""" Sampling quadratures of non-Gaussian states using the bosonic backend ===================================================================== .. note:: This tutorial is second in a series on the ``bosonic`` backend. For an introduction to the backend, see :doc:`part 1 `. Once finished this tutorial, check out :doc:`part 3 ` for a deep dive into using the ``bosonic`` backend to simulate qubits encoded in photonic modes. These tutorials accompany our research paper [[#bourassa2021]_]. In a previous :doc:`tutorial ` we introduced the ``bosonic`` backend, the fourth and latest software backend of Strawberry Fields. In this tutorial we continue investigating some of its functionality by exploring how to sample quadrature (homodyne) measurements in non-Gaussian states. While these simulations could be done in the ``fock`` backend, it is a lot more convenient to use the new backend, since simulations in the ``fock`` backend need to remain under an energy ``cutoff``. We first return to our old friend, the cat state, and see how it looks when projected (measured) along different quadratures of phase space. Then we investigate a useful class of non-Gaussian states known as grid or GKP states. These states are named after `Daniel Gottesman `_ , `Alexei Kitaev `_ and `John Preskill `_ who introduced them in a seminal paper in 2001 [[#gottesman2001]_]. """ ###################################################################### # Cat quadratures # --------------- # In a previous tutorial we investigated the Wigner function of the cat # state. Now, we will explore the distributions of its quadratures. # The distributions of the position :math:`q` and momentum :math:`p` # quadratures can be obtained by integrating the Wigner function: # # .. math:: # \text{Pr}(q=x) = \int_{-\infty}^\infty dp~ W(x,p), \\ # \text{Pr}(p=y) = \int_{-\infty}^\infty dq~ W(q,y). # # Note that even though the Wigner function can be negative (as it is a `quasiprobability distribution `_), the quadrature # distributions :math:`\text{Pr}(q=x)` and :math:`\text{Pr}(p=y)` are proper # probability distributions, i.e., they are non-negative and their integrals # equal unity. # # We will use Strawberry Fields to investigate how these distributions look # for a cat state. # We do the usual imports import strawberryfields as sf import numpy as np import matplotlib.pyplot as plt import matplotlib as mpl from matplotlib import cm # Create cat state prog_cat = sf.Program(1) with prog_cat.context as q: sf.ops.Catstate(a=2) | q eng = sf.Engine("bosonic") cat = eng.run(prog_cat).state ###################################################################### # Strawberry Fields provides handy functions for accessing the quadrature # distributions. These can retrieved using the ``marginal`` method of the # ``state`` object: # Calculate the quadrature distributions scale = np.sqrt(sf.hbar) quad_axis= np.linspace(-6, 6, 1000) * scale cat_prob_x = cat.marginal(0, quad_axis) # This is the q quadrature cat_prob_p = cat.marginal(0, quad_axis, phi=np.pi / 2) # This is the p quadrature ###################################################################### # We can also use Strawberry Fields to generate samples of a given quadrature. # In this case, we do it for the :math:`q` quadrature: # Run the program again, collecting q samples this time shots = 2000 # Number of samples prog_cat_x = sf.Program(1) with prog_cat_x.context as q: sf.ops.Catstate(a=2) | q sf.ops.MeasureX | q eng = sf.Engine("bosonic") cat_samples_x = eng.run(prog_cat_x, shots=shots).samples[:, 0] ###################################################################### # We can also sample the :math:`p` quadrature: # Run the program again, collecting p samples this time prog_cat_p = sf.Program(1) with prog_cat_p.context as q: sf.ops.Catstate(a=2) | q sf.ops.MeasureP | q eng = sf.Engine("bosonic") cat_samples_p = eng.run(prog_cat_p, shots=shots).samples[:, 0] ###################################################################### # We can verify that the samples distribute according to the expected distribution: # Plot the results fig, axs = plt.subplots(1, 2, figsize=(10, 4)) fig.suptitle(r"$|0^{\alpha}\rangle_{cat}$, $\alpha=2$", fontsize=18) axs[0].hist(cat_samples_x / scale, bins=100, density=True, label="Samples", color="cornflowerblue") axs[0].plot(quad_axis/ scale, cat_prob_x * scale, "--", label="Ideal", color="tab:red") axs[0].set_xlabel(r"q (units of $\sqrt{\hbar}$)", fontsize=15) axs[0].set_ylabel("Pr(q)", fontsize=15) axs[1].hist(cat_samples_p / scale, bins=100, density=True, label="Samples", color="cornflowerblue") axs[1].plot(quad_axis/ scale, cat_prob_p * scale, "--", label="Ideal", color="tab:red") axs[1].set_xlabel(r"p (units of $\sqrt{\hbar}$)", fontsize=15) axs[1].set_ylabel("Pr(p)", fontsize=15) axs[1].legend() axs[0].tick_params(labelsize=13) axs[1].tick_params(labelsize=13) fig.tight_layout(rect=[0, 0.03, 1, 0.95]) plt.show() ###################################################################### # As we can see, we obtain a positive-valued probability distribution. # The two lobes arising from the coherent states superposed in the cat state # are clearly visible in :math:`q` distribution. # ###################################################################### # GKP quadratures # --------------- # As mentioned before, the ``bosonic`` backend can easily handle a number # of nonclassical states of light. A particularly useful family of states # are the so-called GKP or grid states. In the same way that cat states # are linear superpositions of two coherent states, GKP states can be understood # as superpositions of multiple squeezed states. # We can generate a GKP state and plot its Wigner function using Strawberry Fields in # the same way we did for cat states. # # Create GKP state epsilon = 0.1 prog_gkp = sf.Program(1) with prog_gkp.context as q: sf.ops.GKP(epsilon=epsilon) | q eng = sf.Engine("bosonic") gkp = eng.run(prog_gkp).state Wgkp = gkp.wigner(mode=0, xvec=quad_axis, pvec=quad_axis) scale = np.max(Wgkp.real) nrm = mpl.colors.Normalize(-scale, scale) plt.axes().set_aspect("equal") plt.contourf(quad_axis, quad_axis, Wgkp, 60, cmap=cm.RdBu, norm=nrm) plt.xlabel(r"q (units of $\sqrt{\hbar}$)", fontsize=15) plt.ylabel(r"p (units of $\sqrt{\hbar}$)", fontsize=15) plt.tight_layout() plt.show() ###################################################################### # We see that GKP states in phase space are made of multiple narrow peaks of # alternating signs. # After getting an idea of how a GKP state looks in phase-space we can now # turn our attention to understanding its quadrature distribution. # As before, we can directly calculate the expected quadrature distributions scale = np.sqrt(sf.hbar * np.pi) quad_axis= np.linspace(-6, 6, 1000) * scale gkp_prob_x = gkp.marginal(0, quad_axis) gkp_prob_p = gkp.marginal(0, quad_axis, phi=np.pi / 2) ###################################################################### # Like before, we can collect samples of the quadratures to # simulate the state being probed with homodyne measurements: # Run the program again, collecting q samples this time prog_gkp_x = sf.Program(1) with prog_gkp_x.context as q: sf.ops.GKP(epsilon=0.1) | q sf.ops.MeasureX | q eng = sf.Engine("bosonic") gkp_samples_x = eng.run(prog_gkp_x, shots=shots).samples[:, 0] # Run the program again, collecting p samples this time prog_gkp_p = sf.Program(1) with prog_gkp_p.context as q: sf.ops.GKP(epsilon=0.1) | q sf.ops.MeasureP | q eng = sf.Engine("bosonic") gkp_samples_p = eng.run(prog_gkp_p, shots=shots).samples[:, 0] ###################################################################### # and can compare the results: # Plot the results fig, axs = plt.subplots(1, 2, figsize=(10, 4)) fig.suptitle(r"$|0^\epsilon\rangle_{GKP}$, $\epsilon=0.1$ (10 dB)", fontsize=18) axs[0].hist(gkp_samples_x / scale, bins=100, density=True, label="Samples", color="cornflowerblue") axs[0].plot(quad_axis/ scale, gkp_prob_x * scale, "--", label="Ideal", color="tab:red") axs[0].set_xlabel(r"q (units of $\sqrt{\pi\hbar}$)", fontsize=15) axs[0].set_ylabel("Pr(q)", fontsize=15) axs[1].hist(gkp_samples_p / scale, bins=100, density=True, label="Samples", color="cornflowerblue") axs[1].plot(quad_axis/ scale, gkp_prob_p * scale, "--", label="Ideal", color="tab:red") axs[1].set_xlabel(r"p (units of $\sqrt{\pi\hbar}$)", fontsize=15) axs[1].set_ylabel("Pr(p)", fontsize=15) axs[1].legend() axs[0].tick_params(labelsize=13) axs[1].tick_params(labelsize=13) fig.tight_layout(rect=[0, 0.03, 1, 0.95]) plt.show() ###################################################################### # The comb of peaks are clearly visible in both the quadratures, as are # visible the Gaussian spread of the individual peaks and the Gaussian # envelope on the height of all the peaks. ###################################################################### # Conclusion and Outlook # ---------------------- # In this tutorial we have explored the sampling functionality # of the ``bosonic`` backend by investigating the quadrature distributions # of both cat and GKP states. For advanced readers, we encourage you to # read :doc:`part 3 ` to take a dive # into how GKP states can be used to encode a qubit in a photonic mode. ###################################################################### # References # ---------- # # .. [#bourassa2021] # # J. Eli Bourassa, Nicolás Quesada, Ilan Tzitrin, Antal Száva, Theodor Isacsson, # Josh Izaac, Krishna Kumar Sabapathy, Guillaume Dauphinais, and Ish Dhand. # Fast simulation of bosonic qubits via Gaussian functions in phase space. # `arXiv:2103.05530 `_, 2021. # # .. [#gottesman2001] # # Daniel Gottesman, Alexei Kitaev, and John Preskill. Encoding a qubit in an oscillator. # doi:10.1103/PhysRevA.64.012310.