4
Circuit
FIG.3. Schematicdiagramofasub-processforcomputingTr{R }withintheTomographyworkflow. Thediagramrunsfrom
wq
lefttoright,withthefirstblackboxontheleftrepresentingthepreparationoftheinitialstate(1)andthemeasurementstate
(2). The blue box on the upper right depicts the placement of the prepared states into the circuit (3), where the Gk gate and
q
the identity gate are inserted. Steps (4) and (5) allow us to obtain the matrix elements of the PTMs corresponding to each of
these gates. Finally, in step (6), Tr{R } is indirectly obtained through a similarity transformation.
wq
the Monte Carlo method. This involves conducting mul- total count of possible arrangements aggregates to αk.
tiple circuit samplings in accordance with their respec- These arrangements are uniquely labeled by the index
tive probabilities and subsequently calculating the aver- q ={1,2,...,αk}. As a result, this algorithm effectively
age. Throughthisprocess,wecanattainthesought-after dissects the trace Tr{Gk} of the higher-order powers of
value of Tr{ρm}. The figure shown in Fig 2 presents G into computations encompassing αk arrangements de-
our computation process through a simplified flowchart, notedasGk. Thecalculationprocessistherebyexecuted
q
starting from the decomposition of Tr{Gk}, transition- on higher-order random quantum states via the applica-
ing to the calculation of each individual Gk, and then tion of a Monte Carlo methodology.
q
proceedingwithaseriesofcomputationswithintheirre- ThecorrespondingprobabilitycombinationQk
p is
spectivesubspaces. Finally,thecomputedresultsTr{Gk} t=1 qt
q represented as P . Therefore, Tr{Gk} can be expressed
q
are weighted and summed to obtain the desired calcula-
as:
tionTr{Gk}. Weillustratethesub-processofcomputing
Tr{R } in Fig 3, which, together with Fig 2, forms the
compw leq
te Tomography computation process.
Xαk
Tr{Gk}= P Tr{Gk}, (15)
q q
q=1
1. Mathematical Treatment:
Where we use Gk to denote Qk G for convenience.
q t=1 qt
We start with the unitary quantum gate G :
0
 
−1 0 ··· 0
 0 1 ··· 0
(G 0) 2n×2n =I 2n −2|0⟩⟨0|⊗n = 

. .
.
. .
.
... . . .  . (14) 2. The matrix representation of G qk
0 0 ··· 1
For each of the αk instances of Gk, a specific Gk is
q q
Through the application of random unitary gates U to chosen for computation where q = 1,2,...,αk. In this
i
the initial gate G , a set of α distinct gates is generated. scenario, the calculation method is provided for arbi-
0
ThesegatesaredenotedasG =U G U†. Thecomposite trary combinations, while the computation process re-
i i 0 i
mains similar for other combinations.
gateGisthendefinedastheweightedsummationofthese
transformed gates: G=Pα p G . Upon choosing a specific combination Gk, a set of k
i=1 i i q
Aninsightfulobservationcanbemadethatinthepres- corresponding G gates is determined, thereby giving
qt
ence of k occurrences of G gates within the circuit, the rise to k specific |ψ ⟩ states, where t=1,2,...,k.
qt