Characterization of quantum dynamics using quantum error correction

Characterizing noisy quantum processes is important to quantum computation and communication (QCC), since quantum systems are generally open. To date, all methods of characterization of quantum dynamics (CQD), typically implemented by quantum process tomography, are off-line, i.e., QCC and CQD are not concurrent, as they require distinct state preparations. Here we introduce a method, “quantum error correction based characterization of dynamics,” in which the initial state is any element from the code space of a quantum error correcting code that can protect the state from arbitrary errors acting on the subsytem subjected to unknown dynamics. The statistics of stabilizer measurements, with possible unitary preprocessing operations, are used to characterize the noise, while the observed syndrome can be used to correct the noisy state. Our method requires at most $2(4^n-1)$ configurations to characterize arbitrary noise acting on $n$ qubits.

Figure 1

Scheme for QECCD. The principal system P (subjected to the uncharacterized noise $\mathcal{E})$ plus CQD ancilla A (assumed to be noiseless) are prepared in a QECC-encoded state. After P is subjected to channel $\mathcal{E}$ (assumed correctable by the QECC used), the stabilizers are measured on the joint system, possibly following unitary operation $U$ or $U{S}^{+}$.

Figure 2

Circuit to measure the stabilizer generators $XIX$ and $YYZ$ for the $\left[\right[3,1\left]\right]$ QECC, (11). The top two wires are the error correction ancillae, while the bottom three wires are the code qubits. Time flows from left to right. The boxes H and M represent a Hadamard and measurement in the computational basis, respectively. The circle with an operation $U\in \{X,Y,Z\}$ represents a control-$U$ operation, with control at the filled circle on the other end of the “stick.” Only two-qubit interactions are used.