Non-Markovianity in Quantum Information Processing: Interplay with Quantum Error Mitigation

TL;DR

The paper reveals that non-Markovianity is an intrinsic aspect of quantum information processing, arising from the intrinsic subsystem structure used in QIP protocols. By introducing a subsystem unitary isomorphism that separates logical and gauge degrees of freedom and leveraging feedback from gauge measurements, it shows that the logical dynamics can be described by non-CPTP maps and canonical master equations, signaling information backflow. It analyzes Pauli-based QEC, bosonic QEC (cat and squeezed cat codes), and quantum teleportation, deriving explicit master equations and decay-rate expressions that quantify non-Markovianity. It further connects this intrinsic non-Markovianity to quantum error mitigation, showing that negativity in the dynamics can reduce QEM sampling overhead, highlighting a practical synergy between QEC and QEM for scalable QIP.

Abstract

Non-Markovian dynamics are typically present in the dynamics of open quantum systems. Despite the rich structure of non-Markovian dynamics, their relevance to quantum information processing (QIP) has been rarely discussed. In this work, we demonstrate that the negativity of the dynamics, a characteristic of non-Markovian dynamics, naturally arises in quantum error correction (QEC) and quantum teleportation. The negativity in open quantum systems is naturally attributed to the information backflow from the environment. We partition the whole Hilbert space into the logical subsystem and the gauge subsystem. The logical subsystem stores the quantum information for QIP, while the gauge subsystem stores the information for recovery of the logical information, i.e., the syndrome measurement outcomes for quantum error correction and Bell measurement outcomes for successful teleportation. We then show that the negativity in quantum information processing appears as a consequence of the feedback operation based on the measurement outcomes of the gauge subsystem. Finally, we show that the negativity of non-Markovianity in QIP reduces the sampling cost of quantum error mitigation (QEM), shedding light on the importance of combination strategies of QEC and QEM in a practical QIP.

Figures (3)

• Figure 1: The conceptual figure illustrating an example of our theory. (a) The subsystem unitary isomorphism operator $V_{\rm S}$ to decompose the state into the logical and gauge subsystems, illustrated by the three-qubit code. In this case, $V_{\rm S}$ is a Clifford unitary operator. (b) The emergence of the non-Markovian processes in QEC. First, the logical state is initialized in $\rho_{\rm L}$ at time $t_1$. The reduced density matrix in the logical subsystem changes into a mixed state $\sum_m d_m Q_m \rho_{\rm L} Q_m^\dag$ at time $t_2$ but correlates with the gauge subsystem. We can perform a feedback operation based on the gauge information, and the state changes into the original state $\rho_{\rm L}$ at time $t_3$ when the KL condition is satisfied. Then, the quantum process from $t_2$ to $t_3$ is generally a non-CPTP map, indicating the presence of non-Markovianity.
• Figure 2: Displacement error and QEC process in physical frame and subsystem frame for the squeezed cat code.
• Figure 3: Dynamics of the QEC process in the squeezed cat code against the displacement error. The expectation value of the logical Pauli $X$ operator is plotted. The solid lines correspond to results from the non-Markovian master equation, while the simulation results of the QEC circuit proposed in Ref. shitara2025exploiting are plotted with crosses. The displacement amplitude is set to be $\Lambda=$1 (red), 0.5 (blue), and 0.25 (red). Other parameters are set to be $\alpha=2$ and $r=1.3$.