[Paper Review] Quantum communication protocols by quantum walks with two coins

Yun Shang, Yu Wang+2 more|arXiv (Cornell University)|Feb 7, 2018

Quantum Computing Algorithms and Architecture3 citations

TL;DR

This paper proposes a novel quantum communication framework using discrete-time quantum walks with two coins to achieve perfect state transfer and generalized teleportation. By encoding information in coin 1 and alternately applying two coin operators, the protocol enables perfect qubit and qudit state transfer on diverse graphs—including lines, N-circles, complete graphs, and regular graphs—without requiring external control during evolution, offering a universal platform for scalable quantum networks.

ABSTRACT

We introduce some new perfect state transfer and teleportation schemes by quantum walks with two coins. Encoding the transferred information in coin 1 state and alternatively using two coin operators, we can perfectly recover the information on coin 1 state at target position only by at most two times of flipping operation. Based on quantum walks with two coins either on the line or on the $N$-circle, we can perfectly transfer any qubit state. In addition, using quantum walks with two coins either on complete graphs or regular graphs, we can first implement perfect qudit state transfer by quantum walks. Compared with existing schemes driven by one coin, more general graph structures can be used to perfectly transfer more general state. Moreover, the external control of coin operator during the transmitting process can be decreased greatly. Selecting coin 1 as the sender and coin 2 as the receiver, we also study how to realize generalized teleportation over long steps of walks by the above quantum walk models. Because quantum walks is an universal quantum computation model, our schemes may provide an universal platform for the design of quantum network and quantum computer.

Motivation & Objective

• To develop a universal quantum communication framework based on quantum walks with two coins for scalable quantum networks.
• To overcome limitations of one-coin quantum walks by enabling perfect state transfer on broader graph structures, including complete and regular graphs.
• To reduce external control complexity during transmission by using alternating coin operations instead of position-dependent operators.
• To extend the protocol to generalized teleportation over long walk steps using the two-coin model.
• To demonstrate that qudit state transfer, previously unachievable with one-coin walks, is feasible on complete and regular graphs.

Proposed method

• Encoding the unknown quantum state in the coin 1 subspace as the initial state of the walk.
• Applying two distinct coin operators alternately at each step to drive the quantum walk evolution on the graph.
• Using the position space as the propagation medium and coin 2 as a relay to preserve and transfer state information.
• Implementing a recovery mechanism via projective measurements on position and coin 1 spaces, followed by local unitary corrections.
• Employing Fourier-transformed bases for coin 1 state measurement to enable state reconstruction at the target.
• Designing specific unitary operations based on measurement outcomes to perfectly recover the original state at the target position.

Experimental results

Research questions

• RQ1 Can perfect qubit state transfer be achieved on infinite lines and N-circles using two-coin quantum walks without position-dependent coin control?
• RQ2 Can perfect qudit state transfer be realized on complete graphs and d-regular graphs using two-coin quantum walks, which is not possible with one-coin walks?
• RQ3 How can generalized teleportation be implemented over long walk steps using the two-coin quantum walk model?
• RQ4 What measurement and correction strategies ensure perfect state recovery at arbitrary target positions?
• RQ5 To what extent does the two-coin model reduce the need for external control during quantum walk evolution compared to existing schemes?

Key findings

• Perfect qubit state transfer is achieved on both the infinite line and N-circles using alternating two-coin operations, with no restriction to opposite sites or even N.
• Perfect qudit state transfer is successfully realized for the first time on complete graphs and d-regular graphs using the two-coin quantum walk model.
• The protocol requires no external control during evolution—only initial encoding and final measurement with local unitary correction—reducing experimental complexity.
• Generalized teleportation over long walk steps is achieved by measuring position and coin 1 states and applying tailored unitary operations based on outcomes.
• The final state after 2t steps on an N-circle is expressed as $ |\phi\rangle^{2t} = \sum_{m=0}^{d-1}\sum_{k=0}^{d-1}a_m|A_i(m+k)\bmod n\rangle|m\rangle|k\rangle/\sqrt{d} $, with perfect recovery possible via measurement and correction.
• For N-circles with $ N $ coprime to $ A_i $, the protocol ensures that different $ m+k $ values map to distinct positions, enabling unambiguous state reconstruction.