{
  "schema": "wai.quantum.exercise/1",
  "id": "toffoli",
  "translations": { "zh-CN": "machine-assisted: to be reviewed by a fluent reader before it is advertised" },
  "title": { "en": "A Toffoli from Clifford+T", "zh-CN": "用 Clifford+T 搭建 Toffoli 门" },
  "task": {
    "en": "Write the Toffoli gate, ccx q[0], q[1], q[2], using only h, t, tdg and cx. The check proves your program the same circuit as the Toffoli, up to a global phase.",
    "zh-CN": "只用 h、t、tdg 和 cx 写出 Toffoli 门 ccx q[0], q[1], q[2]。检查会证明你的程序与 Toffoli 门在相差一个全局相位的意义下是同一条线路。"
  },
  "starter": [
    "OPENQASM 3.0;",
    "include \"stdgates.inc\";",
    "qubit[3] q;",
    "// the Toffoli, which this exercise does not allow as one gate",
    "ccx q[0], q[1], q[2];"
  ],
  "intent": { "equivalent": "OPENQASM 3.0;\ninclude \"stdgates.inc\";\nqubit[3] q;\nccx q[0], q[1], q[2];\n" },
  "constraints": { "allow": ["h", "t", "tdg", "cx"], "t_count": { "max": 7 } },
  "hints": {
    "en": [
      "A Toffoli is a controlled-controlled-Z with a Hadamard on each side of the target.",
      "The controlled-controlled-Z multiplies |111⟩ by −1. Written as phases, that is a product of e^{±iπ/4} factors on the parities x₀, x₁, x₂, x₀⊕x₁, x₀⊕x₂, x₁⊕x₂ and x₀⊕x₁⊕x₂: seven T or T† gates.",
      "Each parity can be brought onto one qubit with controlled-NOTs, a T or T† applied there, and the controlled-NOTs undone."
    ],
    "zh-CN": [
      "Toffoli 门就是在目标比特两侧各加一个 Hadamard 门的受控受控 Z 门。",
      "受控受控 Z 门把 |111⟩ 乘以 −1。写成相位的形式，它是作用在奇偶性 x₀、x₁、x₂、x₀⊕x₁、x₀⊕x₂、x₁⊕x₂ 和 x₀⊕x₁⊕x₂ 上的 e^{±iπ/4} 因子之积：七个 T 或 T† 门。",
      "每种奇偶性都可以用受控非门汇集到一个量子比特上，在那里作用一个 T 或 T†，再把这些受控非门撤销。"
    ]
  },
  "reference": [
    "OPENQASM 3.0;",
    "include \"stdgates.inc\";",
    "qubit[3] q;",
    "h q[2];",
    "cx q[1], q[2];",
    "tdg q[2];",
    "cx q[0], q[2];",
    "t q[2];",
    "cx q[1], q[2];",
    "tdg q[2];",
    "cx q[0], q[2];",
    "t q[1];",
    "t q[2];",
    "h q[2];",
    "cx q[0], q[1];",
    "t q[0];",
    "tdg q[1];",
    "cx q[0], q[1];"
  ],
  "wrong": [
    {
      "fails": "intent",
      "note": "one T where a T† belongs leaves a relative phase",
      "qasm": ["OPENQASM 3.0;", "include \"stdgates.inc\";", "qubit[3] q;", "h q[2];", "cx q[1], q[2];", "tdg q[2];", "cx q[0], q[2];", "t q[2];", "cx q[1], q[2];", "tdg q[2];", "cx q[0], q[2];", "t q[1];", "t q[2];", "h q[2];", "cx q[0], q[1];", "tdg q[0];", "tdg q[1];", "cx q[0], q[1];"]
    },
    {
      "fails": "constraint",
      "note": "the Toffoli as one gate",
      "qasm": ["OPENQASM 3.0;", "include \"stdgates.inc\";", "qubit[3] q;", "ccx q[0], q[1], q[2];"]
    }
  ]
}