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C = {C ,...,C }, (37)
1 K
where C , k = 1,...,K, is the k-th stability class.
k
Theorem 2 TheC(ϕ(cid:126) ), r = 1,...,R stabilityclassofaϕ(cid:126) stabilizedsequence, ϕ(cid:126) = ϕ ,...,ϕ ,
r r r r,1 r,L
of the quantum computer can be learned via φ (ϕ(cid:126) ) = (ν (ϕ(cid:126) ))T fC(ϕ(cid:126) ) quantities in the high-
k r k r k r
dimensional Hilbert space H, where ν (ϕ ) = 1 (ϕ ), and fC(ϕ ) ∈ [0,1] is a probability.
k r,i π r,i k r,i
Proof. Since the gate parameters are stabilized, the gate parameters ϕ(cid:126) and ϕ(cid:126) of the r-th and
r r+1
(r+1)-thsequencesmustbecorrelatedinthestablesystemstate|ϕ(cid:126)(cid:105)(7)ofthequantumcomputer.
Letβ fromequation(11)betheRstabilizedsequences,whereϕ(cid:126) isthestabilizedgateparameter
r
vector of an r-th sequence of the quantum computer, and let S be the set of all sequences of gate
parameters as
(cid:91)
S = {ϕ |ϕ ∈ ϕ(cid:126) }. (38)
r,i r,i r
ϕ(cid:126)r∈β
For an k-th stabilization class C , a probabilistic classifier function fC [31,37] can be defined as
k k
fC : S → [0,1]. (39)
k
The goal is to learn a function that maps any ϕ(cid:126) sequence to the correct stability class. Applying
r
equation (39) on a given sequence ϕ(cid:126) , i.e., fC(ϕ(cid:126) ) therefore maps ϕ(cid:126) to a given stability class via
r k r r
the classification of each L gate parameter of the sequence.
Thus, an i-th stabilized gate parameter ϕ of an r-th sequence ϕ(cid:126) can be also classified into
r,i r
a particular stabilization class from C (37). The fC(ϕ ) ∈ [0,1], k = 1,...,K, classifier (39)
k r,i
is trained to classify [37] each of the ϕ gate parameters of ϕ(cid:126) , i = 1,...,L via outputting a
r,i r
corresponding probability that ϕ belongs to a given C class. For a particular ϕ , the sum of
r,i k r,i
the probabilities yields
K
(cid:88)
fC(ϕ ) = 1, (40)
k r,i
k=1
for all i.
Then, let ν (ϕ ) ≥ 0 be a weight parameter associated with a particular ϕ and k-th class
k r,i r,i
C , defined as
k
ν (ϕ ) = 1 (ϕ ), (41)
k r,i π r,i
which normalizes ϕ into the range of [0,1], ν (ϕ ) ∈ [0,1].
r,i k r,i
For an r-th sequence ϕ(cid:126) , a ν (ϕ(cid:126) ) collection can be defined as
r k r
ν (ϕ(cid:126) ) = [ν (ϕ ),...,ν (ϕ )], (42)
k r k r,1 k r,L
(cid:80)L
where ν (ϕ ) = 1.
i=1 k r,i
8
From equations (39) and (42), the φ (ϕ(cid:126) ) evolution of a particular sequence ϕ(cid:126) with respect to
k r r
a k-th class C is defined as
k
φ (ϕ(cid:126) ) = (ν (ϕ(cid:126) ))TfC(ϕ(cid:126) )
k r k r k r
(43)
= (cid:2) ν (ϕ )fC(ϕ ),...,ν (ϕ )fC(ϕ )(cid:3) .
k r,1 k r,1 k r,L k r,L
Since the φ (ϕ(cid:126) ) term (43) is a non-linear map, the problem of correlation analysis [31,37] between
k r
the inner products of non-linear functions φ (ϕ(cid:126) ) and φ (ϕ(cid:126) ) can be reformulated via a kernel
k r l r
machine K [34–36] as K(φ (ϕ(cid:126) ),φ (ϕ(cid:126) )), which yields a distance in a high-dimensional Hilbert
k r l r
space H. This distance in H can therefore be used as a metric to describe the correlation between
φ (ϕ(cid:126) ) and φ (ϕ(cid:126) ).
k r l r
Let X be the input space and let K be an arbitrary kernel machine, defined for a given x,y ∈ X
via the kernel function
K(x,y) = Γ(x)T Γ(y), (44)
where
Γ : X → H (45)
is a nonlinear map from X to the high-dimensional reproducing kernel Hilbert space (RKHS) H
associated with K. Without a loss of generality, dim(H)(cid:29)dim(X), and we assume that the map
Γ in equation (45) has no inverse.
Then, for a φ (ϕ(cid:126) ) and φ (ϕ(cid:126) ), let ρ(φ (ϕ(cid:126) ),φ (ϕ(cid:126) )) → H be the correlation identifier, as
k r l r k r l r
ρ(φ (ϕ(cid:126) ),φ (ϕ(cid:126) ))
k r l r
= K(φ (ϕ(cid:126) ),φ (ϕ(cid:126) ))
k r l r
(46)
L
= (cid:88) K(cid:0) ν (ϕ )fC(ϕ ),ν (ϕ )fC(ϕ )(cid:1) .
k r,i k r,i l r,i l r,i
i=1
Assuming that K is a Gaussian kernel [34–36] in equation (46), for an i-th gate parameter the
kernel function is