Asymptotic Exceptional Steady States in Dissipative Dynamics

Abstract

Spectral degeneracies in Liouvillian generators of dissipative dynamics generically occur as exceptional points, where the corresponding non-Hermitian operator becomes non-diagonalizable. Steady states, i.e. zero-modes of Liouvillians, are considered a fundamental exception to this rule since a no-go theorem excludes non-diagonalizable degeneracies there. Here, we demonstrate that the crucial issue of diverging timescales in dissipative state preparation is largely tantamount to an asymptotic approach towards the forbidden scenario of an exceptional steady state in the thermodynamic limit. With case studies ranging from NP-complete satisfiability problems encoded in a quantum master equation to the dissipative preparation of a symmetry protected topological phase, we reveal the close relation between the computational complexity of the problem at hand, and the finite size scaling towards the exceptional steady state, exemplifying both exponential and polynomial scaling. Formally treating the weight $W$ of quantum jumps in the Lindblad master equation as a parameter, we show that exceptional steady states at the physical value $W=1$ may be understood as a critical point hallmarking the onset of dynamical instability.

Figures (3)

• Figure 1: (a) Liouvillian spectrum ($W=1$ in Eq. \ref{['eq:lindblad']}) for a satisfiable 3SAT instance with a unique solution, $N=14$ variables, and $M=\text{round}(\alpha_cN)$ clauses, where the satisfiability threshold $\alpha_c\approx4.267$. The two red states form the asymptotic exceptional steady state (AESS). (b,c) Liouvillian gap $\Delta$ and eigenoperator overlap $|\nu|^2$ as a function of $W$ (other parameters as in (a)). The insets in (b,c) show the mean values of $\overline\Delta$ and $\overline{|\nu|^2}$ at $W=1$ change with the number of variables $N$, averaging over $10^3$ random instances for each $N$ with $M=\text{round}(\alpha_cN)$. All instances have a unique solution. Shadow regions indicate the standard deviation among instances.
• Figure 2: Eigenvalues (a) of steady and metastable states and their eigenoperator overlap (b) over $W$ [same instance as in Fig. \ref{['fig1']}]. The shaded region represents $W>1$, where $\text{Re}(\lambda_1)>0$ renders the system dynamically unstable in the thermodynamic limit. We observe an EP at $W_c\approx 1.00278$. (c) Perturbed eigenvalues for $\mathcal{L}_{W_c}+\mathcal{L}_{W_c}^{\text{pert}}$, where $\delta$ is the perturbation strength. (d) The mean $W_c$ decays with $N$. Each point is averaged over $10^3$ instances with one solution and the shaded region marks the standard deviation between instances.
• Figure 3: AESS in quantum systems. (a) The case of including a modified transverse field [Eq. \ref{['eq:PXP']}] in dissipative 3SAT solver. We set $M=\text{round}(\alpha_c N)$ and $W=1$. All points are obtained by averaging over $10^3$ satisfiable instances with a unique solution. (b) The perturbed eigenvalues of $\mathcal{L}_{W_c}+\mathcal{L}_{W_c}^{\text{pert}}$ for a uniquely solvable 3SAT instance with $N=11$. (c) Eigenoperator overlap of $\mathcal{L}_1$ for preparing the AKLT state [Eq. \ref{['eq:AKLT']}]. (d) The perturbed eigenvalues of $\mathcal{L}_{W_c}+\mathcal{L}_{W_c}^{\text{pert}}$ for AKLT state preparation with $N=7$.