too. This shows that the knowledge of the Pauli channel structure is necessary for the application of the optimal experiment design. It follows that if – as it is the case in general – this knowledge is not available, i.e., we have a Pauli channel with
• unknown depolarizing parameters λi, and
• an unknown set of complementary subalgebras in which the channel is depolarizing (see section 2.3.2),
then for such Pauli channels we need fundamentally different approaches to solve the optimal identification problem.
For qubits, this unknown channel can be modeled with the matrixS of the rotated Pauli channel in section 2.3.3. This model contains six independent matrix elements, these can form the parameter vector s = [s1, . . . , s6]T. After channel estimation, the eigenvectors of the estimate Sˆ will be the estimated Bloch vectors vˆi of the channel directions, and its eigenvalues will be the cor-responding estimated scaling valuesλˆi. The Choi matrix of such a generalized channel is
XS = 1 2
1 +s3 s5−is6 s5+ is6 s1+s2
s5+ is6 1−s3 s1−s2+ 2is4 −s5−is6 s5−is6 s1−s2−2is4 1−s3 −s5+ is6
s1+s2 −s5 + is6 −s5−is6 1 +s3
This matrix is linear in its parameters si, thus an exact affine decomposition based parametrization can be made. This implies
Statement 8.1. The parameter estimation of the qubit Pauli channel with unknown channel directions remains a convex optimization problem.
Of course, we could try to simply find the optimal experiment configuration (b,{±m})for this channel similarly as in the known channel structure case in section 7.2.1, namely, by maximizing the Fisher information matrixF(s|b,m) of a single experiment:
F(s|b,m) = ∇s(mTSb)∇Ts(mTSb)
1−(mTSb)2 (8.1)
However, we see that this matrix depends on the full channel matrix S. This clearly shows for qubits that – as we mentioned above based on Statement 7.2 – the optimal configuration surely depends on the channel structure, i.e., the channel directions; thus it can not be determined without knowingSitself!
The aim of this chapter is to overcome this problem. In the following two sections, methods for the estimation of such general qubit Pauli channel will be studied using two fundamentally different approaches.
8.2 Channel direction estimation
The method described in this section estimates the unknown channel di-rections of a rotated Pauli channel from section 2.3.3, while resulting in a first estimate on the λ parameter values too.
83
8. Identification of a Pauli channel with unknown structure
8.2.1 Estimation algorithm for channel directions
Let the three directions in which the qubit Pauli channel is depolarizing be |v1i, |v2i and |v3i. Then quantum state estimation steps can be used to determine these. The proposed method is essentially an adaptation of the power iterations algorithm from linear algebra.
The effect of the channelE for the input pure Bloch vectorb˜ can be written asE(˜b) = Sb˜ =P3
i=1λivTi bv˜ i. In the rest of this section, the words “vector”
and “state” are used as synonyms, both referring to Bloch vectors.
The task is to estimate the three depolarizing directions {|vii} of E by estimating the corresponding Bloch vectors vi. Let the set of found channel direction Bloch vectors be D. Let D ={}and n = 0, this is the initialization step. The following algorithm describes the direction estimation procedure.
Algorithm 1 Direction estimation
1: repeat
2: Prepare a pure state b˜(n) ∈D⊥.
3: repeat
4: Put b˜(n) into the composite channel Ek formed by cascading k in-stances of the channelE, then get the output b(n+1).
5: Perform quantum state tomography on b(n+1) to get the estimate bˆ(n+1).
6: Project bˆ(n+1) to D⊥ to getbˆ(n+1)proj .
7: Normalizebˆ(n+1)proj to get the pure stateb˜(n+1).
8: Increasen by1.
9: untilThe distance
b˜(n)−b˜(n+1)
is smaller than some prescribed value.
10: Putb˜(n+1) into D, set n to 0.
11: until Dimension of D⊥ is0.
We now give the mathematical arguments that support the steps of the above algorithm.
Case of different channel parameter values
Assume that all of theλi channel parameters have different absolute values, and recall that |λi|<1, i= 1,2,3.
• Step 4: Assume we use a pure state b˜(n) as input to the channel and obtain the output b(n1) = Sb˜(n). If we repeat this procedure, i.e., we put the channel output b(n1) back into the channel as input to get the output b(n2), then by the power iterations method, the vector b(nℓ)
kb(nℓ)k
will converge to the axis of vm corresponding to the dominant channel parameter λm, i.e., the parameter with the largest absolute value. The normalization in the above sequence is inevitable, as the output states do not remain pure during the iterated channel effect, i.e., the length
b(nℓ) of the sequence will not remain 1, it will converge to zero instead.
8.2. Channel direction estimation
• Step 5-7: Thus, to avoid the vector sequence from converging to the maximally mixed state, we have to do the normalization of the output Bloch vector b(n+1) = b(nk) = Skb˜(n) manually after each step. This means that we have to exchange the output state with the pure state which points in the same direction. In order to do this we need to perform quantum state tomography (Step 5) to get an estimate bˆ(n+1) and normalize it in Step 7 to obtain the pure stateb˜(n+1)= bˆ(n+1)
kbˆ(n+1)k which can be put again into the channel. This way, the sequence of vectors will indeed converge to vm.
• Step 2,6,10,11: After the first channel direction vm was found using this procedure, we can continue the search in the plane D⊥ orthogonal tovm
(Step 10 and 2). However, due to the inaccuracies in state tomography, the direction we will find will not be exactly vm, rather some vector b˜⋆ ≈vm. Thus convergence is more robust against these inaccuracies if we apply a projection to the output vector in Step 6 onto the subspace D⊥. When the second direction is found, then the third can be easily obtained, as it will be the one orthogonal to both the first and the second direction. Thus the direction estimation procedure is finished using only two iterations (Step 11).
Special cases
In the degenerate cases when some of the channel parametersλi have equal absolute values, then the channel is equally depolarizing in the linear span of those directions, i.e., there are no exact channel directions defined in that subspace. This means that we can use any state inside this subspace as channel direction, so the sequence b˜(n) ∞n=0 of states is only required to converge to an arbitrary state inside this subspace, which is guaranteed by the above procedure.
It follows, that if all the channel parameters have equal absolute values, then the channel is the depolarizing channel, which means that any three orthogonal Bloch vectors can be used to represent channel directions.
In the case of λi = 1 for somei, the above argument differs only in the fact that in theory, the normalization step is not necessary. The iterated channel effect does not make the length of the input Bloch vector tend to zero.
Moreover, in the special case when the starting vector b˜(n) has zero com-ponent along the axis corresponding to the dominant channel parameter λm, then in theory, the algorithm will find the channel direction with the second dominant parameter first.
Accuracy and efficiency
The accuracy of this procedure has of course a limit, set by the accuracy of quantum state tomography. Convergence to a channel direction is guaranteed only until the difference in the input and output state is not comparable with 85
8. Identification of a Pauli channel with unknown structure
the uncertainty of the state estimation procedure. Thus, when the sequence reaches this limit, the searching procedure should stop. It can also occur that we give a good initial guess, and start with an input state which is very close to a channel direction. Then the procedure can finish almost immediately, because slow convergence can only occur close to channel directions.
This algorithm – though rather resource intensive – thus estimates the directions of a qubit Pauli channel. By using the algorithm we can get infor-mation also on the channel parameters which can then be made more accurate using the optimal tomography configurations described in section 7.2.1, thus making a two step Pauli channel estimation procedure.
Finally, an interesting question can be raised. If the channel directions are unknown then how practical is it to try obtaining them, possibly by using the method presented in this section? To answer this question, a compari-son would be necessary from the aspect of resource requirement between our method of direction estimation combined with optimal experiment design and a general channel estimation method that uses no a priori knowledge about the channel structure. This study is not in the scope of this thesis, but the papers [33] and [27] suggest, that in order to achieve an estimation accuracy of order comparable with the results given in section 7.3 for qubit channels without making assumptions on the channel structure, may require a number of measurements of order 104–105. This is at least about the same order as the approximate measurement requirement of our two step procedure.
8.2.2 A simple numerical example
In order to illustrate the operation and properties of the above proposed channel direction estimation algorithm, a simple illustrative numerical example is presented here for a qubit Pauli channel with different parametersλ1 = 0.6, λ2 = 0.3, and λ3 = 0.1.
The three unknown channel directions were chosen to be the eigenvectors of the Pauli matrices. The uncertainty in the estimated channel output state arising from quantum state tomography was simulated using random perturba-tions in the output state. The perturbation of theith Bloch vector component bi is a random term of the form
ξ
r1−bi
N ,
whereξ is a random number taken form the standard normal distribution, N is the number of measurements in the state tomography step, and 1−Nbi is the variance of the estimator bˆi.
The result of the numerical test can be seen in Figure 8.1. The three unknown channel directions are shown by the black axes in the Bloch sphere.
The prime labeled vectors indicate the perturbed and normalized input states in each step, and the number labeled vectors indicate channel outputs. The starting input vector was chosen randomly at the beginning of the search. The