3
The expression for Tr{Gkρ} can be derived, with the ul- required for tomography can be reduced by representing
timate goal of estimating: quantum states as MPS, in the worst case the resources
requiredfortomographystillincreaseexponentiallywith
1 1
Tr{ρm+1}=Tr{( I− G)mρ} thenumberofbits[42,45]. Inthecontextofourcurrent
2 2 research problem, there is a silver lining: the subspace
1 Xm (11) we are investigating maintains a dimension that remains
= Ck(−1)kTr{Gkρ}.
2m m unaffected by the number of qubits. This distinctive fea-
k=0 ture becomes particularly advantageous. In the subse-
quent section, we expound on our utilization of the GST
Let: p = 1 Ck, x =(−1)kTr{Gkρ}. The above equa-
k 2m m k method to extract the pertinent information from this
tion can be transformed into an expectation calculation:
designated subspace.
In comparison with the preceding context, the process
m ofpreparingrandomquantumstatesadherestothesame
X
Tr{ρm+1}= p kx k =E k(x k). (12) approach as the Hadamard Test method. This method-
k=0 ology necessitates the application of an assortment of
stochastically selected gates {(U ) ,i = 1,2,...,α}
To estimate the expectation, we need to generate quan- i 2n×2n
onto the initial quantum state (typically |0⟩⟨0|⊗n), re-
tum circuits using a random sampling method. For each
sulting in the emergence of a random quantum state.
circuit, we sample m times, with a 1/2 probability of
The primary objective revolves around the computation
adding a CG gate to the circuit and a 1/2 probability
of Tr{ρm}, which is achieved through the intermediary
of doing nothing. After generating multiple circuits, we
of Tr{Gm}, where G = I −2ρ. We can express Tr{ρm}
take the average of the results.
as
m
1 X
B. Algorithm for calculating Tr{ρm+1}. Tr{ρm}= Ck(−1)kTr{Gk}. (13)
2m m
k=0
The pseudocode below, referred to as Algorithm 1, is
for calculating Tr{ρm+1}. Similar to the approach employed in the HT-based al-
gorithm, we can estimate this summation by leveraging
Algorithm1
Input:n,p ,U ,N
i i
Output:Tr{ρm+1} PTMTransform
1: Set the initial state to |0⟩⊗n+1. Subspace
2: RandomlyselectU i withprobabilityp i toactonthetar-
get qubits. dim𝑤𝑞′ =dim𝑤𝑞 +1
3: Apply a Hadamard gate to the ancilla qubit.
PTM Transform
4: Sample m times, with a 1/2 probability of adding a CG
gatetothecircuitanda1/2probabilityofdoingnothing.
5: Apply another Hadamard gate to the ancilla qubit.
6: Perform a computational basis measurement on the aux-
iliary qubit circuit.
7: Repeat the above steps N times and take the average.
III. TRACE ESTIMATION OF QUANTUM
TOMOGRAPHY
FIG. 2. Process of calculating Tr{Gk} using Gate Set To-
mography. Firstly, in Step 1, decompose Tr{Gk} into the
A. Theoretical Part of Quantum Tomography
calculationofeachTr{Gk}. Then,takeitssubspacew (Step
q q
2) and transform the matrix representation of the subspace
Quantum tomography denotes a suite of techniques intothecorrespondingPauliTransferMatrix,denotedasR .
wq
aiming to reconstruct an unknown quantum channel or Simultaneously,inStep4,expandthesubspacew byonedi-
q
state through experimental measurements. This process mension. Next,similarly,transformitsmatrixrepresentation
is pivotal for the comprehensive understanding and au- w′ into its corresponding Pauli Transfer Matrix, denoted as
q
thentication of quantum apparatus [40–44]. Neverthe- R wq′. Then,throughStep6,calculatetheirtracesseparately.
less,thescalabilityofquantumtomographyposesachal- By employing certain mathematical techniques in Step 7, we
lenge, as the indispensable measurements and computa- can use the obtained traces to calculate Tr{w q}, and subse-
quentlyobtainTr{Gk}(Step8). Finally,inStep9,weobtain
tionalresourcesexperienceexponentialgrowthintandem q
Tr{Gk}, which can be seen as the inverse process of Step 1.
with qubit numbers. Note that although the resources