{"id":42243,"date":"2025-10-30T21:00:28","date_gmt":"2025-10-30T15:30:28","guid":{"rendered":"https:\/\/tocxten.com\/?page_id=42243"},"modified":"2025-10-30T21:25:39","modified_gmt":"2025-10-30T15:55:39","slug":"representation-of-qubits","status":"publish","type":"page","link":"https:\/\/tocxten.com\/index.php\/representation-of-qubits\/","title":{"rendered":"Representation of Qubits"},"content":{"rendered":"\n

1. Classical vs Quantum Representation<\/strong><\/h2>\n\n\n\n

Classical 2-Bit Example<\/strong><\/h3>\n\n\n\n

In classical computing, information is represented using bits<\/strong>, where each bit can exist in one of two possible states \u2014 0 or 1<\/strong>.
A system with two bits<\/strong> can represent four distinct states<\/strong>:<\/p>\n\n\n\n

Bit Combination<\/th>Binary<\/th>Decimal Equivalent<\/th><\/tr><\/thead>
00<\/td>0<\/td>0<\/td><\/tr>
01<\/td>1<\/td>1<\/td><\/tr>
10<\/td>2<\/td>2<\/td><\/tr>
11<\/td>3<\/td>3<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n

Each bit combination corresponds to a unique decimal value between 0 and 3<\/strong>. Importantly, a classical system can exist in only one of these states at any given time<\/strong>.<\/p>\n\n\n\n

For example:<\/strong>
If we have two classical bits, and their values are 0<\/code> and 1<\/code>, the system is in state 01<\/code>.
It cannot simultaneously represent 10<\/code> or 11<\/code>. This deterministic nature limits classical computing to sequential state exploration<\/strong>.<\/p>\n\n\n\n

Quantum 2-Qubit System<\/strong><\/h3>\n\n\n\n

Quantum computing introduces a new paradigm based on the principles of superposition<\/strong> and quantum parallelism<\/strong>. A qubit (quantum bit)<\/strong> is the fundamental unit of quantum information. Unlike a classical bit that can be either 0 or 1, a qubit can exist in a superposition<\/strong> of both 0 and 1 simultaneously. For a two-qubit system<\/strong>, the possible basis states are the same as the classical combinations:<\/p>\n\n\n\n

\u222300\u27e9,\u222301\u27e9,\u222310\u27e9,\u222311\u27e9<\/strong><\/p>\n\n\n\n

However, in quantum computing, the state of the system<\/strong> can be a linear combination<\/strong> of all these basis states:<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

This means that a two-qubit system can represent all four states simultaneously<\/strong>, until measured.
Upon measurement, the system collapses<\/strong> to one of these states, with probabilities given by the squares of their respective amplitudes.<\/p>\n\n\n\n

2. Representation of Qubits<\/strong><\/h2>\n\n\n\n

To mathematically describe qubits, we use state vectors<\/strong> in a complex Hilbert space<\/strong>.<\/p>\n\n\n\n

State Vector Representation<\/strong><\/h3>\n\n\n\n

A state vector<\/strong> represents the quantum state of a system and contains the probability amplitudes<\/strong> of each possible outcome.<\/p>\n\n\n\n

Example: A Quantum Coin<\/strong><\/h4>\n\n\n\n

Consider a quantum coin that can exist in a superposition of Head (H)<\/strong> and Tail (T)<\/strong>.<\/p>\n\n\n\n

If the coin is in the Head state<\/strong>, we can write:<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

If the coin is in the Tail state<\/strong>, we represent it as:<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Thus, the general state of the coin can be expressed as a superposition<\/strong>:<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

where \u03b1 and \u03b2 are complex numbers<\/strong> that satisfy:<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

This state vector expresses the probability of being in Head or Tail state<\/strong> \u2014 analogous to being in 0 or 1 state in a qubit.<\/p>\n\n\n\n

3 Dirac Notation (Bra-Ket Notation)<\/strong><\/h2>\n\n\n\n

Dirac notation, introduced by Paul Dirac<\/strong>, provides a convenient and elegant way to represent quantum states and their operations.<\/p>\n\n\n\n

It simplifies matrix and vector operations used in quantum mechanics.<\/p>\n\n\n\n

\n\n\n\n

Basic Components<\/strong><\/h3>\n\n\n\n
    \n
  1. Ket (|\u03c8\u27e9)<\/strong> : Represents a column vector<\/strong> \u2014 the quantum state itself.
    Example:<\/li>\n<\/ol>\n\n\n\n
    \"\"<\/figure>\n\n\n\n
    \"\"<\/figure>\n\n\n\n
    \"\"<\/figure>\n\n\n\n

    4. Common Qubit States<\/strong><\/h2>\n\n\n\n

    Here are the most frequently used qubit states<\/strong> in quantum computing:<\/p>\n\n\n\n

    \"\"<\/figure>\n\n\n\n

    5. Measurement in Qubit Notation<\/strong><\/h2>\n\n\n\n

    When a qubit is measured, it collapses<\/strong> from its superposition into one of the basis states<\/strong>.<\/p>\n\n\n\n

    If a qubit is in state:<\/p>\n\n\n\n

    \"\"<\/figure>\n\n\n\n

    then upon measurement:<\/p>\n\n\n\n

    \"\"<\/figure>\n\n\n\n

    The measurement process<\/strong> destroys the superposition and leaves the qubit in the observed state<\/p>\n\n\n\n

    Example<\/strong><\/h3>\n\n\n\n

    Suppose<\/p>\n\n\n\n

    \"\"<\/figure>\n\n\n\n
    \"\"<\/figure>\n\n\n\n

    6. Visualization on the Bloch Sphere<\/strong><\/h2>\n\n\n\n

    Every single qubit state can be visualized on a Bloch Sphere<\/strong>, a 3D representation of the state space.<\/p>\n\n\n\n

    The general qubit state:<\/p>\n\n\n\n

    \"\"<\/figure>\n\n\n\n

    corresponds to a point on the sphere defined by:<\/p>\n\n\n\n

      \n
    • \u03b8 (theta)<\/strong>: polar angle<\/li>\n\n\n\n
    • \u03c6 (phi)<\/strong>: azimuthal angle<\/li>\n<\/ul>\n\n\n\n

      This representation is powerful for visualizing quantum gates<\/strong>, rotations<\/strong>, and state transformations<\/strong>.<\/p>\n\n\n\n

      7. Summary<\/strong><\/h2>\n\n\n\n
        \n
      • Classical bits<\/strong> can exist in one state at a time, while qubits<\/strong> can exist in superposition.<\/li>\n\n\n\n
      • Dirac notation<\/strong> provides a compact mathematical way to describe quantum states.<\/li>\n\n\n\n
      • State vectors<\/strong> and amplitudes<\/strong> define probabilities of measurement outcomes.<\/li>\n\n\n\n
      • Measurement collapses<\/strong> the qubit to one basis state.<\/li>\n\n\n\n
      • The Bloch sphere<\/strong> helps visualize any single-qubit state geometrically.<\/li>\n<\/ul>\n\n\n\n

        8. Qiskit Example<\/strong><\/p>\n\n\n\n

        from qiskit import QuantumCircuit\nfrom qiskit.visualization import plot_bloch_multivector\nfrom qiskit.quantum_info import Statevector\n\n# Create a single qubit quantum circuit\nqc = QuantumCircuit(1)\n\n# Apply Hadamard gate to create superposition\nqc.h(0)\nqc.draw('mpl')\n\n# Get the statevector\nstate = Statevector.from_instruction(qc)\nplot_bloch_multivector(state)\n<\/code><\/pre>\n\n\n\n
        \"\"<\/figure>\n\n\n\n
        <\/p>\n","protected":false},"excerpt":{"rendered":"
        1. Classical vs Quantum Representation Classical 2-Bit Example In classical computing, information is represented using bits, where each bit can exist in one of two possible states \u2014 0 or…<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"om_disable_all_campaigns":false,"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"footnotes":"","_links_to":"","_links_to_target":""},"class_list":["post-42243","page","type-page","status-publish","hentry"],"post_mailing_queue_ids":[],"_links":{"self":[{"href":"https:\/\/tocxten.com\/index.php\/wp-json\/wp\/v2\/pages\/42243","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/tocxten.com\/index.php\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/tocxten.com\/index.php\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/tocxten.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/tocxten.com\/index.php\/wp-json\/wp\/v2\/comments?post=42243"}],"version-history":[{"count":21,"href":"https:\/\/tocxten.com\/index.php\/wp-json\/wp\/v2\/pages\/42243\/revisions"}],"predecessor-version":[{"id":42292,"href":"https:\/\/tocxten.com\/index.php\/wp-json\/wp\/v2\/pages\/42243\/revisions\/42292"}],"wp:attachment":[{"href":"https:\/\/tocxten.com\/index.php\/wp-json\/wp\/v2\/media?parent=42243"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}