Operator Quantum Zeno Effect: Protecting Quantum Information with Noisy Two-Qubit Interactions

The time evolution of some quantum states can be slowed down or even stopped under frequent measurements. This is the usual quantum Zeno effect. Here, we report an operator quantum Zeno effect, in which the evolution of some physical observables is slowed down through measurements even though the quantum state changes randomly with time. Based on the operator quantum Zeno effect, we show how we can protect quantum information from decoherence with two-qubit measurements, realizable with noisy two-qubit interactions.

Figure 1

The scheme of protecting one logical qubit encoded in three qubits from noise. The quantum state $|{\psi }_{\mathrm{in}}⟩$ is encoded into three qubits by initializing the other two qubits in states $|0⟩$ and $|x,0⟩$, respectively. Here, $|x,\nu ⟩$ is an eigenstate of $X$ with eigenvalue $(-1{)}^{\nu }$. To protect the logical state for time $\tau$, $N$ sets of measurements are performed with the frequency $f=N/\tau$. Each set includes measurements of $Z$, $ZZ$, $X$ and $XX$ on corresponding qubits. Here, we use $M_{\sigma }$ to denote the measurement of $\sigma$. The logical qubit is decoded with $M_Z$ and $M_X$, whose outcomes are ${\nu }_Z$ and ${\nu }_X$, respectively. Finally, a single-qubit operation is performed in order to correct the Pauli frame of the output state $|{\psi }_{\mathrm{out}}⟩$.

Figure 2

The probabilities of Pauli errors, $p_X$, $p_Y$, and $p_Z$ (solid line, dashed line, and dotted line, respectively). Here, we consider one-local noise with parameters $\{{\mathbf{a}}_i=(a_{i,X},a_{i,Y},a_{i,Z})\}$, which are uniformly distributed random vectors with $\parallel {\mathbf{a}}_i\parallel \le a$. The unit of time is $a^{-1}$. One can find that $p_X$ and $p_Z$ are coincident. By increasing the measurement frequency (from top to bottom: without measurements, $f=10$, $f=100$ and $f=1000$), one can reduce the probabilities of getting Pauli errors.

Figure 3

The lifetime of a logical qubit protected by both frequent measurements and the surface code. Here, we utilize measurements described by superoperators ${\mathcal{P}}_{\sigma }(\zeta )•=[(1+\zeta )/2]•+[(1-\zeta )/2]\sigma •\sigma$, where $\sigma$ is the measured operator. When $\zeta =0$ the superoperator is a projective measurement investigated in the text. When $\zeta >0$, the superoperator corresponds to a weak measurement. By increasing the measurement frequency, one can extend the lifetime of the logical qubit. Numerical results show that our scheme also works with weak measurements.