0.2.Quantum Computing with Matrices, ok.ipynb

{"nbformat":4,"nbformat_minor":0,"metadata":{"colab":{"provenance":[],"collapsed_sections":["Id1_8joHHF1R","MG4KkBKXccmh","akkdxEaqI6E-"]},"kernelspec":{"display_name":"Python 3 (ipykernel)","language":"python","name":"python3"},"language_info":{"codemirror_mode":{"name":"ipython","version":3},"file_extension":".py","mimetype":"text/x-python","name":"python","nbconvert_exporter":"python","pygments_lexer":"ipython3","version":"3.8.10"}},"cells":[{"cell_type":"markdown","source":["#### **Quantum Computing with Matrices Exercises**"],"metadata":{"id":"IBHGIBs-chJ4"}},{"cell_type":"markdown","source":["##### **Ex.1:Write $V = \\begin{bmatrix} 3+2i \\\\ 4-2i \\end{bmatrix}$ as a sum of $|0\\rangle$ and $|1 \\rangle$**"],"metadata":{"id":"wLjxgdx-pjFa"}},{"cell_type":"markdown","source":["To create a state $|\\psi\\rangle$ equivalent to the state $V$ above we need to normalize it. Remember that for $|\\psi\\rangle = a|0\\rangle + b|1\\rangle$ to represent a qubit vector state, $|a|^2+|b|^2=1$."],"metadata":{"id":"Glk7n-bBdKcH"}},{"cell_type":"markdown","source":["Also, remember that if $a$ is a complex number like $a = 3+4.i$, its module $|a|$ is calculated by $|a|^2 = a^*.a = (3-4i).(3+4i) = 25$. Therefore $|a|=5$. $a^*$ is the complex conjugate of $a$"],"metadata":{"id":"yB0-IygTg6Ao"}},{"cell_type":"code","source":["import numpy as np\n","\n","V = np.array([[3+2j],[4-2j]])\n","modV = np.real(V.T.conjugate().dot(V)[0,0])**0.5\n","Vn = V/modV; Vn"],"metadata":{"colab":{"base_uri":"https://localhost:8080/"},"id":"CJ_iA9pYdvzA","outputId":"de28a297-98b3-4def-9d17-668ec748af87","executionInfo":{"status":"ok","timestamp":1708350564577,"user_tz":180,"elapsed":8,"user":{"displayName":"Gustavo Mirapalheta","userId":"00687033162702398655"}}},"execution_count":1,"outputs":[{"output_type":"execute_result","data":{"text/plain":["array([[0.52223297+0.34815531j],\n","       [0.69631062-0.34815531j]])"]},"metadata":{},"execution_count":1}]},{"cell_type":"code","source":["v0 = np.array([[1,0]]).T\n","v1 = np.array([[0,1]]).T\n","Vn[0,0]*v0 + Vn[1,0]*v1"],"metadata":{"colab":{"base_uri":"https://localhost:8080/"},"id":"QjiJtuIzfnJw","outputId":"2fb015bf-67a8-4d44-8873-c03039a1ad43","executionInfo":{"status":"ok","timestamp":1708350564577,"user_tz":180,"elapsed":7,"user":{"displayName":"Gustavo Mirapalheta","userId":"00687033162702398655"}}},"execution_count":2,"outputs":[{"output_type":"execute_result","data":{"text/plain":["array([[0.52223297+0.34815531j],\n","       [0.69631062-0.34815531j]])"]},"metadata":{},"execution_count":2}]},{"cell_type":"markdown","source":["Let's confirm the probabilities calculating them from the state $V$ normalized amplitudes."],"metadata":{"id":"6vPmM9X2j567"}},{"cell_type":"code","source":["Vn[0,0].conjugate()*Vn[0,0]"],"metadata":{"id":"8ywf-hVQkDSg","executionInfo":{"status":"ok","timestamp":1708350564577,"user_tz":180,"elapsed":5,"user":{"displayName":"Gustavo Mirapalheta","userId":"00687033162702398655"}},"colab":{"base_uri":"https://localhost:8080/"},"outputId":"9f3537ee-fbc6-48f5-cd6d-04fb18f3ed08"},"execution_count":3,"outputs":[{"output_type":"execute_result","data":{"text/plain":["(0.39393939393939403+0j)"]},"metadata":{},"execution_count":3}]},{"cell_type":"code","source":["Vn[1,0].conjugate()*Vn[1,0]"],"metadata":{"id":"2NHkl48Qkr4p","executionInfo":{"status":"ok","timestamp":1708350564577,"user_tz":180,"elapsed":4,"user":{"displayName":"Gustavo Mirapalheta","userId":"00687033162702398655"}},"colab":{"base_uri":"https://localhost:8080/"},"outputId":"11a5ce37-b506-4c6c-8959-fe8ec27f76da"},"execution_count":4,"outputs":[{"output_type":"execute_result","data":{"text/plain":["(0.6060606060606062+0j)"]},"metadata":{},"execution_count":4}]},{"cell_type":"markdown","source":["##### **Ex.2: Using matrices, prove that the CNOT gate is its own inverse. Do the same for the Hadamard gate**"],"metadata":{"id":"2D5Xgh1pCqkw"}},{"cell_type":"code","source":["import numpy as np\n","CNOT = np.array([[1,0,0,0],\n","                 [0,1,0,0],\n","                 [0,0,0,1],\n","                 [0,0,1,0]])\n","CNOT.dot(CNOT)"],"metadata":{"id":"gIv3HpSmCww7","executionInfo":{"status":"ok","timestamp":1708350564577,"user_tz":180,"elapsed":3,"user":{"displayName":"Gustavo Mirapalheta","userId":"00687033162702398655"}},"colab":{"base_uri":"https://localhost:8080/"},"outputId":"0d4645b1-a85d-43d5-f71d-5fd27790150f"},"execution_count":5,"outputs":[{"output_type":"execute_result","data":{"text/plain":["array([[1, 0, 0, 0],\n","       [0, 1, 0, 0],\n","       [0, 0, 1, 0],\n","       [0, 0, 0, 1]])"]},"metadata":{},"execution_count":5}]},{"cell_type":"code","source":["import numpy as np\n","H = np.array([[1,1],[1,-1]])/2**0.5\n","H.dot(H)"],"metadata":{"id":"mnpeGy00ppys","executionInfo":{"status":"ok","timestamp":1708350565044,"user_tz":180,"elapsed":469,"user":{"displayName":"Gustavo Mirapalheta","userId":"00687033162702398655"}},"colab":{"base_uri":"https://localhost:8080/"},"outputId":"18173ec1-d40d-4f56-cbd0-763ec374990d"},"execution_count":6,"outputs":[{"output_type":"execute_result","data":{"text/plain":["array([[ 1.00000000e+00, -2.23711432e-17],\n","       [-2.23711432e-17,  1.00000000e+00]])"]},"metadata":{},"execution_count":6}]},{"cell_type":"markdown","source":["##### **Ex.3a1: Present the X Pauli matrix $ \\begin{bmatrix}\n","0 & 1 \\\\\n","1 & 0\n","\\end{bmatrix}  $ and show that it is its own inverse**"],"metadata":{"id":"7a1XoqY3cHwQ"}},{"cell_type":"code","source":["import numpy as np\n","X = np.array([[0,1],\n","              [1,0]])\n","X"],"metadata":{"id":"cP3SdEVtcOva","executionInfo":{"status":"ok","timestamp":1708350565044,"user_tz":180,"elapsed":14,"user":{"displayName":"Gustavo Mirapalheta","userId":"00687033162702398655"}},"colab":{"base_uri":"https://localhost:8080/"},"outputId":"0eb56610-c88e-4cfc-91ac-d9a2e0ef05a8"},"execution_count":7,"outputs":[{"output_type":"execute_result","data":{"text/plain":["array([[0, 1],\n","       [1, 0]])"]},"metadata":{},"execution_count":7}]},{"cell_type":"code","source":["X.conj().T.dot(X)"],"metadata":{"id":"I1H3EYSYcwpv","executionInfo":{"status":"ok","timestamp":1708350565044,"user_tz":180,"elapsed":12,"user":{"displayName":"Gustavo Mirapalheta","userId":"00687033162702398655"}},"colab":{"base_uri":"https://localhost:8080/"},"outputId":"14330ba0-c975-4eac-baa9-83a44e870204"},"execution_count":8,"outputs":[{"output_type":"execute_result","data":{"text/plain":["array([[1, 0],\n","       [0, 1]])"]},"metadata":{},"execution_count":8}]},{"cell_type":"markdown","source":["##### **Ex.3a2: Present the Y Pauli matrix $ \\begin{bmatrix}\n","0 & -j \\\\\n","j & 0\n","\\end{bmatrix}  $ and show that it is its own inverse**"],"metadata":{"id":"xXOIlxPMHe4j"}},{"cell_type":"code","source":["Y = np.array([[0,-1j],\n","              [1j,0]])\n","Y"],"metadata":{"id":"MqrcZJpjAeD3","executionInfo":{"status":"ok","timestamp":1708350565044,"user_tz":180,"elapsed":11,"user":{"displayName":"Gustavo Mirapalheta","userId":"00687033162702398655"}},"colab":{"base_uri":"https://localhost:8080/"},"outputId":"482eea51-bc12-4181-b773-5bed0eac3325"},"execution_count":9,"outputs":[{"output_type":"execute_result","data":{"text/plain":["array([[ 0.+0.j, -0.-1.j],\n","       [ 0.+1.j,  0.+0.j]])"]},"metadata":{},"execution_count":9}]},{"cell_type":"code","source":["Y.conj().T.dot(Y)"],"metadata":{"id":"D_OzPFgDc5zn","executionInfo":{"status":"ok","timestamp":1708350565044,"user_tz":180,"elapsed":10,"user":{"displayName":"Gustavo Mirapalheta","userId":"00687033162702398655"}},"colab":{"base_uri":"https://localhost:8080/"},"outputId":"768bc35a-c324-4ec1-da4d-261614e6a928"},"execution_count":10,"outputs":[{"output_type":"execute_result","data":{"text/plain":["array([[1.+0.j, 0.+0.j],\n","       [0.+0.j, 1.+0.j]])"]},"metadata":{},"execution_count":10}]},{"cell_type":"markdown","source":["##### **Ex.3a3: Present the Z Pauli matrix $ \\begin{bmatrix}\n","1 & 0 \\\\\n","0 & -1\n","\\end{bmatrix}  $ and show that it is its own inverse**"],"metadata":{"id":"aWdU1bcYHgmN"}},{"cell_type":"code","source":["Z = np.array([[1,0],\n","              [0,-1]])\n","Z"],"metadata":{"id":"xJcqx0U8AhSR","executionInfo":{"status":"ok","timestamp":1708350565044,"user_tz":180,"elapsed":9,"user":{"displayName":"Gustavo Mirapalheta","userId":"00687033162702398655"}},"colab":{"base_uri":"https://localhost:8080/"},"outputId":"fc5063e4-144c-432c-ada9-4673bbe8ee7d"},"execution_count":11,"outputs":[{"output_type":"execute_result","data":{"text/plain":["array([[ 1,  0],\n","       [ 0, -1]])"]},"metadata":{},"execution_count":11}]},{"cell_type":"code","source":["Z.conj().T.dot(Z)"],"metadata":{"id":"Njvb_Ficc7rG","executionInfo":{"status":"ok","timestamp":1708350565045,"user_tz":180,"elapsed":9,"user":{"displayName":"Gustavo Mirapalheta","userId":"00687033162702398655"}},"colab":{"base_uri":"https://localhost:8080/"},"outputId":"e884771b-c039-4404-90a9-d673b673dd39"},"execution_count":12,"outputs":[{"output_type":"execute_result","data":{"text/plain":["array([[1, 0],\n","       [0, 1]])"]},"metadata":{},"execution_count":12}]},{"cell_type":"markdown","source":["##### **Ex.3b: Show that $H = \\frac{1}{\\sqrt{2}}( X + Z)$  through numpy**"],"metadata":{"id":"SGZYT9_EFkEP"}},{"cell_type":"code","source":["H = (X+Z)/np.sqrt(2); H"],"metadata":{"id":"jXvXgtHXeljL","executionInfo":{"status":"ok","timestamp":1708350565045,"user_tz":180,"elapsed":8,"user":{"displayName":"Gustavo Mirapalheta","userId":"00687033162702398655"}},"colab":{"base_uri":"https://localhost:8080/"},"outputId":"a7c6f69f-a31d-49db-9357-7f8acd5f305a"},"execution_count":13,"outputs":[{"output_type":"execute_result","data":{"text/plain":["array([[ 0.70710678,  0.70710678],\n","       [ 0.70710678, -0.70710678]])"]},"metadata":{},"execution_count":13}]},{"cell_type":"markdown","source":["##### **Ex.3c: Show that $X = H.Z.H$**"],"metadata":{"id":"zj8YwVXFGRxU"}},{"cell_type":"code","source":["H.dot(Z).dot(H)"],"metadata":{"id":"fYwXDBZMesEk","executionInfo":{"status":"ok","timestamp":1708350565045,"user_tz":180,"elapsed":7,"user":{"displayName":"Gustavo Mirapalheta","userId":"00687033162702398655"}},"colab":{"base_uri":"https://localhost:8080/"},"outputId":"9411a361-65a3-4814-a210-f4e9f70f3ace"},"execution_count":14,"outputs":[{"output_type":"execute_result","data":{"text/plain":["array([[-2.23711432e-17,  1.00000000e+00],\n","       [ 1.00000000e+00, -2.23711432e-17]])"]},"metadata":{},"execution_count":14}]},{"cell_type":"markdown","source":["##### **Ex.3d: Show that Z = H.X.H**"],"metadata":{"id":"WeYbhB1UGg7i"}},{"cell_type":"code","source":["H.dot(X).dot(H)"],"metadata":{"id":"iUzWW1v-ewW0","executionInfo":{"status":"ok","timestamp":1708350565045,"user_tz":180,"elapsed":6,"user":{"displayName":"Gustavo Mirapalheta","userId":"00687033162702398655"}},"colab":{"base_uri":"https://localhost:8080/"},"outputId":"06061b46-f137-4655-c14a-c3d4a0c7b43a"},"execution_count":15,"outputs":[{"output_type":"execute_result","data":{"text/plain":["array([[ 1.00000000e+00, -2.23711432e-17],\n","       [ 2.23711432e-17, -1.00000000e+00]])"]},"metadata":{},"execution_count":15}]},{"cell_type":"markdown","source":["##### **Ex.3e: Show that Y = -H.Y.H**"],"metadata":{"id":"Id1_8joHHF1R"}},{"cell_type":"code","source":["-H.dot(Y).dot(H)"],"metadata":{"id":"YMbVNd-dfEz4","executionInfo":{"status":"ok","timestamp":1708350565045,"user_tz":180,"elapsed":5,"user":{"displayName":"Gustavo Mirapalheta","userId":"00687033162702398655"}},"colab":{"base_uri":"https://localhost:8080/"},"outputId":"1c22e4a4-adb0-4302-b673-be725e047414"},"execution_count":16,"outputs":[{"output_type":"execute_result","data":{"text/plain":["array([[-0.+2.23711432e-17j, -0.-1.00000000e+00j],\n","       [-0.+1.00000000e+00j, -0.-2.23711432e-17j]])"]},"metadata":{},"execution_count":16}]},{"cell_type":"markdown","source":["##### **Ex.4a1: Present the matrix S = $ \\begin{bmatrix}\n","1 & 0 \\\\\n","0 & j\n","\\end{bmatrix}  $ and show that it is its own inverse**"],"metadata":{"id":"MG4KkBKXccmh"}},{"cell_type":"code","source":["import numpy as np\n","S = np.array([[1,0],\n","              [0,1j]])\n","S"],"metadata":{"id":"F7hoYIveccUO","executionInfo":{"status":"ok","timestamp":1708350565045,"user_tz":180,"elapsed":4,"user":{"displayName":"Gustavo Mirapalheta","userId":"00687033162702398655"}},"colab":{"base_uri":"https://localhost:8080/"},"outputId":"3a9821cf-4ca9-4843-ff42-1022c53f65f9"},"execution_count":17,"outputs":[{"output_type":"execute_result","data":{"text/plain":["array([[1.+0.j, 0.+0.j],\n","       [0.+0.j, 0.+1.j]])"]},"metadata":{},"execution_count":17}]},{"cell_type":"code","source":["S.conj().T.dot(S)"],"metadata":{"id":"BLLWP4TRhRbE","executionInfo":{"status":"ok","timestamp":1708350565045,"user_tz":180,"elapsed":4,"user":{"displayName":"Gustavo Mirapalheta","userId":"00687033162702398655"}},"colab":{"base_uri":"https://localhost:8080/"},"outputId":"7681a6ad-7bba-4785-f744-eb0055c701ae"},"execution_count":18,"outputs":[{"output_type":"execute_result","data":{"text/plain":["array([[1.+0.j, 0.+0.j],\n","       [0.+0.j, 1.+0.j]])"]},"metadata":{},"execution_count":18}]},{"cell_type":"markdown","source":["##### **Ex.4a2: Present the matrix T = $ \\begin{bmatrix}\n","1 & 0 \\\\\n","0 & e^{j\\frac{\\pi}{4}}\n","\\end{bmatrix}  $ and show that it is its own inverse**"],"metadata":{"id":"akkdxEaqI6E-"}},{"cell_type":"code","source":["T = np.array([[1,0],\n","              [0,np.exp(1j*np.pi/4)]])\n","T"],"metadata":{"id":"W1SyENfbBEPb","executionInfo":{"status":"ok","timestamp":1708350565045,"user_tz":180,"elapsed":4,"user":{"displayName":"Gustavo Mirapalheta","userId":"00687033162702398655"}},"colab":{"base_uri":"https://localhost:8080/"},"outputId":"5b00fd69-8263-4577-a2fb-c78722ae2ff9"},"execution_count":19,"outputs":[{"output_type":"execute_result","data":{"text/plain":["array([[1.        +0.j        , 0.        +0.j        ],\n","       [0.        +0.j        , 0.70710678+0.70710678j]])"]},"metadata":{},"execution_count":19}]},{"cell_type":"code","source":["T.conj().T.dot(T)"],"metadata":{"id":"dgFe7xTihVS9","executionInfo":{"status":"ok","timestamp":1708350565635,"user_tz":180,"elapsed":593,"user":{"displayName":"Gustavo Mirapalheta","userId":"00687033162702398655"}},"colab":{"base_uri":"https://localhost:8080/"},"outputId":"5dc02bc1-78fb-44a9-d632-54207fbbf47f"},"execution_count":20,"outputs":[{"output_type":"execute_result","data":{"text/plain":["array([[1.+0.j, 0.+0.j],\n","       [0.+0.j, 1.+0.j]])"]},"metadata":{},"execution_count":20}]},{"cell_type":"markdown","metadata":{"id":"fSAcu_1CxpTf"},"source":["##### **Ex.5:Show the SWAP operator matrix $S = \\begin{bmatrix}\n","1 & 0 & 0 & 0 \\\\\n","0 & 0 & 1 & 0 \\\\\n","0 & 1 & 0 & 0 \\\\\n","0 & 0 & 0 & 1\n","\\end{bmatrix}  $**"]},{"cell_type":"markdown","source":["The SWAP exchanges the first with the second bit. Therefore $S|00\\rangle = |00\\rangle$, $S|01\\rangle = |10\\rangle$, $S|10\\rangle = |01\\rangle$ and $S|11\\rangle = |11\\rangle$"],"metadata":{"id":"dy0amqwOL-U9"}},{"cell_type":"code","execution_count":21,"metadata":{"id":"_QvmtfxDxooo","executionInfo":{"status":"ok","timestamp":1708350565635,"user_tz":180,"elapsed":24,"user":{"displayName":"Gustavo Mirapalheta","userId":"00687033162702398655"}},"colab":{"base_uri":"https://localhost:8080/"},"outputId":"a19da267-1d00-4342-9c66-931dd5d2f372"},"outputs":[{"output_type":"execute_result","data":{"text/plain":["array([[1, 0, 0, 0],\n","       [0, 0, 1, 0],\n","       [0, 1, 0, 0],\n","       [0, 0, 0, 1]])"]},"metadata":{},"execution_count":21}],"source":["S = np.array([[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]); S"]},{"cell_type":"code","execution_count":22,"metadata":{"id":"ewWCZdg2yR7Z","executionInfo":{"status":"ok","timestamp":1708350565635,"user_tz":180,"elapsed":23,"user":{"displayName":"Gustavo Mirapalheta","userId":"00687033162702398655"}},"colab":{"base_uri":"https://localhost:8080/"},"outputId":"301c5463-c60f-47c8-c279-8dd79a432481"},"outputs":[{"output_type":"execute_result","data":{"text/plain":["array([[1],\n","       [0],\n","       [0],\n","       [0]])"]},"metadata":{},"execution_count":22}],"source":["v00 = np.array([[1,0,0,0]]).T\n","S.dot(v00)"]},{"cell_type":"code","execution_count":23,"metadata":{"id":"t0zywe-WyUkj","executionInfo":{"status":"ok","timestamp":1708350565635,"user_tz":180,"elapsed":21,"user":{"displayName":"Gustavo Mirapalheta","userId":"00687033162702398655"}},"colab":{"base_uri":"https://localhost:8080/"},"outputId":"aca0c876-ef65-462c-f056-c437e6a02346"},"outputs":[{"output_type":"execute_result","data":{"text/plain":["array([[0],\n","       [0],\n","       [1],\n","       [0]])"]},"metadata":{},"execution_count":23}],"source":["v01 = np.array([[0,1,0,0]]).T\n","S.dot(v01)"]},{"cell_type":"code","execution_count":24,"metadata":{"id":"opTrLOdMyWbw","executionInfo":{"status":"ok","timestamp":1708350565635,"user_tz":180,"elapsed":19,"user":{"displayName":"Gustavo Mirapalheta","userId":"00687033162702398655"}},"colab":{"base_uri":"https://localhost:8080/"},"outputId":"220691a2-efc5-4e49-bce4-c25d97a45940"},"outputs":[{"output_type":"execute_result","data":{"text/plain":["array([[0],\n","       [1],\n","       [0],\n","       [0]])"]},"metadata":{},"execution_count":24}],"source":["v10 = np.array([[0,0,1,0]]).T\n","S.dot(v10)"]},{"cell_type":"code","execution_count":25,"metadata":{"id":"RqOu4tslyYTf","executionInfo":{"status":"ok","timestamp":1708350565635,"user_tz":180,"elapsed":17,"user":{"displayName":"Gustavo Mirapalheta","userId":"00687033162702398655"}},"colab":{"base_uri":"https://localhost:8080/"},"outputId":"9c3fdacd-c233-4292-ddfd-d81ae77b6810"},"outputs":[{"output_type":"execute_result","data":{"text/plain":["array([[0],\n","       [0],\n","       [0],\n","       [1]])"]},"metadata":{},"execution_count":25}],"source":["v11 = np.array([[0,0,0,1]]).T\n","S.dot(v11)"]},{"cell_type":"markdown","metadata":{"id":"eT58FeMdKNB4"},"source":["##### **Ex.6:Show that $S_{ij} = C_{ij}$ $ \\cdot $ $C_{ji} \\cdot C_{ij}$**"]},{"cell_type":"markdown","metadata":{"id":"V0DVUSDuzdZc"},"source":["Vamos confirmar que $C_{ij}.C_{ji}.C_{ij} = S_{ij}$\n","\n","$$C_{ij}.C_{ji}.C_{ij}.|00\\rangle = C_{ij}.C_{ji}.|00\\rangle = C_{ij}.|00\\rangle = |00\\rangle$$\n","\n","$$C_{ij}.C_{ji}.C_{ij}.|01\\rangle = C_{ij}.C_{ji}.|01\\rangle = C_{ij}.|11\\rangle = |10\\rangle$$\n","\n","$$C_{ij}.C_{ji}.C_{ij}.|10\\rangle = C_{ij}.C_{ji}.|11\\rangle = C_{ij}.|01\\rangle = |01\\rangle$$\n","\n","$$C_{ij}.C_{ji}.C_{ij}.|11\\rangle = C_{ij}.C_{ji}.|10\\rangle = C_{ij}.|10\\rangle = |11\\rangle$$"]},{"cell_type":"code","source":["C = np.array([[1,0,0,0],\n","              [0,1,0,0],\n","              [0,0,0,1],\n","              [0,0,1,0]]);C"],"metadata":{"id":"TMn_JVprKY8C","executionInfo":{"status":"ok","timestamp":1708350565635,"user_tz":180,"elapsed":15,"user":{"displayName":"Gustavo Mirapalheta","userId":"00687033162702398655"}},"colab":{"base_uri":"https://localhost:8080/"},"outputId":"edc7b8f9-17a6-467e-a8ab-ef7660fc6959"},"execution_count":26,"outputs":[{"output_type":"execute_result","data":{"text/plain":["array([[1, 0, 0, 0],\n","       [0, 1, 0, 0],\n","       [0, 0, 0, 1],\n","       [0, 0, 1, 0]])"]},"metadata":{},"execution_count":26}]},{"cell_type":"code","source":["C_ = np.array([[1,0,0,0],\n","               [0,0,0,1],\n","               [0,0,1,0],\n","               [0,1,0,0]]);C_"],"metadata":{"id":"RCQRLN9LMnsi","executionInfo":{"status":"ok","timestamp":1708350565636,"user_tz":180,"elapsed":15,"user":{"displayName":"Gustavo Mirapalheta","userId":"00687033162702398655"}},"colab":{"base_uri":"https://localhost:8080/"},"outputId":"ea4d3648-a98b-48af-a6f6-87b614a81513"},"execution_count":27,"outputs":[{"output_type":"execute_result","data":{"text/plain":["array([[1, 0, 0, 0],\n","       [0, 0, 0, 1],\n","       [0, 0, 1, 0],\n","       [0, 1, 0, 0]])"]},"metadata":{},"execution_count":27}]},{"cell_type":"code","source":["C.dot(C_).dot(C)"],"metadata":{"id":"aRheCPduKavH","executionInfo":{"status":"ok","timestamp":1708350565636,"user_tz":180,"elapsed":13,"user":{"displayName":"Gustavo Mirapalheta","userId":"00687033162702398655"}},"colab":{"base_uri":"https://localhost:8080/"},"outputId":"4e6ad531-e384-4b48-dd79-250ad328e035"},"execution_count":28,"outputs":[{"output_type":"execute_result","data":{"text/plain":["array([[1, 0, 0, 0],\n","       [0, 0, 1, 0],\n","       [0, 1, 0, 0],\n","       [0, 0, 0, 1]])"]},"metadata":{},"execution_count":28}]},{"cell_type":"markdown","source":["##### **Ex.7: Present the projection operators $n$ and $n'$**"],"metadata":{"id":"MGJ6o50INzVc"}},{"cell_type":"markdown","metadata":{"id":"rNQ4kp5b7EeV"},"source":["O operador $n$ projeta um vetor qualquer $|\\psi\\rangle = a|0\\rangle + b|1\\rangle$ em sua componente $|1\\rangle$. O operador $n'$ é o complemento de $n$ (igual a $I-n$) e projeta o vetor na sua componente $|0\\rangle$. $I$ é a matriz identidade de 2a ordem (2x2)."]},{"cell_type":"code","execution_count":29,"metadata":{"id":"_o4uxHT677GA","executionInfo":{"status":"ok","timestamp":1708350565636,"user_tz":180,"elapsed":12,"user":{"displayName":"Gustavo Mirapalheta","userId":"00687033162702398655"}},"colab":{"base_uri":"https://localhost:8080/"},"outputId":"d898d372-5530-4ff7-e770-8ec1812ceef7"},"outputs":[{"output_type":"execute_result","data":{"text/plain":["array([[1],\n","       [1]])"]},"metadata":{},"execution_count":29}],"source":["v = v0 + v1; v"]},{"cell_type":"code","execution_count":30,"metadata":{"id":"lwd-Z2Hx7tUo","executionInfo":{"status":"ok","timestamp":1708350565636,"user_tz":180,"elapsed":11,"user":{"displayName":"Gustavo Mirapalheta","userId":"00687033162702398655"}},"colab":{"base_uri":"https://localhost:8080/"},"outputId":"611f8044-e27e-4a26-a41b-e8b1165d9560"},"outputs":[{"output_type":"execute_result","data":{"text/plain":["array([[0, 0],\n","       [0, 1]])"]},"metadata":{},"execution_count":30}],"source":["n = np.array([[0,0],[0,1]]); n"]},{"cell_type":"code","execution_count":31,"metadata":{"id":"3sgcaSsb8C7A","executionInfo":{"status":"ok","timestamp":1708350565636,"user_tz":180,"elapsed":10,"user":{"displayName":"Gustavo Mirapalheta","userId":"00687033162702398655"}},"colab":{"base_uri":"https://localhost:8080/"},"outputId":"6b91f3fd-21a5-4b68-d687-dbbbdf20d33c"},"outputs":[{"output_type":"execute_result","data":{"text/plain":["array([[0],\n","       [1]])"]},"metadata":{},"execution_count":31}],"source":["n.dot(v)"]},{"cell_type":"code","execution_count":32,"metadata":{"id":"vJfqRfVT7zvo","executionInfo":{"status":"ok","timestamp":1708350565636,"user_tz":180,"elapsed":9,"user":{"displayName":"Gustavo Mirapalheta","userId":"00687033162702398655"}},"colab":{"base_uri":"https://localhost:8080/"},"outputId":"531825fd-efaf-4150-f775-5b8e7cc3c054"},"outputs":[{"output_type":"execute_result","data":{"text/plain":["array([[1, 0],\n","       [0, 0]])"]},"metadata":{},"execution_count":32}],"source":["n_ = np.array([[1,0],[0,0]]);n_"]},{"cell_type":"code","execution_count":33,"metadata":{"id":"2HRJt0b38Mby","executionInfo":{"status":"ok","timestamp":1708350565636,"user_tz":180,"elapsed":8,"user":{"displayName":"Gustavo Mirapalheta","userId":"00687033162702398655"}},"colab":{"base_uri":"https://localhost:8080/"},"outputId":"7751d8cc-c817-464a-e061-e2f9d4076221"},"outputs":[{"output_type":"execute_result","data":{"text/plain":["array([[1., 0.],\n","       [0., 1.]])"]},"metadata":{},"execution_count":33}],"source":["I2 = np.identity(2); I2"]},{"cell_type":"code","execution_count":34,"metadata":{"id":"WGvOhW-D8VEb","executionInfo":{"status":"ok","timestamp":1708350565636,"user_tz":180,"elapsed":7,"user":{"displayName":"Gustavo Mirapalheta","userId":"00687033162702398655"}},"colab":{"base_uri":"https://localhost:8080/"},"outputId":"c2b30417-85f1-4469-d956-779127ea50f0"},"outputs":[{"output_type":"execute_result","data":{"text/plain":["array([[1., 0.],\n","       [0., 0.]])"]},"metadata":{},"execution_count":34}],"source":["I2 - n"]},{"cell_type":"code","execution_count":35,"metadata":{"id":"CgJT6_ek8FcP","executionInfo":{"status":"ok","timestamp":1708350565636,"user_tz":180,"elapsed":6,"user":{"displayName":"Gustavo Mirapalheta","userId":"00687033162702398655"}},"colab":{"base_uri":"https://localhost:8080/"},"outputId":"d850268c-53a9-4b35-f0e6-e90834c0426f"},"outputs":[{"output_type":"execute_result","data":{"text/plain":["array([[1],\n","       [0]])"]},"metadata":{},"execution_count":35}],"source":["n_.dot(v)"]},{"cell_type":"markdown","metadata":{"id":"ZTbai-XW8rJv"},"source":["##### **Ex.7b: Prove that: $n^2 = n$ and $n'^2 = n'$**"]},{"cell_type":"code","execution_count":36,"metadata":{"id":"80BdWi-69ucB","executionInfo":{"status":"ok","timestamp":1708350565636,"user_tz":180,"elapsed":5,"user":{"displayName":"Gustavo Mirapalheta","userId":"00687033162702398655"}},"colab":{"base_uri":"https://localhost:8080/"},"outputId":"48d08259-1fdb-4581-816e-c4ffea61a68b"},"outputs":[{"output_type":"execute_result","data":{"text/plain":["array([[0, 0],\n","       [0, 1]])"]},"metadata":{},"execution_count":36}],"source":["n.dot(n)"]},{"cell_type":"code","execution_count":37,"metadata":{"id":"NTXcfS_P9xIo","executionInfo":{"status":"ok","timestamp":1708350566090,"user_tz":180,"elapsed":458,"user":{"displayName":"Gustavo Mirapalheta","userId":"00687033162702398655"}},"colab":{"base_uri":"https://localhost:8080/"},"outputId":"93dcfc09-c31a-488d-d87c-e5d189280f5b"},"outputs":[{"output_type":"execute_result","data":{"text/plain":["array([[1, 0],\n","       [0, 0]])"]},"metadata":{},"execution_count":37}],"source":["n_.dot(n_)"]},{"cell_type":"markdown","metadata":{"id":"-NHN7Bhe-5Vm"},"source":["##### **Ex.7c: Prove that $nn'=0$ and $n'n=0$**"]},{"cell_type":"code","execution_count":38,"metadata":{"id":"6jIHz2Ry9zjc","executionInfo":{"status":"ok","timestamp":1708350566090,"user_tz":180,"elapsed":33,"user":{"displayName":"Gustavo Mirapalheta","userId":"00687033162702398655"}},"colab":{"base_uri":"https://localhost:8080/"},"outputId":"26899d23-e157-4c93-8b76-44b0c4b101bf"},"outputs":[{"output_type":"execute_result","data":{"text/plain":["array([[0, 0],\n","       [0, 0]])"]},"metadata":{},"execution_count":38}],"source":["n.dot(n_)"]},{"cell_type":"code","execution_count":39,"metadata":{"id":"tPpUw9uA92FA","executionInfo":{"status":"ok","timestamp":1708350566091,"user_tz":180,"elapsed":33,"user":{"displayName":"Gustavo Mirapalheta","userId":"00687033162702398655"}},"colab":{"base_uri":"https://localhost:8080/"},"outputId":"5d7ca1b8-77f5-4576-c52e-25653d11b46b"},"outputs":[{"output_type":"execute_result","data":{"text/plain":["array([[0, 0],\n","       [0, 0]])"]},"metadata":{},"execution_count":39}],"source":["n_.dot(n)"]},{"cell_type":"markdown","metadata":{"id":"74orZ86E-_Dd"},"source":["##### **Ex.7d: Prove that $n+n'=I$**"]},{"cell_type":"code","execution_count":40,"metadata":{"id":"Mk94IQc-933H","executionInfo":{"status":"ok","timestamp":1708350566091,"user_tz":180,"elapsed":32,"user":{"displayName":"Gustavo Mirapalheta","userId":"00687033162702398655"}},"colab":{"base_uri":"https://localhost:8080/"},"outputId":"9f523a41-3d44-4e56-af4e-feab25a46fd7"},"outputs":[{"output_type":"execute_result","data":{"text/plain":["array([[1, 0],\n","       [0, 1]])"]},"metadata":{},"execution_count":40}],"source":["n+n_"]},{"cell_type":"markdown","metadata":{"id":"dZHO-jrv_DkH"},"source":["##### **Ex.7e: Prove that $nX=Xn'$ and $Xn = n'X$**"]},{"cell_type":"code","execution_count":41,"metadata":{"id":"SktU-hYe96IB","executionInfo":{"status":"ok","timestamp":1708350566091,"user_tz":180,"elapsed":31,"user":{"displayName":"Gustavo Mirapalheta","userId":"00687033162702398655"}},"colab":{"base_uri":"https://localhost:8080/"},"outputId":"38f1fd5d-758c-4dd5-d7e9-8876869d66ca"},"outputs":[{"output_type":"execute_result","data":{"text/plain":["array([[0, 0],\n","       [1, 0]])"]},"metadata":{},"execution_count":41}],"source":["n.dot(X)"]},{"cell_type":"code","execution_count":42,"metadata":{"id":"iKFn16u198OQ","executionInfo":{"status":"ok","timestamp":1708350566091,"user_tz":180,"elapsed":30,"user":{"displayName":"Gustavo Mirapalheta","userId":"00687033162702398655"}},"colab":{"base_uri":"https://localhost:8080/"},"outputId":"4a0e1452-c1e2-492e-aa38-bf0e8799365e"},"outputs":[{"output_type":"execute_result","data":{"text/plain":["array([[0, 0],\n","       [1, 0]])"]},"metadata":{},"execution_count":42}],"source":["X.dot(n_)"]},{"cell_type":"markdown","metadata":{"id":"6un9f7n7_m1h"},"source":["##### **Ex.8: Analize the expression: $S_{ij} = n_in_j + n'_in'_j$ $+ X_iX_j(n_in'_j + n'_in_j)$**"]},{"cell_type":"code","source":["import numpy as np\n","n = np.array([[0,0],[0,1]]); n"],"metadata":{"id":"_udPDZWFQnQB","executionInfo":{"status":"ok","timestamp":1708350566091,"user_tz":180,"elapsed":29,"user":{"displayName":"Gustavo Mirapalheta","userId":"00687033162702398655"}},"colab":{"base_uri":"https://localhost:8080/"},"outputId":"fafc16af-5093-45f6-8a1a-f8f28164fd83"},"execution_count":43,"outputs":[{"output_type":"execute_result","data":{"text/plain":["array([[0, 0],\n","       [0, 1]])"]},"metadata":{},"execution_count":43}]},{"cell_type":"code","source":["n_ = np.array([[1,0],[0,0]]);n_"],"metadata":{"id":"Y3QaDBb0QseC","executionInfo":{"status":"ok","timestamp":1708350566091,"user_tz":180,"elapsed":28,"user":{"displayName":"Gustavo Mirapalheta","userId":"00687033162702398655"}},"colab":{"base_uri":"https://localhost:8080/"},"outputId":"121963c8-9bac-4198-aaca-1c4ed146eb44"},"execution_count":44,"outputs":[{"output_type":"execute_result","data":{"text/plain":["array([[1, 0],\n","       [0, 0]])"]},"metadata":{},"execution_count":44}]},{"cell_type":"code","execution_count":45,"metadata":{"id":"1EgGCP3aAAmX","executionInfo":{"status":"ok","timestamp":1708350566091,"user_tz":180,"elapsed":27,"user":{"displayName":"Gustavo Mirapalheta","userId":"00687033162702398655"}},"colab":{"base_uri":"https://localhost:8080/"},"outputId":"3d9c7b04-8d63-4ca4-cdc2-5195b4ef3743"},"outputs":[{"output_type":"execute_result","data":{"text/plain":["array([[0, 0, 0, 0],\n","       [0, 0, 0, 0],\n","       [0, 0, 0, 0],\n","       [0, 0, 0, 1]])"]},"metadata":{},"execution_count":45}],"source":["# ni nj operam em bits distintos. Como cada operador n atua em um vetor de\n","#  2 dimensões, ni nj é um operador de 4 dimensões. Sendo assim, fica implícito\n","#  pelos subscritos que ni nj é um produto tensorial.\n","np.kron(n,n)"]},{"cell_type":"code","execution_count":46,"metadata":{"id":"Q7v4CcwzAd4o","executionInfo":{"status":"ok","timestamp":1708350566091,"user_tz":180,"elapsed":26,"user":{"displayName":"Gustavo Mirapalheta","userId":"00687033162702398655"}},"colab":{"base_uri":"https://localhost:8080/"},"outputId":"f4133c2e-51ef-4ca4-8ef1-e85597e5c7f4"},"outputs":[{"output_type":"execute_result","data":{"text/plain":["array([[1, 0, 0, 0],\n","       [0, 0, 0, 0],\n","       [0, 0, 0, 0],\n","       [0, 0, 0, 0]])"]},"metadata":{},"execution_count":46}],"source":["# O mesmo vale para n'i n'j\n","np.kron(n_,n_)"]},{"cell_type":"code","execution_count":47,"metadata":{"id":"lx-h8vqZAl-F","executionInfo":{"status":"ok","timestamp":1708350566091,"user_tz":180,"elapsed":25,"user":{"displayName":"Gustavo Mirapalheta","userId":"00687033162702398655"}},"colab":{"base_uri":"https://localhost:8080/"},"outputId":"7716cdce-ca58-4552-e41d-dbac303d2f0f"},"outputs":[{"output_type":"execute_result","data":{"text/plain":["array([[0, 0, 0, 0],\n","       [0, 0, 0, 0],\n","       [0, 0, 1, 0],\n","       [0, 0, 0, 0]])"]},"metadata":{},"execution_count":47}],"source":["# Para ni n'j\n","np.kron(n,n_)"]},{"cell_type":"code","execution_count":48,"metadata":{"id":"qINDYhQdArB5","executionInfo":{"status":"ok","timestamp":1708350566091,"user_tz":180,"elapsed":24,"user":{"displayName":"Gustavo Mirapalheta","userId":"00687033162702398655"}},"colab":{"base_uri":"https://localhost:8080/"},"outputId":"7e940edd-a7de-4c18-ff53-6935f1076e56"},"outputs":[{"output_type":"execute_result","data":{"text/plain":["array([[0, 0, 0, 0],\n","       [0, 1, 0, 0],\n","       [0, 0, 0, 0],\n","       [0, 0, 0, 0]])"]},"metadata":{},"execution_count":48}],"source":["# E para n'i nj\n","np.kron(n_,n)"]},{"cell_type":"code","execution_count":49,"metadata":{"id":"68l0O0YvAvp1","executionInfo":{"status":"ok","timestamp":1708350566091,"user_tz":180,"elapsed":23,"user":{"displayName":"Gustavo Mirapalheta","userId":"00687033162702398655"}},"colab":{"base_uri":"https://localhost:8080/"},"outputId":"c602effa-1298-444d-9297-d3a093b34be8"},"outputs":[{"output_type":"execute_result","data":{"text/plain":["array([[0, 0, 0, 1],\n","       [0, 0, 1, 0],\n","       [0, 1, 0, 0],\n","       [1, 0, 0, 0]])"]},"metadata":{},"execution_count":49}],"source":["# Xi Xj são dois operadores 2x2 de inversão, cada um atuando em um bit\n","#  distinto. Sendo assim, fazemos o produto tensorial entre ambos\n","#  para obter XiXj\n","np.kron(X,X)"]},{"cell_type":"code","execution_count":50,"metadata":{"id":"BgfrO-PmBEVQ","executionInfo":{"status":"ok","timestamp":1708350566091,"user_tz":180,"elapsed":22,"user":{"displayName":"Gustavo Mirapalheta","userId":"00687033162702398655"}},"colab":{"base_uri":"https://localhost:8080/"},"outputId":"0572cd4a-9613-49a1-c4c1-2e550f340f98"},"outputs":[{"output_type":"execute_result","data":{"text/plain":["array([[0, 0, 0, 0],\n","       [0, 0, 1, 0],\n","       [0, 1, 0, 0],\n","       [0, 0, 0, 0]])"]},"metadata":{},"execution_count":50}],"source":["# No caso da expressão XiXj(ninj' + ni'nj) temos entre parênteses duas\n","#  matrizes 4x4 (ninj'+ni'nj) e fora dos parênteses uma matriz 4x4 (XiXj)\n","#  Sendo assim fazemos neste caso o produto matricial normal para\n","#  calcular esta expressão.\n","np.kron(X,X).dot(np.kron(n,n_)+np.kron(n_,n))"]},{"cell_type":"code","execution_count":51,"metadata":{"id":"OirlR5ZABn2I","executionInfo":{"status":"ok","timestamp":1708350566092,"user_tz":180,"elapsed":22,"user":{"displayName":"Gustavo Mirapalheta","userId":"00687033162702398655"}},"colab":{"base_uri":"https://localhost:8080/"},"outputId":"842fd127-0c31-460d-afab-b991919d2788"},"outputs":[{"output_type":"execute_result","data":{"text/plain":["array([[1, 0, 0, 0],\n","       [0, 0, 1, 0],\n","       [0, 1, 0, 0],\n","       [0, 0, 0, 1]])"]},"metadata":{},"execution_count":51}],"source":["# E por último somamos com o resultado inicial de ninj + ni'nj'\n","#  Como pode-se ver, esta expressão gera o operador de SWAP\n","np.kron(n,n) + np.kron(n_,n_) + np.kron(X,X).dot(np.kron(n,n_)+np.kron(n_,n))"]},{"cell_type":"code","execution_count":52,"metadata":{"id":"iIaZi5jtCKuu","executionInfo":{"status":"ok","timestamp":1708350566092,"user_tz":180,"elapsed":21,"user":{"displayName":"Gustavo Mirapalheta","userId":"00687033162702398655"}},"colab":{"base_uri":"https://localhost:8080/"},"outputId":"7215672c-038d-4ce9-d51c-daef8ae63039"},"outputs":[{"output_type":"execute_result","data":{"text/plain":["array([[0, 0, 0, 0],\n","       [0, 0, 0, 0],\n","       [0, 0, 0, 0],\n","       [0, 0, 0, 1]])"]},"metadata":{},"execution_count":52}],"source":["# É importante observar que ninj forma um operador que projeta um vetor de\n","#  4 dimensões (00, 01, 10, 11) em sua componente 11\n","np.kron(n,n)"]},{"cell_type":"code","execution_count":53,"metadata":{"id":"FdSW5E92CV0K","executionInfo":{"status":"ok","timestamp":1708350566092,"user_tz":180,"elapsed":20,"user":{"displayName":"Gustavo Mirapalheta","userId":"00687033162702398655"}},"colab":{"base_uri":"https://localhost:8080/"},"outputId":"873fa9c0-52f2-4441-e0ea-cef8671656ec"},"outputs":[{"output_type":"execute_result","data":{"text/plain":["array([[1, 0, 0, 0],\n","       [0, 0, 0, 0],\n","       [0, 0, 0, 0],\n","       [0, 0, 0, 0]])"]},"metadata":{},"execution_count":53}],"source":["# De maneira similar ni'nj' projeta um vetor de 4 dimensões em sua componente 00\n","np.kron(n_,n_)"]},{"cell_type":"code","execution_count":54,"metadata":{"id":"1KJx0JBLCtXD","executionInfo":{"status":"ok","timestamp":1708350566092,"user_tz":180,"elapsed":19,"user":{"displayName":"Gustavo Mirapalheta","userId":"00687033162702398655"}},"colab":{"base_uri":"https://localhost:8080/"},"outputId":"05f027d3-68a4-4937-a5cf-13a92894bc6f"},"outputs":[{"output_type":"execute_result","data":{"text/plain":["array([[0, 0, 0, 0],\n","       [0, 1, 0, 0],\n","       [0, 0, 0, 0],\n","       [0, 0, 0, 0]])"]},"metadata":{},"execution_count":54}],"source":["# ni'nj projeta em 01\n","np.kron(n_,n)"]},{"cell_type":"code","execution_count":55,"metadata":{"id":"LqReWdQoCf7r","executionInfo":{"status":"ok","timestamp":1708350566092,"user_tz":180,"elapsed":18,"user":{"displayName":"Gustavo Mirapalheta","userId":"00687033162702398655"}},"colab":{"base_uri":"https://localhost:8080/"},"outputId":"68cc139f-a641-478a-d5db-56720201e1fe"},"outputs":[{"output_type":"execute_result","data":{"text/plain":["array([[0, 0, 0, 0],\n","       [0, 0, 0, 0],\n","       [0, 0, 1, 0],\n","       [0, 0, 0, 0]])"]},"metadata":{},"execution_count":55}],"source":["# Assim como ninj' projeta em 10\n","np.kron(n,n_)"]},{"cell_type":"code","execution_count":56,"metadata":{"id":"1_XPnI0nC4X7","executionInfo":{"status":"ok","timestamp":1708350566092,"user_tz":180,"elapsed":18,"user":{"displayName":"Gustavo Mirapalheta","userId":"00687033162702398655"}},"colab":{"base_uri":"https://localhost:8080/"},"outputId":"a1cd443c-e8f2-41fa-9b98-9e634763df69"},"outputs":[{"output_type":"execute_result","data":{"text/plain":["array([[1, 0, 0, 0],\n","       [0, 1, 0, 0],\n","       [0, 0, 1, 0],\n","       [0, 0, 0, 1]])"]},"metadata":{},"execution_count":56}],"source":["# E por último vemos que ni'nj' + ni'nj + ninj' + ninj = I\n","nn = np.kron(n,n)\n","nn_ = np.kron(n,n_)\n","n_n = np.kron(n_,n)\n","n_n_ = np.kron(n_,n_)\n","nn + nn_ + n_n + n_n_"]},{"cell_type":"markdown","source":["##### **Ex.9: Prove that $(a \\otimes b)$ $.(c \\otimes d)$ $= (a.c)$ $\\otimes(b.d)$**"],"metadata":{"id":"-goLmNeMQS5j"}},{"cell_type":"code","source":[],"metadata":{"id":"JPSJbfHrqcLP","executionInfo":{"status":"ok","timestamp":1708350566092,"user_tz":180,"elapsed":17,"user":{"displayName":"Gustavo Mirapalheta","userId":"00687033162702398655"}}},"execution_count":56,"outputs":[]},{"cell_type":"markdown","source":["##### **Ex.10: Prove that $(n \\otimes n)$ $.(n' \\otimes n') = 0$**"],"metadata":{"id":"rgApWdmDR0_6"}},{"cell_type":"code","source":["import numpy as np\n","n = np.array([[0,0],[0,1]]); n"],"metadata":{"id":"As8qjjIbRGoy","executionInfo":{"status":"ok","timestamp":1708350566092,"user_tz":180,"elapsed":16,"user":{"displayName":"Gustavo Mirapalheta","userId":"00687033162702398655"}},"colab":{"base_uri":"https://localhost:8080/"},"outputId":"2d907783-c3c4-409b-d2a6-375fa5ca2729"},"execution_count":57,"outputs":[{"output_type":"execute_result","data":{"text/plain":["array([[0, 0],\n","       [0, 1]])"]},"metadata":{},"execution_count":57}]},{"cell_type":"code","source":["n_ = np.array([[1,0],[0,0]]);n_"],"metadata":{"id":"hHr7hbezRLEp","executionInfo":{"status":"ok","timestamp":1708350566092,"user_tz":180,"elapsed":16,"user":{"displayName":"Gustavo Mirapalheta","userId":"00687033162702398655"}},"colab":{"base_uri":"https://localhost:8080/"},"outputId":"b5d386b1-be59-4e09-e219-52dd2e321b52"},"execution_count":58,"outputs":[{"output_type":"execute_result","data":{"text/plain":["array([[1, 0],\n","       [0, 0]])"]},"metadata":{},"execution_count":58}]},{"cell_type":"code","execution_count":59,"metadata":{"id":"f0Jcih3LEPyD","executionInfo":{"status":"ok","timestamp":1708350566092,"user_tz":180,"elapsed":15,"user":{"displayName":"Gustavo Mirapalheta","userId":"00687033162702398655"}},"colab":{"base_uri":"https://localhost:8080/"},"outputId":"4bde57b3-5b85-400d-b0bb-d082f6d5f33e"},"outputs":[{"output_type":"execute_result","data":{"text/plain":["array([[0, 0, 0, 0],\n","       [0, 0, 0, 0],\n","       [0, 0, 0, 0],\n","       [0, 0, 0, 0]])"]},"metadata":{},"execution_count":59}],"source":["# (n x n).(n_ x n_)\n","np.kron(n,n).dot(np.kron(n_,n_))"]},{"cell_type":"code","execution_count":60,"metadata":{"id":"F1dixl-HMMAt","executionInfo":{"status":"ok","timestamp":1708350566092,"user_tz":180,"elapsed":14,"user":{"displayName":"Gustavo Mirapalheta","userId":"00687033162702398655"}}},"outputs":[],"source":["# Faz sentido que (ni x nj).(ni_ x nj_) seja o vetor nulo pois:\n","#   ni_ x nj_ é um operador que recebe um vetor de 4 dimensões e\n","#   produz como resultado apenas sua componente |00>. Por sua\n","#   vez ni x nj é um operador que recebe um vetor de 4 dimensões\n","#   e produz como resultado apenas sua componente |11>. Esta\n","#   componente foi zerada pelo primeiro operador, logo o resultado\n","#   será nulo. Isto vai acontecer sempre que a componente não\n","#   zerada do primeiro operador for diferente da do segundo em\n","#   outras palavras sempre que ij do primeiro for diferente de ij\n","#   do segundo."]},{"cell_type":"code","execution_count":61,"metadata":{"id":"xO2ylsb-NhsQ","executionInfo":{"status":"ok","timestamp":1708350566093,"user_tz":180,"elapsed":15,"user":{"displayName":"Gustavo Mirapalheta","userId":"00687033162702398655"}},"colab":{"base_uri":"https://localhost:8080/"},"outputId":"15e70009-2366-4685-a6e4-d850f245040e"},"outputs":[{"output_type":"execute_result","data":{"text/plain":["array([[0, 0, 0, 0],\n","       [0, 0, 0, 0],\n","       [0, 0, 0, 0],\n","       [0, 0, 0, 0]])"]},"metadata":{},"execution_count":61}],"source":["np.kron(n,n).dot(np.kron(n_,n_))"]},{"cell_type":"code","execution_count":62,"metadata":{"id":"Mahd2WQgNr3X","executionInfo":{"status":"ok","timestamp":1708350566093,"user_tz":180,"elapsed":14,"user":{"displayName":"Gustavo Mirapalheta","userId":"00687033162702398655"}}},"outputs":[],"source":["# Outra forma de entender é transformar (n x n).(n_ x n_) em\n","#  (n.n_) x (n.n_). Como n é ortogonal a n_, a projeção de n\n","#  em n_ dará zero também."]},{"cell_type":"code","execution_count":63,"metadata":{"id":"SA9ey4gdD3uk","executionInfo":{"status":"ok","timestamp":1708350566093,"user_tz":180,"elapsed":14,"user":{"displayName":"Gustavo Mirapalheta","userId":"00687033162702398655"}},"colab":{"base_uri":"https://localhost:8080/"},"outputId":"98715d53-7f53-4ce1-a5f0-99eb80e99661"},"outputs":[{"output_type":"execute_result","data":{"text/plain":["array([[0, 0, 0, 0],\n","       [0, 0, 0, 0],\n","       [0, 0, 0, 0],\n","       [0, 0, 0, 0]])"]},"metadata":{},"execution_count":63}],"source":["np.kron(n.dot(n_),n.dot(n_))"]},{"cell_type":"code","execution_count":64,"metadata":{"id":"BNOUks1BJsm3","executionInfo":{"status":"ok","timestamp":1708350566093,"user_tz":180,"elapsed":13,"user":{"displayName":"Gustavo Mirapalheta","userId":"00687033162702398655"}},"colab":{"base_uri":"https://localhost:8080/"},"outputId":"651638dc-47cb-4c20-963a-357f7d4079d4"},"outputs":[{"output_type":"execute_result","data":{"text/plain":["array([[0, 0, 0, 0],\n","       [0, 0, 0, 0],\n","       [0, 0, 0, 0],\n","       [0, 0, 0, 0]])"]},"metadata":{},"execution_count":64}],"source":["nn.dot(n_n_)"]},{"cell_type":"code","execution_count":65,"metadata":{"id":"RZ5aTbAzFSzL","executionInfo":{"status":"ok","timestamp":1708350566093,"user_tz":180,"elapsed":12,"user":{"displayName":"Gustavo Mirapalheta","userId":"00687033162702398655"}},"colab":{"base_uri":"https://localhost:8080/"},"outputId":"48ba65e0-4c6b-4405-bf8c-60bc4356b3b5"},"outputs":[{"output_type":"execute_result","data":{"text/plain":["array([[0, 0, 0, 0],\n","       [0, 0, 0, 0],\n","       [0, 0, 0, 0],\n","       [0, 0, 0, 0]])"]},"metadata":{},"execution_count":65}],"source":["# (n_ x n).(n x n_)\n","np.kron(n_,n).dot(np.kron(n,n_))"]},{"cell_type":"code","execution_count":66,"metadata":{"id":"JWcCcbqBJ3MU","executionInfo":{"status":"ok","timestamp":1708350566093,"user_tz":180,"elapsed":11,"user":{"displayName":"Gustavo Mirapalheta","userId":"00687033162702398655"}},"colab":{"base_uri":"https://localhost:8080/"},"outputId":"3e0dae8d-31be-43e7-ee24-f89d0ae47140"},"outputs":[{"output_type":"execute_result","data":{"text/plain":["array([[0, 0, 0, 0],\n","       [0, 0, 0, 0],\n","       [0, 0, 0, 0],\n","       [0, 0, 0, 0]])"]},"metadata":{},"execution_count":66}],"source":["n_n.dot(nn_)"]},{"cell_type":"code","execution_count":67,"metadata":{"id":"Koh_X7ULFc5c","executionInfo":{"status":"ok","timestamp":1708350566093,"user_tz":180,"elapsed":10,"user":{"displayName":"Gustavo Mirapalheta","userId":"00687033162702398655"}},"colab":{"base_uri":"https://localhost:8080/"},"outputId":"1e4f7c25-5b41-4c78-b762-0753a42ad72d"},"outputs":[{"output_type":"execute_result","data":{"text/plain":["array([[0, 0, 0, 0],\n","       [0, 0, 0, 0],\n","       [0, 0, 0, 0],\n","       [0, 0, 0, 0]])"]},"metadata":{},"execution_count":67}],"source":["# (n x n_).(n_ x n)\n","np.kron(n, n_).dot(np.kron(n_,n))"]},{"cell_type":"code","execution_count":68,"metadata":{"id":"xnq3xru4J7SI","executionInfo":{"status":"ok","timestamp":1708350566093,"user_tz":180,"elapsed":9,"user":{"displayName":"Gustavo Mirapalheta","userId":"00687033162702398655"}},"colab":{"base_uri":"https://localhost:8080/"},"outputId":"680334fa-42bc-4b75-c98d-3057408ef146"},"outputs":[{"output_type":"execute_result","data":{"text/plain":["array([[0, 0, 0, 0],\n","       [0, 0, 0, 0],\n","       [0, 0, 0, 0],\n","       [0, 0, 0, 0]])"]},"metadata":{},"execution_count":68}],"source":["nn_.dot(n_n)"]},{"cell_type":"code","execution_count":69,"metadata":{"id":"B76EhqUZFpLK","executionInfo":{"status":"ok","timestamp":1708350566093,"user_tz":180,"elapsed":8,"user":{"displayName":"Gustavo Mirapalheta","userId":"00687033162702398655"}},"colab":{"base_uri":"https://localhost:8080/"},"outputId":"635f41f3-b17a-432b-ddf0-f87a8f8a3f72"},"outputs":[{"output_type":"execute_result","data":{"text/plain":["array([[0, 0, 0, 0],\n","       [0, 0, 0, 0],\n","       [0, 0, 0, 0],\n","       [0, 0, 0, 0]])"]},"metadata":{},"execution_count":69}],"source":["# (n_ x n_).(n x n)\n","np.kron(n_, n_).dot(np.kron(n, n))"]},{"cell_type":"code","execution_count":70,"metadata":{"id":"guRcvrXcKBKa","executionInfo":{"status":"ok","timestamp":1708350566093,"user_tz":180,"elapsed":7,"user":{"displayName":"Gustavo Mirapalheta","userId":"00687033162702398655"}},"colab":{"base_uri":"https://localhost:8080/"},"outputId":"2b13fd36-6e78-4560-e4ea-45624ca33444"},"outputs":[{"output_type":"execute_result","data":{"text/plain":["array([[0, 0, 0, 0],\n","       [0, 0, 0, 0],\n","       [0, 0, 0, 0],\n","       [0, 0, 0, 0]])"]},"metadata":{},"execution_count":70}],"source":["n_n_.dot(nn)"]}]}