Discrete-Time Quantum Walk and Feed-Forward Coins

• Discrete-Time Quantum Walks (DTQWs) are quantum processes on a lattice using coin and shift operations to facilitate unitary evolution and underpin quantum algorithms and simulation.
• The feed-forward coin model introduces nonlinearity by making the coin operation depend on neighboring amplitudes, leading to sub-ballistic, q-Gaussian anomalous diffusion.
• This nonlinear DTQW framework enables tuning of transport properties and provides a test-bed for simulating classical nonlinear diffusion, such as dynamics described by the porous-medium equation.

A discrete-time quantum walk (DTQW) is a unitary evolution on a Hilbert space composed of position and internal "coin" degrees of freedom, featuring a sequence of coin and shift operations similar in spirit to a quantum-mechanical cellular automaton. DTQWs are foundational for quantum algorithms, quantum simulation, and the paper of quantum transport phenomena. Standard DTQWs employ coin operators that are static or at most site-dependent, leading to ballistic spreading of the walker's probability distribution. The feed-forward quantum coin model represents a nonlinear generalization where the coin operator at each site and step depends on neighboring coin amplitudes from the previous time, fundamentally altering the transport and introducing anomalous diffusion. The model connects quantum walk dynamics with nonlinear classical diffusion equations, notably the porous medium equation, yielding rich theoretical and practical implications for quantum computation and quantum simulation.

1. Structural Generalization: Feed-Forward Quantum Coin

In the feed-forward DTQW, the standard unitary update with a fixed coin is replaced by a nonlinear, memory-dependent operator. The state at time $t$, with amplitudes $a_{j}^{t}$ and $b_{j}^{t}$ corresponding to the two coin basis components at position $j$, evolves according to: $$\begin{align*} a_{j-1}^{t+1} &= g_{j}^{t} \, a_{j}^t - \sqrt{1 - (g_{j}^{t})^{2}} \, b_{j}^t, \ b_{j+1}^{t+1} &= \sqrt{1 - (g_{j}^{t})^{2}} \, a_{j}^t + (g_{j}^{t})^* \, b_{j}^t, \end{align*}$$ where the rate function (or "feed-forward coin") is

$g_{j}^{t} = a_{j-1}^{t} + i b_{j+1}^{t}.$

This construction introduces strong, time-dependent nonlinearity and spatial nonlocality, as the coin operation at $(j, t+1)$ explicitly depends on amplitudes at neighboring positions and the previous step. When $g_{j}^{t}$ is set to a constant, the standard homogeneous DTQW is recovered. The system thus transitions from linear, site-local quantum walks to a correlated, history-dependent regime.

2. Mathematical Description and Nonlinear PDE Limit

The state space of the feed-forward DTQW is $\mathcal{H}_{s} \otimes \mathcal{H}_{c}$, with $|j\rangle$ labeling lattice positions and a two-dimensional coin. The nonlinear update above, together with the normalization

$$P_{j}^{t} = |a_{j}^{t}|^{2} + |b_{j}^{t}|^{2}, \qquad \sum_{j=-\infty}^{\infty} P_{j}^{t} = 1,$$

defines the system's evolution. In the continuum and long-time limit, the probability density $\rho(x,t)$ is shown to approximately satisfy the (asymptotic) nonlinear diffusion equation: $$\partial_{t}\rho(x,t) \approx \frac{1}{4} \partial_{x}^{2} [\rho(x,t)]^{2}.$$ This is a specific case (with $m=2$) of the porous-medium equation (PME),

$\partial_{t} p(x,t) = \partial_{x}^{2} [p(x,t)]^m,$

which is well-known in classical nonlinear diffusion contexts. The quantum model, however, retains interference terms not present in the classical Markovian approximation. These terms are responsible for deviations from pure PME-type dynamics, notably in the scaling exponent of the evolving distribution's width.

3. Anomalous Diffusion Behavior

The crucial effect of the feed-forward coin is a dramatic change in transport properties. Standard DTQWs propagate ballistically, i.e., the standard deviation $\sigma(t) \sim t$. By contrast, simulations of the feed-forward model report:

• The probability distribution is well fit by a $q$-Gaussian:

$$p_{q}(x,t) \propto [1 - (1-q) (x^{2}/\sigma_{q}^{2}(t))]^{1/(1-q)},$$

with $q \approx 0.5$.

• The standard deviation grows sublinearly:

$\sigma_{q=0.5}(t) \sim t^{0.4}.$

• In the associated Markovian (interference-free) model, diffusion is even slower: $\sigma_{q=0}(t) \sim t^{1/3}$, in precise correspondence to the PME with $m=2$.

In summary, interference in the quantum model "enhances" the spread relative to the Markovian limit, but the process remains sub-ballistic and strongly non-Gaussian.

4. Implications for Quantum Simulation and Computation

The introduction of feed-forward, history-dependent coin operators in DTQWs provides new degrees of control and complexity in quantum walk models. This includes:

• The ability to tune the speed and character of quantum transport, which may prove useful for constructing algorithms sensitive to anomalous diffusion or for simulating classical nonlinear systems within a quantum framework.
• Insights into how quantum interference, nonlinearity, and memory can combine to yield classical-like anomalous diffusion from unitary quantum dynamics.
• The potential for experimental realization in optical systems, using spatial/polarization encoding for position/coin and programmable feed-forward feedback based on measurement outcomes.

5. Analytical and Numerical Validation

The findings are substantiated through a combination of analytical and extensive numerical studies:

• Numerical simulations (with up to $10^{8}$ steps) confirm the anomalously slow, $q$-Gaussian broadening and extract the relevant exponents.
• Analytically, the nonlinear DTQW dynamics are mapped to a nonlinear PDE by separating the evolution into a Markovian component and interference terms. The continuum limit is systematically derived.
• Asymptotic Lie symmetry analysis demonstrates that the long-time solution is a $q$-Gaussian, consistent with the porous-medium equation's fundamental solutions under the prescribed nonlinearity.
• Practical implementation is discussed in terms of an optical network: spatial modes for the walker, polarization for the coin, and feed-forward coin control realized via intensity-dependent wave-plate and beam splitter networks. Despite introducing nonlinearity, the scheme remains within reach of current experimental techniques.

6. Context within the Broader Landscape and Future Directions

The feed-forward quantum coin DTQW offers a bridge between discrete nonlinear quantum systems and classical nonlinear transport models. Key distinctions from linear DTQWs include:

• Memory and nonlinearity as intrinsic physical resources for engineering quantum transport phenomena that mimic or generalize classical anomalous diffusion.
• A mathematical and experimental framework for connecting quantum cellular automata to nonlinear partial differential equations and generalized entropy formalisms (Tsallis statistics).
• New platforms for exploring the impact of coin nonlinearity in quantum computation, possibly allowing for resource-efficient simulation of systems with classical nonlinear dynamics.

The model invites extensions to higher dimensions, alternative nonlinear feedback forms, and experimental exploration, with significant implications for quantum information processing and fundamental studies in quantum transport and simulation.