Non-unitary enhanced transfer efficiency in quantum walk search on complex networks

TL;DR

The paper investigates transfer efficiency of a quantum walk to a trapping sink on complex networks by introducing a sink attached to the target in a spatial-search setting. It develops the Stochastic Quantum Walk Search (SQWS), a hybrid dynamics framework that blends unitary CTQW with dissipative CTRW via a tunable parameter $\omega$, plus a sink with rate $\Gamma$. Numerical results show that a low-noise hybrid regime (approximately 90% coherence) can outperform purely quantum or classical dynamics on certain sparse graphs, while dense graphs favor quantum dynamics; the presence of the sink is essential for this advantage. The study links performance to entropy decay rates and local graph metrics, suggesting that dissipation can serve as a resource for transport in open quantum systems and offering guidance for designing dissipation-aware quantum networks and hardware.

Abstract

The task of finding an element in an unstructured database is known as spatial search and can be expressed as a quantum walk evolution on a graph. In this article, we modify the usual search problem by adding an extra trapping vertex to the graph, which is only connected to the target element. We study the transfer efficiency of the walker to a trapping site, using the search problem as a case study. Thus, our model offers no computational advantage for the search problem, but focuses on information transport in an open environment with a search Hamiltonian. The walker evolution is a mix between classical and quantum walk search dynamics. The balance between unitary and non-unitary dynamics is tuned with a parameter, and we numerically show that depending on the graph topology and the connectivity of the target element, this hybrid approach can outperform a purely classical or quantum evolution for reaching the trapping site. We show that this behavior is only observed in the presence of an extra trapping site, and that depending on the topology and a tunable parameter controlling the strength of the oracle, a hybrid regime composed of 90% coherent dynamics can lead to either the highest or worst transfer efficiency to the trapping site. We also relate the performance of an hybrid regime to the entropy's decay rate. As the introduction of non-unitary operations may be considered as noise, we interpret this phenomena as a noisy-assisted quantum evolution.

Figures (13)

• Figure 1: Modified cycle graph $C_6$ on which we add the extra sink vertex $\phi$ connected to the target vertex $m$. The irreversible transition from $m$ to $\phi$ is weighted with the sink rate $\Gamma$. The classical and quantum walk search dynamics only act on the vertices of the initial graph, not on the sink vertex. The search dynamics guides the walker to the target vertex $m$ for it to reach the sink $\phi$, and “escape” the graph.
• Figure 2: Stochastic Quantum Walk Search (SQWS) performances with different values of $\gamma\in [0,30]$ on graphs of 64 vertices, namely, the complete graph ($K_{64}$), the 6-dimensional hypercube graph ($Q_6$), the cycle graph ($C_{64}$), the path graph ($P_{65}$), a maze graph ($M_{73}$) and the tadpole graph ($T_{32,32}$). The time evolution is set to $t=640$ units of time. The quantum walk dynamics is recovered for $\omega=0$, the classical random walk for $\omega=1$, and a linear combination of the two when $\omega\in]0,1[$.
• Figure 3: Instances $T_{8,8}$ and $L_{8,8}$ of the tadpole (upper) and lollipop (lower) graphs, that are respectively a fusion between the cycle graph $C_8$ or the complete graph $K_8$ with the path graph $P_8$. We respectively refer to the locations of the red, green and blue vertices as cycle, shared and path vertices for the tadpole graph, and complete, shared and path for the lollipop graph.
• Figure 4: Von Neumann entropy $S(\rho)$ as a function of time $t$ for different values of $\gamma$ for the complete graph $K_{64}$, the 6-hypercube $Q_6$, the cycle graph $C_{64}$ and the tadpole graph $T_{32,32}$ for the cycle vertex. The markers indicate the time $t_S$ at which the entropy reaches its maximum value before decreasing down to zero.
• Figure 5: Evolution of the duration $t_S$ needed to reach maximum entropy as a function of $\gamma$ for the complete graph $K_{64}$, the 6-hypercube $Q_6$, the cycle graph $C_{64}$ and the tadpole graph $T_{32,32}$ (with cycle vertex) for different values of interpolation $\omega\in[0,1]$.