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quantum

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r s rs
F∗ Objective function of the stabilization procedure.
c Regularization constant.
ω Weight coefficient for the r-th and s-th sequences, s > r,
rs
 (cid:18) (cid:107)∆(θ(cid:126)∗)−∆(θ(cid:126)∗)(cid:107)2(cid:19)
  exp − r s ,if (s−r) ≤ κ
ζ
ω = ,
rs
 0, otherwise
where κ and ζ are nonzero parameters.
W Weight matrix, W = ω .
rs rs
21
(cid:80)
η Diagonal matrix of the weight coefficients, η = ω , with rela-
rr s rs
tion (∆β)T η∆β = I.
σ Matrix, σ = I +c(η−W).
Ω Parameter, Ω = Tr(ST(∆αη(∆α)T)S).
λ Diagonal matrix of eigenvalues.
T Training set of random gate parameters of the QG-structure of the
quantum computer, T = (X ,...,X ), where X is a d-dimensional
1 q i
random vector.
T¯ Mean of all training samples.
Y Learned output for an r-th sequence.
r
(r)
y Learned j-th output for an i-th unitary of an r-th sequence.
i,j
(r) (r) (r) (r)
∆y˜ Difference, ∆y˜ = |y˜ −y˜ |.
i i i i+1
C A k-th stability class, k = 1,...,K, for the classification of the sta-
k
bility of the stabilized sequences of β.
C Set of K stability classes, C = {C ,...,C }.
1 K
C(ϕ(cid:126) ) Stability class of a stabilized sequence ϕ(cid:126) .
r r
C(β) Stability classes of all ϕ(cid:126) stabilized sequences, r = 1,...,R, of β,
r
C(β) = [C(ϕ(cid:126) ),...,C(ϕ(cid:126) )]T.
1 R
δK Space of K ×K symmetric positive semi-definite matrices.
+
X Input space.
K Kernel machine.
H Reproducing Kernel Hilbert Space (RKHS) associated with the ker-
nel machine K.
Γ A nonlinear map, Γ : X → H, from X to the high-dimensional
Hilbert space H associated with K.
f (x,y) L2 distance in H , f (x,y) = (cid:107)x−y(cid:107)2.
d d 2
fC Probabilistic classifier function, fC : S → [0,1], where S =
k k
(cid:83) {ϕ |ϕ ∈ ϕ(cid:126) }, (cid:80)K fC(ϕ ) = 1.
ϕ(cid:126)r∈β r,i r,i r k=1 k r,i
ν (ϕ ) Parameter, associated with a particular ϕ and k-th class C ,
k r,i r,i k
ν (ϕ ) = 1 (ϕ ).
k r,i π r,i
ν (ϕ(cid:126) ) CollectionofLparameters, ν (ϕ(cid:126) ) = [ν (ϕ ),...,ν (ϕ )], where
k r k r k r,1 k r,L
(cid:80)L
ν (ϕ ) = 1.
i=1 k r,i
22
φ (ϕ(cid:126) ) Non-linear map in H, defined for a stabilized sequence ϕ(cid:126) as
k r r
φ (ϕ(cid:126) ) = (ν (ϕ(cid:126) ))T fC(ϕ(cid:126) ), where ν (ϕ ) = 1 (ϕ ), and
k r k r k r k r,i π r,i
fC(ϕ ) ∈ [0,1] outputs a probability.
k r,i
ι(·) Function, returns an inner product.
Z A parameter of procedure P .
S
B A parameter of procedure P .
S
(cid:96) (ϕ(cid:126) ) A parameter of algorithm A .
k r C
ξ(ϕ(cid:126) ) A parameter of algorithm A .
r C
23
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Authorized quantum computation
Yu Tanaka
Department of Physics, Graduate School of Science, University of Tokyo, Tokyo 113-0033 Japan
Advanced Materials Laboratories, Sony, Kanagawa 243-0021 Japan
Mio Murao
Department of Physics, Graduate School of Science, University of Tokyo, Tokyo 113-0033 Japan
PRESTO, JST, Kawaguchi, Saitama 332-0012, Japan
Institute for Nano Quantum Information Electronics, University of Tokyo, Tokyo 153-8505, Japan