Bidirectional quantum teleportation using quantum walks

TL;DR

The paper develops bidirectional quantum teleportation protocols based on coined quantum walks on independent $1$-D lattices and $4$-cycles, enabling simultaneous teleportation in both directions without pre-shared entanglement. It provides explicit single-qubit and two-qubit schemes using either nearest-neighbor moves with a single coin or mixed moves with two coins (including next-nearest-neighbor steps), and demonstrates that single-step and two-step walk implementations are equivalent through detailed measurement-to-unitary mappings. The contributions include concrete walk definitions ($W_{1}$–$W_{4}$), conditional shift operators, measurement bases, and correction tables that guarantee faithful state reconstruction with calculable success probabilities. This work enriches quantum-information processing by showing how quantum-walk dynamics can generate and harness entanglement during computation, with potential extensions to more qubits, albeit with practical constraints in distributed, remote scenarios where joint operations are challenging.

Abstract

We present a method for bidirectional teleportation of a single qubit using quantum walks on two independent one dimensional lattices and two independent cycles with four vertices, employing nearest neighbor jumps with coin outcomes. In addition, we discuss two different methods for two qubit teleportation by employing nearest neighbor jumps and next nearest neighbor jumps with a single coin and two coins, respectively. Finally, we show that the two qubit single jump quantum walk and the two jump quantum walk teleportation schemes yield the same results.

Figures (6)

• Figure 1: The quantum walk on a one dimensional line. If the coin toss results in $0$, the walker moves from position $i$ to $i+1$ (rightward). If the coin toss result in $1$, the walker moves from $i$ to $i-1$ (leftward).
• Figure 2: Circuit diagram for bidirectional single qubit teleportation using quantum walk.
• Figure 3: Two independent cyclic position basis ($4$ cycle) on a one dimensional line, figure (a) for Alice and figure (b) for Bob.
• Figure 4: Circuit diagram for bidirectional teleportation of two-qubit states using single step quantum walk.
• Figure 5: The quantum walk on a one dimensional line using two coins. If coins outcome is $00$, the walker moves from $i$ to $i+2$; if coins outcomes is $11$, the walker moves from $i$ to $i-2$; if coins outcomes is $10$, the walker moves from $i$ to $i+1$; if coin outcomes is $01$, the walker moves from $i$ to $i-1$.