t b a b a

m index qubits to control the job-machine information encoding. As each index qubit is
prepared in state 1/ √ 2(| 0〉+ | 1〉), the target qubits will be prepared in superpositions of
all possible job-machine states, e j
i , 1 ≤ i ≤ n, 1 ≤ j ≤ m.
Example: Table 4.1 summarizes the processing time of 8 jobs on 4 machines, where 0 <
Tij < 2 4 = 16. Thus n = 3, m = 2, and q = 4. The running time of job J1 on machines
M1, M2, M3 and M4 are, respectively, 1, 3, 7, and 15 units of time.
Table 4.1: An example of 8 jobs running on 4 machines
Job/Machine M1 M2 M3 M4
J1 1 3 7 15
J2 2 1 9 3
J3 6 2 5 8
J4 11 13 7 4
J5 15 12 3 10
J6 10 7 8 14
J7 5 2 3 9
J8 1 10 11 13
The four vectors used to encode the processing time of job J1 on machines M1, M2, M3
and M4 are respectively:
| e 1 1〉 =| 000001〉, | e 2 1〉 =| 010011〉, | e 3 1〉 =| 100111〉, and | e 4 1〉 =| 111111〉.
For J2 the basis vectors are | 000010〉, | 010001〉, | 101001〉, | 110011〉, and so on.
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