QuTiPv5 Paper Notebook: Floquet formalism and speed comparison

tutorials-v5/time-evolution/024_v5_paper-floquet-speed-test.md

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+
+# QuTiPv5 Paper Example: Floquet Speed Test
+
+Authors: Maximilian Meyer-Mölleringhof (m.meyermoelleringhof@gmail.com), Marc Gali (galilabarias.marc@aist.go.jp), Neill Lambert (nwlambert@gmail.com), Paul Menczel (paul@menczel.net)
+
+## Introduction
+
+In this example we will discuss Hamiltonians with periodic time-dependence and how we can solve the time evolution of the system in QuTiP.
+A natural approach to this is using the Floquet theorem [\[1, 2\]](#References).
+Similar to the Bloch theorem (for spatial periodicity), this method helps to drastically simplify the problem.
+Let $H$ be a time-dependent and periodic Hamiltonian with period $T$, such that
+
+$H(t) = H(t + T)$
+
+Following Floquet's theorem, there exist solutions to the Schrödinger equation that take the form
+
+$\ket{\psi_\alpha (t)} = \exp(-i \epsilon_\alpha t / \hbar) \ket{\Psi_\alpha (t)}$,
+
+where $\ket{\Psi_\alpha (t)} = \ket{\Psi_\alpha (t + T)}$ are the Floquet modes and $\epsilon_\alpha$ are the quasi-energies.
+Any solution of the Schrödinger equation can then be written as a linear combination
+
+$\ket{\psi(t)} = \sum_\alpha c_\alpha \ket{\psi_\alpha(t)}$,
+
+where the constants $c_\alpha$ are determined by the initial conditions.
+
+Inserting the Floquet states above into the Schrödinger equation, we can define the Floquet Hamiltonian,
+
+$H_F(t) = H(t) - i \hbar \dfrac{\partial}{\partial t}$,
+
+which renders the problem time-independent,
+
+$H_F(t) \ket{\Psi_\alpha(t)} = \epsilon_\alpha \ket{\Psi_\alpha(t)}$.
+
+Solving this equation then allows us to find $\ket{\psi (t)}$ at any large $t$.
+
+Our goal in this notebook is to show how this procedure is done in QuTiP and how it compares to the standard `sesolve` method - specifically in terms of computation time.
+
+```python
+import time
+
+import matplotlib.pyplot as plt
+import numpy as np
+from qutip import (CoreOptions, FloquetBasis, about, basis, expect, qeye,
+                   qzero_like, sesolve, sigmax, sigmay, sigmaz, tensor)
+
+%matplotlib inline
+```
+
+## Two-level system with periodic drive
+
+We consider a two-level system that is exposed to a periodic drive:
+
+$H(t) = - \dfrac{\epsilon}{2} \sigma_z - \dfrac{\Delta}{2} \sigma_x + \dfrac{A}{2} \sin(\omega_d t) \sigma_x$,
+
+where $\epsilon$ is the energy splitting, $\Delta$ the coupling strength and $A$ the drive amplitude.
+
+```python
+delta = 0.2 * 2 * np.pi
+epsilon = 1.0 * 2 * np.pi
+A = 2.5 * 2 * np.pi
+omega = 1.0 * 2 * np.pi
+T = 2 * np.pi / omega
+
+H0 = -1 / 2 * (epsilon * sigmaz() + delta * sigmax())
+H1 = A / 2 * sigmax()
+args = {"w": omega}
+H = [H0, [H1, lambda t, w: np.sin(w * t)]]
+```
+
+```python
+psi0 = basis(2, 0)  # initial condition
+
+# Numerical parameters
+dt = 0.01
+N_T = 10  # Number of periods
+tlist = np.arange(0.0, N_T * T, dt)
+```
+
+```python
+computational_time_floquet = np.zeros_like(tlist)
+computational_time_sesolve = np.zeros_like(tlist)
+
+expect_floquet = np.zeros_like(tlist)
+expect_sesolve = np.zeros_like(tlist)
+```
+
+```python
+# Running the simulation
+for n, t in enumerate(tlist):
+    # Floquet basis
+    # --------------------------------
+    tic_f = time.perf_counter()
+    floquetbasis = FloquetBasis(H, T, args)
+    # Decomposing inital state into Floquet modes
+    f_coeff = floquetbasis.to_floquet_basis(psi0)
+    # Obtain evolved state in the original basis
+    psi_t = floquetbasis.from_floquet_basis(f_coeff, t)
+    p_ex = expect(sigmaz(), psi_t)
+    toc_f = time.perf_counter()
+
+    # Saving data
+    computational_time_floquet[n] = toc_f - tic_f
+    expect_floquet[n] = p_ex
+
+    # sesolve
+    # --------------------------------
+    tic_f = time.perf_counter()
+    psi_se = sesolve(H, psi0, tlist[: n + 1], e_ops=[sigmaz()], args=args)
+    p_ex_ref = psi_se.expect[0]
+    toc_f = time.perf_counter()
+
+    # Saving data
+    computational_time_sesolve[n] = toc_f - tic_f
+    expect_sesolve[n] = p_ex_ref[-1]
+```
+
+```python
+fig, axs = plt.subplots(1, 2, figsize=(13.6, 4.5))
+axs[0].plot(tlist / T, computational_time_floquet, "-")
+axs[0].plot(tlist / T, computational_time_sesolve, "--")
+
+axs[0].set_yscale("log")
+axs[0].set_yticks(np.logspace(-3, -1, 3))
+axs[1].plot(tlist / T, np.real(expect_floquet), "-")
+axs[1].plot(tlist / T, np.real(expect_sesolve), "--")
+
+axs[0].set_xlabel(r"$t \, / \, T$")
+axs[1].set_xlabel(r"$t \, / \, T$")
+axs[0].set_ylabel(r"Computational Time [$s$]")
+axs[1].set_ylabel(r"$\langle \sigma_z \rangle$")
+axs[0].legend(("Floquet", "sesolve"))
+
+xticks = np.rint(np.linspace(0, N_T, N_T + 1, endpoint=True))
+axs[0].set_xticks(xticks)
+axs[0].set_xlim([0, N_T])
+axs[1].set_xticks(xticks)
+axs[1].set_xlim([0, N_T])
+
+plt.show()
+```
+
+## One Dimensional Ising Chain
+
+As a second use case for the Floquet method, we look at the one-dimensional Ising chain with a periodic drive.
+We describe such a system using the Hamiltonian
+
+$H(t) = g_0 \sum_{n=1}^N \sigma_z^{(n)} - J_0 \sum_{n=1}^{N - 1} \sigma_x^{(n)} \sigma_x^{(n+1)} + A \sin (\omega_d t) \sum_{n=1}^N \sigma_x^{(n)}$,
+
+where $g_0$ is the level splitting, $J_0$ is the nearest-neighbour coupling constant and $A$ is the drive amplitude.
+
+As outlined in the QuTiPv5 paper, it's educational to study the dimensional scaling of the Floquet method compared to the standard `sesolve`.
+Specifically how the crossing time (when the Floquet method becomes faster) depends on the dimensions $N$.
+Although we will not reproduce the whole plot for computation time reasons, we implement the solution for one specific choice of $N$.
+Feel free to play around with the parameters!
+
+```python
+N = 4  # number of spins
+g0 = 1  # energy-splitting
+J0 = 1.4  # coupling strength
+A = 1.0  # drive strength
+omega = 1.0 * 2 * np.pi  # drive frequency
+T = 2 * np.pi / omega  # drive period
+```
+
+```python
+# For Hamiltonian Setup
+def setup_Ising_drive(N, g0, J0, A, omega, data_type="CSR"):
+    """
+    # N    : number of spins
+    # g0   : splitting,
+    # J0   : couplings
+    # A    : drive amplitude
+    # omega: drive frequency
+    """
+    with CoreOptions(default_dtype=data_type):
+
+        sx_list, sy_list, sz_list = [], [], []
+        for i in range(N):
+            op_list = [qeye(2)] * N
+            op_list[i] = sigmax().to(data_type)
+            sx_list.append(tensor(op_list))
+            op_list[i] = sigmay().to(data_type)
+            sy_list.append(tensor(op_list))
+            op_list[i] = sigmaz().to(data_type)
+            sz_list.append(tensor(op_list))
+
+        # Hamiltonian - Energy splitting terms
+        H_0 = 0.0
+        for i in range(N):
+            H_0 += g0 * sz_list[i]
+
+        # Interaction terms
+        H_1 = qzero_like(H_0)
+        for n in range(N - 1):
+            H_1 += -J0 * sx_list[n] * sx_list[n + 1]
+
+        # Driving terms
+        if A > 0:
+            H_d = 0.0
+            for i in range(N):
+                H_d += A * sx_list[i]
+            args = {"w": omega}
+            H = [H_0, H_1, [H_d, lambda t, w: np.sin(w * t)]]
+        else:
+            args = {}
+            H = [H_0, H_1]
+
+        # Defining initial conditions
+        state_list = [basis(2, 1)] * (N - 1)
+        state_list.append(basis(2, 0))
+        psi0 = tensor(state_list)
+
+        # Defining expectation operator
+        e_ops = sz_list
+        return H, psi0, e_ops, args
+```
+
+```python
+H, psi0, e_ops, args = setup_Ising_drive(N, g0, J0, A, omega)
+```
+
+```python
+# Simulation parameters
+N_T = 10
+dt = 0.01
+tlist = np.arange(0, N_T * T, dt)
+tlist_0 = np.arange(0, T, dt)  # One period tlist
+
+options = {"progress_bar": False, "store_floquet_states": True}
+```
+
+```python
+computational_time_floquet = np.ones(tlist.shape) * np.nan
+computational_time_sesolve = np.ones(tlist.shape) * np.nan
+
+expect_floquet = np.zeros((N, len(tlist)))
+expect_sesolve = np.zeros((N, len(tlist)))
+```
+
+```python
+# Running the simulation
+for n, t in enumerate(tlist):
+    # Floquet basis
+    # --------------------------------
+    tic_f = time.perf_counter()
+    if t < T:
+        # find the floquet modes for the time-dependent hamiltonian
+        floquetbasis = FloquetBasis(H, T, args)
+    else:
+        floquetbasis = FloquetBasis(H, T, args, precompute=tlist_0)
+
+    # Decomposing inital state into Floquet modes
+    f_coeff = floquetbasis.to_floquet_basis(psi0)
+    # Obtain evolved state in the original basis
+    psi_t = floquetbasis.from_floquet_basis(f_coeff, t)
+    p_ex = expect(e_ops, psi_t)
+    toc_f = time.perf_counter()
+
+    # Saving data
+    computational_time_floquet[n] = toc_f - tic_f
+
+    # sesolve
+    # --------------------------------
+    tic_f = time.perf_counter()
+    output = sesolve(H, psi0, tlist[: n + 1], e_ops=e_ops, args=args)
+    p_ex_r = output.expect
+    toc_f = time.perf_counter()
+
+    # Saving data
+    computational_time_sesolve[n] = toc_f - tic_f
+
+    for i in range(N):
+        expect_floquet[i, n] = p_ex[i]
+        expect_sesolve[i, n] = p_ex_r[i][-1]
+```
+
+```python
+fig, axs = plt.subplots(1, 2, figsize=(12, 5))
+ax = axs[0]
+axs[0].plot(tlist / T, computational_time_floquet, "-")
+axs[0].plot(tlist / T, computational_time_sesolve, "--")
+
+axs[0].set_yscale("log")
+axs[0].set_yticks(np.logspace(-3, -1, 3))
+lg = []
+for i in range(N):
+    axs[1].plot(tlist / T, np.real(expect_floquet[i, :]), "-")
+    lg.append(f"n={i+1}")
+for i in range(N):
+    axs[1].plot(tlist / T, np.real(expect_sesolve[i, :]), "--", color="black")
+lg.append("sesolve")
+
+axs[0].set_xlabel(r"$t \, / \, T$")
+axs[1].set_xlabel(r"$t \, / \, T$")
+axs[0].set_ylabel(r"$Computational \; Time,\; [s]$")
+axs[1].set_ylabel(r"$\langle \sigma_z^{{(n)}} \rangle$")
+axs[0].legend(("Floquet", "sesolve"))
+axs[1].legend(lg)
+xticks = np.rint(np.linspace(0, N_T, N_T + 1, endpoint=True))
+axs[0].set_xticks(xticks)
+axs[0].set_xlim([0, N_T])
+axs[1].set_xticks(xticks)
+axs[1].set_xlim([0, N_T])
+txt = f"$N={N}$, $g_0={g0}$, $J_0={J0}$, $A={A}$"
+axs[0].text(0.0, 1.05, txt, transform=axs[0].transAxes)
+
+plt.show()
+```
+
+## References
+
+[1] [Floquet, Annales scientifiques de l’École Normale Supérieure (1883)](http://www.numdam.org/articles/10.24033/asens.220/)
+
+[2] [Shirley, Phys.Rev. (1965)](https://link.aps.org/doi/10.1103/PhysRev.138.B979)
+
+[3] [QuTiP 5: The Quantum Toolbox in Python](https://arxiv.org/abs/2412.04705)
+
+
+## About
+
+```python
+about()
+```
+
+## Testing
+
+```python
+for i in range(N):
+    assert np.allclose(
+        np.real(expect_floquet[i, :]), np.real(expect_sesolve[i, :]), atol=1e-5
+    ), f"floquet and sesolve solutions for Ising chain element {i} deviate."
+```