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Concentrates for Typical Instances, arXiv:1812.04170 (2018).
[43] Zhou, L., Wang, S.-T., Choi, S., Pichler, H. and Lukin, M. D. Quantum Approximate Opti-
mization Algorithm: Performance, Mechanism, and Implementation on Near-Term Devices,
arXiv:1812.01041 (2018).
[44] Lechner, W. Quantum Approximate Optimization with Parallelizable Gates,
arXiv:1802.01157v2 (2018).
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Maximum Cut Problem, arXiv:1811.08419 (2018).
[46] Ho, W. W., Jonay, C. and Hsieh, T. H. Ultrafast State Preparation via the Quantum Approx-
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Circuit, Physical Review Letters, vol. 119, no. 18, p. 180511 (2017).
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A Appendix
A.1 Abbreviations
QG Quantum Gate structure of a gate-model quantum computer
RKHS Reproducing Kernel Hilbert Space
A.2 Notations
The notations of the manuscript are summarized in Table A.1.
Table A.1: Summary of notations.
QG Quantum gate structure of a gate-model quantum computer.
L Number of unitaries in the QG structure of the quantum computer.
U (θ ) An i-th unitary gate, U (θ ) = exp(−iθ P), where P is a generalized
i i i i i
Pauli operator formulated by a tensor product of Pauli operators
{X,Y,Z}, while θ is referred to as the gate parameter associated to
i
U (θ ).
i i
|θ(cid:126)(cid:105) System state of the quantum computer, |θ(cid:126)(cid:105) =
U (θ )U (θ )...U (θ ), where U (θ ) identifies an i-th
L L L−1 L−1 1 1 i i
unitary gate.
θ(cid:126) Gate parameter vector, a collection of gate parameters of the L uni-
taries, θ(cid:126)= [θ ,...,θ ,θ ]T.
1 L−1 L
C Classical objective function of a computational problem fed into the
quantum computer.
f(θ(cid:126)) Objective function of the quantum computer.
|θ(cid:126)∗(cid:105) Optimal state of the quantum computer.
θ(cid:126)∗ Gate parameter vector in the |θ(cid:126)∗(cid:105) system state, θ(cid:126)∗ = [θ∗,...,θ∗]T.
1 L
f(θ(cid:126)∗) Objective function value in the |θ(cid:126)∗(cid:105) system state.
P GeneralizedPaulioperatorformulatedbythetensorproductofPauli
operators {X,Y,Z}.
|ϕ(cid:126)(cid:105) Stable system state with objective function f(ϕ(cid:126)) = (cid:104)ϕ(cid:126)|C|ϕ(cid:126)(cid:105) = f(θ(cid:126)∗).
ϕ(cid:126) Gate parameter vector associated to the stable system state |ϕ(cid:126)(cid:105),
ϕ(cid:126) = [ϕ ,...,ϕ ]T.
1 L
θ(cid:126)∗ Gate parameter vector, identifies the quantum state |θ(cid:126)∗(cid:105) of an r-th
r r
running sequence, r = 1,...,R, of the quantum computer, θ(cid:126)∗ =
r
[θ∗ ,...,θ∗ ]T.
r,1 r,L
20
ϕ(cid:126) Gate parameter vector, identifies the stabilized quantum state |ϕ(cid:126) (cid:105)
r r
of an r-th sequence of the quantum computer, ϕ(cid:126) = [ϕ ,...,ϕ ]T.
r r,1 r,L
α Matrix, formulated via the R sequences of the quantum computer,
α = [θ(cid:126)∗,...,θ(cid:126)∗].
1 R
β Matrix, formulated via the R stabilized sequences of the quantum
computer, β = [ϕ(cid:126) ,...,ϕ(cid:126) ].
1 R
S Stabilizer matrix, yields β from α as β = STα, STS = I, where I is
the identity matrix.
F Solution framework.
P Stabilization procedure.
S
A Stabilization algorithm.
S
A Classification algorithm.
C
C(β) Stability-class of β.
∆(θ(cid:126)∗) Vector, defined for an r-th sequence of the quantum computer,
r
∆(θ(cid:126)∗) = θ(cid:126)∗−θ(cid:126)∗ .
r r r+1
∆(ϕ(cid:126) ) Vector, defined for an r-th stabilized sequence of the quantum com-
r
puter, ∆(ϕ(cid:126) ) = ϕ(cid:126) −ϕ(cid:126) .
r r r+1
∆α A collection of ∆(θ(cid:126)∗) vectors, ∆α = [∆(θ(cid:126)∗),...,∆(θ(cid:126)∗ )].
r 1 R−1
∆β A collection of ∆(ϕ(cid:126) ) vectors, ∆β = [∆(ϕ(cid:126) ),...,∆(ϕ(cid:126) )].
r 1 R−1
χ Sum defined via ∆β as χ = (cid:80)R−1(cid:107)∆(ϕ(cid:126) )(cid:107)2, where (cid:107)·(cid:107)2 is the
r r 2 2
squared L2-norm.
γ Parameter, defined as γ = ω (cid:107)∆(ϕ(cid:126) )−∆(ϕ(cid:126) )(cid:107)2, where ∆(ϕ(cid:126) )
rs rs rs r s 2 r
and ∆(ϕ(cid:126) ) are derived for an r-th and s-th sequences, s > r, while
s
ω is a weight coefficient.
rs
τ A sum, defined for the r = 1,...,R−1 sequences of the quantum
(cid:80)R−1(cid:80)R−1γ
computer, τ = .