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Discrete-Time Quantum Walks

Updated 16 August 2025
  • Discrete-Time Quantum Walks are unitary quantum analogs of classical random walks that use coin operators to conditionally shift the walker's position.
  • They simulate diverse quantum phenomena including quantum transport, topological states, and gauge field interactions, demonstrating transitions from ballistic spreading to diffusive behavior.
  • Analytical frameworks provide precise evolution equations and diffusion coefficients that quantify decoherence effects from random artificial gauge fields.

Discrete-Time Quantum Walks (DTQWs) are unitary quantum analogs of classical random walks, constructed on discrete lattices where a walker possesses an internal degree of freedom (typically called the "coin"). At each discrete time step, a coin operator acts on the internal state, followed by a conditional position shift. DTQWs exhibit ballistic spreading, strong quantum interference effects, and can be engineered to simulate diverse quantum phenomena including relativistic equations, quantum transport, topological states, and gauge field interactions.

1. Mathematical Definition and Structure

A standard DTQW on a 1D lattice is defined by iteratively applying a unitary evolution operator, typically of the form

U=S(CI),U = S \cdot (C \otimes I),

where CC is the coin unitary (often SU(2)SU(2)), and SS is the conditional position-shift operator. For a walker’s state Ψj,m\Psi_{j,m} at time jj and position mm, the update rule is

[ψj+1,mL ψj+1,mR]=B(θj,ξj)[ψj,m+1L ψj,m1R],\begin{bmatrix} \psi^L_{j+1,m} \ \psi^R_{j+1,m} \end{bmatrix} = \mathcal{B}(\theta_j, \xi_j) \begin{bmatrix} \psi^L_{j,m+1} \ \psi^R_{j,m-1} \end{bmatrix},

with B(θ,ξ)=[e<sup>iξ</sup>cosθamp;isinθ isinθamp;e<sup>iξ</sup>cosθ]\mathcal{B}(\theta, \xi) = \begin{bmatrix} e<sup>{i\xi}</sup> \cos\theta &amp; i \sin\theta \ i \sin\theta &amp; e<sup>{-i\xi}</sup> \cos\theta \end{bmatrix}.

Generalizations include higher spatial dimensions, more complex coin spaces, time- and position-dependent coins, and incorporation of artificial gauge fields. The quantum amplitude of the walker spreads and interferes on the graph, a property that underpins its use in quantum simulation and quantum computation.

2. Gauge Fields, Disorder, and Decoherence in DTQWs

DTQWs serve as platforms for studying the impact of artificial gauge fields and disorder on quantum transport. The model can include random gauge-field fluctuations by letting the coin parameters (such as angle θ\theta or phase ξ\xi) be stochastic variables, sampled independently at each time step with a uniform law (noise amplitude σ\sigma).

The averaged evolution equation for the density operator is then: ρˉ^N(K,p)=RˉN(K,p,σ)ρˉ^0(K,p),\hat{\bar{\rho}}_N(K,p) = \bar{\mathcal R}^N(K,p,\sigma)\,\hat{\bar{\rho}}_0(K,p), with

Rˉ(K,p,σ)=1σIσR(ω,K,p)dω.\bar{\mathcal R}(K,p,\sigma) = \frac{1}{\sigma}\int_{I_\sigma} \mathcal R(\omega, K, p) d\omega.

Randomness in these fields induces decoherence; the averaged evolution becomes nonunitary, and the decay of off-diagonal elements in the density operator (as quantified by spin coherence Cj=maxm,mρj,m,mLRC_j = \max_{m,m'} | \rho_{j,m,m'}^{LR} |) accelerates with increasing noise. As jj \to \infty, the quantum walk transitions from ballistic to diffusive behavior, with the density profile becoming Gaussian and mean-square displacement growing linearly in time: m2e/g(j,σ)=2De/g(σ)j,\overline{m^2}^{\,e/g}(j,\sigma) = 2D^{e/g}(\sigma)\, j, with explicit expressions for De/g(σ)D^{e/g}(\sigma) determined analytically.

This framework provides an exact, deterministic description for the ensemble-averaged DTQW with decoherence from random gauge noise, bypassing stochastic simulation averaging.

3. Long-Time and Continuous Limits

The asymptotic behavior of DTQWs with random gauge fields admits a central limit theorem, indicating that at large times, the spatial density profile converges to a Gaussian: ρ^je/g(Kj,p)12exp(αe/g(p,σ)jKj2)u1,\hat{\rho}_j^{e/g}(K_j,p) \sim \frac{1}{2} \exp\left(-\alpha^{e/g}(p,\sigma)\, j\, K_j^2 \right) u_1, where Kj=K/jK_j = K_*/\sqrt{j} and u1u_1 is a basis spinor. The diffusion coefficient and the prefactor αe/g(p,σ)\alpha^{e/g}(p,\sigma) are explicitly computed (see section 5 below for their formulas).

A continuous limit is obtained by grouping steps and expanding for small parameters (weak disorder, long wavelengths): tρ^(t,K,p)=(Se/g(K,p,σ)1)ρ^(t,K,p),\partial_t \hat{\rho}(t, K, p) = \left( \mathcal{S}^{e/g}(K,p,\sigma)-1 \right) \hat{\rho}(t,K,p), with Se/g(K,p,σ)\mathcal{S}^{e/g}(K,p,\sigma) expanded to second order in KK, pp, and σ\sigma (see detailed matrix in the source). This limit connects the averaged discrete dynamics to a partial differential equation, interpolating between Dirac-like transport and diffusion depending on the disorder.

4. Effects of Random Artificial Electric and Gravitational Fields

The analysis distinguishes two physical scenarios:

  • Random electric field: Coin phase ξ\xi is random, angle θ\theta is fixed (π/4\pi/4).
  • Random gravitational field: Coin angle θ\theta is random, phase ξ\xi is fixed (π/2\pi/2).

In both cases, the average dynamics is nonunitary, coherence is rapidly lost as a function of the noise amplitude, and the system approaches classical, diffusive transport. The continuous limit for the pure (coherent) DTQW recovers the Dirac equation coupled to deterministic gauge fields, but once randomness in the coin is included, the continuous limit models a diffusive system, with noise manifesting as a diffusion term for the slow degrees of freedom.

5. Explicit Analytical Results

Key analytical results for the asymptotic dynamics include:

  • Density operator evolution (Fourier space):

ρ^je/g(Kj,p)12exp(αe/g(p,σ)jKj2)u1,\hat{\rho}_j^{e/g}(K_j,p) \sim \frac{1}{2} \exp\big( -\alpha^{e/g}(p,\sigma)\, j\, K_j^2 \big) u_1,

with the functions, for the electric case,

αe(p,σ)=23+(sinc(σ))2+2(sinc(σ/2))2(1+sinc(σ))+4cos(2p)(sinc(σ)+(sinc(σ/2))2)3+(sinc(σ))22(sinc(σ/2))2(1+sinc(σ))+4cos(2p)(sinc(σ)(sinc(σ/2))2)\alpha^{e}(p,\sigma)=2\,\frac{3+(\text{sinc}(\sigma))^2+2\,(\text{sinc}(\sigma/2))^2\,(1+\text{sinc}(\sigma))+4\,\cos(2p)(\text{sinc}(\sigma)+(\text{sinc}(\sigma/2))^2)}{3+(\text{sinc}(\sigma))^2-2\,(\text{sinc}(\sigma/2))^2\,(1+\text{sinc}(\sigma))+4\,\cos(2p)(\text{sinc}(\sigma)-(\text{sinc}(\sigma/2))^2)}

and for the gravitational case,

αg(p,σ)=21+(sinc(σ))21(sinc(σ))2,\alpha^{g}(p,\sigma)=2\,\frac{1+(\text{sinc}(\sigma))^2}{1-(\text{sinc}(\sigma))^2},

where sinc(σ)=sinσσ\text{sinc}(\sigma) = \frac{\sin \sigma}{\sigma}.

  • Mean-square displacement growth:

m2e/g(j,σ)=2De/g(σ)j,\overline{m^2}^{\,e/g}(j,\sigma) = 2D^{e/g}(\sigma) \,j,

with, for gravitational case,

Dg(σ)=21+(sinc(σ))21(sinc(σ))2,D^g(\sigma) = 2 \, \frac{1+(\text{sinc}(\sigma))^2}{1-(\text{sinc}(\sigma))^2},

and an explicit, more intricate expression for De(σ)D^e(\sigma) involving sinc\text{sinc} functions.

  • Full coin+shift evolution (coin matrix in physical basis):

B(θ,ξ)=[eiξcosθisinθ isinθeiξcosθ]\mathcal{B}(\theta,\xi) { = \begin{bmatrix} e^{i\xi} \cos\theta & i \sin\theta \ i \sin\theta & e^{-i\xi} \cos\theta \end{bmatrix} }

  • Ensemble-averaged evolution:

ρˉ^N(K,p)=RˉN(K,p,σ)ρˉ^0(K,p)\hat{\bar{\rho}}_N(K,p) = \bar{\mathcal{R}}^N(K,p,\sigma) \hat{\bar{\rho}}_0(K,p)

with

Rˉ(K,p,σ)=1σIσR(ω,K,p)dω\bar{\mathcal{R}}(K,p,\sigma) = \frac{1}{\sigma} \int_{I_\sigma} \mathcal{R}(\omega, K, p) d\omega

6. Broader Significance and Outlook

The paper establishes a deterministic, exact analytical framework for describing DTQWs exposed to random artificial gauge fields, mapping random discrete dynamics into a tractable, closed-form evolution for the averaged density operator. The shift from ballistic (quantum) to diffusive (classical-like) behavior, emergence of Gaussian profiles, and explicit computation of diffusion constants are key advances.

This approach offers analytical access to quantum transport under decoherence, providing precise characterization when randomness is present in gauge-like field parameters—relevant for quantum simulation, condensed-matter emulation, and studying quantum-to-classical transitions.

Potential extensions include generalization to walks where gauge fields vary in both space and time, and application of this method to more intricate topological and interacting scenarios where noise or disorder play essential roles. The robust technique has implications for efficient numerical simulation, as exact ensemble-averaged dynamics can be computed without Monte Carlo sampling.

Table: Key Results for DTQWs in Random Gauge Fields

Physical Scenario Random Variable Asymptotic Behavior Diffusion Coefficient
Random electric field Coin phase ξ\xi Gaussian, diffusive De(σ)D^e(\sigma) (analytic, see above)
Random gravitational field Coin angle θ\theta Gaussian, diffusive Dg(σ)=2(1+sinc2(σ))/(1sinc2(σ))D^g(\sigma) = 2(1+\text{sinc}^2(\sigma))/(1-\text{sinc}^2(\sigma))

Decoherence is driven by time-dependent random fluctuations in coin parameters, progressively destroying quantum interference and leading to classical diffusion in the probability distribution. This transition is quantitatively captured by analytically derived diffusion constants and the evolution equations for averaged density operators.