{
  "schema": "wai.quantum.exercise/1",
  "id": "phase-estimation",
  "translations": { "zh-CN": "machine-assisted: to be reviewed by a fluent reader before it is advertised" },
  "title": { "en": "Phase estimation", "zh-CN": "相位估计" },
  "task": {
    "en": "The T gate multiplies |1⟩ by e^{2πi/8}: its phase is 1/8. Three counting qubits have picked up that phase, written as Σₓ e^{2πi·x/8}|x⟩, with q[2] the highest bit. Finish with the inverse quantum Fourier transform so that the counting register reads 001, that is 8 × 1/8, every time.",
    "zh-CN": "T 门把 |1⟩ 乘以 e^{2πi/8}：它的相位是 1/8。三个计数量子比特已经获得了这个相位，写作 Σₓ e^{2πi·x/8}|x⟩，其中 q[2] 是最高位。最后加上逆量子傅里叶变换，使计数寄存器每次都读出 001，即 8 × 1/8。"
  },
  "starter": [
    "OPENQASM 3.0;",
    "include \"stdgates.inc\";",
    "qubit[4] q;",
    "bit[3] c;",
    "// the eigenstate |1⟩ of the T gate, on q[3]",
    "x q[3];",
    "// the counting register",
    "h q[0];",
    "h q[1];",
    "h q[2];",
    "// controlled T to the powers 1, 2 and 4",
    "cp(pi/4) q[0], q[3];",
    "cp(pi/2) q[1], q[3];",
    "cp(pi) q[2], q[3];",
    "// the inverse quantum Fourier transform: your gates here",
    "c[0] = measure q[0];",
    "c[1] = measure q[1];",
    "c[2] = measure q[2];"
  ],
  "intent": { "distribution": { "001": 1.0 } },
  "constraints": { "allow": ["h", "x", "cp", "swap"], "counts": { "x": { "max": 1 }, "cp": { "min": 6, "max": 6 }, "h": { "min": 6, "max": 6 } } },
  "hints": {
    "en": [
      "The previous lesson's inverse transform is close, but it was written for q[0] as the highest bit. Here q[2] is.",
      "Undo the order first: a swap of the outer qubits, then a Hadamard on each qubit from q[0] up, each preceded by cp(−π/2ᵏ) from the qubits already done."
    ],
    "zh-CN": [
      "上一课的逆变换已经很接近了，但它是以 q[0] 为最高位写的。这里最高位是 q[2]。",
      "先把顺序倒回来：交换最外侧的两个量子比特；然后从 q[0] 开始依次对每个量子比特作用 Hadamard 门，在每个 Hadamard 之前，从已经处理过的量子比特作用 cp(−π/2ᵏ)。"
    ]
  },
  "reference": [
    "OPENQASM 3.0;",
    "include \"stdgates.inc\";",
    "qubit[4] q;",
    "bit[3] c;",
    "x q[3];",
    "h q[0];",
    "h q[1];",
    "h q[2];",
    "cp(pi/4) q[0], q[3];",
    "cp(pi/2) q[1], q[3];",
    "cp(pi) q[2], q[3];",
    "swap q[0], q[2];",
    "h q[0];",
    "cp(-pi/2) q[1], q[0];",
    "h q[1];",
    "cp(-pi/4) q[2], q[0];",
    "cp(-pi/2) q[2], q[1];",
    "h q[2];",
    "c[0] = measure q[0];",
    "c[1] = measure q[1];",
    "c[2] = measure q[2];"
  ],
  "wrong": [
    {
      "fails": "intent",
      "note": "Hadamards alone do not read a phase of 1/8",
      "qasm": ["OPENQASM 3.0;", "include \"stdgates.inc\";", "qubit[4] q;", "bit[3] c;", "x q[3];", "h q[0];", "h q[1];", "h q[2];", "cp(pi/4) q[0], q[3];", "cp(pi/2) q[1], q[3];", "cp(pi) q[2], q[3];", "h q[0];", "h q[1];", "h q[2];", "c[0] = measure q[0];", "c[1] = measure q[1];", "c[2] = measure q[2];"]
    }
  ]
}