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the quantum computer in the optimal state for an arbitrary number of running sequences that is
determined by the actual environment or by the current problem. Another challenge connected to
the problem of stabilization of the system state of a quantum computer is the classification of the
∗School of Electronics and Computer Science, University of Southampton, Southampton SO17 1BJ, U.K., and
DepartmentofNetworkedSystemsandServices,BudapestUniversityofTechnologyandEconomics,1117Budapest,
Hungary, and MTA-BME Information Systems Research Group, Hungarian Academy of Sciences, 1051 Budapest,
Hungary.
†Department of Networked Systems and Services, Budapest University of Technology and Economics, 1117 Bu-
dapest, Hungary.
1
sequences of the stabilized quantum states into stability-classes. Practically, a solution to these
problems can be covered by an unsupervised learning method.
Here, we propose a method for the stabilization of an optimal quantum state of a quantum
computer through an arbitrary number of running sequences. We define a solution that utilizes
unsupervised learning algorithms to determine the stable quantum states of the quantum computer
and to classify the stable quantum states into stability classes. The proposed results are useful for
experimental gate-based quantum computations and near-term quantum computer architectures.
The novel contributions of our manuscript are as follows:
1. Weproposeamethodforthestabilizationofanoptimalquantumstateofaquantumcomputer
through an arbitrary number of running sequences.
2. We define a solution that utilizes unsupervised learning algorithms to determine the stable
quantum states of the quantum computer.
3. We evaluate a solution to classify the stable system states into stability classes.
This paper is organized as follows. Section 2 provides the problem statement. Section 3 discusses
the stabilization procedure of an optimal quantum state of a quantum computer. Section 4 defines
an unsupervised learning method to find the stable quantum states and the stability classes of the
stabilized quantum states. In Section 5, a numerical evaluation is proposed. Finally, Section 6
concludes with the results. Supplemental information is included in the Appendix.
2 Problem Statement
Let QG be the quantum gate structure of a gate-model quantum computer with a sequence of L
unitaries [12–15] with an n-length input system |ψ(cid:105),
dn−1
(cid:88)
|ψ(cid:105) = α |i(cid:105), (1)
i
i=0
where d is the dimension (d=2 for a qubit system), (cid:80)dn−1|α |2 = 1, and let
i=0 i
|θ(cid:126)∗(cid:105) = U (θ∗)U (cid:0) θ∗ (cid:1) ...U (θ∗)|ψ(cid:105) (2)
L L L−1 L−1 1 1
betheoptimalsystemstateofthequantumcomputerthatmaximizesaparticularobjectivefunction
f(θ(cid:126)∗),
f(θ(cid:126)∗) = (cid:104)θ(cid:126)∗|C|θ(cid:126)∗(cid:105) (3)
of an arbitrary problem fed into the quantum computer, where C is the classical value of the
objective function, while θ(cid:126)∗ is the gate parameter vector,
θ(cid:126)∗ = [θ∗,...,θ∗]T (4)
1 L
that identifies the L unitaries, U (θ∗),...,U (θ∗), of the QG quantum circuit of the quantum
1 1 1 L
computer in the optimal state |θ(cid:126)∗(cid:105), such that an i-th unitary, U (θ∗) is as [13]
i i
U (θ∗) = exp(−iθ∗P), (5)
i i i
2
where θ∗ is the gate parameter (real continuous variable) of unitary U , P is a generalized Pauli
i i
operator formulated by the tensor product of Pauli operators {X,Y,Z} [13,14].
The aim is to stabilize the |θ(cid:126)∗(cid:105) optimal state of the quantum computer through R running
sequences via unsupervised learning of the evolution of the unitaries in the quantum computer.
TheR runningsequencesreferstoR inputsystemsfedintotheinputofthequantumcomputer,
such that in an r-th running sequence, r = 1,...,R, an r-th input system, |ψ (cid:105) (defined as in (1)),
r
is evolved via the sequence of the L uniaries of the quatum computer. The R running sequences
identify an input system, |ψ (cid:105), formulated via R, n-length quantum systems, as
in
|ψ (cid:105) = |ψ (cid:105)⊗...⊗|ψ (cid:105), (6)
in 1 R
where it is considered that the R input systems are unentangled.
Let ϕ(cid:126) be the gate parameter vector associated with the stable system state |ϕ(cid:126)(cid:105),
|ϕ(cid:126)(cid:105) = U (ϕ )U (ϕ )...U (ϕ )|ψ(cid:105) (7)
L L L−1 L−1 1 1
as
ϕ(cid:126) = [ϕ ,...,ϕ ]T , (8)
1 L
where ϕ ∈ [0,π] is the gate parameter of unitary U in the stabilized system state |ϕ(cid:126)(cid:105), such that
i i
the objective function value is stabilized into
f(ϕ(cid:126)) = (cid:104)ϕ(cid:126)|C|ϕ(cid:126)(cid:105) = f(θ(cid:126)∗). (9)
For the R sequences of the quantum computer, we define matrices α and β as
α = [θ(cid:126)∗,...,θ(cid:126)∗], (10)
1 R
where θ(cid:126)∗ = [θ∗ ,...,θ∗ ]T identifies the quantum state |θ(cid:126)∗(cid:105) of an r-th running sequence of the
r r,1 r,L r
quantum computer, while
β = [ϕ(cid:126) ,...,ϕ(cid:126) ], (11)
1 R
where ϕ(cid:126) = [ϕ ,...,ϕ ]T, identifies the stabilized quantum state |ϕ(cid:126) (cid:105) of an r-th sequence of the
r r,1 r,L r
quantum computer.
The problem therefore is to find β from α that stabilizes the |θ(cid:126)∗(cid:105) optimal state of the quantum
computer through R sequences as
β = STα, (12)
where S is a stabilizer matrix,
STS = I, (13)
and I is the identity matrix.
The problems to be solved are therefore summarized as follows.
Problem 1 Find S to construct β (11) from α (10) to stabilize the quantum computer in |θ(cid:126)∗(cid:105) via
|ϕ(cid:126)(cid:105) for all running sequences.
Problem 2 Describe the stability of β via unsupervised learning of the stability levels of the |ϕ(cid:126) (cid:105)
r