Quantum Magic in Discrete-Time Quantum Walk

Abstract

Quantum magic, which accounts for the non-stabilizer content of a state, is essential for universal quantum computation beyond classically simulable resources. We investigate the generation and evolution of quantum magic in discrete-time quantum walks (DTQWs) using the Stabilizer Renyi Entropy as a measure of quantum magic. We investigate single- and two-walker quantum walks on a one-dimensional lattice, considering a wide range of initial coin states. Our results reveal that DTQWs can dynamically generate significant magic, with the amount and structure strongly dependent on the initial state of the coin. In the case of a single walker, the relationship between magic and entanglement is found to be nontrivial and complementary at long times. These findings position DTQWs as accessible and controllable platforms for producing quantum magic, offering a new perspective on their role in quantum information processing and reliable quantum computation.

Figures (10)

• Figure 1: The SRE as a function of time for the initial state of coin to be (a) $\ket{\Psi(0)}_C = \ket{\uparrow}$, (b) $\ket{\Psi(0)}_C = (\ket{\uparrow} + e^{i \pi/4} \ket{\downarrow})/\sqrt{2}$ while lattice part is localized in both the cases at $\ket{x = 0}$. The plots in the inset show the growth of non-stabilizerness at initial time steps. The system size is taken to be $1001$ for both plots.
• Figure 2: The SRE as a function of time and the parameter $\theta$ in the initial state of the coin for (a) $\phi = 0$, (b) $\phi = \pi/4$, (c) $\phi = \pi/2$. We can see the oscillating patterns in the density plots for almost all the values of the parameters $\theta$ and $\phi$; however, they have different amplitudes and oscillation rates. The system size is taken to be $1001$ for all the plots.
• Figure 3: Top Row: The SRE for different initial coin states, parameterized using the Bloch sphere representation, at $t = 0$, and in Middle $\&$ Bottom row, the SRE and von-Neumann entanglement entropy in the asymptotic limit as a function of the $\theta$ and $\phi$ from Eq. \ref{['eq:initialstate']}, respectively. We also plot the projection of the Bloch sphere in $x-y$, $y-z$, and $z-x$ planes for clarity. We observe a complementary behavior of SRE and the von-Neumann entropy in the long-time limit.
• Figure 4: The time evolution of the SRE for a fixed initial coin state, $\ket{\Psi(0)}_C = \ket{\uparrow}$, under varying decoherence strengths $\lambda$. (a) Long-time behavior of $\mathcal{M}(\rho)$ up to $t = 1000$ for weak decoherence values. A small but nonzero values of $\lambda$ lead to gradual damping of oscillations, while strong decoherence suppresses magic more rapidly (b) Short-time dynamics of $\mathcal{M}(\rho)$ for a broader range of decoherence strengths. While strong decoherence quickly kills magic, weak and moderate decoherence still allow nontrivial oscillations and a delayed decay of magic, indicating robustness at early times. The lattice part is localized at $\ket{x = 0}$.
• Figure 5: Stabilizer Rényi Entropy $\mathcal{M}(\rho)$ at time $t = 50$ as a function of the initial coin state parameterized by $\theta \in [0, \pi]$ and $\phi \in [0, 2\pi)$ for decoherence strengths $\lambda = 0$, $0.01$, $0.10$, and $0.20$. While the maximum SRE decreases with increasing $\lambda$, the overall landscape pattern remains qualitatively similar across these cases, indicating robustness in the structure of magic generation. For example, for $\lambda = 0.1$ we see a reduction of only $\approx 20 \%$ in the peak value of the SRE.