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1 2
≡
Here, σ =I⊗3m, we have used the relation
0
σ σ σ ρσ σ σ =0
α β α α γ α
Xα
for any ρ and β =γ, and defined
6
Ck 2 =D .
| β| β
Xk
Note that
D = Ck 2 =1.
β | β|
Xβ Xβ,k
It is obvious that
Tr I⊗m ψ 0 ⊗m + ⊗m ρ =0.
1
h(cid:0) − (cid:1)⊗ ⊗ × i
7
Furthermore,
Tr I⊗m ψ 0 ⊗m + ⊗m ρ
2
h(cid:0) − (cid:1)⊗ ⊗ × i
= 1 D Tr I⊗m ψ 0 ⊗m + ⊗m P†σ P ψ 0 ⊗m + ⊗m P†σ P
β β β
(3m)! PX,β6=0 h(cid:0) − (cid:1)⊗ ⊗ × (cid:0) ⊗ ⊗ (cid:1) i
1
D (2m (3m 1)!)
β
≤ (3m)! × −
βX6=0
2m (3m 1)!
× −
≤ (3m)!
2
= .
3
Therefore,
Tr I⊗m ψ 0 ⊗m + ⊗m ρ 2 ,
before
h(cid:0) − (cid:1)⊗ ⊗ × i≤ 3
which means
Tr ψ 0 ⊗m + ⊗m ρ Tr I⊗m 0 ⊗m + ⊗m ρ 2
before before
h ⊗ ⊗ × i ≥ h ⊗ ⊗ × i− 3
2
1 ǫ
≥ − − 3
1
= ǫ,
3 −
where we have used the assumption that
p =Tr I⊗m 0 ⊗m + ⊗m ρ 1 ǫ.
ψpass before
h ⊗ ⊗ × i≥ −
Therefore
1 ⊗m ⊗m 1
ψ 0 + ρ 1 ǫ
2(cid:13) ⊗ ⊗ − before (cid:13)1 ≤ r −(cid:16)3 − (cid:17)
(cid:13) (cid:13)
(cid:13) (cid:13) 2
= +ǫ.
r3
--- End of pdfs/document_7.pdf ---
--- Start of pdfs/document_8.pdf ---
State Stabilization for Gate-Model Quantum Computers
Laszlo Gyongyosi∗ Sandor Imre†
Abstract
9102
Gate-modelquantumcomputerscanallowquantumcomputationsinnear-termimplementa-
tions. Thestabilizationofanoptimalquantumstateofaquantumcomputerisachallenge,since
itrequiresstablequantumevolutionsviaaprecisecalibrationoftheunitaries. Here,wepropose
peS a method for the stabilization of an optimal quantum state of a quantum computer through an
arbitrary number of running sequences. The optimal state of the quantum computer is set to
maximize an objective function of an arbitrary problem fed into the quantum computer. We
3 also propose a procedure to classify the stabilized quantum states of the quantum computer
into stability classes. The results are convenient for gate-model quantum computations and
]hp-tnauq[
near-term quantum computers.
1 Introduction
Quantum computers can make possible quantum computations for efficient problem solving [4–22].
Gate-based quantum computations represent a way to construct gate-model quantum computers.
In a gate-model quantum computer architecture, computations are implemented via sequences
1v44010.9091:viXra
of unitary operations [12–15,22–30]. Gate-model quantum computers allow establishing experi-
mental quantum computations in near-term architectures [1–3,38–47]. Practical demonstrations
of gate-model quantum computers have been already proposed [4–15] and several physical-layer
developments are currently in progress.
Finding a stable quantum state of a quantum computer is a challenge, since it requires precise
unitaries that yield stable quantum evolutions in the quantum computer. The problem is further
increased if the stable system state must be available for a pre-determined time or for a pre-
determined number of running sequences. Particularly, the quantum state of a quantum computer
subject to stabilization also coincides with the optimal quantum state. The optimal quantum state
of a quantum computer maximizes a particular objective function of an arbitrary computational
problem fed into the quantum computer. The problem therefore is to fix the quantum state of