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Figure 2: The f (r) relative entropy values between the gate parameters of the stabilized
D(β(cid:107)β∗)
system |ϕ(cid:126) (cid:105) and target system |ϕ(cid:126)∗(cid:105) for R running sequences, r = 1,...R, R = 10, E(D(β(cid:107)β∗)) =
r r
0.1. (a) N = 1. (b) N = 2. (c) N = 3. (d). Stability parameter δ(r) for the different relative
entropy values.
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where we set X as X = 2C2N24π2(cid:14) R2, while C > 0 is a constant, thus (66) is evaluated as
R
(cid:90)
F (f(ϕ(cid:126) )) = 1 2C2N24π2 cos2(cid:0) CN2πr(cid:1) dr
r R R2 R
1
N (cid:90)2π (68)
= 2C2N24π2 1 2cos2(cid:0) r(cid:48)(cid:1) dr(cid:48)
R2 N2π
1
C2N24π2
= .
R2
For the target system |ϕ(cid:126)∗(cid:105), the constant C∗ is set as
r
f(ϕ(cid:126)∗) = 2(C∗)2N24π2 cos2(cid:0) C∗N2πr(cid:1) , (69)
r R2 R
while for |ϕ(cid:126) (cid:105), we set C as C > C∗, thus (65) can be evaluated as
r
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:32)(cid:18) C2N24π2(cid:19)(cid:32) (C∗)2N24π2(cid:33)(cid:33)
(cid:12) (cid:12)
(cid:12) (cid:116)F(cid:32)F f(ϕ(cid:126)f r( )ϕ(cid:126) −r) C− Ff ( (cid:32)ϕ(cid:126)∗ f) (− (cid:12)
µ(β,β∗) = (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)(cid:118) (cid:117) (cid:117) (cid:18) 2N R2 24R π2 2(cid:19)2(cid:33)  r r)−(CR ∗2 )2 RN 224π2(cid:33)2 (cid:12) (cid:12) (cid:12) (cid:12) (cid:12), (70)
(cid:117) ϕ(cid:126)∗
(cid:12) (cid:12)
where
(cid:16)(cid:16) C2N24π2(cid:17)(cid:16) (C∗)2N24π2(cid:17)(cid:17)
F f(ϕ(cid:126) )− f(ϕ(cid:126)∗)−
r R2 r R2
= F (cid:16) f(ϕ(cid:126) )f(ϕ(cid:126)∗)−f(ϕ(cid:126) ) (C∗)2N24π2 − C2N24π2 f(ϕ(cid:126)∗)+ C2N24π2(C∗)2N24π2(cid:17)
r r r R2 R2 r R2 R2 (71)
(cid:18) (cid:19)
= 1 2π3C2(C∗)2N3((C∗−C)sin(N4π(C∗+C))+(C∗+C)sin(N4π(C∗−C))) ,
R ((C∗)2−C2)R3
and
(cid:18) (cid:19) (cid:18) (cid:19)
(cid:16) (cid:17)2 (cid:16) (cid:17)2
F f(ϕ(cid:126) )− C2N24π2 = F f(ϕ(cid:126) )2−2f(ϕ(cid:126) ) C2N24π2 + C2N24π2
r R2 r r R2 R2
(cid:16) C4N48π4(cid:17) (72)
= 1
R R3
C4N48π4
= ,
R4
thus (70) is simplified as
(cid:12) (cid:12)
(cid:12) (cid:12)
µ(β,β∗) = (cid:12) (cid:12)2π3C2(C∗)2N3((C∗−C)sin(N4 (cid:114)π(C∗+C))+(C∗+C)sin(N4π(C∗−C)))(cid:12) (cid:12). (73)
(cid:12) C4N48π4(C∗)4N48π4 (cid:12)
(cid:12) (((C∗)2−C2)R4) (cid:12)
R4 R4
The values of (73) are depicted in Fig. 3.
InFig.4thedistributionofµ(β,β∗)infunctionofC andC∗ aredepicted, C ∈ [0,1], C∗ ∈ [0,1]
for different values of N, f(ϕ(cid:126) ) and f(ϕ(cid:126)∗) are evaluated as given in (67) and (69), L = 1, and
r r
R = 10 .
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Figure3: Gateparametervaluesoff(ϕ(cid:126) )andf(ϕ(cid:126)∗)infunctionofrunningsequencer,r = 1,...,R,
r r
R = 10, L = 1, for different N, C and C∗, C ∈ [0,1], C∗ ∈ [0,1]. (a-b) N = 1, C = 0.125, C∗ = 0.1.
(c-d) N = 1, C = 0.3, C∗ = 0.2 (e-f) N = 2, C = 0.5, C∗ = 0.3. (g) The µ(β,β∗) correlation
values between the gate parameters of (a-f), S is an indexing parameter.
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(cid:78)(cid:9) (cid:67),(cid:67)(cid:11)(cid:10) (cid:78)(cid:9) (cid:67),(cid:67)(cid:11)(cid:10) (cid:78)(cid:9) (cid:67),(cid:67)(cid:11)(cid:10)
1 1 1
0.9 0.9 0.9
0.8 0.8 0.8
0.7 0.7 0.7
0.6 0.6 0.6 0,8-1
C C C 0,6-0,8
0.5 0.5 0.5 0,4-0,6
0,2-0,4
0.4 0.4 0.4 0-0,2
0.3 0.3 0.3
0.2 0.2 0.2
0.1 0.1 0.1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
C(cid:11) C(cid:11) C(cid:11)
(a) (b) (c)
Figure 4: The distribution of µ(β,β∗) in function of C and C∗ at f(ϕ(cid:126) ) and f(ϕ(cid:126)∗), L = 1, R = 10.
r r
(a) N = 1. (b) N = 2. (c) N = 3.
6 Conclusions
Here, we defined a method for the learning of stable quantum evolutions in gate-model quantum
computer architectures. The model stabilizes an optimal state of a quantum computer to maximize
theparticularobjectivefunctionofanarbitraryproblemfedintothequantumcomputer. Themodel
learns a stabilizer matrix that stabilizes the state of the quantum computer through an arbitrary
number of run sequences. We also defined a scheme to characterize the stability of the stabilized
states via unsupervised learning of the stability classes of the stabilized sequences. The results