Single Qubit States and Measurements

\[ \newcommand{\braket}[2]{\langle{#1}|{#2}\rangle} \]

1 QI2 Practice

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;" --> -->
<!-- }) -->

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

  const initialStates = [
    "\\frac{1}{\\sqrt{2}}|0\\rangle + \\frac{1}{\\sqrt{2}}|1\\rangle",
    "\\frac{1}{\\sqrt{2}}|0\\rangle + \\frac{i}{\\sqrt{2}}|1\\rangle",
    "\\frac{1}{\\sqrt{2}}|0\\rangle - \\frac{i}{\\sqrt{2}}|1\\rangle",
    "\\frac{1}{\\sqrt{5}}|0\\rangle + \\frac{2}{\\sqrt{5}}|1\\rangle",
    "\\frac{1}{\\sqrt{5}}|0\\rangle + \\frac{2i}{\\sqrt{5}}|1\\rangle",
    "\\frac{3}{5}|0\\rangle + \\frac{4i}{5}|1\\rangle",
    "\\frac{1}{\\sqrt{3}}|0\\rangle + \\frac{\\sqrt{2}i}{\\sqrt{3}}|1\\rangle",
    "\\frac{2}{3}|0\\rangle - \\frac{\\sqrt{5}i}{3}|1\\rangle"
  ];

  const bases = [
    [
      "\\frac{1}{\\sqrt{2}}|0\\rangle + \\frac{1}{\\sqrt{2}}|1\\rangle",
      "\\frac{1}{\\sqrt{2}}|0\\rangle - \\frac{1}{\\sqrt{2}}|1\\rangle"
    ],
    [
      "\\frac{1}{\\sqrt{2}}|0\\rangle + \\frac{i}{\\sqrt{2}}|1\\rangle",
      "\\frac{1}{\\sqrt{2}}|0\\rangle - \\frac{i}{\\sqrt{2}}|1\\rangle"
    ],
    [
      "\\frac{1}{\\sqrt{5}}|0\\rangle + \\frac{2i}{\\sqrt{5}}|1\\rangle",
      "\\frac{2i}{\\sqrt{5}}|0\\rangle + \\frac{1}{\\sqrt{5}}|1\\rangle"
    ],
    [
      "\\frac{3}{5}|0\\rangle + \\frac{4}{5}|1\\rangle",
      "\\frac{4}{5}|0\\rangle - \\frac{3}{5}|1\\rangle"
    ],
    [
      "\\frac{1}{\\sqrt{3}}|0\\rangle + \\frac{\\sqrt{2}i}{\\sqrt{3}}|1\\rangle",
      "\\frac{\\sqrt{2}i}{\\sqrt{3}}|0\\rangle + \\frac{1}{\\sqrt{3}}|1\\rangle"
    ],
    [
      "|0\\rangle",
      "|1\\rangle"
    ]
  ];

  const state = initialStates[Math.floor(Math.random() * initialStates.length)];
  const basis = bases[Math.floor(Math.random() * bases.length)];
  const outcome = basis[Math.floor(Math.random() * basis.length)];

  return html`<p>Suppose you have a qubit in the state ${tex`|\psi\rangle = ${state}`}, and you make a measurement in the basis ${tex`M = \{${basis[0]},${basis[1]}\}`}.</p>
  <ul>
    <li>What is the probability of outcome ${tex`${outcome}`}? (Your probability need not be totally simplified, but it should not contain any absolute values or imaginary numbers.)</li>
    <li>In this case, what does ${tex`|\psi\rangle`} collapse to?</li>
  </ul>`;
}

html`<div style="font-size: 1.1em; line-height: 1.5; padding: 1em; border-left: 4px solid #4a90e2; background: #f8f9fa;">
  ${problem}
</div>`

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.