Figure 1
Quantum-circuit diagrams illustrating the principle of deferred measurement. In the circuit on the left, a measurement
on the lower ancilla yields result
; this result controls a subsequent unitary
, applied to the probe and the upper ancilla, and determines a conditional measurement
on the upper ancilla, which has result
. The two measurement results then control a unitary
applied to the probe and a conditional measurement
on the probe. The left-hand circuit is equivalent to the circuit on the right, in which the controls are applied coherently (boxed gates
and
), and the measurements, deferred to the end of the circuit, tell one which unitary was applied. Without loss of generality, we can assume the measurements are described by orthogonal projectors
,
, and
, because any generalized measurement can be modeled by a projection-valued measurement on an extended system. The unitary transformations in the left-hand circuit,
and
, are evolution operators generated by the Hamiltonians
and
, whereas the corresponding coherent controlled unitaries in the circuit on the right,
and
, are generated by the Hamiltonians
and
. It is easy to verify from the evolution equations that the controlled unitaries in the right-hand circuit are given by
and
. Thus the principle of deferred measurement can be rendered algebraically in the following way: if we use the left-hand circuit, the probability for obtaining results
,
, and
takes the form
, where
is the initial state of the probe and ancillas and
; pulling the measurement projectors to the left in
changes the unitaries to the corresponding coherent controlled operations, i.e.,
, which gives the form of the probability obtained from the right-hand circuit.
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