6
is the PTM of w in the Hilbert-Schmidt space. QPT
q
assumes that the initial state and final measurements
are known. In practice, these states and measurements
must be prepared using quantum gates, and these gates
{F ,F′} themselves may have imperfections [40]: r s
⟨⟨ρ |=⟨⟨0|F ,
r r
|ρ ⟩⟩=F′|0⟩⟩.
s s
Indeed, the initial states and final measurements that
werepreparedusinggatesarenotdirectlyknownandcan
introduce errors in the estimation process. GST aims to
characterizethefullyunknownsetofgatesandstates[46].
R={|ρ⟩⟩,⟨⟨E|,R ,...,R ,...},q =1,2,...,αk.
w1 wq
(20)
GST has similar requirements to QPT: the ability to
measure the set of gates R={|ρ⟩⟩,⟨⟨E|,R ,...,R }
w1 w αk
in the form of expectation values:
p=Tr{ρ†w (ρ )}=⟨⟨ρ |R |ρ ⟩⟩=Tr{ρ†w ρ w†}.
r q s r wq s r q s q
To simplify the expression, we use F ,F′, where (r,s =
r s 1,2,...,d2), to denote the quantum gates used for
preparing quantum states and measurements (as shown
in Fig 4), which are G (θ )U and U† G† (θ ), respec-
qr r qr qs qs s
tively. The density matrices of these prepared quantum
statesarelinearlyindependent. PleaserefertoAppendix
A for details. By constructing a quantum circuit, it is
possible to compute the d2 matrix elements of the PTM
matrix R . We define (R ) as follows:
wq wq rs
(R ) =⟨⟨ρ |R |ρ ⟩⟩=⟨⟨0|F R F |0⟩⟩. (21) wq rs r wq s r wq s
Inserting the completeness state into it yields:
p =(R ) =⟨⟨ρ |R |ρ ⟩⟩
rs wq rs r wq s
=⟨⟨0|F R F |0⟩⟩
r wq s
X
= ⟨⟨0|F |a⟩⟩⟨⟨a|R |b⟩⟩⟨⟨b|F |0⟩⟩
r wq s
a,b
X
= A (R )B .
ra wq bs
a,b
…
…
…
… …
It can be easily verified that:
p =(AR B) , rs wq rs
X
A= |r⟩⟩⟨⟨0|F ,
r (22) r
X
B = F |0⟩⟩⟨⟨s|.
s
s
Let R˜ = AR B, the identity matrix I also serves
wq wq
as a mapping, and its PTM has the following properties:
R |ρ ⟩⟩=|I(ρ )⟩⟩=|Iρ I†⟩⟩=|ρ ⟩⟩.
I s s s s
It can be observed that the action of R is similar to the
I
identitymatrixanditsmatrixelementscanbeexpressed
as follows:
(R ) =⟨⟨ρ |R |ρ ⟩⟩=Tr{ρ†Iρ I†}=Tr{ρ†ρ }.
I rs r I s r s r s
Denoting g = (R ) = ⟨⟨ρ |ρ ⟩⟩ = ⟨⟨0|F F |0⟩⟩, we
rs I rs r s r s
can insert the completeness state and obtain:
X
g = ⟨⟨0|F |a⟩⟩⟨⟨a|b⟩⟩⟨⟨b|F |0⟩⟩=(AB) . (23)
rs r s rs
a,b
For a given combination Gk, let R˜ =AB, where A and
q Iq
Barematrices. Theexperimentalmeasurementvaluep
rs
correspondstotherscomponentofthematrixA(R )B,
wq
while g corresponds to the rs component of the matrix
rs
(AB). The quantum channel we reconstruct will differ
from the real quantum channel by a similarity transfor-
mation:
(R˜ )−1R˜ =B−1A−1AR B =B−1R B. I wq wq wq
Therefore, we can estimate the trace of R :
wq
Tr{(R˜ )−1R˜ }=Tr{B−1R B}=Tr{R }. (24)
I wq wq wq
Based on the calculations mentioned earlier, the value of
Tr{R } = |Tr{w }|2 can be determined. However, this
wq q
is not the final result for Tr{w }.
q
Note: Inordertoavoidill-conditionedmatricesR ,it
Iq
is necessary to perform subspace selection. For detailed
analysis, please refer to Section IVA.
5. Mathematical Processing of Results
TocalculateTr{w },anoperationinvolvingtakingthe
q
square root is required: |Tr{w }|2 = Tr{R }. In gen- q wq
eral, Tr{w } can be decomposed into real and imaginary
q
parts:
Tr{w }=Re[Tr{w }]+i·Im[Tr{w }].
q q q
FIG. 4. This is a quantum circuit with n-Qubits used to
compute Tr{ρ†Gkρ Gk}. First, we add the prepared state ρ†
r q s q r then
tothecircuit. Next,weapplytheoperatorGk. Then,weadd
q
ρ s to the circuit. Tr{R }=(Re[Tr{w }])2+(Im[Tr{w }])2.
wq q q