10
… …
TABLE II. The numerical value of the parameter
…
HadamardTest Tomography
… Tr{ρ2} 0.650 0.650
Tr{ρ3} 0.486 0.486
…
Tr{ρ4} 0.375 0.375
*The computational results obtained using the two algorithms
…
inthisarticlearepresentedwiththreedecimalplaces.
FIG. 5. n-qubits Quantum Circuit: The first line repre- andGSTareveryclose. Moredataconsideringshotnoise
sents an ancillary circuit, with the initial state set to |0⟩ . can be found in Appendix B.
a
It then undergoes a Hadamard gate, resulting in a super- However, compared to HT, the GST method doesn’t
position state. The three lines below represent the prepa- require an additional ancillary qubit, reducing the im-
ration and measurement of the target quantum states, with
plementation cost of the quantum circuit. This implies
initial states all set to |0⟩. After applying the first random
thatpracticalapplicationscouldbenefitfromusingfewer
gate U , the target quantum state ρ = U |0⟩⟨0|U† is
y1 y1 y1 y1 physical qubits and controlled gates, which is a critical
obtained through tensor product. Next, controlled quan-
factor to consider. Reducing the number of controlled
tumgatesG ,G ···G areapplied,followedbyanother
y2 y3 yk+1
gates could lower the error rate of the quantum circuit,
Hadamardgateontheancillarycircuit. Finally,measurement
in the computational basis is performed on the ancillary cir- thereby enhancing the system’s reliability.
cuit. Based on the obtained Tr{ρm}, it is possible to cal-
culate Tr{ρlnρ}, Tr{eρt}, Tr{eiρt}, etc. For example
Tr{ρlnρ}: theexpansionofρlnρwithrespecttoGyields
Similarly,byfollowingthesameprocedurewecanobtain the following expression:
α Xk+1"k Y+1 # ρlnρ≈−1 ln2(I−G)− 1 G+ 1 {(1− 1 )G2+
P(1)= p P (1). (44) 2 2 2 2
ys y
1 1 1 1 1
y=1 s=1 ( − )G3+···+( − )Gn+ Gn+1}.
2 3 n−1 n n
With Eq.(9) and Eq.(10), we can deduce Tr{Gkρ}, sub- (45)
sequently leading us to the derivation of Tr{ρm+1}. For Tr{ρlnρ}, the theoretical value can be obtained
using the method of matrix multiplication. After ex-
pansion, it can be calculated using the two methods de-
scribed in this paper.
C. Obtaining the data and exploring potential
applications As shown in Table III, it’s evident that as the power
mincreases,thetheoreticalandexperimentalvaluescon-
verge, yielding a diminishing relative error. Upon juxta-
Inthisstudy,therandomgatesinthecircuitaresimu-
posing Table II, one can discern the remarkable similar-
latedbyselectingdifferentparametersforU(θ,ϕ,λ). For
ity between the outcomes derived from both algorithms.
generality, the parameter selection is based on random
The code can be found in HT and GST.
numbers. The parameters for the random gates used in
thisstudyareshowninTableI. Theprobabilitiesforthe
four random gates are 0.1, 0.2, 0.3, and 0.4. Based on
VI. CONCLUSION
the above content, the values of Tr{ρm} obtained using
two different methods are shown in Table II.
In this article, we present two algorithms for comput-
From the analysis of the data above, we can see that
ingthepowerfunctionofthedensitymatrixbyencoding
the numerical values of Tr{ρm} obtained using the HT
the quantum state into a quantum channel. The first al-
gorithmisbasedontheHT. Intheabsenceofnoise, this
algorithm provides an unbiased estimate of the power
TABLE I. The values of the parameters
function. However, it requires a significant number of
θys ϕys λys control-G gates, which is not favorable for current hard-
i=1 0.29π 0.07π 0.11π warelimitations. Therefore,weproposeanalternativeal-
gorithmbasedonGST.TheoriginalGSTisnotscalable,
i=2 0.46π 0.62π 0.82π
but for our specific problem, we can perform GST only
i=3 0.41π 0.59π 0.53π
within the non-trivial subspace and extract the neces-
i=4 0.55π 0.31π 0.60π
saryinformation. Theadvantageofthisalgorithmliesin
*Theparametersoftherandomgatesusedtogeneratetheabove significantly reducing the utilization of two-qubit gates.
data. Nevertheless, due to the need to mitigate the impact of