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quantum states of β.
TheresolutionsofProblems1and2areproposedinTheorems1and2. Thesolutionframework
F is defined via a P stabilization procedure with an embedded stabilization algorithm A (see
S S
Theorem 1), and via an A classification algorithm that characterizes the stability class of the
C
results of P (see Theorem 2). Fig. 1 depicts the system model.
S
3
(cid:30)
(cid:40)
S
(cid:90)
1
C (cid:9)(cid:67)(cid:10)
(cid:66) S (cid:67)
(cid:35) QG (cid:25) (cid:25)
C
S
(cid:90)
R
Figure 1: The framework F for the stabilization of the optimal state of the quantum computer and
the stability-class determination. In the R running sequences, R input systems are fed into the
input of the quantum computer, in an r-th running sequence, r = 1,...,R, an r-th input system,
(cid:80)
|ψ (cid:105) = α |i(cid:105), is evolved via the sequence of the L uniaries of the quatum computer. The R
r i i
running sequences identify an input system |ψ (cid:105) = |ψ (cid:105)⊗...⊗|ψ (cid:105) (considering that the R input
in 1 R
systems are unentangled). The R running sequences of the QG structure of the quantum computer
produces α = [θ(cid:126)∗,...,θ(cid:126)∗], where θ(cid:126)∗ = [θ∗ ,...,θ∗ ]T. The P stabilization procedure outputs
1 R r r,1 r,L S
β = [ϕ(cid:126) ,...,ϕ(cid:126) ], where ϕ(cid:126) = [ϕ ,...,ϕ ]T, via an embedded stabilization algorithm A that
1 R r r,1 r,L S
determines the S stabilizer matrix. The C(β) stability-level of the resulting β is determined via a
classification algorithm A . The P and A methods are realized as unsupervised learning.
C S S
4
3 Stabilization of the Optimal State of the Quantum Computer
Theorem 1 The S matrix for the stabilization of the |θ(cid:126)∗(cid:105) optimal state of the quantum computer
via β = STα, can be determined via the minimization of an objective function F∗.
Proof. For an r-th sequence of the quantum computer, define ∆(θ(cid:126)∗) and ∆(ϕ(cid:126) ) as
r r
∆(θ(cid:126)∗) = θ(cid:126)∗−θ(cid:126)∗ (14)
r r r+1
and
∆(ϕ(cid:126) ) = ϕ(cid:126) −ϕ(cid:126) , (15)
r r r+1
respectively. These vectors formulate ∆α and ∆β as
∆α = [∆(θ(cid:126)∗),...,∆(θ(cid:126)∗ )] (16)
1 R−1
and
∆β = [∆(ϕ(cid:126) ),...,∆(ϕ(cid:126) )], (17)
1 R−1
respectively. Then, using equations (16) and (17) for the r = 1,...,R−1 sequences, let χ be a
sum defined as
R−1
(cid:88)
χ = (cid:107)∆(ϕ(cid:126) )(cid:107)2
r 2
r
(cid:16) (cid:17) (18)
= Tr ∆β(∆β)T
(cid:16) (cid:16) (cid:17) (cid:17)
= Tr ST ∆α(∆α)T S ,
where (cid:107)·(cid:107)2 is the squared L2-norm, Tr(·) is the trace operator, ∆α is as given in equation (16),
2
and ∆β is as in equation (17).
For the r-th and s-th sequences, s > r, with ∆(ϕ(cid:126) ) and ∆(ϕ(cid:126) ), let γ be defined as
r s rs
γ = ω (cid:107)∆(ϕ(cid:126) )−∆(ϕ(cid:126) )(cid:107)2, (19)
rs rs r s 2
where ω is a weight coefficient defined as
rs
 (cid:18) (cid:107)∆(θ(cid:126)∗)−∆(θ(cid:126)∗)(cid:107)2(cid:19)
 exp − r s ,if (s−r) ≤ κ
ω = ζ , (20)
rs
 0, otherwise
where κ and ζ are nonzero parameters.
A sum is defined for the r = 1,...,R−1 sequences of the quantum computer as
R−1R−1
(cid:88) (cid:88)
τ = γ . (21)
rs
r s
At a particular S in equations (18) and (21), the stabilization of the optimal state of the quantum
computer through the R sequences can be reformulated via an objective function F∗, subject to a
minimization as
F∗ = argmin (χ+cτ)
S
(cid:16) (cid:16) (cid:16) (cid:17) (cid:17) (cid:17) (22)
= argmin Tr ST ∆α(∆α)T S +cτ ,
S
5
where c is a regularization constant [31,32]. The F∗ objective function therefore stabilizes the
optimal state via the minimization of χ, while the term cτ achieves stabilization between the
sequences.
Then, let W be the weight matrix formulated via the coefficients (20) with W = ω , and let
rs rs