Deferred measurement principle

The deferred measurement principle is a result in quantum computing which states that delaying measurements until the end of a quantum computation doesn't affect the probability distribution of outcomes.^[2][3]

Two equivalent quantum logic circuits. One where measurement happens first, and one where an operation conditioned on the to-be-measured qubit happens first.

Measurement is performed early and the resulting classical bits are sent. The classical bits control if the 1-qubit X and Z gates are executed, allowing teleportation.^[1]

By moving the measurement to the end, the 2-qubit controlled-X and -Z gates need to be applied, which requires both qubits to be near (i.e. at a distance where 2-qubit quantum effects can be controlled), and thus limits the distance of the teleportation. While logically equivalent, deferring the measurement have physical implications.

Example: Two variants of the teleportation circuit. The 2-qubit states ${\displaystyle |\Phi ^{+}\rangle }$ and ${\displaystyle |\beta _{00}\rangle }$ refer to the same Bell state.

A consequence of the deferred measurement principle is that measuring commutes with conditioning. The choice of whether to measure a qubit before, after, or during an operation conditioned on that qubit will have no observable effect on a circuit's final expected results.

Thanks to the deferred measurement principle, measurements in a quantum circuit can often be shifted around so they happen at better times. For example, measuring qubits as early as possible can reduce the maximum number of simultaneously stored qubits; potentially enabling an algorithm to be run on a smaller quantum computer or to be simulated more efficiently. Alternatively, deferring all measurements until the end of circuits allows them to be analyzed using only pure states.