t b a b a

Table 4.2: The truth table for Max quantum circuit with two 2-qubit input registers | x〉, | y〉.
| x〉 | y〉 | Max〉 | x〉 | y〉 | Max〉
00 00 00 00 10 10
01 00 01 01 10 10
10 00 10 10 10 10
11 00 11 11 10 11
00 01 01 00 11 11
01 01 01 01 11 11
10 01 10 10 11 11
11 01 11 11 11 11
The output of a quantum circuit as in Figure 4.4 is an entangled state, rather than the
tensor product | T1〉⊗ | T2〉 ⊗ · · · ⊗ | TM〉⊗ | Cmax〉. This means that once we make a mea-
surement on the system, we project all the vectors T1, T2, . . . TM onto one special schedule
with its makespan Cmax. Recall that the | T 〉 was prepared in an equal superposition of run-
ning times of the machines under all possible schedules. Thus, such a Max operation prepares
the Cmax register in an equal superposition of the makespans of all possible schedules.
4.3.3 Searching the Schedule
Up to now, we discussed the implementation of the quantum circuits presented in Figure
4.1:
• We prepare each job vector | Ji〉 in a superposition state which includes the running
times on all machines. The first m qubits of a job vector are used for the index of
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