Discrete-Time Photonic Quantum Walk

• Discrete-time photonic quantum walks are stroboscopic evolutions where a photonic walker traverses a discrete lattice via repeated coin and shift operations.
• They are implemented using various platforms—bulk optics, time-multiplexed fibers, integrated photonics, and synthetic lattices—to study quantum transport and topological phases.
• Advanced methods leverage momentum-space analysis, non-Hermitian dynamics, and multiphoton interference to enable precise quantum simulation and photonic information processing.

Discrete-time photonic quantum walks are stroboscopic quantum evolutions in which a photonic “walker” propagates on a discrete lattice under repeated application of a unitary step operator that combines an internal two-level “coin” transformation with a conditional shift in position. In photonic implementations, the walker has been encoded in path modes, time bins, orbital angular momentum, transverse momentum, and other synthetic coordinates, while the coin has commonly been encoded in polarization. This stepwise Floquet structure distinguishes discrete-time quantum walks from continuous-time quantum walks generated by static Hamiltonians, and it underlies their use in quantum transport, multiphoton interference, Floquet band engineering, non-Hermitian scattering, and topological phenomena (Neves et al., 2019).

1. Formal definition and operator structure

A one-dimensional discrete-time quantum walk acts on a Hilbert space that factorizes as $H = H_{\text{coin}} \otimes H_{\text{pos}}$, with $$H_{\text{coin}} = \mathrm{span}\{|0\rangle,|1\rangle\}$$ and $H_{\text{pos}} = \mathrm{span}\{|x\rangle\}$ (Sansoni et al., 2011). In photonic realizations, the identification $|0\rangle \to |H\rangle$ and $|1\rangle \to |V\rangle$ is common, so polarization provides the coin degree of freedom while the walker occupies a discrete set of photonic modes or positions (Neves et al., 2019).

The standard single-step unitary is

$U = S (C \otimes I),$

where $C$ is a local $SU(2)$ coin and $S$ is a conditional translation (Neves et al., 2019). In one common convention,

$$S = \sum_x \left( |x+1\rangle\langle x| \otimes |0\rangle\langle 0| + |x-1\rangle\langle x| \otimes |1\rangle\langle 1| \right),$$

while in polarization language this becomes a routing of $$H_{\text{coin}} = \mathrm{span}\{|0\rangle,|1\rangle\}$$0 and $$H_{\text{coin}} = \mathrm{span}\{|0\rangle,|1\rangle\}$$1 into opposite directions (Neves et al., 2019). After $$H_{\text{coin}} = \mathrm{span}\{|0\rangle,|1\rangle\}$$2 steps the evolution is $$H_{\text{coin}} = \mathrm{span}\{|0\rangle,|1\rangle\}$$3 (Sansoni et al., 2011).

Several photonic implementations use alternative but equivalent operator orderings. In the time-multiplexed topological walk of the supersymmetric polarization-anomaly experiment, the single-step operator is written

$$H_{\text{coin}} = \mathrm{span}\{|0\rangle,|1\rangle\}$$4

with

$$H_{\text{coin}} = \mathrm{span}\{|0\rangle,|1\rangle\}$$5

and

$$H_{\text{coin}} = \mathrm{span}\{|0\rangle,|1\rangle\}$$6

(Barkhofen et al., 2018). Split-step walks refine the same principle by decomposing the full translation into two half-shifts,

$$H_{\text{coin}} = \mathrm{span}\{|0\rangle,|1\rangle\}$$7

thereby exposing Floquet topological structure more directly (Neves et al., 2019).

Momentum-space analysis is central for translationally invariant walks. The Floquet operator $$H_{\text{coin}} = \mathrm{span}\{|0\rangle,|1\rangle\}$$8 has eigenvalues $$H_{\text{coin}} = \mathrm{span}\{|0\rangle,|1\rangle\}$$9, where $H_{\text{pos}} = \mathrm{span}\{|x\rangle\}$0 is the quasienergy (Neves et al., 2019). In the single-rotation optical walk reviewed there,

$H_{\text{pos}} = \mathrm{span}\{|x\rangle\}$1

whereas in split-step walks

$H_{\text{pos}} = \mathrm{span}\{|x\rangle\}$2

(Neves et al., 2019). This Floquet description permits independent gaps at quasienergies $H_{\text{pos}} = \mathrm{span}\{|x\rangle\}$3, $H_{\text{pos}} = \mathrm{span}\{|x\rangle\}$4, and, in some symmetry classes, also $H_{\text{pos}} = \mathrm{span}\{|x\rangle\}$5 (Barkhofen et al., 2018).

2. Photonic encodings and experimental architectures

Photonic DTQWs have been realized with bulk optics, integrated photonics, time-multiplexed fiber loops, single-beam orbital-angular-momentum platforms, and synthetic lattices in transverse momentum (Neves et al., 2019). The choice of platform determines how the abstract position basis $H_{\text{pos}} = \mathrm{span}\{|x\rangle\}$6 is embedded into physical modes and how the coin and shift are implemented.

Architecture	Walker encoding	Coin encoding
Bulk/spatial multiplexing	distinct paths or spatial modes	polarization
Time multiplexing	time bins	polarization
OAM single-beam walks	orbital angular momentum $H_{\text{pos}} = \mathrm{span}\{|x\rangle\}$7	polarization
Transverse-momentum synthetic lattice	discrete momentum modes	polarization

In free-space spatial multiplexing, calcite beam displacers implement the conditional shift and waveplates implement the coin; concatenating such modules yields multiple steps (Neves et al., 2019). Integrated photonic implementations instead use directional couplers or cascaded interferometers. In a four-step 3D integrated circuit, balanced directional couplers realize the Hadamard coin and the conditional routing simultaneously, with a transition rule

$H_{\text{pos}} = \mathrm{span}\{|x\rangle\}$8

$H_{\text{pos}} = \mathrm{span}\{|x\rangle\}$9

(Sansoni et al., 2011). That device used femtosecond-laser-written waveguides, a $|0\rangle \to |H\rangle$0 tilt angle in the coupling region to equalize $|0\rangle \to |H\rangle$1 and $|0\rangle \to |H\rangle$2 coupling, a waveguide separation of $|0\rangle \to |H\rangle$3, interaction length $|0\rangle \to |H\rangle$4 for balanced splitting, and total chip length $|0\rangle \to |H\rangle$5 (Sansoni et al., 2011).

Time-multiplexed architectures encode position in arrival-time bins. A polarizing beam splitter and differential delays implement the shift, while waveplates or electro-optic modulators implement the coin (Neves et al., 2019). In the topological fiber-loop experiment, a polarizing beam splitter separated $|0\rangle \to |H\rangle$6 and $|0\rangle \to |H\rangle$7 components into fibers of different lengths so that $|0\rangle \to |H\rangle$8 and $|0\rangle \to |H\rangle$9, and a Soleil–Babinet compensator together with a fast electro-optic modulator implemented the position-dependent coin $|1\rangle \to |V\rangle$0 (Barkhofen et al., 2018). More generally, time multiplexing routinely achieves $|1\rangle \to |V\rangle$1 steps, with coupling per roundtrip exceeding $|1\rangle \to |V\rangle$2, overall per-roundtrip survival $|1\rangle \to |V\rangle$3, and outcoupling at $|1\rangle \to |V\rangle$4 per step (Neves et al., 2019).

Single-beam realizations replace spatial interferometer networks with structured-light modes. In one implementation, the walker is the orbital angular momentum state $|1\rangle \to |V\rangle$5 and the coin is circular polarization $|1\rangle \to |V\rangle$6 (Cardano et al., 2014). A $|1\rangle \to |V\rangle$7-plate with $|1\rangle \to |V\rangle$8 implements the conditional OAM shift

$|1\rangle \to |V\rangle$9

after compensation of the spin flip induced by the plate (Cardano et al., 2014). A closely related proposal had already described the same SAM–OAM mapping and showed how tuned $U = S (C \otimes I),$0-plates with $U = S (C \otimes I),$1 produce $U = S (C \otimes I),$2 and $U = S (C \otimes I),$3, with a compensating half-wave plate restoring the conventional shift operator (Zhang et al., 2010).

Synthetic lattices in transverse momentum provide another route. In the 2D two-photon DTQW, a site $U = S (C \otimes I),$4 was encoded as a Gaussian beam with transverse tilt $U = S (C \otimes I),$5, and geometric-phase polarization gratings implemented conditional translations $U = S (C \otimes I),$6 and $U = S (C \otimes I),$7 (Esposito et al., 2022). A plausible implication is that such synthetic-space platforms bridge path-based and mode-based realizations while keeping the walk in a compact optical footprint.

3. Symmetries, bands, and topological structure

The Floquet character of DTQWs makes symmetry analysis especially important. In split-step and related walks, chiral symmetry yields winding numbers and bulk–boundary correspondence (Neves et al., 2019). In the supersymmetric polarization-anomaly experiment, the Bloch operator in a symmetry-adapted basis obeys

$U = S (C \otimes I),$8

which protects the gaps at $U = S (C \otimes I),$9 corresponding to quasienergies $C$0 and $C$1 (Barkhofen et al., 2018). An additional independent symmetry acts on the sublattice degree of freedom,

$C$2

producing paired eigenvalues $C$3 and protecting the gaps at $C$4 (Barkhofen et al., 2018). The paper identifies this as a “unitary supersymmetry,” and the squared evolution factorizes as

$C$5

in analogy with supersymmetric Hamiltonians (Barkhofen et al., 2018).

For the alternating two-site unit cell of that walk, the quasienergy bands satisfy

$C$6

with four bands related by

$C$7

(Barkhofen et al., 2018). Bulk states with $C$8 obey the symmetry-enforced constraints $C$9, $SU(2)$0, and $SU(2)$1, which confine them to wind on a 3-torus defined by the angles $SU(2)$2 (Barkhofen et al., 2018). Interchanging the order of $SU(2)$3 and $SU(2)$4 in the unit cell changes that winding, so a domain wall binds midgap states at $SU(2)$5 (Barkhofen et al., 2018).

Topological DTQWs also extend naturally beyond one dimension. In a study of 2D Zak-phase landscapes, separable photonic DTQWs preserving spatial inversion symmetry and time-reversal symmetry were shown to have vanishing Berry curvature yet nontrivial Zak-phase structure because the Berry connection remains nonzero (Puentes, 2023). For the Hadamard walk, non-commuting-rotation walk, and split-step walk, the dispersions are

$SU(2)$6

$SU(2)$7

$SU(2)$8

respectively (Puentes, 2023). The extended 2D Zak phase was written as

$SU(2)$9

and a protocol was proposed to break time-reversal symmetry while keeping the Berry curvature zero by enforcing $S$0 (Puentes, 2023).

Higher-order topology has also been formulated in coinless DTQWs. A two-dimensional coinless walk with a four-substep Floquet operator

$S$1

simulates a second-order topological insulator with zero-dimensional corner states (Meng et al., 2020). The effective Floquet Hamiltonian obeys sublattice symmetry

$S$2

and the quadrupole phase is diagnosed by a $S$3 invariant

$S$4

computed from nested Wilson loops (Meng et al., 2020). This suggests that DTQWs support not only conventional edge topology but also higher-order bulk–boundary correspondence.

4. Polarization, tomography, and anomalies

Because many photonic DTQWs encode the coin in polarization, polarization-resolved detection gives direct access to internal-state dynamics that are often inaccessible in other quantum-walk platforms. In the supersymmetric anomaly experiment, the polarization Pauli operators $S$5 were mapped onto Stokes-like observables, with circular polarization corresponding to finite $S$6 in the symmetry-adapted basis $S$7 (Barkhofen et al., 2018).

Bulk and midgap states have sharply different polarization signatures in that walk. For bulk bands with $S$8, the constraints enforce $S$9, so bulk states are linearly polarized in the $$S = \sum_x \left( |x+1\rangle\langle x| \otimes |0\rangle\langle 0| + |x-1\rangle\langle x| \otimes |1\rangle\langle 1| \right),$$0 basis (Barkhofen et al., 2018). For the interface states at $$S = \sum_x \left( |x+1\rangle\langle x| \otimes |0\rangle\langle 0| + |x-1\rangle\langle x| \otimes |1\rangle\langle 1| \right),$$1, the constraint is lifted because $$S = \sum_x \left( |x+1\rangle\langle x| \otimes |0\rangle\langle 0| + |x-1\rangle\langle x| \otimes |1\rangle\langle 1| \right),$$2, allowing

$$S = \sum_x \left( |x+1\rangle\langle x| \otimes |0\rangle\langle 0| + |x-1\rangle\langle x| \otimes |1\rangle\langle 1| \right),$$3

(Barkhofen et al., 2018). This fixes an alternating circular polarization pattern: even sites are right-handed with $$S = \sum_x \left( |x+1\rangle\langle x| \otimes |0\rangle\langle 0| + |x-1\rangle\langle x| \otimes |1\rangle\langle 1| \right),$$4, and odd sites are left-handed with $$S = \sum_x \left( |x+1\rangle\langle x| \otimes |0\rangle\langle 0| + |x-1\rangle\langle x| \otimes |1\rangle\langle 1| \right),$$5 (Barkhofen et al., 2018).

The corresponding experiment used a bulk configuration with $$S = \sum_x \left( |x+1\rangle\langle x| \otimes |0\rangle\langle 0| + |x-1\rangle\langle x| \otimes |1\rangle\langle 1| \right),$$6 and $$S = \sum_x \left( |x+1\rangle\langle x| \otimes |0\rangle\langle 0| + |x-1\rangle\langle x| \otimes |1\rangle\langle 1| \right),$$7, and an interface configuration obtained by swapping the coin order at $$S = \sum_x \left( |x+1\rangle\langle x| \otimes |0\rangle\langle 0| + |x-1\rangle\langle x| \otimes |1\rangle\langle 1| \right),$$8 (Barkhofen et al., 2018). Trapping at $$S = \sum_x \left( |x+1\rangle\langle x| \otimes |0\rangle\langle 0| + |x-1\rangle\langle x| \otimes |1\rangle\langle 1| \right),$$9 after 13 steps varied strongly with the input polarization in the interface case, from approximately $$H_{\text{coin}} = \mathrm{span}\{|0\rangle,|1\rangle\}$$00 to $$H_{\text{coin}} = \mathrm{span}\{|0\rangle,|1\rangle\}$$01 as a quarter-wave-plate angle $$H_{\text{coin}} = \mathrm{span}\{|0\rangle,|1\rangle\}$$02 was scanned, while varying only weakly in the bulk configuration (Barkhofen et al., 2018). Full tomography of the trapped state at step 17 and $$H_{\text{coin}} = \mathrm{span}\{|0\rangle,|1\rangle\}$$03 yielded

$$H_{\text{coin}} = \mathrm{span}\{|0\rangle,|1\rangle\}$$04

consistent with $$H_{\text{coin}} = \mathrm{span}\{|0\rangle,|1\rangle\}$$05; on odd sites the observed state was $$H_{\text{coin}} = \mathrm{span}\{|0\rangle,|1\rangle\}$$06 (Barkhofen et al., 2018). The central point is that the anomalous circular polarization is detected directly, without an external symmetry-breaking probe.

The same platform logic reappears in other photonic DTQWs. Time-multiplexed reviews emphasize that arrival-time histograms reconstruct position distributions while polarization analysis reconstructs the coin state (Neves et al., 2019). In OAM-based single-beam walks, polarization analysis is combined with OAM projective measurements by waveplates, polarizers, and spatial-light-modulator holograms (Cardano et al., 2014). A plausible implication is that photonic DTQWs are unusual among Floquet simulators in permitting simultaneous access to walker and coin observables at the single-step level.

5. Multiphoton walks, exchange statistics, and interactions

Photonic DTQWs are not limited to single-particle transport. In a 3D integrated photonic circuit, two-photon polarization entanglement was used to emulate bosonic, fermionic, and anyonic exchange statistics in a four-step DTQW (Sansoni et al., 2011). The key mechanism was polarization-independent spatial evolution: if the linear optical network acts identically on $$H_{\text{coin}} = \mathrm{span}\{|0\rangle,|1\rangle\}$$07 and $$H_{\text{coin}} = \mathrm{span}\{|0\rangle,|1\rangle\}$$08, then the symmetry of the polarization state controls the exchange symmetry of the spatial amplitudes (Sansoni et al., 2011). The Bell states

$$H_{\text{coin}} = \mathrm{span}\{|0\rangle,|1\rangle\}$$09

generated effective bosonic-like and fermionic-like behavior, while

$$H_{\text{coin}} = \mathrm{span}\{|0\rangle,|1\rangle\}$$10

interpolated continuously between them (Sansoni et al., 2011).

Experimentally, the integrated device exhibited polarization-insensitive single-photon behavior with similarity $$H_{\text{coin}} = \mathrm{span}\{|0\rangle,|1\rangle\}$$11, and two-photon distributions with $$H_{\text{coin}} = \mathrm{span}\{|0\rangle,|1\rangle\}$$12 for $$H_{\text{coin}} = \mathrm{span}\{|0\rangle,|1\rangle\}$$13, $$H_{\text{coin}} = \mathrm{span}\{|0\rangle,|1\rangle\}$$14 for $$H_{\text{coin}} = \mathrm{span}\{|0\rangle,|1\rangle\}$$15, and anyonic-state similarities up to $$H_{\text{coin}} = \mathrm{span}\{|0\rangle,|1\rangle\}$$16 (Sansoni et al., 2011). Bosonic-like inputs enhanced diagonal and near-diagonal correlations, while fermionic-like inputs suppressed same-mode coincidences (Sansoni et al., 2011).

The extension to two-dimensional discrete-time dynamics was demonstrated in a synthetic transverse-momentum lattice for two correlated photons (Esposito et al., 2022). There the one-step unitary was

$$H_{\text{coin}} = \mathrm{span}\{|0\rangle,|1\rangle\}$$17

with polarization as the coin and transverse momentum indices $$H_{\text{coin}} = \mathrm{span}\{|0\rangle,|1\rangle\}$$18 as the walker coordinates (Esposito et al., 2022). The experiment reached up to three DTQW steps, for which the number of accessible sites grew as $$H_{\text{coin}} = \mathrm{span}\{|0\rangle,|1\rangle\}$$19, $$H_{\text{coin}} = \mathrm{span}\{|0\rangle,|1\rangle\}$$20, and $$H_{\text{coin}} = \mathrm{span}\{|0\rangle,|1\rangle\}$$21 after steps $$H_{\text{coin}} = \mathrm{span}\{|0\rangle,|1\rangle\}$$22, $$H_{\text{coin}} = \mathrm{span}\{|0\rangle,|1\rangle\}$$23, and $$H_{\text{coin}} = \mathrm{span}\{|0\rangle,|1\rangle\}$$24 (Esposito et al., 2022). Indistinguishability was calibrated by Hong–Ou–Mandel interference with $$H_{\text{coin}} = \mathrm{span}\{|0\rangle,|1\rangle\}$$25, and the observed distributions had similarity $$H_{\text{coin}} = \mathrm{span}\{|0\rangle,|1\rangle\}$$26 for a single photon and $$H_{\text{coin}} = \mathrm{span}\{|0\rangle,|1\rangle\}$$27 for two photons (Esposito et al., 2022). A nonclassical correlation witness was violated by up to $$H_{\text{coin}} = \mathrm{span}\{|0\rangle,|1\rangle\}$$28 standard deviations at step 1 (Esposito et al., 2022).

Most photonic DTQWs remain linear-optical. A distinct theoretical direction introduces strong interactions by replacing passive beam splitters with single-atom beam splitters (Zheng et al., 2022). In that construction, the one-photon scattering amplitudes are

$$H_{\text{coin}} = \mathrm{span}\{|0\rangle,|1\rangle\}$$29

while the two-photon $$H_{\text{coin}} = \mathrm{span}\{|0\rangle,|1\rangle\}$$30 matrix acquires a nonlinear term $$H_{\text{coin}} = \mathrm{span}\{|0\rangle,|1\rangle\}$$31 that produces correlated transport and two-photon bound states (Zheng et al., 2022). For the example $$H_{\text{coin}} = \mathrm{span}\{|0\rangle,|1\rangle\}$$32, $$H_{\text{coin}} = \mathrm{span}\{|0\rangle,|1\rangle\}$$33, and $$H_{\text{coin}} = \mathrm{span}\{|0\rangle,|1\rangle\}$$34, the statistical pattern was boson-like at $$H_{\text{coin}} = \mathrm{span}\{|0\rangle,|1\rangle\}$$35, fermion-like at $$H_{\text{coin}} = \mathrm{span}\{|0\rangle,|1\rangle\}$$36, and linear for $$H_{\text{coin}} = \mathrm{span}\{|0\rangle,|1\rangle\}$$37 (Zheng et al., 2022). Since this is a proposal rather than a completed experiment, it is best read as an interacting extension of the photonic DTQW framework.

6. Computation, non-Hermitian dynamics, and broader roles

Photonic DTQWs occupy an intermediate position between interferometric linear optics and programmable Floquet simulation. On the computational side, arbitrary graph walks can be described in second-quantized form by creation operators that transform linearly under the walk unitary, while multiphoton coincidence patterns reveal exchange interference (Rohde et al., 2010). For general graphs, the step is again $$H_{\text{coin}} = \mathrm{span}\{|0\rangle,|1\rangle\}$$38, with local coins $$H_{\text{coin}} = \mathrm{span}\{|0\rangle,|1\rangle\}$$39 acting on the neighborhood of each vertex and a shift that permutes source and destination labels (Rohde et al., 2010). This framework is directly aligned with layered photonic interferometers built from beamsplitters, phase shifters, and deterministic mode routing (Rohde et al., 2010).

A more recent time-multiplexed hybrid architecture translates arbitrary linear transformations into DTQW coin and step operators and maps them to experimental parameters of a fiber-loop platform (Lammers et al., 26 Sep 2025). In that work the per-step unitary is written

$$H_{\text{coin}} = \mathrm{span}\{|0\rangle,|1\rangle\}$$40

with coin states encoded in polarization, position in time bins, and an unbalanced Mach–Zehnder implementing the shift (Lammers et al., 26 Sep 2025). The compiler reduces the target transformation to nearest-neighbor couplers arranged in staggered layers, with worst-case depth $$H_{\text{coin}} = \mathrm{span}\{|0\rangle,|1\rangle\}$$41 for a $$H_{\text{coin}} = \mathrm{span}\{|0\rangle,|1\rangle\}$$42-mode target and total programmed couplers on the order of $$H_{\text{coin}} = \mathrm{span}\{|0\rangle,|1\rangle\}$$43 (Lammers et al., 26 Sep 2025). The same paper states that the architecture is “highly resilient against experimental imperfections” and that, in the local-network model analyzed there, similarity remains $$H_{\text{coin}} = \mathrm{span}\{|0\rangle,|1\rangle\}$$44 under random phase noise while fidelity is independent of average loss in the reported simulations (Lammers et al., 26 Sep 2025).

Algorithmic use with a single walker has also been made explicit. A single-particle DTQW using polarization and path degrees of freedom was proposed for the Deutsch–Jozsa and Bernstein–Vazirani algorithms, retaining the one-query structure of both algorithms while reducing spatial-mode requirements by using polarization for one qubit and paths for the remaining qubits (Sangwan et al., 20 Aug 2025). The implementation uses HWP-based Hadamard and Pauli-$$H_{\text{coin}} = \mathrm{span}\{|0\rangle,|1\rangle\}$$45 coins, $$H_{\text{coin}} = \mathrm{span}\{|0\rangle,|1\rangle\}$$46 beam splitters for path Hadamards, PBS-based conditional shifts, and phase shifters for oracle phases (Sangwan et al., 20 Aug 2025). This suggests that DTQW-based photonic information processing can interpolate between algorithm-specific interferometers and more general universal processors.

Non-Hermitian DTQWs form another major branch. In coupled fiber loops with short- and long-loop amplitudes $$H_{\text{coin}} = \mathrm{span}\{|0\rangle,|1\rangle\}$$47, the one-step update in the presence of a complex potential $$H_{\text{coin}} = \mathrm{span}\{|0\rangle,|1\rangle\}$$48 is

$$H_{\text{coin}} = \mathrm{span}\{|0\rangle,|1\rangle\}$$49

$$H_{\text{coin}} = \mathrm{span}\{|0\rangle,|1\rangle\}$$50

(Longhi, 2022). For drifting Kramers–Kronig potentials $$H_{\text{coin}} = \mathrm{span}\{|0\rangle,|1\rangle\}$$51, invisibility is achieved when $$H_{\text{coin}} = \mathrm{span}\{|0\rangle,|1\rangle\}$$52 is analytic in one half-plane, decays faster than $$H_{\text{coin}} = \mathrm{span}\{|0\rangle,|1\rangle\}$$53, and the drift satisfies $$H_{\text{coin}} = \mathrm{span}\{|0\rangle,|1\rangle\}$$54 in the slowly drifting regime (Longhi, 2022). Under those conditions the walk supports

$$H_{\text{coin}} = \mathrm{span}\{|0\rangle,|1\rangle\}$$55

for arbitrary incident wave packets (Longhi, 2022).

A separate free-space platform realizes non-unitary DTQWs in transverse momentum using dichroic liquid-crystal metasurfaces (Savarese et al., 7 Mar 2025). There the non-unitary shift is

$$H_{\text{coin}} = \mathrm{span}\{|0\rangle,|1\rangle\}$$56

and the step operator is $$H_{\text{coin}} = \mathrm{span}\{|0\rangle,|1\rangle\}$$57 with a uniform quarter-wave-plate coin $$H_{\text{coin}} = \mathrm{span}\{|0\rangle,|1\rangle\}$$58 (Savarese et al., 7 Mar 2025). Experiments reached five steps at six dichroism settings $$H_{\text{coin}} = \mathrm{span}\{|0\rangle,|1\rangle\}$$59 with complex quasienergies

$$H_{\text{coin}} = \mathrm{span}\{|0\rangle,|1\rangle\}$$60

(Savarese et al., 7 Mar 2025). Ballistic spreading persisted despite loss, indicating coherent non-unitary evolution rather than decoherence (Savarese et al., 7 Mar 2025).

Across these directions, a recurring pattern is that discrete-time photonic quantum walks are valued not simply as optical analogues of random walks, but as programmable Floquet systems in which coin, shift, symmetry class, dimensionality, particle number, and even non-unitarity can be engineered directly in the optical hardware. The specific implementations differ substantially, but the operator language $$H_{\text{coin}} = \mathrm{span}\{|0\rangle,|1\rangle\}$$61 and its variants remain the common framework linking topology, tomography, multiphoton interference, and photonic information processing (Neves et al., 2019).