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1538 JOURNAL OF COMPUTERS, VOL. 8, NO. 6, JUNE 2013
the visibility at X
i
. The basic framework of BIQACA
described as follows:
1) Selecting the target position of ant movement
By applying the principle of randomness, a number of
qubits in the current position were randomly selected to
constitute a position update vector S. The transition rule
and transition probability of ant k from position X to
position
where q∈[0, 1] is even-distributed random number, q 0
∈[0, 1] is probability parameter, P is the set of occupied
points for ant in unit space, X
s
tion as per formula (8) ; α is the update parameter of
pheromone, β is the update parameter of visibility.
2) Realizing the movement of ant towards target position
via quantum rotation gate
After the ant has selected the target position, its
movement process can be realized by changing the phase
of qubit it brought for quantum rotation gate. In unit
space, suppose the current position for ant at time t is P
i
,
selected target position is P k , update vector of P i is S,
then the update of phase angle increment at P
i
is
where )}
t+ 1 t+ 1 t t t t+ 1 t t+
1
⎡cosϕij sinθ ⎤ ⎡
ij
cosϕij sinθ ⎤ ⎡
ij
cos( ϕij +Δ ϕij )sin( θij +Δθ
) ⎤
ij
⎢ t+ 1 t+ 1⎥ ⎢ t t ⎥ ⎢ t t+ 1 t t+
1 ⎥
⎢sinϕij sinθij ⎥ = U ⎢sinϕij sinθij ⎥ = ⎢sin( ϕij +Δ ϕij )sin( θij +Δθij
) ⎥
⎢
t+ 1 t t t 1
cosθ ⎥ ⎢
ij
cosθ ⎥ ⎢
+
ij
cos( θij θ ) ⎥
⎣ ⎦ ⎣ ⎦ ⎣ +Δ
ij ⎦ (15)
Apparently, U-gate can rotate the phase of qubit by
t 1
ϕ +
t 1
Δ
ij
and Δ θ +
ij
.
3) Adjustment strategy of search space
i
In BIQACA, the search space for each qubit is designed
as [ lowBd ij , upBdij
] , the search space at initializa-
X
s
are:
α
β
⎧arg max{ τ ( X
s) iη
( Xs)} q ≤ q
tion is [ 0.25π
,0.75π
] , during optimizing process of ants,
0
⎪ Xs∈P
X
s
= ⎨ these search spaces are related with the contraction level
⎪
⎩ Xs
q > q0
(8) of each qubit, and decrease exponentially, which can significantly
improve the solution accuracy of the algorithm.
α
β
τ ( Xs) iη
( Xs)
t+
1
t
pX (
s
) =
α
β
τ ( X
u) η ( X
u)
X
∑ i ,
(9)
⎡lowBd
⎤ ⎡ 1 ⎤⎡ ij
lowBd ⎤
ij
⎢ t+
1 ⎥ = ⎢ t ⎥
ij
⎢ t ⎥ (16)
nL
⎢
s,
Xu∈P
⎣ upBdij
⎥⎦ ⎢⎣nf
⎥⎦⎢⎣upBdij
⎥⎦
where j ∈ { S(1),
S(2),
,
S(
sm)}
, S is the update vector
of ants, nf = 2 is the constriction factor, nL t represents
ij
is the selected target loca-
the contraction level of t-th iteration.
4) Processing of ant position variation
Suppose the current position is P i , update vector of P i
is S, the search space of P i is [ lowBd ij , upBdij
] . Then the
update of phase angle increment at P
i
is:
⎧ Δϕij
= (2rand
−1)(
upBdij
− lowBdij
)
⎨
⎩
Δϕij
= sign(
Δϕij
)( abs(
Δϕij
) + lowBdij
) (17)
⎧ Δθij
= (2rand
−1)(
upBdij
− lowBdij
)
⎨
⎩
Δθij
= sign(
Δθij
)( abs(
Δθij
) + lowBdij
) (18)
( φkj − φij ) × rand
t
⎧⎪
j
φkj ≠φij
Δ φij
= ⎨
(10) 5) Random behavior of ants
⎪⎩ Δ φij φkj = φij
If P i is not improved after continuous limited-time cycles,
the position should be abandoned, the ants will generate
a new P '
t
t
⎧Δ ϕij
+ 2π Δ ϕij
< −π
t+
1 ⎪
i through random behavior to substtute P i .
t t
Δ ϕij = ⎨ Δ ϕij −π ≤ Δϕij
≤ π (11)
'
⎪ Δ
t
t
⎩ ϕij
− 2π Δ ϕij
> π
ϕij = mean( ϕi
) + Δϕ ij
(19)
'
( θkj − θij ) × rand
t
⎧⎪
j
θkj ≠θ
θij = mean( θi
) + Δθ ij
ij
(20)
Δ θij
= ⎨
(12)
⎪⎩ Δ θij θkj = θij
where i ∈ { 1,2, ,
n}
, j ∈ { S(1),
S(2),
,
S(
sm)}
, S is the
update vector of current position, Δ ϕij
and Δ θij
are updated
using (17), (18), mean( θ
t
t
⎧Δ θij
+ π Δ θij
< −π
/2
t+
1 ⎪ t t
i ) is the mean value of
Δ θij = ⎨Δ θij −π /2 ≤ Δθij
≤ π /2 (13)
vector of phase angle θ
⎪ Δ
t
t
i at P i .
⎩ θij
− π Δ θij
> π /2
6) Update rules for pheromone intensity and visibility
When the ant completes a traverse, the current position
j ∈ { S(1),
S(2),
,
S(
sm , rand
j
is random num-
is mapped into the solution space of optimal problem
Δ ϕij
, Δ θij
can be obtained using from unit space, fitness function is calculated, and the
intensity and visibility of pheromone at current position
should be updated.
⎧τ( X
i) = (1 − ρ) τ( Xi) + ρτ( Xi)
t+ 1 t+ 1 t+ 1 t+ 1 t+ 1 t t+
1
⎨
⎡cos Δϕij cos Δθij −sin Δϕij cos Δθij sin Δ θij cos( ϕij +Δϕij
) ⎤
⎢ t+ 1 t+ 1 t+ 1 t+ 1 t t+
1 ⎥
⎩ τ ( X
i) = Qfit( X
i)
(21)
U = ⎢sin Δϕij cos Δθij cos Δϕij cos Δθij sin Δ θsin( ϕij +Δϕij
) ⎥
⎢
t+ 1 t t+ 1 t+
1
−sin Δθij −tan( ϕij / 2)sin Δθij cos Δθ
⎥
⎣ ij ⎦ (14)
ber between [0, 1];
(17), (18).
Update of probability amplitude of qubit based on
quantum rotation gate
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