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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
(47)
= exp(cid:0) −1f (cid:0) ν (ϕ )fC(ϕ ),ν (ϕ )fC(ϕ )(cid:1)(cid:1) ,
c d k r,i k r,i l r,i l r,i
where c = 2σ2, while f (·) yields the L2-distance in H,
d
f (cid:0) ν (ϕ )fC(ϕ ),ν (ϕ )fC(ϕ )(cid:1)
d k r,i k r,i l r,i l r,i
(48)
=(cid:13) (cid:13)ν k(ϕ r,i)f kC(ϕ r,i)−ν l(ϕ r,i)f lC(ϕ r,i)(cid:13) (cid:13)2 2.
For a given φ (ϕ(cid:126) ) and φ (ϕ(cid:126) ), an f (φ (ϕ(cid:126) ),φ (ϕ(cid:126) )) → δK average is yielded as
k r l r A c r c r +
ς(φ (ϕ(cid:126) ),φ (ϕ(cid:126) )) = (φ (ϕ(cid:126) ))Tφ (ϕ(cid:126) )
c r c r c r c r
L L (49)
= (cid:88) ν2(ϕ )(cid:0) fC(ϕ )(cid:1)2 ≤ (cid:88) ν (ϕ )fC(ϕ ),
c r,i c r,i c r,i c r,i
i=1 i=1
9
where δK refers to the space of K×K symmetric positive semi-definite matrices [35–37], while the
+
ι inner products of φ (ϕ(cid:126) ) and φ (ϕ(cid:126) ) are represented in δK via ι(φ (ϕ(cid:126) ),φ (ϕ(cid:126) )) → δK as
k r l r + k r l r +
ι(φ (ϕ(cid:126) ),φ (ϕ(cid:126) )) = (φ (ϕ(cid:126) ))Tφ (ϕ(cid:126) )
k r l r k r l r
L (50)
= (cid:88) ν (ϕ )ν (ϕ )(cid:0) fC(ϕ )(cid:1)(cid:0) fC(ϕ )(cid:1) .
k r,i l r,i k r,i l r,i
i=1
The ϕ(cid:126) sequence is classified into a given class from set C, as given in Algorithm 2 (A ).
r C
Algorithm 2 Learning the Classification of the Stabilized Quantum States of the Quantum
Computer
Step 1. Let ϕ(cid:126) be the r-th sequence of the quantum computer, with the L stabilized gate
r
parameters ϕ ,...,ϕ .
r,1 r,L
Step 2. Define set C of the K stability classes via (37).
Step 3. Select k that identifies k-th stability class C , and learn function ρ(φ (ϕ(cid:126) ),φ (ϕ(cid:126) ))
k k r l r
(46) using the K kernel machine (44) for all l, l (cid:54)= k.
Step 4. Determine (cid:96) (ϕ(cid:126) ) = maxρ(φ (ϕ(cid:126) ),φ (ϕ(cid:126) )).
k r k r l r
∀l
Step 5. Repeat steps 3-4 for all k.
Step 6. Determine ξ(ϕ(cid:126) ) = maxφ (ϕ(cid:126) ).
r k r
∀k
Step 7. Classify ϕ(cid:126) into stability class C(ϕ(cid:126) ) via the set C as
r r
C(ϕ(cid:126) ) = ξ(ϕ(cid:126) )C +(cid:96) (ϕ(cid:126) )C ,
r r p k r q
where p indexes the maximal φ (ϕ(cid:126) ) in ξ(ϕ(cid:126) ), while q indexes the maximal φ (ϕ(cid:126) ) in
k r r l r
ξ(ϕ(cid:126) ).
r
Step 8. Repeat steps 1-8 for all r.
Step 9. Output the stability classes C(β) = [C(ϕ(cid:126) ),...,C(ϕ(cid:126) )]T of the stabilized
1 R
quantum states |ϕ(cid:126) (cid:105), r = 1,...,R of the quantum computer.
r
5 Numerical Evaluation
5.1 System Stability
Let |φ(cid:105) be the stabilized system state of the quantum computer formulated by R output systems,
|ϕ(cid:126) (cid:105), r = 1,...,R, as
r
|φ(cid:105) = |ϕ(cid:126) (cid:105)⊗···⊗|ϕ(cid:126) (cid:105), (51)
1 R
with gate parameters β, as given in (11).
Then, let |φ∗(cid:105) be a target stabilized system of the quantum computer, as
|φ∗(cid:105) = |ϕ(cid:126)∗(cid:105)⊗···⊗|ϕ(cid:126)∗(cid:105), (52)
1 R
with target gate parameters β∗, as
β∗ = [ϕ(cid:126)∗,...,ϕ(cid:126)∗] (53)
1 R
10
(cid:104) (cid:105)T
where ϕ(cid:126)∗ = ϕ∗ ,...,ϕ∗ .
r r,1 r,L
Then, let [β] refer to the gate parameter ϕ of an l-th unitary of an r-th running sequence of
rl r,l
the quantum computer, l = 1,...,L, r = 1,...,R, in the state |φ(cid:105), and let [β∗] identify the target
rl
gate parameter ϕ∗ in state |φ∗(cid:105).
r,l
Then, let D(β(cid:107)β∗) be the relative entropy between β and β∗, as
(cid:88)(cid:16) (cid:17)
D(β(cid:107)β∗) = [β] log [β] rl +[β∗] −[β] . (54)
rl [β∗] rl rl
rl
r,l
where D(β(cid:107)β∗) ≥ 0, and let f (r) ≥ 0 be a function that returns the value of the relative
D(β(cid:107)β∗)
entropy function for an r-th running sequence as
(cid:88)(cid:16) (cid:17)
f (r) = [β] log [β] rl +[β∗] −[β] . (55)
D(β(cid:107)β∗) rl [β∗] rl rl
rl
l