Asymmetric quantum error correction via code conversion

In many physical systems it is expected that environmental decoherence will exhibit an asymmetry between dephasing and relaxation that may result in qubits experiencing discrete phase errors more frequently than discrete bit errors. In the presence of such an error asymmetry, an appropriately asymmetric quantum code—that is, a code that can correct more phase errors than bit errors—will be more efficient than a traditional, symmetric quantum code. Here we construct fault tolerant circuits to convert between an asymmetric subsystem code and a symmetric subsystem code. We show that, for a moderate error asymmetry, the failure rate of a logical circuit can be reduced by using a combined symmetric asymmetric system and that doing so does not preclude universality.

Figure 1

Examples of group elements for each relevant subsystem of the asymmetric subsystem code $\mathcal{C}\left(5,3\right)$. Each vertex point represents one of the 15 physical qubits needed to encode a single logical qubit. (a) and (b) illustrate $X$ and $Z$ elements from the stabilizer group $\mathcal{S}$. (c) illustrates two commuting elements from the non-Abelian gauge group, $\mathcal{T}$. Finally, (d) and (e) illustrate the two Pauli gates for the single logically encoded qubit from the group $\mathcal{T}$.

Figure 2

$Z$ syndrome extraction under $\mathcal{C}\left(5,3\right)$ for column $j$. To ensure fault tolerance it is necessary to repeat this circuit or to perform other redundant parity checks [20].

Figure 3

$X$ syndrome extraction under $\mathcal{C}\left(5,3\right)$ for row $i$ [20].

Figure 4

Conversion from $\mathcal{C}\left(3,3\right)$ to $\mathcal{C}\left(5,3\right)$ for column $j$.

Figure 5

Conversion from $\mathcal{C}\left(5,3\right)$ to $\mathcal{C}\left(3,3\right)$. $\mathcal{P}$ is the gate sequence in Fig. 3.

Figure 6

Converting from $\mathcal{C}\left(5,3\right)$ to $\mathcal{C}\left(3,3\right)$ requires eliminating this type of gauge state. Here, the gauge part of the subsystem code is stabilized with respect to $X_{2,1}{X}_{4,1}$ and $X_{3,3}{X}_{5,3}$. If the six qubits in the last two rows are measured, then residual $X$ errors remain at grid locations (2,1) and (3,3). To protect against this type of error, the system is stabilized with respect to $Z_{4,1}{Z}_{4,2}$ and $Z_{5,2}{Z}_{5,3}$, this ensures that these $X$ gauge stabilizers are removed (since they do not commute with $Z_{4,1}{Z}_{4,2}$ and $Z_{5,2}{Z}_{5,3}$) and errors are not induced during the conversion.

Figure 7

Converting from $\mathcal{C}\left(5,3\right)$ to $\mathcal{C}\left(3,3\right)$ can also enact a logical $Z$ operation. Prior to the conversion, the system is stabilized with respect to $⟨\phantom{|}{Z}_{i,1}{Z}_{i,2};Z_{i,2}{Z}_{i,3}|i∊\left\{4,5\right\}⟩$. Consider the case when the state is a $+1$ eigenstate of $Z_{\ast ,1}$, (i.e., a $|{+}_L⟩$ state), then (a) represents the $Z$ operators for which the state is a $+1$ eigenstate. In the absence of errors, after the conversion there are four possible measurement results of the six measured qubits. Situation (b) is where one row is measured $|111⟩$ and the other $|000⟩$. In this case the converted state flips from a $+1$ eigenstate of $Z_{\ast ,1}$ to a $-1$ eigenstate, therefore a logical $Z$ operation has been performed. In situation (c), both rows have been measured to be $|111⟩$, since the factors of $-1$ cancel, the down converted state is still a $+1$ eigenstate of $Z_{\ast ,1}$, therefore no logical operation has been performed.

Figure 8

Using the combined symmetric asymmetric system for a period of logical memory. Blocks of qubits are labeled by the number of qubits in the block and by the role of the block at that part of the circuit, where $D$ indicates data, $A$ indicates ancilla, and $C$ indicates additional qubits that are required to convert to $\mathcal{C}\left(5,3\right)$. $N$ is the number of consecutive error correction cycles in the period. Estimates of the performance of this circuit relative to correction under $\mathcal{C}\left(3,3\right)$ are contained in Table .