Decoherence time maximization and partial isolation for open quantum harmonic oscillator memory networks

TL;DR

The paper tackles the problem of maximizing the memory decoherence time for a network of open quantum harmonic oscillators connected by direct energy and field-mediated couplings and driven by external quantum fields. It models the network with linear quantum stochastic differential equations on a graph, defines a fidelity-based memory measure via a weighted mean-square deviation $\Delta(t)$ of a linear readout $\varphi(t)=F X(t)$, and derives a high-fidelity asymptotic expression for the decoherence time $\tau(\epsilon)$. The main result is a necessary-and-sufficient condition for optimality in the form of Sylvester-like linear equations for the blocks of the direct energy coupling matrix, under which the approximate decoherence time is maximized; a convex-quadratic structure enables efficient computation and reveals how graph sparsity governs the solution. The work also introduces a partially isolated subnetwork framework that can achieve longer decoherence times by enforcing $FB=0$, yielding a quadratic short-horizon growth of $\Delta(t)$ and a closed-form asymptotic for $\tau(\epsilon)$. Together, these contributions provide a tractable approach to engineering quantum memories in networked OQHOs with clear pathways to numerical algorithms and potential extensions beyond short horizons and to finite-level systems.

Abstract

This paper considers a network of open quantum harmonic oscillators which interact with their neighbours through direct energy and field-mediated couplings and also with external quantum fields. The position-momentum dynamic variables of the network are governed by linear quantum stochastic differential equations associated with the nodes of a graph whose edges specify the interconnection of the component oscillators. Such systems can be employed as Heisenberg picture quantum memories with an engineered ability to approximately retain initial conditions over a bounded time interval. We use the quantum memory decoherence time defined previously in terms of a fidelity threshold on a weighted mean-square deviation for a subset (or linear combinations) of network variables from their initial values. This approach is applied to maximizing a high-fidelity asymptotic approximation of the decoherence time over the direct energy coupling parameters of the network. The resulting optimality condition is a set of linear equations for blocks of a sparse matrix associated with the edges of the direct energy coupling graph of the network. We also discuss a setting where the quantum network has a subset of dynamic variables which are affected by the external fields only indirectly, through a complementary ``shielding'' system. This holds under a rank condition on the network-field coupling matrix and can be achieved through an appropriate field-mediated coupling between the component oscillators. The partially isolated subnetwork has a longer decoherence time in the high-fidelity limit, thus providing a particularly relevant candidate for a quantum memory.

Figures (3)

• Figure 1: An illustration of a network of OQHOs (depicted as numbered circles) with external quantum Wiener processes $W_j$ and output fields $Y_{jk}$ which mediate the coupling between the component OQHOs. This field-mediated coupling is represented by arrows, while dashed lines indicate the direct energy coupling.
• Figure 2: Two component OQHOs $j$ and $k$ (with $j,k \in V$ such that $j\ne k$) belong to the neighbourhoods $N_k^-$ and $N_j^+$ (shown as dashed boxes), respectively, if and only if they are coupled through the output field $Y_{jk}$ from $j$ to $k$.
• Figure 3: A schematic representation of (\ref{['phipsi']}) as an interconnection of quantum subnetworks $\Phi$, $\Psi$, where $\Phi$ is affected by the external fields $W$ only through $\Psi$ which interacts with $W$.

Theorems & Definitions (8)

• Theorem 1
• proof
• Theorem 2
• proof
• Lemma 1
• Lemma 2
• Theorem 3
• proof