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Let f∗ (r) be a target value for function (55), and let ∆(cid:0) f (r)(cid:1) be the difference [33]
D(β(cid:107)β∗) D(β(cid:107)β∗)
between f∗ (r) and (55), as
D(β(cid:107)β∗)
R
(cid:90)
∆(cid:0) f (r)(cid:1) = 1 ∂2 (r)dr, (56)
D(β(cid:107)β∗) R D(β(cid:107)β∗)
1
where ∂ (r) is the derivative of f (r).
D(β(cid:107)β∗) D(β(cid:107)β∗)
Using (56), we define a stability parameter δ to quantify the variation of the |ϕ(cid:126) (cid:105) stabilized
r
system state of the r-th running sequence of the quantum computer, as
(cid:18) (cid:113) (cid:19)−1
δ(r) = R ∆(cid:0) f (r)(cid:1) . (57)
2π D(β(cid:107)β∗)
For analytical purposes, let us assume that f (r) oscillates between a minimal value γ ≥ 0,
D(β(cid:107)β∗)
and a maximal value γ ≤ λ ≤ 1, defined as
(cid:0) (cid:1)
γ = argmin f (r) , (58)
D(β(cid:107)β∗)
∀r
and
(cid:0) (cid:1)
λ = argmax f (r) , (59)
D(β(cid:107)β∗)
∀r
therefore f (r) can be rewritten as
D(β(cid:107)β∗)
f (r) = csin(cid:0) N2πr(cid:1) +E(D(β(cid:107)β∗)), (60)
D(β(cid:107)β∗) R
where c is a constant, set as
c = 1 (λ−γ), (61)
2
while 0 ≤ E(D(β(cid:107)β∗)) ≤ 1 is an expected value of (54), set as
E(D(β(cid:107)β∗)) = c+γ, (62)
while N is the number of oscillations.
11
Therefore, (56) can be evaluated as
R
(cid:90)
∆(cid:0) f (r)(cid:1) = 1 2N24π2 cos2(cid:0) N2πr(cid:1) dr
D(β(cid:107)β∗) R R2 R
1
N (cid:90)2π (63)
= 2N24π2 1 2cos2(cid:0) r(cid:48)(cid:1) dr(cid:48)
R2 N2π
1
N24π2
= ,
R2
where r ∈ [r ,r +R], with r = 1.
0 0 0
Then, by using (63), the quantity in (57) is as
(cid:18) (cid:113) (cid:19)−1
δ(r) = R N24π2 = 1, (64)
2π R2 N
that identifies the inverse of the number of oscillations.
Therefore, (64) identifies the stability of the system state |ϕ(cid:126) (cid:105) of the quantum computer in
r
the r-th running sequence if f (r) has the form of (60). For an arbitrary f (r), the
D(β(cid:107)β∗) D(β(cid:107)β∗)
stability parameter δ(r) is evaluated via (57). The high value of δ(r) indicates that the stabilized
system |ϕ(cid:126) (cid:105) in (51) changes slowly. Particularly, if δ(r) ≥ δ∗(r), where δ∗(r) is a target value for
r
δ(r), then the system state |ϕ(cid:126) (cid:105) of the quantum computer is considered as stable.
r
The values of f (r) (60) and δ(r) (64) for R running sequences are depicted in Fig. 2.
D(β(cid:107)β∗)
5.2 Gate Parameter Correlations
Let |ϕ(cid:126) (cid:105) be the stabilized state of the quantum computer in the r-th running sequence, with
r
ϕ(cid:126) = [ϕ ,...,ϕ ]T,andlet|ϕ(cid:126)∗(cid:105)bethetargetstabilizedsystemstateinther-thrunningsequence,
r r,1 r,L r
(cid:104) (cid:105)T
with ϕ(cid:126)∗ = ϕ∗ ,...,ϕ∗ .
r r,1 r,L
Then, let µ be a correlation coefficient [33] that measures the correlation of the gate parameters
β and β∗ of |φ(cid:105) (51) and |φ∗(cid:105) (11), defined as
(cid:12) (cid:12)
(cid:12) (cid:12)
µ(β,β∗) = (cid:12) F((f(ϕ(cid:126)r)−F(f(ϕ(cid:126)r)))(f(ϕ(cid:126)∗ r)−F(f(ϕ(cid:126)∗ r)))) (cid:12), (65)
(cid:12)(cid:113) (cid:12)
(cid:12) F((f(ϕ(cid:126)r)−F(f(ϕ(cid:126)r)))2)F((f(ϕ(cid:126)∗ r)−F(f(ϕ(cid:126)∗ r)))2)(cid:12)
where |·| is the absolute value, f(ϕ(cid:126) ) is a function of r that represents the values of the gate
r
parameter vector ϕ(cid:126) , while F (·) is defined over r ∈ [1,R], as
r
R
(cid:90)
F (f(x)) = 1 f(x)dr. (66)
R
1
For illustration purposes, let us assume that L = 1, and f(ϕ(cid:126) ) is as
r
Xcos2(cid:0) CN2πr(cid:1)
f(ϕ(cid:126) ) = , (67)
r R