{ "cells": [ { "cell_type": "markdown", "id": "9a2bf45c", "metadata": {}, "source": [ "# Tensor product between QOperations\n", "\n", "This note describes tensor products between Qoperations and their samples." ] }, { "cell_type": "markdown", "id": "3ad23cfb", "metadata": {}, "source": [ "## Tensor Product\n", "\n", "Tensor product is an operation in the **spatial direction**. To compute the tensor product, use `tensor_product()` in the `quara.objects.operators` module.\n", "\n", "As a simple example, let's look at how to compute the tensor product between the following States.\n", "\n", "**formula:**\n", "\n", "$\\rho_{z0} {\\otimes } \\rho_{z1}$\n", "\n", "**quantum circuit diagram:**\n", "\n", "── $\\rho_{z0}$ ── \n", "── $\\rho_{z1}$ ──" ] }, { "cell_type": "markdown", "id": "86f0a40d", "metadata": {}, "source": [ "First, prepare the State to be used for the operation. Note that the IDs of the ElementalSystems between QOperations to be calculated with tensor products must be different." ] }, { "cell_type": "code", "execution_count": 1, "id": "2c29e024", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "vec of state_z0: [0.70710678 0. 0. 0.70710678]\n", "vec of state_z1: [ 0.70710678 0. 0. -0.70710678]\n" ] } ], "source": [ "from quara.objects.composite_system_typical import generate_composite_system\n", "from quara.objects.qoperation_typical import generate_qoperation\n", "\n", "# Prepare State\n", "c_sys_0 = generate_composite_system(\"qubit\", 1, ids_esys=[0])\n", "c_sys_1 = generate_composite_system(\"qubit\", 1, ids_esys=[1])\n", "state_z0 = generate_qoperation(mode=\"state\", name=\"z0\", c_sys=c_sys_0)\n", "state_z1 = generate_qoperation(mode=\"state\", name=\"z1\", c_sys=c_sys_1)\n", "\n", "print(f\"vec of state_z0: {state_z0.vec}\")\n", "print(f\"vec of state_z1: {state_z1.vec}\")" ] }, { "cell_type": "markdown", "id": "42e79d53", "metadata": {}, "source": [ "$\\rho_{z0} {\\otimes } \\rho_{z1}$ can be written as follows." ] }, { "cell_type": "code", "execution_count": 2, "id": "53c5e050", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Type:\n", "State\n", "\n", "Dim:\n", "4\n", "\n", "Vec:\n", "[ 0.5 0. 0. -0.5 0. 0. 0. 0. 0. 0. 0. 0. 0.5 0.\n", " 0. -0.5]\n" ] } ], "source": [ "from quara.objects.operators import tensor_product\n", "\n", "# Tensor product\n", "result = tensor_product(state_z0, state_z1)\n", "\n", "# Result\n", "print(result)" ] }, { "cell_type": "markdown", "id": "fd5e018c", "metadata": {}, "source": [ "The result of the tensor product between 1-qubit States is a 2-qubit State." ] }, { "cell_type": "markdown", "id": "09687445", "metadata": {}, "source": [ "### Order of operations" ] }, { "cell_type": "markdown", "id": "ab5672de", "metadata": {}, "source": [ "The order of operations within `tensor_product()` follows the **order of the ElementalSystem IDs of each QOperation** in ascending order.\n", "\n", "In the following cases, the tensor product of $\\rho_{z0}$ and $\\rho_{z1}$ is computed. Since the ElementalSystem ID of $\\rho_{z0}$ is `0` and the ElementalSystem ID of $\\rho_{z0}$ is `1`, the tensor product is calculated in the order $\\rho_{z0} {\\otimes } {\\rho_{z1}}$. Because the calculation is based on the order of the IDs in the ElementalSystem, the result will be the same even if the order passed to the `tensor_product()` argument is different." ] }, { "cell_type": "code", "execution_count": 3, "id": "a6abdf4e", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Case 1:\n", "[ 0.5 0. 0. -0.5 0. 0. 0. 0. 0. 0. 0. 0. 0.5 0.\n", " 0. -0.5]\n", "Case 2:\n", "[ 0.5 0. 0. -0.5 0. 0. 0. 0. 0. 0. 0. 0. 0.5 0.\n", " 0. -0.5]\n" ] } ], "source": [ "# Prepare State\n", "c_sys_0 = generate_composite_system(\"qubit\", 1, ids_esys=[0])\n", "c_sys_1 = generate_composite_system(\"qubit\", 1, ids_esys=[1])\n", "state_0 = generate_qoperation(mode=\"state\", name=\"z0\", c_sys=c_sys_0)\n", "state_1 = generate_qoperation(mode=\"state\", name=\"z1\", c_sys=c_sys_1)\n", "\n", "# Case 1:\n", "result = tensor_product(state_z0, state_z1)\n", "print(\"Case 1:\")\n", "print(result.vec)\n", "\n", "# Case 2: Same result as Case 1\n", "result = tensor_product(state_z1, state_z0)\n", "print(\"Case 2:\")\n", "print(result.vec)" ] }, { "cell_type": "markdown", "id": "a2c6803f", "metadata": {}, "source": [ "Different ElementalSystem IDs will change the results as follows. Since the ElementalSystem ID of $\\rho_{z0}$ is `1` and the ElementalSystem ID of $\\rho_{z0}$ is `0`, the tensor product is calculated in the order $\\rho_{z1} {\\otimes } {\\rho_{z0}}$, not $\\rho_{z0} {\\otimes } {\\rho_{z1}}$." ] }, { "cell_type": "code", "execution_count": 4, "id": "ef8fef8e", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Case 3:\n", "[ 0.5 0. 0. 0.5 0. 0. 0. 0. 0. 0. 0. 0. -0.5 0.\n", " 0. -0.5]\n" ] } ], "source": [ "# Prepare State\n", "c_sys_0 = generate_composite_system(\"qubit\", 1, ids_esys=[1]) # change id_esys\n", "c_sys_1 = generate_composite_system(\"qubit\", 1, ids_esys=[0]) # change id_esys\n", "state_z0 = generate_qoperation(mode=\"state\", name=\"z0\", c_sys=c_sys_0)\n", "state_z1 = generate_qoperation(mode=\"state\", name=\"z1\", c_sys=c_sys_1)\n", "\n", "# Case 3: Different results from Case 1 and Case 2\n", "result = tensor_product(state_z0, state_z1)\n", "print(\"Case 3:\")\n", "print(result.vec)" ] }, { "cell_type": "markdown", "id": "b9592598", "metadata": {}, "source": [ "### Tensor product of three or more QOperations" ] }, { "cell_type": "markdown", "id": "7681b787", "metadata": {}, "source": [ "For three or more QOepration operations, add to the argument. The following code computes the tensor product of three States ( $\\rho_{z0} {\\otimes } \\rho_{z1} {\\otimes } \\rho_{x0}$ ) " ] }, { "cell_type": "code", "execution_count": 5, "id": "18169f14", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Type:\n", "State\n", "\n", "Dim:\n", "8\n", "\n", "Vec:\n", "[ 0.35355339 0.35355339 0. 0. 0. 0.\n", " 0. 0. 0. 0. 0. 0.\n", " -0.35355339 -0.35355339 0. 0. 0. 0.\n", " 0. 0. 0. 0. 0. 0.\n", " 0. 0. 0. 0. 0. 0.\n", " 0. 0. 0. 0. 0. 0.\n", " 0. 0. 0. 0. 0. 0.\n", " 0. 0. 0. 0. 0. 0.\n", " 0.35355339 0.35355339 0. 0. 0. 0.\n", " 0. 0. 0. 0. 0. 0.\n", " -0.35355339 -0.35355339 0. 0. ]\n" ] } ], "source": [ "# Prepare States\n", "c_sys_0 = generate_composite_system(\"qubit\", 1, ids_esys=[0])\n", "c_sys_1 = generate_composite_system(\"qubit\", 1, ids_esys=[1])\n", "c_sys_2 = generate_composite_system(\"qubit\", 1, ids_esys=[2])\n", "\n", "state_z0 = generate_qoperation(mode=\"state\", name=\"z0\", c_sys=c_sys_0)\n", "state_z1 = generate_qoperation(mode=\"state\", name=\"z1\", c_sys=c_sys_1)\n", "state_x0 = generate_qoperation(mode=\"state\", name=\"x0\", c_sys=c_sys_2)\n", "\n", "# Tensor product of three or more QOperations\n", "result = tensor_product(state_z0, state_z1, state_x0)\n", "print(result)" ] }, { "cell_type": "markdown", "id": "aea5914f", "metadata": {}, "source": [ "It is also possible to specify QOperations as a list. For example, the following two expressions have the same meaning." ] }, { "cell_type": "code", "execution_count": 6, "id": "e53cf1ff", "metadata": {}, "outputs": [], "source": [ "# Specify by appending to the argument\n", "result = tensor_product(state_z0, state_z1, state_x0)\n", "\n", "# Specify by list\n", "result = tensor_product([state_z0, state_z1, state_x0])" ] }, { "cell_type": "markdown", "id": "497217e1", "metadata": {}, "source": [ "### Utility for typical tensor product\n", "\n", "Tensor products between typical QOperations can also be computed with `generate_qoperation()`. The tensor product of Qoperation names \"a\" and \"b\" is written \"a_b\".\n", "\n", "For example, the tensor product of two States, z0 and z1, can be written as follows" ] }, { "cell_type": "code", "execution_count": 7, "id": "cc295789", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Type:\n", "State\n", "\n", "Dim:\n", "4\n", "\n", "Vec:\n", "[ 0.5 0. 0. -0.5 0. 0. 0. 0. 0. 0. 0. 0. 0.5 0.\n", " 0. -0.5]\n" ] } ], "source": [ "from quara.objects.qoperation_typical import generate_qoperation\n", "\n", "# Using utility function\n", "# Tensor product of z0 and z1\n", "c_sys_2qubit = generate_composite_system(\"qubit\", 2, ids_esys=[0, 1])\n", "result = generate_qoperation(mode=\"state\", name=\"z0_z1\", c_sys=c_sys_2qubit)\n", "print(result)" ] }, { "cell_type": "markdown", "id": "8d7ded79", "metadata": {}, "source": [ "## Supported operations" ] }, { "cell_type": "markdown", "id": "5ca4739c", "metadata": {}, "source": [ "The `tensor_product()` supports the following combinations of QOperations.\n", "\n", "| Input | Output |\n", "|:-------|:---------|\n", "| State ${\\otimes }$ State | State|\n", "| Povm ${\\otimes }$ Povm | Povm |\n", "|Gate ${\\otimes }$ Gate | Gate |\n", "| State ${\\otimes }$ StateEnsemble | StateEnsemble |\n", "| StateEnsemble ${\\otimes }$ State | StateEnsemble |\n", "| StateEnsemble ${\\otimes }$ StateEnsemble | StateEnsemble |\n", "|Gate ${\\otimes }$ MProcess | MProcess |\n", "| MProcess ${\\otimes }$ Gate | MProcess |\n", "| MProcess ${\\otimes }$ MProcess | MProcess |\n", "| MatrixBasis ${\\otimes }$ MatrixBasis | MatrixBasis |" ] }, { "cell_type": "markdown", "id": "5ff18b8a", "metadata": {}, "source": [ "## Examples" ] }, { "cell_type": "markdown", "id": "0a4b32d3", "metadata": {}, "source": [ "### State ${\\otimes }$ State -> State" ] }, { "cell_type": "code", "execution_count": 8, "id": "cf1ab8d6", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Type:\n", "State\n", "\n", "Dim:\n", "4\n", "\n", "Vec:\n", "[ 0.5 0. 0. -0.5 0. 0. 0. 0. 0. 0. 0. 0. 0.5 0.\n", " 0. -0.5]\n" ] } ], "source": [ "# Prepare\n", "c_sys_0 = generate_composite_system(\"qubit\", 1, ids_esys=[0])\n", "c_sys_1 = generate_composite_system(\"qubit\", 1, ids_esys=[1])\n", "state_0 = generate_qoperation(mode=\"state\", name=\"z0\", c_sys=c_sys_0)\n", "state_1 = generate_qoperation(mode=\"state\", name=\"z1\", c_sys=c_sys_1)\n", "\n", "# Tensor product\n", "result = tensor_product(state_0, state_1)\n", "print(result) # State" ] }, { "cell_type": "markdown", "id": "0508d8cb", "metadata": {}, "source": [ "### Povm ${\\otimes }$ Povm -> Povm" ] }, { "cell_type": "code", "execution_count": 9, "id": "89c717ef", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Type:\n", "Povm\n", "\n", "Dim:\n", "4\n", "\n", "Number of outcomes:\n", "4\n", "\n", "Vecs:\n", "[[ 0.5 0. 0. 0.5 0.5 0. 0. 0.5 0. 0. 0. 0. 0. 0.\n", " 0. 0. ]\n", " [ 0.5 0. 0. -0.5 0.5 0. 0. -0.5 0. 0. 0. 0. 0. 0.\n", " 0. 0. ]\n", " [ 0.5 0. 0. 0.5 -0.5 0. 0. -0.5 0. 0. 0. 0. 0. 0.\n", " 0. 0. ]\n", " [ 0.5 0. 0. -0.5 -0.5 0. 0. 0.5 0. 0. 0. 0. 0. 0.\n", " 0. 0. ]]\n" ] } ], "source": [ "# Prepare\n", "povm_0 = generate_qoperation(mode=\"povm\", name=\"x\", c_sys=c_sys_0)\n", "povm_1 = generate_qoperation(mode=\"povm\", name=\"z\", c_sys=c_sys_1)\n", "\n", "# Tensor product\n", "result = tensor_product(povm_0, povm_1)\n", "print(result) # Povm" ] }, { "cell_type": "markdown", "id": "d9a0e023", "metadata": {}, "source": [ "### Gate ${\\otimes }$ Gate -> Gate" ] }, { "cell_type": "code", "execution_count": 10, "id": "7a938582", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Type:\n", "Gate\n", "\n", "Dim:\n", "4\n", "\n", "HS:\n", "[[ 1. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.]\n", " [ 0. -1. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.]\n", " [ 0. 0. 1. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.]\n", " [ 0. 0. 0. -1. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.]\n", " [ 0. 0. 0. 0. 1. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.]\n", " [ 0. 0. 0. 0. 0. -1. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.]\n", " [ 0. 0. 0. 0. 0. 0. 1. 0. 0. 0. 0. 0. 0. 0. 0. 0.]\n", " [ 0. 0. 0. 0. 0. 0. 0. -1. 0. 0. 0. 0. 0. 0. 0. 0.]\n", " [ 0. 0. 0. 0. 0. 0. 0. 0. -1. 0. 0. 0. 0. 0. 0. 0.]\n", " [ 0. 0. 0. 0. 0. 0. 0. 0. 0. 1. 0. 0. 0. 0. 0. 0.]\n", " [ 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. -1. 0. 0. 0. 0. 0.]\n", " [ 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 1. 0. 0. 0. 0.]\n", " [ 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. -1. 0. 0. 0.]\n", " [ 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 1. 0. 0.]\n", " [ 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. -1. 0.]\n", " [ 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 1.]]\n" ] } ], "source": [ "# Prepare\n", "gate_0 = generate_qoperation(mode=\"gate\", name=\"x\", c_sys=c_sys_0)\n", "gate_1 = generate_qoperation(mode=\"gate\", name=\"y\", c_sys=c_sys_1)\n", "\n", "# Tensor product\n", "result = tensor_product(gate_0, gate_1)\n", "print(result) # Gate" ] }, { "cell_type": "markdown", "id": "65eb3f5c", "metadata": {}, "source": [ "### State ${\\otimes }$ StateEnsemble -> StateEnsemble" ] }, { "cell_type": "code", "execution_count": 11, "id": "36c2dc50", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Type:\n", "StateEnsemble\n", "\n", "States:\n", "states[0]: [0.5 0. 0. 0.5 0. 0. 0. 0. 0. 0. 0. 0. 0.5 0. 0. 0.5]\n", "states[1]: [ 0.5 0. 0. -0.5 0. 0. 0. 0. 0. 0. 0. 0. 0.5 0.\n", " 0. -0.5]\n", "\n", "MultinomialDistribution:\n", "shape = (2,)\n", "ps = [1. 0.]\n" ] } ], "source": [ "# Prepare\n", "state_0 = generate_qoperation(mode=\"state\", name=\"z0\", c_sys=c_sys_0)\n", "state_ensemble_0 = generate_qoperation(mode=\"state_ensemble\", name=\"z0\", c_sys=c_sys_1)\n", "\n", "# Tensor product\n", "result = tensor_product(state_0, state_ensemble_0)\n", "print(result) # StateEnsemble" ] }, { "cell_type": "markdown", "id": "3306bd02", "metadata": {}, "source": [ "### StateEnsemble ${\\otimes }$ State -> StateEnsemble" ] }, { "cell_type": "code", "execution_count": 12, "id": "15b4df29", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Type:\n", "StateEnsemble\n", "\n", "States:\n", "states[0]: [0.5 0. 0. 0.5 0. 0. 0. 0. 0. 0. 0. 0. 0.5 0. 0. 0.5]\n", "states[1]: [ 0.5 0. 0. -0.5 0. 0. 0. 0. 0. 0. 0. 0. 0.5 0.\n", " 0. -0.5]\n", "\n", "MultinomialDistribution:\n", "shape = (2,)\n", "ps = [1. 0.]\n" ] } ], "source": [ "# Tensor product\n", "result = tensor_product(state_ensemble_0, state_0)\n", "print(result) # StateEnsemble" ] }, { "cell_type": "markdown", "id": "d341850d", "metadata": {}, "source": [ "### StateEnsemble ${\\otimes }$ StateEnsemble -> StateEnsemble" ] }, { "cell_type": "code", "execution_count": 13, "id": "d6ec9787", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Type:\n", "StateEnsemble\n", "\n", "States:\n", "states[0]: [0.5 0. 0. 0.5 0. 0. 0. 0. 0. 0. 0. 0. 0.5 0. 0. 0.5]\n", "states[1]: [ 0.5 0. 0. -0.5 0. 0. 0. 0. 0. 0. 0. 0. 0.5 0.\n", " 0. -0.5]\n", "states[2]: [ 0.5 0. 0. 0.5 0. 0. 0. 0. 0. 0. 0. 0. -0.5 0.\n", " 0. -0.5]\n", "states[3]: [ 0.5 0. 0. -0.5 0. 0. 0. 0. 0. 0. 0. 0. -0.5 0.\n", " 0. 0.5]\n", "\n", "MultinomialDistribution:\n", "shape = (2, 2)\n", "ps = [0. 1. 0. 0.]\n" ] } ], "source": [ "# Prepare\n", "state_ensemble_0 = generate_qoperation(mode=\"state_ensemble\", name=\"z0\", c_sys=c_sys_0)\n", "state_ensemble_1 = generate_qoperation(mode=\"state_ensemble\", name=\"z1\", c_sys=c_sys_1)\n", "\n", "# Tensor product\n", "result = tensor_product(state_ensemble_0, state_ensemble_1)\n", "print(result) # StateEnsemble" ] }, { "cell_type": "markdown", "id": "f3938276", "metadata": {}, "source": [ "### Gate ${\\otimes }$ MProcess -> MProcess" ] }, { "cell_type": "code", "execution_count": 14, "id": "104c34d6", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Type:\n", "MProcess\n", "\n", "Dim:\n", "4\n", "\n", "HSs:\n", "[array([[ 0.5, 0.5, 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. ,\n", " 0. , 0. , 0. , 0. , 0. ],\n", " [ 0.5, 0.5, 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. ,\n", " 0. , 0. , 0. , 0. , 0. ],\n", " [ 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. ,\n", " 0. , 0. , 0. , 0. , 0. ],\n", " [ 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. ,\n", " 0. , 0. , 0. , 0. , 0. ],\n", " [ 0. , 0. , 0. , 0. , 0.5, 0.5, 0. , 0. , 0. , 0. , 0. ,\n", " 0. , 0. , 0. , 0. , 0. ],\n", " [ 0. , 0. , 0. , 0. , 0.5, 0.5, 0. , 0. , 0. , 0. , 0. ,\n", " 0. , 0. , 0. , 0. , 0. ],\n", " [ 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. ,\n", " 0. , 0. , 0. , 0. , 0. ],\n", " [ 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. ,\n", " 0. , 0. , 0. , 0. , 0. ],\n", " [ 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , -0.5, -0.5, 0. ,\n", " 0. , 0. , 0. , 0. , 0. ],\n", " [ 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , -0.5, -0.5, 0. ,\n", " 0. , 0. , 0. , 0. , 0. ],\n", " [ 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. ,\n", " 0. , 0. , 0. , 0. , 0. ],\n", " [ 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. ,\n", " 0. , 0. , 0. , 0. , 0. ],\n", " [ 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. ,\n", " 0. , -0.5, -0.5, 0. , 0. ],\n", " [ 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. ,\n", " 0. , -0.5, -0.5, 0. , 0. ],\n", " [ 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. ,\n", " 0. , 0. , 0. , 0. , 0. ],\n", " [ 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. ,\n", " 0. , 0. , 0. , 0. , 0. ]]), array([[ 0.5, -0.5, 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. ,\n", " 0. , 0. , 0. , 0. , 0. ],\n", " [-0.5, 0.5, 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. ,\n", " 0. , 0. , 0. , 0. , 0. ],\n", " [ 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. ,\n", " 0. , 0. , 0. , 0. , 0. ],\n", " [ 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. ,\n", " 0. , 0. , 0. , 0. , 0. ],\n", " [ 0. , 0. , 0. , 0. , 0.5, -0.5, 0. , 0. , 0. , 0. , 0. ,\n", " 0. , 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0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. ,\n", " 0. , 0. , 0. , 0. , 0. ]])]\n", "\n", "ModeSampling:\n", "False\n" ] } ], "source": [ "# Prepare\n", "gate_0 = generate_qoperation(mode=\"gate\", name=\"x\", c_sys=c_sys_0)\n", "mprocess_0 = generate_qoperation(mode=\"mprocess\", name=\"x-type1\", c_sys=c_sys_1)\n", "\n", "# Tensor product\n", "result = tensor_product(gate_0, mprocess_0)\n", "print(result) # MProcess" ] }, { "cell_type": "markdown", "id": "b59afb1e", "metadata": {}, "source": [ "### MProcess ${\\otimes }$ Gate -> MProcess" ] }, { "cell_type": "code", "execution_count": 15, "id": "b14e3541", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Type:\n", "MProcess\n", "\n", "Dim:\n", "4\n", "\n", "HSs:\n", "[array([[ 0.5, 0.5, 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. ,\n", " 0. , 0. , 0. , 0. , 0. ],\n", " [ 0.5, 0.5, 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. ,\n", " 0. , 0. , 0. , 0. , 0. ],\n", " [ 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. ,\n", " 0. , 0. , 0. , 0. , 0. ],\n", " [ 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. ,\n", " 0. , 0. , 0. , 0. , 0. ],\n", " [ 0. , 0. , 0. , 0. , 0.5, 0.5, 0. , 0. , 0. , 0. , 0. ,\n", " 0. , 0. , 0. , 0. , 0. ],\n", " [ 0. , 0. , 0. , 0. , 0.5, 0.5, 0. , 0. , 0. , 0. , 0. ,\n", " 0. , 0. , 0. , 0. , 0. ],\n", " [ 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. ,\n", " 0. , 0. , 0. , 0. , 0. ],\n", " [ 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. ,\n", " 0. , 0. , 0. , 0. , 0. ],\n", " [ 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , -0.5, -0.5, 0. ,\n", " 0. , 0. , 0. , 0. , 0. ],\n", " [ 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , -0.5, -0.5, 0. ,\n", " 0. , 0. , 0. , 0. , 0. ],\n", " [ 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. ,\n", " 0. , 0. , 0. , 0. , 0. ],\n", " [ 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. ,\n", " 0. , 0. , 0. , 0. , 0. ],\n", " [ 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. ,\n", " 0. , -0.5, -0.5, 0. , 0. 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generate_qoperation(mode=\"gate\", name=\"x\", c_sys=c_sys_0)\n", "\n", "# Tensor product\n", "result = tensor_product(mprocess_0, gate_0)\n", "print(result) # MProcess" ] }, { "cell_type": "markdown", "id": "23edeb2b", "metadata": {}, "source": [ "### MProcess ${\\otimes }$ MProcess -> MProcess" ] }, { "cell_type": "code", "execution_count": 16, "id": "591dc759", "metadata": { "scrolled": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Type:\n", "MProcess\n", "\n", "Dim:\n", "4\n", "\n", "HSs:\n", "[array([[0.25, 0.25, 0. , 0. , 0. , 0. , 0. , 0. , 0.25, 0.25, 0. ,\n", " 0. , 0. , 0. , 0. , 0. ],\n", " [0.25, 0.25, 0. , 0. , 0. , 0. , 0. , 0. , 0.25, 0.25, 0. ,\n", " 0. , 0. , 0. , 0. , 0. ],\n", " [0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. ,\n", " 0. , 0. , 0. , 0. , 0. ],\n", " [0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. ,\n", " 0. , 0. , 0. , 0. , 0. ],\n", " [0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. ,\n", " 0. , 0. , 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0. , 0. , 0. , 0. ],\n", " [ 0.25, -0.25, 0. , 0. , 0. , 0. , 0. , 0. , -0.25,\n", " 0.25, 0. , 0. , 0. , 0. , 0. , 0. ],\n", " [ 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. ,\n", " 0. , 0. , 0. , 0. , 0. , 0. , 0. ],\n", " [ 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. ,\n", " 0. , 0. , 0. , 0. , 0. , 0. , 0. ],\n", " [ 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. ,\n", " 0. , 0. , 0. , 0. , 0. , 0. , 0. ],\n", " [ 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. ,\n", " 0. , 0. , 0. , 0. , 0. , 0. , 0. ],\n", " [ 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. ,\n", " 0. , 0. , 0. , 0. , 0. , 0. , 0. ],\n", " [ 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. ,\n", " 0. , 0. , 0. , 0. , 0. , 0. , 0. ]])]\n", "\n", "ModeSampling:\n", "False\n" ] } ], "source": [ "# Prepare\n", "mprocess_0 = generate_qoperation(mode=\"mprocess\", name=\"x-type1\", c_sys=c_sys_1)\n", "mprocess_1 = generate_qoperation(mode=\"mprocess\", name=\"y-type1\", c_sys=c_sys_0)\n", "\n", "# Tensor product\n", "result = tensor_product(mprocess_0, mprocess_1)\n", "print(result) # MProcess" ] }, { "cell_type": "markdown", "id": "b613d4ce", "metadata": {}, "source": [ "### MatrixBasis ${\\otimes }$ MatrixBasis -> MatrixBasis" ] }, { "cell_type": "code", "execution_count": 17, "id": "5013ec23", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "(array([[1.+0.j, 0.+0.j, 0.+0.j, 0.+0.j],\n", " [0.+0.j, 1.+0.j, 0.+0.j, 0.+0.j],\n", " [0.+0.j, 0.+0.j, 1.+0.j, 0.+0.j],\n", " [0.+0.j, 0.+0.j, 0.+0.j, 1.+0.j]]), array([[0.+0.j, 1.+0.j, 0.+0.j, 0.+0.j],\n", " [1.+0.j, 0.+0.j, 0.+0.j, 0.+0.j],\n", " [0.+0.j, 0.+0.j, 0.+0.j, 1.+0.j],\n", " [0.+0.j, 0.+0.j, 1.+0.j, 0.+0.j]]), array([[0.+0.j, 0.-1.j, 0.+0.j, 0.-0.j],\n", " [0.+1.j, 0.+0.j, 0.+0.j, 0.+0.j],\n", " [0.+0.j, 0.-0.j, 0.+0.j, 0.-1.j],\n", " [0.+0.j, 0.+0.j, 0.+1.j, 0.+0.j]]), array([[ 1.+0.j, 0.+0.j, 0.+0.j, 0.+0.j],\n", " [ 0.+0.j, -1.+0.j, 0.+0.j, -0.+0.j],\n", " [ 0.+0.j, 0.+0.j, 1.+0.j, 0.+0.j],\n", " [ 0.+0.j, -0.+0.j, 0.+0.j, -1.+0.j]]), array([[0.+0.j, 0.+0.j, 1.+0.j, 0.+0.j],\n", " [0.+0.j, 0.+0.j, 0.+0.j, 1.+0.j],\n", " [1.+0.j, 0.+0.j, 0.+0.j, 0.+0.j],\n", " [0.+0.j, 1.+0.j, 0.+0.j, 0.+0.j]]), array([[0.+0.j, 0.+0.j, 0.+0.j, 1.+0.j],\n", " [0.+0.j, 0.+0.j, 1.+0.j, 0.+0.j],\n", " [0.+0.j, 1.+0.j, 0.+0.j, 0.+0.j],\n", " [1.+0.j, 0.+0.j, 0.+0.j, 0.+0.j]]), array([[0.+0.j, 0.-0.j, 0.+0.j, 0.-1.j],\n", " [0.+0.j, 0.+0.j, 0.+1.j, 0.+0.j],\n", " [0.+0.j, 0.-1.j, 0.+0.j, 0.-0.j],\n", " [0.+1.j, 0.+0.j, 0.+0.j, 0.+0.j]]), array([[ 0.+0.j, 0.+0.j, 1.+0.j, 0.+0.j],\n", " [ 0.+0.j, -0.+0.j, 0.+0.j, -1.+0.j],\n", " [ 1.+0.j, 0.+0.j, 0.+0.j, 0.+0.j],\n", " [ 0.+0.j, -1.+0.j, 0.+0.j, -0.+0.j]]), array([[0.+0.j, 0.+0.j, 0.-1.j, 0.-0.j],\n", " [0.+0.j, 0.+0.j, 0.-0.j, 0.-1.j],\n", " [0.+1.j, 0.+0.j, 0.+0.j, 0.+0.j],\n", " [0.+0.j, 0.+1.j, 0.+0.j, 0.+0.j]]), array([[0.+0.j, 0.+0.j, 0.-0.j, 0.-1.j],\n", " [0.+0.j, 0.+0.j, 0.-1.j, 0.-0.j],\n", " [0.+0.j, 0.+1.j, 0.+0.j, 0.+0.j],\n", " [0.+1.j, 0.+0.j, 0.+0.j, 0.+0.j]]), array([[ 0.+0.j, 0.-0.j, 0.-0.j, -1.+0.j],\n", " [ 0.+0.j, 0.+0.j, 1.-0.j, 0.-0.j],\n", " [ 0.+0.j, 1.-0.j, 0.+0.j, 0.-0.j],\n", " [-1.+0.j, 0.+0.j, 0.+0.j, 0.+0.j]]), array([[ 0.+0.j, 0.+0.j, 0.-1.j, 0.-0.j],\n", " [ 0.+0.j, -0.+0.j, 0.-0.j, 0.+1.j],\n", " [ 0.+1.j, 0.+0.j, 0.+0.j, 0.+0.j],\n", " [ 0.+0.j, -0.-1.j, 0.+0.j, -0.+0.j]]), array([[ 1.+0.j, 0.+0.j, 0.+0.j, 0.+0.j],\n", " [ 0.+0.j, 1.+0.j, 0.+0.j, 0.+0.j],\n", " [ 0.+0.j, 0.+0.j, -1.+0.j, -0.+0.j],\n", " [ 0.+0.j, 0.+0.j, -0.+0.j, -1.+0.j]]), array([[ 0.+0.j, 1.+0.j, 0.+0.j, 0.+0.j],\n", " [ 1.+0.j, 0.+0.j, 0.+0.j, 0.+0.j],\n", " [ 0.+0.j, 0.+0.j, -0.+0.j, -1.+0.j],\n", " [ 0.+0.j, 0.+0.j, -1.+0.j, -0.+0.j]]), array([[ 0.+0.j, 0.-1.j, 0.+0.j, 0.-0.j],\n", " [ 0.+1.j, 0.+0.j, 0.+0.j, 0.+0.j],\n", " [ 0.+0.j, 0.-0.j, -0.+0.j, 0.+1.j],\n", " [ 0.+0.j, 0.+0.j, -0.-1.j, -0.+0.j]]), array([[ 1.+0.j, 0.+0.j, 0.+0.j, 0.+0.j],\n", " [ 0.+0.j, -1.+0.j, 0.+0.j, -0.+0.j],\n", " [ 0.+0.j, 0.+0.j, -1.+0.j, -0.+0.j],\n", " [ 0.+0.j, -0.+0.j, -0.+0.j, 1.-0.j]]))\n" ] } ], "source": [ "from quara.objects.matrix_basis import get_pauli_basis\n", "\n", "# Prepare\n", "basis_0 = get_pauli_basis()\n", "basis_1 = get_pauli_basis()\n", "\n", "# Tensor product\n", "result = tensor_product(basis_0, basis_1)\n", "print(result) # MatrixBasis" ] } ], "metadata": { "hide_input": false, "kernelspec": { "display_name": "Python 3", "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" }, "toc": { "base_numbering": 1, "nav_menu": {}, "number_sections": true, "sideBar": true, "skip_h1_title": false, "title_cell": "Table of Contents", "title_sidebar": "Contents", "toc_cell": false, "toc_position": {}, "toc_section_display": true, "toc_window_display": true } }, "nbformat": 4, "nbformat_minor": 5 }