--- jupytext: formats: ipynb,md:myst text_representation: extension: .md format_name: myst format_version: 0.13 jupytext_version: 1.14.0 kernelspec: display_name: Python 3 (phys-555) language: python name: phys-555 --- # Universal Quantum Computing II ```{code-cell} ipython3 :tags: [hide-cell] import mmf_setup; mmf_setup.nbinit() ``` ## Two-Qubit Operations In the previous assignment, you learned how to implement arbitrary single-qubit operations by controlling magnetic fields. A universal set of gates can be constructed out of these and a single two-qubit operation such as the ᴄɴᴏᴛ gate: \begin{gather*} \mathsf{\small CNOT} = \begin{pmatrix} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 1\\ 0 & 0 & 1 & 0 \end{pmatrix}. \end{gather*} Implementing this gate is significantly more challenging. Here we will explore some possible ways of doing this. Continuing with out example spin-½ qubits, consider the dipole-dipole interaction: \begin{gather*} \op{V} = \mu_{D} \vect{\op{S}}_{A} \cdot \vect{\op{S}}_{B} = \mu_{D}\begin{pmatrix} 1\\ & -1 & 2\\ & 2 & -1\\ &&& 1 \end{pmatrix}. \end{gather*} This interaction describes the behavior of one qubit in the presence of the magnetic field generated by the spin of the other qubit and vice-versa. To simplify our results, we shall choose units so that $\hbar = \mu_D = 1$. :::{margin} \begin{gather*} \begin{pmatrix} \mat{σ}_x\\ & \mat{σ}_x \end{pmatrix} \begin{pmatrix} & \mat{1}\\ \mat{1} \end{pmatrix} + \\ \begin{pmatrix} \mat{σ}_y\\ & \mat{σ}_y \end{pmatrix} \begin{pmatrix} & -\I\mat{1}\\ \I\mat{1} \end{pmatrix} + \\ \begin{pmatrix} \mat{σ}_z\\ & \mat{σ}_z \end{pmatrix} \begin{pmatrix} \mat{1}\\ & - \mat{1} \end{pmatrix}\\ = \begin{pmatrix} & & & 1\\ & & 1 \\ & 1\\ 1\\ \end{pmatrix}\\ +\begin{pmatrix} &&& -1\\ && 1\\ & 1\\ -1\\ \end{pmatrix}\\ +\begin{pmatrix} 1\\ & -1 \\ && -1\\ && & 1\\ \end{pmatrix} \end{gather*} ::: 1. Show that, in these simplified units, \begin{gather*} \op{V} = \frac{1}{4} \begin{pmatrix} 1\\ & -1 & 2\\ & 2 & -1\\ &&& 1 \end{pmatrix} \end{gather*} by explicitly computing this from the Pauli matrices: \begin{gather*} \vec{\op{S}_{A}} = \frac{\vec{\mat{σ}}}{2}\otimes \mat{1}, \qquad \vec{\op{S}_{B}} = \mat{1}\otimes \frac{\vec{\mat{σ}}}{2}. \end{gather*} I.e. show that \begin{gather*} 4\op{V} = \sum_{i=\{x, y, z\}} (\mat{σ}_i\otimes \mat{1})(\mat{1}\otimes \mat{σ}_i)\\ = \begin{pmatrix} 0 & 1\\ 1 & 0 \\ && 0 & 1\\ && 1 & 0\\ \end{pmatrix} \begin{pmatrix} && 1 & 0\\ && 0 & 1 \\ 1 & 0\\ 0 & 1\\ \end{pmatrix} + \cdots\\ = \begin{pmatrix} 1\\ & -1 & 2\\ & 2& -1\\ &&& 1 \end{pmatrix} \end{gather*} 2. Diagonalize this matrix so that you can compute the matrix exponential and find the form of the corresponding 2-qubit unitary operations. *(Hint: you only need to diagonalize a 2×2 block, and this block is diagonalized by the matrix corresponding to the Hadamard gate.)* :::{dropdown} Answer The answer can be expressed in terms of the Hadamard gate $H$: \begin{gather*} \begin{pmatrix} -1 & 2\\ 2& -1 \end{pmatrix} = H \begin{pmatrix} 1\\ & -3 \end{pmatrix} H \end{gather*} ::: I.e. show that \begin{gather*} \mat{U} = \exp\left(\mat{V}t/\I\hbar\right) = \begin{pmatrix} e^{\I\theta}\\ & H\begin{pmatrix} e^{\I\theta}\\ & e^{-3\I\theta} \end{pmatrix} H\\ && e^{\I\theta} \end{pmatrix}, \end{gather*} where $\theta = \hbar \mu_D t/4$. 3. Can you combine this with single qubit gates to form ᴄɴᴏᴛ? ```python # Brute-force optimization to see if we can do this. # Apparently not. import mmf_setup; mmf_setup.nbinit() import numpy as np import scipy.linalg from scipy.linalg import expm, logm, eigh from scipy.optimize import minimize from numpy import kron from phys_555_2022.utils import sigmas H = np.array([[1, 1], [1, -1]]) / np.sqrt(2) X, Y, Z = sigmas CNOT = np.eye(4) CNOT[2:, 2:] = X.real CZ = np.eye(4) CZ[2:, 2:] = Z.real V = np.diag([1, -1, -1, 1]) V[1,2] = V[2, 1] = 2 I = np.eye(2) sigma_mus = np.asarray([I] + sigmas.tolist()) def unpack(y): theta = y[-1] bs = np.reshape(y[:-1], (4, 4)) Hs = [np.einsum('a,a...', _b, sigma_mus) for _b in bs] Us = [expm(_H/2j) for _H in Hs] return Us, theta def get_residue(y): (Ui1, Ui2, Uo1, Uo2), theta = unpack(y) M = np.zeros((4, 4), dtype=complex) M[1:3, 1:3] = H @ np.diag([np.exp(1j*theta), np.exp(-3j*theta)]) @ H M[0, 0] = M[-1, -1] = np.exp(1j*theta) U = kron(Ui1, Ui2) @ M @ kron(Uo1, Uo2) #U *= np.exp(-1j*np.angle(U[0,0])) assert np.allclose(U.conj().T @ U, np.eye(4)) return (abs(U - CNOT)**2).sum() def get_res(rng=np.random.default_rng(seed=1)): y0 = rng.random(17) * 2*np.pi res = minimize(get_residue, x0=y0) return res.fun def find_match(U_goal, H, rng=np.random.default_rng(seed=1)): """Try to find a set of 4 single-qubit gates transforming U to U_goal. Arguments --------- U_goal : 4x4 array Target 2-qubit gate. H : 4x4 array Starting 2-qubit Hamiltonian. Single qubit operations will be applied to the inputs and outputs to try to match U_goal with exponentiation of H. """ assert np.allclose(H, H.T.conj()) D, V = eigh(H) assert np.allclose(V @ np.diag(D) @ V.T.conj(), H) def unpack(y): theta = y[-1] bs = np.reshape(y[:-1], (4, 4)) Hs = [np.einsum('a,a...', _b, sigma_mus) for _b in bs] Us = [expm(_H/2j) for _H in Hs] return Us, theta def get_U(y): (Ui1, Ui2, Uo1, Uo2), theta = unpack(y) M = V @ np.diag(np.exp(1j*theta*D)) @ V.T.conj() U = kron(Ui1, Ui2) @ M @ kron(Uo1, Uo2) assert np.allclose(U.conj().T @ U, np.eye(4)) return U def get_residue(y): return (abs(get_U(y) - U_goal)**2).sum() y0 = rng.random(17) * 2*np.pi res = minimize(get_residue, x0=y0) U = get_U(res.x) H = logm(U)/1j return res.fun, U, H ``` ```{code-cell} ipython3 import numpy as np import scipy.linalg from scipy.linalg import expm, logm, eigh from scipy.optimize import minimize from numpy import kron from phys_555_2022.utils import sigmas H = np.array([[1, 1], [1, -1]]) / np.sqrt(2) X, Y, Z = sigmas CNOT = np.eye(4) CNOT[2:, 2:] = X.real CZ = np.eye(4) CZ[2:, 2:] = Z.real I = np.eye(2) sigma_mus = np.asarray([I] + sigmas.tolist()) def find_match(U_goal, H, y0=None, rng=np.random.default_rng(seed=1)): """Try to find a set of 4 single-qubit gates transforming U to U_goal. Arguments --------- U_goal : 4x4 array Target 2-qubit gate. H : 4x4 array Starting 2-qubit Hamiltonian. Single qubit operations will be applied to the inputs and outputs to try to match U_goal with exponentiation of H. """ assert np.allclose(H, H.T.conj()) D, V = eigh(H) assert np.allclose(V @ np.diag(D) @ V.T.conj(), H) def unpack(y): theta = y[-1] bs = np.reshape(y[:-1], (4, 4)) Hs = [np.einsum('a,a...', _b, sigma_mus) for _b in bs] Us = [expm(_H/2j) for _H in Hs] return Us, theta def get_U(y): (Ui1, Ui2, Uo1, Uo2), theta = unpack(y) M = V @ np.diag(np.exp(1j*theta*D)) @ V.T.conj() U = kron(Ui1, Ui2) @ M @ kron(Uo1, Uo2) assert np.allclose(U.conj().T @ U, np.eye(4)) return U def get_residue(y): return (abs(get_U(y) - U_goal)**2).sum() if y0 is None: y0 = rng.random(17) * 2*np.pi res = minimize(get_residue, x0=y0) U = get_U(res.x) H = logm(U)/1j return res, U, H ``` ```{code-cell} ipython3 rng = np.random.default_rng(seed=3) V = rng.normal(size=(4, 4*2)).view(complex) V += V.T.conj() V = V.reshape((2, 2, 2, 2)) V = (V + np.einsum('abcd->badc', V)).reshape((4, 4)) V = np.diag([1, -1, -1, 1]) V[1,2] = V[2, 1] = 2 res, U, UH = find_match(CNOT, V, rng=rng) for n in range(10): res, U, UH = find_match(CNOT, UH, rng=rng) print(res.fun) #res, U, UH = find_match(CNOT, UH, y0=res.x, rng=rng) #print(res.fun) #for n in range(10): # res, U, UH = find_match(CNOT, UH, y0=0*res.x, rng=rng) # print(res.fun) ``` ```{code-cell} ipython3 (abs(U - CNOT)**2).sum() ``` ```{code-cell} ipython3 ```