A joint measurement extracts information about correlations among multiple qubits rather than the state of each qubit separately. A useful special case is the parity measurement, which reports whether the basis states have even or odd parity. section 4.6.1 section 4.9
Perform a parity measurement by:
- Allocating an auxiliary qubit in the |0⟩ state. section 4.6.1
- Applying one CNOT per input qubit, using each input qubit as control and the auxiliary qubit as target. section 4.6.1
- Measuring the auxiliary qubit to read the parity bit. section 4.6.1
Two-qubit parity
For 0.8|00⟩ + 0.6|11⟩, the auxiliary qubit ends in |0⟩, indicating even parity. For 0.8|01⟩ + 0.6|10⟩, it ends in |1⟩, indicating odd parity. Because every basis state in each superposition has the same parity, the auxiliary qubit does not remain entangled with the input and its measurement preserves the input state. section 4.6.1
Table 4.2 The steps of performing a parity measurement on a two-qubit state
| The input state | \(0.8 |00\rangle + 0.6 |11\rangle\) | \(0.8 |01\rangle + 0.6 |10\rangle\) |
|---|---|---|
| Parity of the input state | 0 | 1 |
| Allocate auxiliary qubit (last bit) | \(0.8 |00\rangle|0\rangle + 0.6 |11\rangle|0\rangle\) | \(0.8 |01\rangle|0\rangle + 0.6 |10\rangle|0\rangle\) |
| Apply CNOT (first qubit as control) | \(0.8 |00\rangle|0\rangle + 0.6 |11\rangle|1\rangle\) | \(0.8 |01\rangle|0\rangle + 0.6 |10\rangle|1\rangle\) |
| Apply CNOT (second qubit as control) | \(0.8 |00\rangle|0\rangle + 0.6 |11\rangle|0\rangle\) | \(0.8 |01\rangle|1\rangle + 0.6 |10\rangle|1\rangle\) |
| State of the auxiliary qubit | \(|0\rangle\) | \(|1\rangle\) |
Trace the auxiliary qubit through allocation, each CNOT, and measurement for even- and odd-parity superpositions. section 4.6.1
