonian_matrix) times = np.linspace(0, 4, 21) W, V = calc.local_pauli("Z", 0), calc.local_pauli("X", 2) exact = np.real(calc.compute_otoc_exact(W, V, times)) circuit = calc.create_otoc_circuit("Z", 0, "X", 2) # T = inf via Bell-pair purification measured = np.array([circuit(t) for t in times[::4]]) print("max |circuit - exact| =", np.max(np.abs(measured - exact[::4]))) fig, ax = plt.subplots(figsize=(6, 3.5)) ax.plot(times, exact, label="exact") ax.plot(times[::4], measured, "o", label="Hadamard-test circuit") ax.set_xlabel("t") ax.set_ylabel(r"Re $F(t)$") ax.legend() plt.show()

max |circuit - exact| = 1.7763568394002505e-15

2. T1/T2 relaxation¶

In [3]:

model = NoiseModel(depolarizing_rate=0.0, t1=40.0, t2=30.0)
sim = NISQSimulator(1, noise_model=model)
plus = np.full((2, 2), 0.5, dtype=complex)
for n_steps in (1, 5, 50):
    rho = sim.evolve_with_noise(plus, np.zeros((2, 2)), total_time=6.0, n_steps=n_steps)
    print(f"n_steps={n_steps:3d}: |rho01| = {abs(rho[0, 1]):.6f}")
print(f"expected 0.5 exp(-t/T2) = {0.5 * np.exp(-6.0 / 30.0):.6f}")

model = NoiseModel(depolarizing_rate=0.0, t1=40.0, t2=30.0) sim = NISQSimulator(1, noise_model=model) plus = np.full((2, 2), 0.5, dtype=complex) for n_steps in (1, 5, 50): rho = sim.evolve_with_noise(plus, np.zeros((2, 2)), total_time=6.0, n_steps=n_steps) print(f"n_steps={n_steps:3d}: |rho01| = {abs(rho[0, 1]):.6f}") print(f"expected 0.5 exp(-t/T2) = {0.5 * np.exp(-6.0 / 30.0):.6f}")

n_steps=  1: |rho01| = 0.409365
n_steps=  5: |rho01| = 0.409365
n_steps= 50: |rho01| = 0.409365
expected 0.5 exp(-t/T2) = 0.409365

3. Noisy Trotter circuit¶

In [4]:

terms = [(0.6, "XX"), (0.4, "ZI"), (-0.3, "YZ")]
model = NoiseModel(depolarizing_rate=0.01, t1=50.0, t2=40.0, gate_time=0.5)
sim = NISQSimulator(2, noise_model=model)
psi0 = np.array([1, 0, 0, 0], dtype=complex)
for n_steps in (2, 8):
    rho = sim.create_pennylane_noisy_circuit(terms, total_time=1.0, n_steps=n_steps)(psi0)
    purity = np.real(np.trace(rho @ rho))
    print(f"{n_steps} Trotter layers: purity = {purity:.4f}")

terms = [(0.6, "XX"), (0.4, "ZI"), (-0.3, "YZ")] model = NoiseModel(depolarizing_rate=0.01, t1=50.0, t2=40.0, gate_time=0.5) sim = NISQSimulator(2, noise_model=model) psi0 = np.array([1, 0, 0, 0], dtype=complex) for n_steps in (2, 8): rho = sim.create_pennylane_noisy_circuit(terms, total_time=1.0, n_steps=n_steps)(psi0) purity = np.real(np.trace(rho @ rho)) print(f"{n_steps} Trotter layers: purity = {purity:.4f}")

2 Trotter layers: purity = 0.9246
8 Trotter layers: purity = 0.7832