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η be a diagonal matrix of the weight coefficients (20) with
(cid:88)
η = ω , (23)
rr rs
s
such that
(∆β)T η∆β = I. (24)
Using W and η, the F∗ objective function in equation (22) can be rewritten as
(cid:16) (cid:16) (cid:16) (cid:17) (cid:16) (cid:17)(cid:17)(cid:17)
F∗ = argmin 1 Tr ∆β(∆β)T +cTr ∆β(η−W)(∆β)T
Ω
S
(cid:16) (cid:16) (cid:16) (cid:17)(cid:17)(cid:17)
= argmin 1 Tr ∆β(I +c(η−W))(∆β)T
Ω
S
(25)
(cid:16) (cid:16) (cid:16) (cid:17)(cid:17)(cid:17)
= argmin 1 Tr ∆βσ(∆β)T
Ω
S
(cid:16) (cid:16) (cid:16) (cid:16) (cid:17) (cid:17)(cid:17)(cid:17)
= argmin 1 Tr ST ∆ασ(∆α)T S ,
Ω
S
where σ is as
σ = I +c(η−W), (26)
and
(cid:16) (cid:16) (cid:17) (cid:17)
Ω = Tr ST ∆αη(∆α)T S . (27)
At a particular ∆α (16) and σ (26), the S stabilizer matrix in equation (25) is evaluated via
(cid:16) (cid:17) (cid:16) (cid:17)
∆ασ(∆α)T S = λ ∆αη(∆α)T S, (28)
where λ is a diagonal matrix of eigenvalues [31,32].
AlgorithmA.1(A )givesthemethodforstabilizingtheoptimalstateofthequantumcomputer.
S
Algorithm 1 Stabilization of the Optimal State of the Quantum Computer
Step 1. Set the R number of sequences for the quantum state stabilization. Formulate α
(10) via R gate parameter vectors θ(cid:126)∗, r = 1,...,R.
r
Step 2. Set κ, and determine the ω weight coefficients via equation (20) for all r and s.
rs
Step 3. Set W, η, and σ (26).
Step 4. Compute the S stabilizer matrix via equation (28).
Step 5. Output β = STα via equation (12) for the stabilization of the optimal quantum
state |θ(cid:126)∗(cid:105) via the stable state |ϕ(cid:126)(cid:105) (7) through R sequences of the quantum computer.
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4 Learning the Stable Quantum State and Stability Class
Lemma 1 The stabilized sequences of the quantum computer can be determined via unsupervised
learning.
Proof. Algorithm 1 with the objective function (25) can be used to formulate an unsupervised
learning framework to find the stabilized unitaries. The steps are detailed in Procedure 1 (P ).
S
Procedure 1 Unsupervised Learning of Stable Quantum Evolutions
Step 1. Construct a T training set of random gate parameters of the QG-structure of the
quantum computer, as
T = (X ,...,X ), (29)
1 q
where X is a K-dimensional random vector, d ≤ R, formulated as
i
X = [θ ,...,θ ], (30)
i i,1 i,K
where θ is the gate parameter of U in X , and j is a random number.
i,j j i
Step 2. Determine the S stabilizer matrix via Algorithm 1.
Step 3. Compute Z = [z ,...,z ] as
1 q
(cid:0) STTT(cid:1)T
Z = = TS, (31)
and set B = [b ,...,b ] as
1 q
B = −ZTT¯, (32)
where T¯ is the mean of all training samples [32].
Step 4. For a given θ(cid:126)∗ of an r-th sequence, learn output Y as
r r
Y = ZTθ(cid:126)∗+B = (TS)T θ(cid:126)∗−(TS)T T¯. (33)
r r r
Step 5. For an i-th gate parameter θ∗ , learn the j-th output y(r) as
r,i i,j
y(r) = z ⊗θ∗ +b , (34)
i,j j r,i j
from which a statistical average for a given i, i = 1,...,L, is
q
y˜(r) = 1 (cid:88) |y(r) |, (35)
i q i,j
j=1
(r)
with difference ∆y˜ as
i
(r) (r) (r)
∆y˜ = |y˜ −y˜ |. (36)
i i i+1
Step 6. Repeat step 5 for all i.
Step 7. Repeat steps 1-5 for the R sequences.
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4.1 Learning the Sequence Stability of Stabilized Quantum States
Proposition 1 The stability of a given sequence ϕ(cid:126) can be characterized via K stability levels. The
r
sequence ϕ(cid:126) can be classified into K stability classes from set C,
r