As we have seen, standard QEC is a completely noise independent proce-dure, i.e., it makes no assumptions on the type of the noise process, rather its aim is to perfectly correct a fixed group of commonly probable errors. In contrast, if the type of the noise process is known, it is more reasonable to try protecting the information against that specific process type. This approach is called the optimization approach of QEC. Its goal is to find the optimal recovery operation R⋆ given a code C and a noise channel E. In the litera-ture authors mostly follow this approach [22, 23, 24] although there are also alternate attempts [25, 26].
A recovery operation R⋆ is said to be optimal, if it maximizes the chan-nel fidelity Fch. The reason for this choice of objective function is apparent
23When there are more than one error operator in the same coset ofSwhich is correctable by someRrecovery operator, then it is the case of degenerate QEC.
3.2. Optimal QEC
based on section 2.4.2; error correction has to take into account the possibility that it is only applied to a part of a larger quantum system, i.e., it not only has to preserve the state on which it is applied, but also has to preserve the entanglement between the state and parts of its environment.
More formally, the following optimization problem has to be solved:
R⋆ = arg max
{R} Fch(R ◦ E). (3.3) In contrast to a Kraus operator element set the Choi matrix is a unique channel representation in the sense that it has no unitary freedom (see section 2.2.2), which makes it suitable for optimization purposes. Using this, (2.20), (2.10) and (2.11), the objective of (3.3) turns into
Fch(R ◦ E) = X
i,j
hhρcs|RiEjiihhRiEj|ρcsii=hhρcs|XR◦E|ρcsii= Tr(XRCρcs,E) . where the Ri are the Kraus elements of R, Cρcs,E := P
i|ρcsEi∗iihhρcsEi∗|, and ρcs = UC21mUC∗ = 2PmC ∈ B(Hcs) is the codeword of the maximally entangled message state 21m.
Taking into account that R must be CPTP, the final form of (3.3):
XR⋆ = arg max
X Tr(XCPC
2m,E) , (3.4)
so thatX ≥0, and Tr2(X) =1 .
This problem is a semidefinite programming problem, a class of convex op-timization problems for which efficient solvers exist (see section A.6). The properties of such a problem can be seen in appendix A.2.3.
Note that for a [[c, m]] code, (3.4) is a 24c −22c (real) dimensional opti-mization problem. In practice the number of dimensions can be reduced to 22(m+c)−2m+c by merging the noise and recovery operator elements with the encoder as Ei′ := EiUC, R′i := UC∗Ri and operating directly on the message space Hms.
3.2.1 Pauli case
In the case of Pauli channels and stabilizer codes the optimal correction operation R⋆ can be generated analytically [19, 25]. This is a very essential result, as it applies to many important channels; furthermore, it may also help understand more complex cases where only numerical methods are available.
As stated in section 3.1.2, only one Pauli error class Ep,µ = √ap,µApWµ
can be corrected for each µ. It is then apparent that the correction will be the most effective if for each µ we choose and correct the most probable error. In contrast, standard QEC chooses to correct the minimum weight (more often only the single-qubit) errors. This means that for stabilizer codes and Pauli channels, standard QEC will be optimal except when the error probabilities are sufficiently unequal. Further comparison of standard and optimal QEC with examples can be seen in appendix A.5.5.
29
3. Theory of quantum error corretion
Formalizing the above, let the index p of the most probable error oper-ator Ap for each WµN(S) coset be denoted by p⋆µ. Now the operator ele-ment set of the optimal R⋆ recovery operation can be obtained as {Wµ∗Ap⋆µ}, µ= 0, . . . ,2c−m−1. Using this, the channel fidelity turns into
Fch(R⋆◦ E) = X
p,µ
X
µ′
Tr
Wµ∗′Ap⋆µ′
√ap,µApWµ
PC 2m
2
=X
µ
ap⋆µ,µ . We used that [Wµ, Ap] = 0, Wµ∗′Wµ = δµ,µ′PC and the logical Pauli operators are mutually Hilbert–Schmidt orthogonal, i.e., Tr ApAp′
=δp,p′2m.
By this formula, the channel fidelity for Pauli channels is thus the sum of the apµ,µ probabilities of the errors selected for correction in each syndrome subspace. It is evident that we get the maximal channel fidelity if we designR to correct the error with the greatest probability in each syndrome subspace.
Strictly speaking, we have proved only that among the(22m)2c−m different QEC operations of the form
R={Wµ∗Apµ} (3.5)
there does not exist any better than the one with pµ =p⋆µ. However, there is not any other kind of QEC operation which is better (see [19] for a complete proof).
Chapter 4
Robustness of quantum error correction
This chapter summarizes a set of new results. The robustness of QEC pro-cedures against channel perturbations is first defined. Using this definition, the robustness of recovery operations for the class of Pauli channels is analyzed in detail. The robustness domains splitting up the set of Pauli channels are explicitly characterized. Furthermore, several case studies on the robustness of recovery operations against general non-Pauli perturbating channels is also studied using the same definitions, with the aim of finding possible general-ization of the results for Pauli channels, in particular the general robustness domains and domain borders.
In section 4.1 the general definitions of robustness and related notions is given. Section 4.2 discusses robustness for the case of Pauli channels, while section 4.3 deals with the general case. Finally, 4.4 summarizes the results.