r""" Quantum state learning ====================== This demonstration works through the process used to produce the state preparation results presented in `"Machine learning method for state preparation and gate synthesis on photonic quantum computers" `__. This tutorial uses the TensorFlow backend of Strawberry Fields, giving us access to a number of additional functionalities including: GPU integration, automatic gradient computation, built-in optimization algorithms, and other machine learning tools. Variational quantum circuits ---------------------------- A key element of machine learning is optimization. We can use TensorFlow's automatic differentiation tools to optimize the parameters of variational quantum circuits constructed using Strawberry Fields. In this approach, we fix a circuit architecture where the states, gates, and/or measurements may have learnable parameters :math:`\vec{\theta}` associated with them. We then define a loss function based on the output state of this circuit. In this case, we define a loss function such that the fidelity of the output state of the variational circuit is maximized with respect to some target state. .. note:: For more details on the TensorFlow backend in Strawberry Fields, please see :ref:`machine_learning_tutorial`. For arbitrary state preparation using optimization, we need to make use of a quantum circuit with a layer structure that is **universal** - that is, by 'stacking' the layers, we can guarantee that we can produce *any* CV state with at-most polynomial overhead. Therefore, the architecture we choose must consist of layers with each layer containing parameterized Gaussian *and* non-Gaussian gates. **The non-Gaussian gates provide both the nonlinearity and the universality of the model.** To this end, we employ the CV quantum neural network architecture as described in `Killoran et al. `__: .. figure:: https://i.imgur.com/NEsaVIX.png :alt: layer Here, - :math:`\mathcal{U}_i(\theta_i,\phi_i)` is an N-mode linear optical interferometer composed of two-mode beamsplitters :math:`BS(\theta,\phi)` and single-mode rotation gates :math:`R(\phi)=e^{i\phi\hat{n}}`, - :math:`\mathcal{D}(\alpha_i)` are single mode displacements in the phase space by complex value :math:`\alpha_i`, - :math:`\mathcal{S}(r_i, \phi_i)` are single mode squeezing operations of magnitude :math:`r_i` and phase :math:`\phi_i`, and - :math:`\Phi(\lambda_i)` is a single mode non-Gaussian operation, in this case chosen to be the Kerr interaction :math:`\mathcal{K}(\kappa_i)=e^{i\kappa_i\hat{n}^2}` of strength :math:`\kappa_i`. Hyperparameters --------------- First, we must define the **hyperparameters** of our layer structure: - ``cutoff``: the simulation Fock space truncation we will use in the optimization. The TensorFlow backend will perform numerical operations in this truncated Fock space when performing the optimization. - ``depth``: The number of layers in our variational quantum circuit. As a general rule, increasing the number of layers (and thus, the number of parameters we are optimizing over) increases the optimizer's chance of finding a reasonable local minimum in the optimization landscape. - ``reps``: the number of steps in the optimization routine performing gradient descent Some other optional hyperparameters include: - The standard deviation of initial parameters. Note that we make a distinction between the standard deviation of *passive* parameters (those that preserve photon number when changed, such as phase parameters), and *active* parameters (those that introduce or remove energy from the system when changed). """ import numpy as np import strawberryfields as sf from strawberryfields.ops import * from strawberryfields.utils import operation # Cutoff dimension cutoff = 9 # Number of layers depth = 15 # Number of steps in optimization routine performing gradient descent reps = 200 # Learning rate lr = 0.05 # Standard deviation of initial parameters passive_sd = 0.1 active_sd = 0.001 ###################################################################### # The layer parameters :math:`\vec{\theta}` # ----------------------------------------- # # We use TensorFlow to create the variables corresponding to the gate # parameters. Note that we focus on a single mode circuit where # each variable has shape ``(depth,)``, with each # individual element representing the gate parameter in layer :math:`i`. import tensorflow as tf # set the random seed tf.random.set_seed(42) # squeeze gate sq_r = tf.random.normal(shape=[depth], stddev=active_sd) sq_phi = tf.random.normal(shape=[depth], stddev=passive_sd) # displacement gate d_r = tf.random.normal(shape=[depth], stddev=active_sd) d_phi = tf.random.normal(shape=[depth], stddev=passive_sd) # rotation gates r1 = tf.random.normal(shape=[depth], stddev=passive_sd) r2 = tf.random.normal(shape=[depth], stddev=passive_sd) # kerr gate kappa = tf.random.normal(shape=[depth], stddev=active_sd) ###################################################################### # For convenience, we store the TensorFlow variables representing the # weights as a tensor: weights = tf.convert_to_tensor([r1, sq_r, sq_phi, r2, d_r, d_phi, kappa]) weights = tf.Variable(tf.transpose(weights)) ###################################################################### # Since we have a depth of 15 (so 15 layers), and each layer takes # 7 different types of parameters, the final shape of our weights # array should be :math:`\text{depth}\times 7` or ``(15, 7)``: print(weights.shape) ###################################################################### # Constructing the circuit # ------------------------ # # We can now construct the corresponding # single-mode Strawberry Fields program: # Single-mode Strawberry Fields program prog = sf.Program(1) # Create the 7 Strawberry Fields free parameters for each layer sf_params = [] names = ["r1", "sq_r", "sq_phi", "r2", "d_r", "d_phi", "kappa"] for i in range(depth): # For the ith layer, generate parameter names "r1_i", "sq_r_i", etc. sf_params_names = ["{}_{}".format(n, i) for n in names] # Create the parameters, and append them to our list ``sf_params``. sf_params.append(prog.params(*sf_params_names)) ###################################################################### # ``sf_params`` is now a nested list of shape ``(depth, 7)``, matching # the shape of ``weights``. sf_params = np.array(sf_params) print(sf_params.shape) ###################################################################### # Now, we can create a function to define the :math:`i`\ th layer, acting on qumode ``q``. We add # the :class:`~strawberryfields.utils.operation` decorator so that the layer can be used as a single # operation when constructing our circuit within the usual Strawberry Fields Program context # layer architecture @operation(1) def layer(i, q): Rgate(sf_params[i][0]) | q Sgate(sf_params[i][1], sf_params[i][2]) | q Rgate(sf_params[i][3]) | q Dgate(sf_params[i][4], sf_params[i][5]) | q Kgate(sf_params[i][6]) | q return q ###################################################################### # # Now that we have defined our gate parameters and our layer structure, we # can construct our variational quantum circuit. # Apply circuit of layers with corresponding depth with prog.context as q: for k in range(depth): layer(k) | q[0] ###################################################################### # Performing the optimization # --------------------------- # # :math:`\newcommand{ket}[1]{\left|#1\right\rangle}` With the Strawberry # Fields TensorFlow backend calculating the resulting state of the circuit # symbolically, we can use TensorFlow to optimize the gate parameters to # minimize the cost function we specify. With state learning, the measure # of distance between two quantum states is given by the fidelity of the # output state :math:`\ket{\psi}` with some target state # :math:`\ket{\psi_t}`. This is defined as the overlap between the two # states: # # .. math:: F = \left|\left\langle{\psi}\mid{\psi_t}\right\rangle\right|^2 # # where the output state can be written # :math:`\ket{\psi}=U(\vec{\theta})\ket{\psi_0}`, with # :math:`U(\vec{\theta})` the unitary operation applied by the variational # quantum circuit, and :math:`\ket{\psi_0}=\ket{0}` the initial state. # # Let's first instantiate the TensorFlow backend, making sure to pass # the Fock basis truncation cutoff. eng = sf.Engine("tf", backend_options={"cutoff_dim": cutoff}) ###################################################################### # Now let's define the target state as the single photon state # :math:`\ket{\psi_t}=\ket{1}`: import numpy as np target_state = np.zeros([cutoff]) target_state[1] = 1 print(target_state) ###################################################################### # Using this target state, we calculate the fidelity with the state # exiting the variational circuit. We must use TensorFlow functions to # manipulate this data, as well as a ``GradientTape`` to keep track of the # corresponding gradients! # # We choose the following cost function: # # .. math:: C(\vec{\theta}) = \left| \langle \psi_t \mid U(\vec{\theta})\mid 0\rangle - 1\right| # # By minimizing this cost function, the variational quantum circuit will # prepare a state with high fidelity to the target state. def cost(weights): # Create a dictionary mapping from the names of the Strawberry Fields # free parameters to the TensorFlow weight values. mapping = {p.name: w for p, w in zip(sf_params.flatten(), tf.reshape(weights, [-1]))} # Run engine state = eng.run(prog, args=mapping).state # Extract the statevector ket = state.ket() # Compute the fidelity between the output statevector # and the target state. fidelity = tf.abs(tf.reduce_sum(tf.math.conj(ket) * target_state)) ** 2 # Objective function to minimize cost = tf.abs(tf.reduce_sum(tf.math.conj(ket) * target_state) - 1) return cost, fidelity, ket ####################################################################### # Now that the cost function is defined, we can define and run the # optimization. Below, we choose the Adam # optimizer that is built into TensorFlow: opt = tf.keras.optimizers.Adam(learning_rate=lr) ###################################################################### # We then loop over all repetitions, storing the best predicted fidelity # value. fid_progress = [] best_fid = 0 for i in range(reps): # reset the engine if it has already been executed if eng.run_progs: eng.reset() with tf.GradientTape() as tape: loss, fid, ket = cost(weights) # Stores fidelity at each step fid_progress.append(fid.numpy()) if fid > best_fid: # store the new best fidelity and best state best_fid = fid.numpy() learnt_state = ket.numpy() # one repetition of the optimization gradients = tape.gradient(loss, weights) opt.apply_gradients(zip([gradients], [weights])) # Prints progress at every rep if i % 1 == 0: print("Rep: {} Cost: {:.4f} Fidelity: {:.4f}".format(i, loss, fid)) ###################################################################### # Results and visualisation # ------------------------- # # Plotting the fidelity vs. optimization step: from matplotlib import pyplot as plt plt.rcParams["font.family"] = "serif" plt.rcParams["font.sans-serif"] = ["Computer Modern Roman"] plt.style.use("default") plt.plot(fid_progress) plt.ylabel("Fidelity") plt.xlabel("Step") ###################################################################### # We can use the following function to plot the Wigner function of our # target and learnt state: import matplotlib.pyplot as plt from mpl_toolkits.mplot3d import Axes3D def wigner(rho): """This code is a modified version of the 'iterative' method of the wigner function provided in QuTiP, which is released under the BSD license, with the following copyright notice: Copyright (C) 2011 and later, P.D. Nation, J.R. Johansson, A.J.G. Pitchford, C. Granade, and A.L. Grimsmo. All rights reserved.""" import copy # Domain parameter for Wigner function plots l = 5.0 cutoff = rho.shape[0] # Creates 2D grid for Wigner function plots x = np.linspace(-l, l, 100) p = np.linspace(-l, l, 100) Q, P = np.meshgrid(x, p) A = (Q + P * 1.0j) / (2 * np.sqrt(2 / 2)) Wlist = np.array([np.zeros(np.shape(A), dtype=complex) for k in range(cutoff)]) # Wigner function for |0><0| Wlist[0] = np.exp(-2.0 * np.abs(A) ** 2) / np.pi # W = rho(0,0)W(|0><0|) W = np.real(rho[0, 0]) * np.real(Wlist[0]) for n in range(1, cutoff): Wlist[n] = (2.0 * A * Wlist[n - 1]) / np.sqrt(n) W += 2 * np.real(rho[0, n] * Wlist[n]) for m in range(1, cutoff): temp = copy.copy(Wlist[m]) # Wlist[m] = Wigner function for |m>`__, # (2019). # # 2. Nathan Killoran, Thomas R. Bromley, Juan Miguel Arrazola, Maria Schuld, # Nicolas Quesada, and Seth Lloyd. Continuous-variable quantum neural networks. # `Physical Review Research, 1(3), 033063. # `__, # (2019).