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Author: jkuruzovich

site/notebooks/nb-01-01-linear-algebra-foundations-expanded.ipynb

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+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "699607db",
+   "metadata": {},
+   "source": [
+    "\n",
+    "# Linear Algebra Foundations for Quantum Computing — Chapter 1\n",
+    "\n",
+    "Quantum computing is built on the language of linear algebra. Every quantum state, transformation, \n",
+    "and measurement can be described using vectors, matrices, and specific algebraic operations. \n",
+    "\n",
+    "In this notebook, we review the key concepts of linear algebra and explain how they form the \n",
+    "foundation for quantum mechanics and quantum information. For each idea, we will also see how \n",
+    "to implement the operations in Python using `numpy`, so that the mathematics can be both \n",
+    "visualized and practiced computationally.\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "d52bb644",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "import numpy as np\n",
+    "np.set_printoptions(precision=4, suppress=True)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "7e445cdf",
+   "metadata": {},
+   "source": [
+    "\n",
+    "## Vectors\n",
+    "\n",
+    "A **vector** is one of the most fundamental mathematical objects we will encounter. \n",
+    "In quantum computing, a vector represents the **state of a quantum system**. \n",
+    "\n",
+    "- For a qubit, this state is written as a column vector in a two-dimensional complex vector space.  \n",
+    "  The “zero” state of a qubit is represented as the vector {math}`[1, 0]^T`, and the “one” state \n",
+    "  as {math}`[0, 1]^T`. \n",
+    "- More generally, a quantum state can be a linear combination of these basis states, such as \n",
+    "  {math}`\\alpha|0\\rangle + \\beta|1\\rangle`, which is simply a vector with complex entries.  \n",
+    "\n",
+    "The length, or **norm**, of a vector plays a critical role in quantum mechanics. The squared \n",
+    "magnitudes of the vector entries represent **probabilities**, which must always sum to 1. For \n",
+    "this reason, all valid quantum states are **normalized vectors** with unit length. Normalization \n",
+    "ensures that when a measurement is performed, the total probability of all possible outcomes is \n",
+    "exactly one.  \n",
+    "\n",
+    "Vectors also provide a geometric picture of quantum states. In the case of a single qubit, \n",
+    "we can think of the vector as pointing in some direction in two-dimensional space (or \n",
+    "equivalently on the surface of the Bloch sphere). For higher-dimensional systems, this \n",
+    "geometric picture becomes abstract, but the idea that vectors encode the “configuration” \n",
+    "of the system remains the same.\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "82e29645",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "\n",
+    "x = np.array([3, 4], dtype=float)\n",
+    "print(\"x =\", x)\n",
+    "print(\"Norm of x =\", np.linalg.norm(x))\n",
+    "print(\"Normalized x =\", x/np.linalg.norm(x))\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "02c6543d",
+   "metadata": {},
+   "source": [
+    "\n",
+    "## Complex Numbers and Conjugation\n",
+    "\n",
+    "Unlike classical probabilities, quantum amplitudes are expressed as **complex numbers**. \n",
+    "A complex number has both a real part and an imaginary part, and can be written as \n",
+    "{math}`z = a + ib`. \n",
+    "\n",
+    "- The **magnitude** or modulus {math}`|z|` tells us the length of this number in the complex plane.  \n",
+    "- The **angle** (or phase) encodes additional information that has no classical counterpart.  \n",
+    "- The **complex conjugate** flips the sign of the imaginary part, and the conjugate transpose \n",
+    "  (Hermitian adjoint) is crucial in quantum mechanics because it ensures probabilities are \n",
+    "  always non-negative.  \n",
+    "\n",
+    "The reason complex numbers are essential is that they allow **interference**. When two amplitudes \n",
+    "with phases interact, they can cancel or reinforce each other. This is what makes phenomena like \n",
+    "the double-slit experiment possible, and it is what gives quantum computing its unique power.\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "e71a28e0",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "z = 3 + 4j\n",
+    "print('|z| =', abs(z), 'conjugate =', np.conj(z))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "1d1490b0",
+   "metadata": {},
+   "source": [
+    "\n",
+    "## Inner Product (Dot Product)\n",
+    "\n",
+    "The dot product of two vectors, also known as the **inner product**, measures the degree of \n",
+    "“overlap” between them. In Euclidean geometry, the dot product tells us whether two vectors are \n",
+    "perpendicular, and how much one vector points in the direction of another. \n",
+    "\n",
+    "In quantum mechanics, the inner product of two states {math}`|u\\rangle` and {math}`|v\\rangle` \n",
+    "is written {math}`\\langle u, v \\rangle`.  \n",
+    "\n",
+    "- If the result is zero, the states are **orthogonal**, meaning that they represent mutually \n",
+    "  exclusive outcomes.  \n",
+    "- More generally, the probability of finding a system prepared in state {math}`|v\\rangle` \n",
+    "  when measured in the basis that includes {math}`|u\\rangle` is proportional to \n",
+    "  {math}`|\\langle u, v \\rangle|^2`.  \n",
+    "\n",
+    "This makes the dot product one of the most important tools in quantum mechanics. \n",
+    "It is the bridge between abstract vector states and physical probabilities that \n",
+    "can be observed in the laboratory.\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "43ffc98e",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "\n",
+    "u = np.array([1, 1j])\n",
+    "v = np.array([1, -1j])\n",
+    "print(\"<u,v> =\", np.vdot(u,v))\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "d88095f6",
+   "metadata": {},
+   "source": [
+    "\n",
+    "## Matrices\n",
+    "\n",
+    "Matrices represent **linear transformations**, and in quantum computing, they describe how \n",
+    "quantum states evolve or how they are manipulated by gates. \n",
+    "\n",
+    "- **Hermitian matrices** are equal to their own conjugate transpose. In quantum mechanics, \n",
+    "  Hermitian matrices represent **observables** — physical quantities like energy, spin, \n",
+    "  or position. Their eigenvalues correspond to the possible results of a measurement.  \n",
+    "\n",
+    "- **Unitary matrices** satisfy {math}`U^\\dagger U = I`. These matrices preserve vector \n",
+    "  length (and therefore probability) and represent **valid quantum evolutions**. Every \n",
+    "  quantum gate is described by a unitary matrix.  \n",
+    "\n",
+    "The special properties of these matrices ensure that quantum mechanics is both \n",
+    "mathematically consistent and physically meaningful.\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "845ffaba",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "\n",
+    "H = (1/np.sqrt(2))*np.array([[1,1],[1,-1]])\n",
+    "print(\"Hadamard gate H:\\n\", H)\n",
+    "print(\"Check H^†H = I:\", np.allclose(H.conj().T @ H, np.eye(2)))\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "44f90777",
+   "metadata": {},
+   "source": [
+    "\n",
+    "## Tensor Product\n",
+    "\n",
+    "One of the most striking differences between classical and quantum systems is how they combine. \n",
+    "Classically, two systems are described by pairing their states. In quantum mechanics, we use the \n",
+    "**tensor product**. \n",
+    "\n",
+    "- If one system is described by a vector of dimension {math}`m` and another by a vector of \n",
+    "  dimension {math}`n`, the joint system lives in a space of dimension {math}`mn`.  \n",
+    "- A single qubit is described by a 2-dimensional vector. Two qubits together require a \n",
+    "  4-dimensional vector, and three qubits require 8 dimensions.  \n",
+    "\n",
+    "This exponential growth is one reason quantum computing is so powerful: the state space grows \n",
+    "extremely quickly with the number of qubits.  \n",
+    "\n",
+    "Tensor products are also the source of **entanglement**, one of the most profound quantum \n",
+    "phenomena. An entangled state cannot be written as a simple tensor product of two smaller states. \n",
+    "This property has no classical analogue and is the basis for quantum communication and \n",
+    "quantum cryptography.\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "1c54e7d5",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "\n",
+    "zero = np.array([1,0])\n",
+    "one = np.array([0,1])\n",
+    "print(\"|0>⊗|1> =\", np.kron(zero,one))\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "95fb491f",
+   "metadata": {},
+   "source": [
+    "\n",
+    "## Unitarity\n",
+    "\n",
+    "A central principle of quantum mechanics is that **probability is conserved**. When a quantum \n",
+    "state evolves over time or is manipulated by gates, the total probability of all outcomes \n",
+    "must remain equal to one.  \n",
+    "\n",
+    "This requirement is exactly captured by **unitary matrices**. A unitary transformation is \n",
+    "essentially a rotation in a high-dimensional complex vector space. It changes the direction \n",
+    "of the state vector but never its length.  \n",
+    "\n",
+    "This is why every valid quantum gate, from the Hadamard gate to the more complex controlled-NOT \n",
+    "gate, is represented by a unitary matrix. Without unitarity, the mathematical structure of \n",
+    "quantum mechanics would not correspond to the physical reality of conserved probability.\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "a36dece1",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "\n",
+    "X = np.array([[0,1],[1,0]],dtype=complex)\n",
+    "print(\"X gate:\\n\", X)\n",
+    "print(\"Unitary check:\", np.allclose(X.conj().T @ X, np.eye(2)))\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "00c0e352",
+   "metadata": {},
+   "source": [
+    "\n",
+    "## Exercises\n",
+    "\n",
+    "1. Normalize the vector {math}`w = (3, 4i)` and verify its norm is 1.  \n",
+    "2. Show that {math}`a = (1, i)` and {math}`b = (1, -i)` are orthogonal by computing their inner product.  \n",
+    "3. Construct the projector onto {math}`u = \\frac{1}{\\sqrt{2}}(1, 1)` and apply it to the vector {math}`x = (2, 1)`.  \n",
+    "4. Compute the tensor product {math}`|0\\rangle \\otimes |1\\rangle`. Which basis vector of the two-qubit system does this correspond to?  \n",
+    "5. Verify that the Pauli-X matrix {math}`\\begin{bmatrix}0&1\\\\1&0\\end{bmatrix}` is unitary.\n"
+   ]
+  }
+ ],
+ "metadata": {},
+ "nbformat": 4,
+ "nbformat_minor": 5
+}