5
For example, consider the cases: (2n−d)×(2n−d)identitymatrixI . There-
(2n−d)×(2n−d)
fore,w isactuallythematrixrepresentationofGk inthe
q =1,{Gk :G G ···G G }. q q
1
|
1 1
{z
1 }1 d-dimensional invariant subspace V q1. Through this ex-
pression, the calculation of Tr{Gk} can be performed:
k layers q
q =2,{G 2k :G 1G 1···G 1G 2}. Tr{Gk}=Tr{w )+Tr{I }
| {z } q q 2n−d (18)
k layers =Tr{w }+2n−d.
q
.
.
.
q =αk,{G αk
k
:G αG α···G αG α}. 3. The PTM representation of w
q
| {z }
k layers
AftercharacterizingGk,weneedtoobtainthesolution
q
Therefore, under the matrix background denoted as Gk, for Tr{w }. However, the matrix w is unknown. w as
q q q q
the subspace dimension d is not constant. Hence, the amapping,findingthesolutionforw requirestheuseof
q
determination of the subspace dimension relies entirely the Pauli Transfer Matrix(PTM). We denote the PTM
on the count of unique gate types present in the given corresponding to w as R .
q wq
orderq. Letusdefinedasthedimensionofthenontrivial Based on the previous discussion, an important rela-
subspace corresponding to the simplified merge of Gk := tionship can be used:
q
Q tok t p= r1 eG paqt r, ew Ghile asre Upre :se Gntin =g Uthe Gra Und †o .m Quga at ne tus mets stu as te ed
s
R wq|ρ s⟩⟩=|w q(ρ s)⟩⟩=|w qρ sw q†⟩⟩
prepared
byq dt ifferenqt
t
ranqt
dom
q gt ate0
s
q at
re represented as:
r,s∈{1,2,...,d2}.
|ψ ⟩,ζ =1,2,...,d.
qζ For simplicity, the subscript q indicating that ρ belongs
In order to ensure completeness, a set of state vectors to the ordering Gk will be omitted below.
{|ϕ ⟩,η = 1,2,··· ,2n −d} is introduced, which are or- q
η Although R is an (d2×d2) matrix, the vector |ρ ⟩⟩
thogonal to all the state vectors |ψ ⟩. wq s
qζ hasdimensionsof(2n)2×1. Since|ρ ⟩⟩onlyhasnon-zero
The rationale behind this is as follows: s
elements in the d2-dimensional subspace, the remaining
Due to the condition ⟨ψ |ϕ ⟩ = 0, it can be inferred
qζ η part of the vector is trivial, consisting of all zeros except
that:
for this subspace.
  By left-multiplying the above equation by ⟨⟨ρ |, the
d r
Y
G qk(|ϕ η⟩)= (I−2(|ψ qζ⟩⟨ψ qζ|)|ϕ η⟩=|ϕ η⟩. matrix elements of the PTM matrix (R wq) d2×d2 are
given as follows:
ζ=1
⟨⟨ρ |R |ρ ⟩⟩=⟨⟨ρ |w ρ w†⟩⟩. (19)
Itisevidentthat{|ϕ ⟩,η =1,2,...,2n−d}formsasetof r wq s r q s q
η
eigenstatesofGk witheigenvalue1. Intherepresentation To calculate the matrix elements (R ) of R using
q wq rs wq
with 2n state vectors: this method, d2 quantum states in the Hilbert-Schmidt
  space are required, represented by vectors |ρ r⟩⟩. The
  proof of the completeness of these states can be found in
|ψ q1⟩,|ψ q2⟩,...,|ψ qd⟩,|ϕ 1⟩,|ϕ 2⟩,...,|ϕ η⟩ . (16) Appendix A.
| {z } | {z } We need to calculate the trace of PTM. Although we
d terms (2n−d) terms
can obtain each matrix element of the PTM sequen-
as the basis in the V space, the matrix Gk can be ex- tially through the circuit, we cannot directly compute
q q
its trace because the basis vectors in the subspace we
pressed as:
generate are not orthogonal. There are two feasible ap-
 w w ... w 0 0 ... proaches:Schmidt decomposition to orthogonalize its ba-
11 12 1d
w w ... w 0 0 ... sis vectors, and then directly calculate the trace by sum-
 21 22 2d 
 ... ... ... ... 0 0 ... ming the main diagonal elements;Calculate its trace in-
 
w w ... w 0 0 ... (17) directly through a matrix similarity transformation.This
 d1 d2 dd 
 0 0 0 0 1 0 ... paper adopts the GST (Gate Set Tomography) method,
 
 0 0 0 0 0 ... ... which is the latter approach of indirectly calculating the
... ... ... ... ... ... 1 trace of PTM through similarity transformations.
The top-left d×d matrix w can be considered as com-
q
posed of the eigenvalues of the d-dimensional invariant 4. GST
subspace V spanned by d states |ψ ⟩ in the basis. The
q1 qζ
remaining part of Gk is composed of the eigenvectors In Quantum Process Tomography (QPT), the infor-
q
in the complement space with basis |ϕ ⟩, correspond- mation required to reconstruct each gate R is con-
η wq
ing to (2n −d) eigenvalues. These eigenvalues form an tained in the measurements of ⟨⟨ρ |R |ρ ⟩⟩, and R
r wq s wq