Parity-Based Time-Bin Encoding Enabling SWAP Between Polarization and Time-Bin Qubits

The gist: This work introduces a parity-based time-bin encoding that enables a deterministic SWAP between polarization and time-bin qubits on a single photon using standard optical components.

System Definition and Encoding

The system defines two qubit encodings, polarization and time-bin, for a single photon in the Hilbert space H = Hpol ⊗ Htime The polarization bases are defined as Horizontal and Vertical using the standard notation: H⟩, V ⟩ and the time-bin basis as even and odd multiples of a time-bin spacing ∆t: even⟩, odd⟩ For compact notation, the logical basis states are identified as 0⟩P ≡ H⟩, 1⟩P ≡ V ⟩, and 0⟩T ≡even⟩, and 1⟩T ≡odd⟩ A general single-photon state in the two-qubit Hilbert space is then given by the expression: ψ⟩ = (α H⟩ + β V ⟩) ⊗ (γ even⟩ + δ odd⟩)

Parity-Based Time-Bin Encoding

The parity-based time-bin encoding maps even multiples of ∆t to 0⟩T and odd multiples of ∆t to 1⟩T Under this encoding, a temporal shift of ∆t implements a logical qubit-flip: 0⟩T ↔ 1⟩T This discrete encoding is chosen because it allows for a continuous grid of even and odd time bins that the photon can fall under, while allowing a delay of ∆t to move it into the other time bin Furthermore, this continuous parity-based time-bin encoding allows us to align the time-bin sections with an EOM voltage waveform or, alternatively, with a Kerr-switching control waveform

SWAP Gate Composition

The SWAP gate can be decomposed into three sequential CNOT operations, which is a standard circuit identity in quantum computing The decomposition is given as SWAP = CNOTP→T CNOTT→P CNOTP→T Alternatively, the CNOT gates can be rearranged as SWAP = CNOTT→P CNOTP→T CNOTT→P The controlled-NOT operations act on the computational basis as CNOTP→T: p⟩P t⟩T 7→ p⟩P t ⊕ p⟩T and CNOTT→P: p⟩P t⟩T 7→ p ⊕ t⟩P t⟩T The basis-state proof shows that the sequence implements the standard SWAP operator USWAP = (1 0 0 0; 0 0 1 0; 0 1 0 0; 0 0 t⟩P p⟩T)

Physical Implementation

The CNOTT→P operation is implemented using an Electro-Optic Modulator, where a time-dependent voltage applied to the EOM induces a polarization rotation via the Pockels effect The EOM implements polarization flips when an applied voltage induces a voltage-controlled birefringence so the device behaves as a tunable waveplate The CNOTP→T operation is implemented using a Polarization Beam Splitter (PBS) and delay, where the V arm experiences a delay ∆t This system implements a polarization controlled delay in the form: H⟩ even⟩ 7→ H⟩ even⟩, V ⟩ even⟩ 7→ V ⟩ odd⟩ The full SWAP architecture is implemented as three sequential stages: (i) a PBS-delay-PBS stage that maps polarization onto a time-bin shift, (ii) an EOM that applies the time-controlled polarization flip, and (iii) a second PBS-delay-PBS stage that completes the polarization-controlled delay

Experimental Constraints

The timing requirements dictate that the time-bin separation ∆t must satisfy: ∆t ≫ τswitch and ∆t > τopen A useful engineering rule is ∆t ≳ τswitch + τopen + τdet + σjitter, where σjitter is the combined RMS timing jitter of the source trigger, EOM drive, and detector chain For a practical first design point, a practical first design point is therefore ∆t = 1 ns This results in an overall gate time of approximately 2.5 ns

Conclusion

The central result of this work is the parity-based time-bin encoding that makes each CNOT physically realizable with standard optics By labeling logical time bins by parity rather than by early/late arrival, a fixed polarization-dependent delay implements the controlled time-bin flip required for CNOTP→T, while the periodic ∆t grid synchronizes EOM-driven polarization rotations for CNOTT→P Together, these operations compose a deterministic SWAP between polarization and timebin qubits on a single photon, providing a concrete routing primitive for multi-degree-of-freedom photonic circuits such as the polarization Toffoli gate This scheme provides a primitive SWAP gate for multi-degree-of-freedom photonic circuits.

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Parity-Based Time-Bin Encoding Enabling SWAP Between Polarization and Time-Bin Qubits Adam Sultan Connor Kupchak Carleton University Ottawa Ontario Canada Multi-degree-of-freedom photonic quantum processing requires routing between degree of freedom DOF qubit encodings on a single photon A SWAP between polarization and time-bin qubits is an advantageous primitive for such architectures however conventional early/late time-bin encoding does not support bidirectional logical time-bin flips from late to early which limits the ability to implement certain quantum operations We introduce a parity-based time-bin encoding in which logical 0⟩T and 1⟩T correspond to even and odd multiples of a spacing ∆t, so that a physical delay of ∆t implements 0⟩T ↔ 1⟩T This encoding is the enabling ingredient that makes a polarization-controlled delay line implement CNOTP→T and aligns naturally with periodic refractive index modulation for CNOTT→P Composing three such CNOT operations sequentially results in a deterministic SWAP between polarization and time-bin degrees of freedom We analyze fieldbased modulation polarization-rotation error probability and timing-resolution constraints set by both EOM drive electronics and photon detection I.

1. A single photon can perform a deterministic SWAP between polarization and time-bin degrees of freedom, enabling routing primitives for multi-degree-of-freedom photonic circuits such as a polarization Toffoli gate [4].

2. The parity-based time-bin encoding allows for a fixed delay of ∆t implements 0⟩T ↔ 1⟩T, which is crucial because conventional early/late time-bin encoding does not support bidirectional logical time-bin flips from late to early.

3. The system supports CNOT operations that are physically realized using standard optical components: CNOTP→T via a polarization-controlled delay implemented by a PBS-delay-PBS stage, and CNOTT→P via an EOM implementing time-conditioned polarization rotation.

4. The architecture provides a path to implement the SWAP operation by composing three sequential CNOT operations: SWAP = CNOTP→T CNOTT→P CNOTP→T (2), offering a coherent routing primitive for connecting different encodings within an architecture.

5. The system's timing constraints specify that the time-bin separation must satisfy ∆t ≳ τopen +τswitch, allowing the design of practical gate times, such as an overall gate time of approximately 2.5 ns for a three-CNOT sequence before accounting for inefficiencies.

Multi-degree-of-freedom photonic quantum processing requires routing between degree-of-freedom (DOF) qubit encodings on a single photon. A SWAP between polarization and time-bin qubits is an advantageous primitive for such architectures, however conventional early/late time-bin encoding does not support bidirectional logical time-bin flips from late to early which limits the ability to implement certain quantum operations. We introduce a parity-based time-bin encoding in which logical T and T correspond to even and odd multiples of a spacing Δt, so that a physical delay of Δt implements T T. This encoding is the enabling ingredient that makes a polarization-controlled delay line implement CNOT P to T and aligns naturally with periodic refractive index modulation for CNOT T to P. Composing three such CNOT operations sequentially results in a deterministic SWAP between polarization and time-bin degrees of freedom. We quantify how EOM voltage errors reduce the polarization-flip probability and estimate timing requirements from modulator switching, detector resolution, and timing jitter, giving an illustrative timing budget.