# 5.4.2 Measurements

Measurements are different from gate operations in a very important way, since rather than transforming a qubit from one definite state to another definite state , the final state after measurement is one of the two eigenstates of the measurement operator, which are obtained with some probability. Therefore, measurements are not reversible. The results of measurements are real numbers, so they can be stored as bits (rather than qubits) in a modern memory cell.

The circuit symbol for the measurement operation is shown in Figure 12.

Figure 12 The symbol for a measurement operation

The probabilistic nature of measurement is a feature of quantum computing that must be accounted for when evaluating the performance of a quantum algorithm.

Example

A circuit is set up as shown in Figure 13 and the input qubits are both . Calculate the output qubits and hence the possible results of the measurements and their probabilities.

Figure 13 A circuit incorporating both a single-qubit and a two-qubit gate

### Answer

Writing the sequence of operations applied to the input qubits and using subscripts to label the qubits and the operations to show which qubit the gates are operating on, gives

acts on so

so now

Note that is the control qubit and is the target qubit. Consequently, when operates on , look at to decide whether is flipped. Again, adding subscripts to identify the qubits,

This is the final state which is measured. It is an entangled state. There are two possible outcomes; either is measured as and is measured as or is measured as and is measured as . From the coefficients, the conclusion is that each outcome has a probability of 1/2.

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