Ignorance Is Bliss: General and Robust Cancellation of Decoherence via No-Knowledge Quantum Feedback

A “no-knowledge” measurement of an open quantum system yields no information about any system observable; it only returns noise input from the environment. Surprisingly, performing such a no-knowledge measurement can be advantageous. We prove that a system undergoing no-knowledge monitoring has reversible noise, which can be canceled by directly feeding back the measurement signal. We show how no-knowledge feedback control can be used to cancel decoherence in an arbitrary quantum system coupled to a Markovian reservoir that is being monitored. Since no-knowledge feedback does not depend on the system state or Hamiltonian, such decoherence cancellation is guaranteed to be general and robust, and can operate in conjunction with any other quantum control protocol. As an application, we show that no-knowledge feedback could be used to improve the performance of dissipative quantum computers subjected to local loss.

Figure 1

(a) Schematic for a measurement-based feedback control protocol. In general, information about the system ${\rho }_t$ is extracted by monitoring the decoherence channel $L$. The optimal estimate ${\pi }_t$ is conditioned on the resulting measurement signal $y_{\theta }(t)$. The system is then controlled with some feedback Hamiltonian $H_{\text{fb}}$. For our no-knowledge feedback protocol, the feedback is simply a modulation of the no-knowledge measurement signal. (b)–(e) Particular example of a driven qubit undergoing dephasing with $H=\mathrm{\Omega }{\sigma }_x$, $L=\sqrt{\gamma }{\sigma }_z$, and $\mathrm{\Omega }/\gamma =1$. (b),(c) Conditional trajectories for $⟨{\sigma }_y{⟩}_c$ when the channel is being monitored (no feedback) with perfect homodyne detection at angles $\theta =4\pi /5$ and $\pi /2$, respectively. Solid red lines represent the dynamics starting from the underlying initial state ${\rho }_0=[I+({\sigma }_x+{\sigma }_y)/\sqrt{2}]/2$, while dashed blue lines represent the filter evolution from the (incorrect) initial estimate ${\pi }_0=[I+({\sigma }_x-{\sigma }_y)/\sqrt{2}]/2$. Although the estimate ${\pi }_t$ converges to ${\rho }_t$ in (b), in the no-knowledge case (c) ${\rho }_t$ and ${\pi }_t$ never converge. (d) Dephasing effect for the unmonitored system [cf. Eq. (1)]. (e) Dephasing is canceled by directly feeding back the no-knowledge measurement via the Hamiltonian $H=\mathrm{\Omega }{\sigma }_x+\sqrt{\gamma }{\sigma }_z{y}_{\pi /2}(t)$. Despite the filter’s inaccurate estimate of ${\rho }_t$, decoherence is completely removed, demonstrating that accurate knowledge of the system is not required for effective decoherence cancellation.

Figure 2

Scheme for engineering measurements of $L_{\pm }$ from the outputs of couplings $L$ and $L^{†}$.

Figure 3

(a) A DQC setup with an $N$ qubit chain coupled to quasilocal operators $Q_i$ and local loss operators $L_i$. As demonstrated in (c), a loss rate of $\gamma /\alpha =10$ decreases the fidelity between the target cluster state and the system steady state (green triangles), and decreases it more severely for a larger number of qubits. (b, i) The errors introduced by the local loss are corrected by applying our no-knowledge feedback protocol on each qubit. (b, ii) For each qubit a no-knowledge measurement is constructed by coupling an additional reservoir $\sqrt{\gamma }{\sigma }_{+}^i$ and measuring ${\sigma }_x^i$ and ${\sigma }_y^i$ at a homodyne angle of $\pi /2$ as summarized in Fig. 2. Decoherence is canceled by feeding back $H=\sqrt{\eta \gamma }\sum _i[{\sigma }_x^i{y}_{\pi /2}^{(i,+)}(t)+{\sigma }_y^i{y}_{\pi /2}^{(i,-)}(t)]$. (c) The fidelity as a function of system size for no feedback (green triangles), and no-knowledge feedback with detection efficiency $\eta =0.9$ (yellow diamonds), $\eta =0.99$ (red squares), and $\eta =1$ (blue circles).