Certifying steady-state properties of open quantum systems

TL;DR

The paper tackles certifying steady-state properties of open quantum systems described by the Lindblad equation by deriving guaranteed upper and lower bounds on expectation values of observables, i.e., bounding $\langle O \rangle_{ss}$ where $\mathcal{L}(\rho_{ss})=0$ and $\langle O \rangle_{ss}=\mathrm{tr}(O\rho_{ss})$. It builds scalable SDP relaxations using Pauli-string moment matrices with elements $M_{ij}=\mathrm{tr}(\rho\,\theta_i^\dagger\theta_j)$, enforces positivity of reduced density matrices, and imposes linear constraints from the adjoint Lindbladian $\mathcal{L}^\dagger$, together with an automatic constraint-generation routine to identify the most informative Pauli operators. Benchmarking on a two-qubit nonequilibrium setup, a periodic 1D chain, and a 2D ladder, the method yields tight bounds for a 12-site chain (within about 1% of the exact value) and informative bounds for systems with hundreds of qubits, with computation times scaling modestly. Overall, this work provides the first general numerical tool to certify steady-state properties of open quantum dynamics, offering guaranteed bounds where variational or tensor-network methods yield only estimates and opening avenues for phase-transition diagnostics and hybrid methods.

Abstract

Estimating the steady-state properties of open many-body quantum systems is a fundamental challenge in quantum science and technologies. In this work, we present a scalable approach based on semi-definite programming to derive certified bounds on the expectation value of an arbitrary observable in the steady state of Lindbladian dynamics. We illustrate our method on a series of many-body systems, including paradigmatic spin-1/2 chains and two-dimensional ladders, considering both equilibrium and nonequilibrium steady-states. We benchmark our method with state-of-the-art tensor-network approaches that, unlike our method, are only able to provide estimates, with no guarantee, on steady-state quantities. For the tested models, only modest computational effort is needed to obtain certified non-trivial bounds for system sizes intractable by exact methods. Our method introduces the first general numerical tool for bounding steady-state properties of open quantum dynamics, opening a new avenue in the understanding of stable configurations in many-body systems.

Figures (4)

• Figure 1: (a) Two-qubit example: Here the left qubit is connected to a hot bath and the right qubit is connected to a cold bath. The system is defined by parameters $\gamma_h$, $\gamma_c$, coupling of the qubits with their respective baths, $\epsilon_h$, $\epsilon_c$, energy gaps of the qubits, $T_c$, $T_h$, the temperatures of the baths, and $g$, coupling between the two qubits. (b) 1D periodic chain example: Here each qubit is coupled to the dissipative bath with coefficient $\gamma$ and coupled to its nearest neighbours with coefficient $J$. The final qubit is also coupled to the first qubit. A field of coefficient $\eta$ is also applied to each qubit in the $X$ direction. (c) 2D ladder example: Here the end qubits are connected to the respective heat bath with coefficient $\gamma_h$and$\gamma_c$, and each qubit is coupled to its nearest neighbours with coefficient $J$. A field of coefficient $\eta$ is also applied to each qubit in the $Z$ direction.
• Figure 2: Upper and lower bounds on the magnetization at the steady state as a function of the system size for the periodic linear chain. The plotted intervals represent the certified region in which the true value of the observable lies. The y-axis is scaled to $[-1,1]$ to demonstrate the fraction of the trivial bounds that we restrict to. We apply method \ref{['sdp-notation']} imposing a limited number of linear constraints and moment matrix size (set 1 - 10000 linear constraints and a $40\times 40$ moment matrix, set 2 - 30000 constraints and a $100\times 100$ matrix, set 3 - 70000 constraints and a $250\times 250$ matrix). Symmetry between all qubits is also imposed. The time to optimize per point was roughly the same within each constraint set (set 1$\approx 10$ seconds, set 2$\approx 1$ minute, set 3$\approx 1$ hour).
• Figure 3: Schematic on the two possible cases: (a) one steady state vs. (b) multiple steady states. $\langle O\rangle_{\max/\min}$ are the outcomes of the exact problem \ref{['sdp-dm-O']}. $\langle {O}\rangle_{\rm ub/lb}$ are the outcomes of problem \ref{['sdp-notation']}, which is a relaxation of \ref{['sdp-dm-O']}. In full generality, we have $\langle {O}\rangle_{\rm ub} \geq \langle {O}\rangle_{\max} \geq \langle {O}\rangle_{\min}\geq \langle {O}\rangle_{\rm lb}.$ Observing $\langle O\rangle_{\max}>\langle O\rangle_{\min}$ implies the existence of multiple steady states.
• Figure 4: Schematic on the difference between estimating with an ansatz and bounding with an SDP relaxation the steady-state properties of open quantum systems undergoing Lindbladian dynamics. The set $\cal D$ is the convex set of density matrices, that contains the convex set $SS$ of steady states. If the Lindbladian admits a single steady state the set $SS$ is geometrically just a single point. (a)-(b) An estimation method based on an ansatz aims at minimizing a certain cost-function over the set $\Omega$ to find an estimation of ${\rm tr}(O \rho_{ss})$. The latter problem is generally non-convex and the obtained result contains an uncontrolled error. (c) An SDP relaxation of the problem implies minimizing (maximizing) on the set $(SS)'\supseteq SS$, finding global optima in $(SS)'$. As a consequence the obtained value bounds the possible values ${\rm tr}(O \rho_{ss})$ from below (above).