Liouvillian Exceptional Points in Quantum Brickwork Circuits

Abstract

We demonstrate that Liouvillian exceptional points (LEPs), previously explored only in continuous Lindbladian dynamics, also emerge in discrete brickwork completely positive trace-preserving (CPTP) circuits. By analytically solving a minimal two-qubit brickwork model, we identify the conditions under which discrete-time LEPs arise and show that they retain the hallmark square-root eigenvalue splitting and linear-in-time sensitivity enhancement. These results establish a direct bridge between continuous non-Hermitian physics and discrete quantum-circuit architectures, opening a path toward the realization of exceptional-point-based sensing on near-term quantum processors.

Figures (6)

• Figure 1: Schematic of the two-step brickwork protocol. Even steps implement a two-qubit unitary gate $U$ (XXZ-type), while odd steps consist of a Kraus relaxation map on qubit 1 and a local unitary channel $V$ on qubit 2. Periodic repetition of this protocol defines the brickwork CPTP map.
• Figure 2: Superoperator exceptional points manifold in parameter space $\{x\equiv \log \lambda, \gamma, \epsilon\}$ with $\epsilon\in[0,1]$ and $q=e^{i\gamma}$. Coordinates of representative points on the LEP manifold are: $a=\{0.3293,\pi/4,0.4\}$, $b=\{0.3466,\pi/2,0.5\}$, and $c=\{0.3013,\pi/9,0.2\}$. Bifurcation diagrams of $\mathcal{T}$ eigenvalues for EP $a$ are shown in Fig. \ref{['FigU1pi4']}, while those for EPs $b$ and $c$ are reported in Fig. S-1 of SM.
• Figure 3: Real part of $\mathcal{T}$ eigenvalues as a function of $\epsilon$ for fixed $\lambda=1.39$ (top panel), and as a function of $\log \lambda$ for fixed $\epsilon=0.4$ (bottom panel). Parameters: $q=e^{i\pi/4}$, $V=I$. Red and blue curves correspond to $\mathcal{T}_+$ and $\mathcal{T}_-$, respectively, while dashed lines refer to analytic (non-containing EP) eigenvalues $\lambda_j=1,\epsilon,\epsilon^2$. Bifurcation points occur at the EP (red dot), located on the EP manifold shown in Fig. \ref{['FigEPmanifold']}.
• Figure 4: Renormalized value of $|\langle \hat{\mathbf{e}}_3[n] \rangle|$ versus discrete time $n$, below EP $\epsilon\equiv \epsilon_0 = 0.32, 0.32 \pm \delta$ (upper row), at EP $\epsilon=0.4, 0.4\pm \delta$ ( middle row), and above EP $\epsilon=0.48, 0.48\pm \delta$ with $\delta=0.01$. Black, red and blue curves correspond to $\epsilon=\epsilon_0$, $\epsilon_0 +\delta$, $\epsilon_0 -\delta$ respectively. Value of $\mu$ is chosen as $\mu = \max(|\mu_9(\epsilon)|,|\mu_{10}(\epsilon)|)$ at the respective $\epsilon$. Linear increase of $|\mu^{-n}\langle \hat{\mathbf{e}}_3[n] \rangle|$ at EP (black curve in the middle row) is due to non-diagonalizability of Liouvillean at EP, see text. Other parameters: $q=\exp(i \pi/4), \lambda = 1.39016$. The EP is located at $\epsilon_{EP} = 0.4$.
• Figure 5: $\mu_9,\mu_{10}$, entering Eq. (\ref{['eq:e3evolution']}) versus $\epsilon$. EP is located at $\epsilon_{EP} = 0.4$. Bottom Panel shows absolute values $|\mu_9|,|\mu_{10}|$ while the Top panel shows their real part (solid lines ) and imaginary part (dashed lines). Other parameters are as in Fig. \ref{['FigEP']}: $q=\exp(i \pi/4), \lambda \approx 1.39$.