Single Qubit States and Measurements

1 QI3 Practice

Code

viewof generateButton = {
  const button = html`<button style="
    background-color: #2563eb;
    color: #ffffff;
    font-size: 15px;
    font-weight: 600;
    padding: 10px 20px;
    border: none;
    border-radius: 6px;
    cursor: pointer;
    box-shadow: 0 2px 4px rgba(0, 0, 0, 0.1);
  ">Generate New Problem</button>`;

  button.value = 0;
  button.onclick = () => {
    button.value++;
    button.dispatchEvent(new CustomEvent("input", { bubbles: true }));
  };

  return button;
}
<!-- viewof generateButton = Inputs.button("Generate New Problem", {
  <!-- style: "background-color: #0066cc; color: white; border-radius: 4px; padding: 8px 16px; font-weight: bold; cursor: pointer;" --> -->
<!-- }) -->

Code

// Cell 2: Problem Generator Logic & Display
problem = {
  generateButton; // Listens to button clicks to re-run cell

  const numBasisOptions = [1,2,2,3,4,4,4];

  const numBasis = numBasisOptions[Math.floor(Math.random() * numBasisOptions.length)];

  // Define available basis states and select 'numBasis' of them randomly
  const allBasis = ["00", "01", "10", "11"];
  
  // Shuffle array and take the required number of basis states
  const shuffledBasis = allBasis.sort(() => 0.5 - Math.random());
  const selectedBasis = shuffledBasis.slice(0, numBasis).sort(); // Sorted for standard ordering

  // Helper to construct the LaTeX coefficient based on the number of basis terms and a random phase
  const getCoeff = (numBasis, phase) => {
    // Phases: '+1', '-1', '+i', '-i'
    
    // For numBasis = 1, magnitude is 1, so we just return the phase sign/imaginary unit
    if (numBasis === 1) {
      if (phase === '+1') return "";
      if (phase === '-1') return "-";
      if (phase === '+i') return "i";
      if (phase === '-i') return "-i";
    }

    // For numBasis > 1, determine the denominator
    let denom = "";
    if (numBasis === 2) denom = "\\sqrt{2}";
    if (numBasis === 3) denom = "\\sqrt{3}";
    if (numBasis === 4) denom = "2";

    // Determine numerator and overall sign
    let numStr = "";
    let sign = "";
    if (phase === '+1') { sign = ""; numStr = "1"; }
    if (phase === '-1') { sign = "-"; numStr = "1"; }
    if (phase === '+i') { sign = ""; numStr = "i"; }
    if (phase === '-i') { sign = "-"; numStr = "i"; }

    return `${sign}\\frac{${numStr}}{${denom}}`;
  };

  const phases = ['+1', '-1', '+i', '-i'];
  let terms = [];

  // Construct each term
  selectedBasis.forEach((basis, index) => {
    const randomPhase = phases[Math.floor(Math.random() * phases.length)];
    const coeff = getCoeff(numBasis, randomPhase);
    const ket = `|${basis}\\rangle`;
    
    let term = `${coeff}${ket}`;

    // Format addition/subtraction signs nicely for joining
    if (index > 0) {
      if (term.startsWith("-")) {
        term = ` - ${term.slice(1)}`;
      } else {
        term = ` + ${term}`;
      }
    }
    
    terms.push(term);
  });

  const state3 = terms.join("");

  return html`<p>Prove either that ${tex`|\psi\rangle = ${state3}`} is entangled or that it is a product state. Your proof should use the definition of a product state/entangled state.</>`;
}

Gemini was used to assist in generating the code that produces the randomized problems on this page. While AI contributed to this content, it was designed, reviewed, and approved by Prof. Shelby Kimmel.