⚛️ quantum physics

Teleportation transition of surface codes on a superconducting quantum processor

Using a 125-qubit superconducting processor, researchers demonstrated the robust teleportation of topological rotated surface codes with distances up to 7 and showed that coherent qubit rotations can boost the entangling threshold to restore topological phase duality, advancing the path toward distributed fault-tolerant quantum computation.

Original authors: Yiren Zou, Hong-Kuan Xia, Aosai Zhang, Xuhao Zhu, Feitong Jin, Qingyuan Wang, Yu Gao, Chuanyu Zhang, Ning Wang, Zhengyi Cui, Fanhao Shen, Zehang Bao, Zitian Zhu, Jiarun Zhong, Gongyu Liu, Jia-Nan Yang

Published 2026-02-26

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Yiren Zou, Hong-Kuan Xia, Aosai Zhang, Xuhao Zhu, Feitong Jin, Qingyuan Wang, Yu Gao, Chuanyu Zhang, Ning Wang, Zhengyi Cui, Fanhao Shen, Zehang Bao, Zitian Zhu, Jiarun Zhong, Gongyu Liu, Jia-Nan Yang, Yihang Han, Yiyang He, Jiayuan Shen, Han Wang, Yanzhe Wang, Jiahua Huang, Xinrong Zhang, Sailang Zhou, Hang Dong, Jinfeng Deng, Yaozu Wu, Zixuan Song, Hekang Li, Zhen Wang, Chao Song, Qiujiang Guo, Pengfei Zhang, Guo-Yi Zhu, H. Wang

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have a very fragile, precious message written on a piece of paper. You want to send it to a friend across the room, but the room is filled with wind, dust, and people bumping into things. If you just hand the paper over, it will likely get torn or lost.

To solve this, you don't send the paper itself. Instead, you wrap the message in a super-strong, self-repairing bubble (this is the "Surface Code"). This bubble is made of many tiny, interconnected threads (qubits). If one thread breaks, the others hold the shape together, and the message remains safe inside.

Now, imagine you want to send this entire bubble to your friend, but you don't have a perfect, magical teleportation beam. You only have a "noisy" connection—a shaky bridge that sometimes lets things fall through.

This paper is about a team of scientists who successfully built a shaky bridge to teleport these magical bubbles between two groups of quantum bits (qubits) on a supercomputer. Here is the story of what they did, explained simply:

1. The Setup: Alice, Bob, and the Shaky Bridge

Think of two groups of people: Alice (who has the message) and Bob (who needs to receive it).

Alice's Side: She has a giant, complex bubble (a "logical qubit") made of 49 tiny threads.
Bob's Side: He starts with a pile of empty, disconnected threads.
The Goal: Move the entire bubble from Alice to Bob without breaking the message inside.

Usually, to teleport something perfectly, you need a "perfectly entangled" connection (like a super-strong, invisible rope) between Alice and Bob. But in the real world, that rope is often frayed or weak. The scientists wanted to see: How weak can the rope get before the teleportation fails?

2. The Experiment: Tuning the "Shakiness"

The team used a 125-qubit superconducting processor (a very advanced quantum computer). They didn't just try to teleport once; they played a game of "Goldilocks."

They created a dial (a parameter called $t$ ) that controlled how "shaky" the connection was.
Dial at 0: The connection is perfect. The bubble teleports flawlessly.
Dial turned up: The connection gets noisier. The bubble starts to wobble.
Dial turned all the way up: The connection is broken. The bubble shatters, and the message is lost.

They found a tipping point (a threshold). As long as the "shakiness" was below this point, the bubble survived the journey. If they turned the dial past that point, the teleportation failed. This is like finding the exact speed limit where a car can still drive on a bumpy road without crashing.

3. The Magic Trick: The "Electric-Magnetic" Dance

Here is where it gets really clever. The scientists discovered that the "shakiness" wasn't just one thing; it could come from different directions (like wind hitting the car from the side vs. from the front).

The Problem: When the noise came from one direction (let's call it the "X-axis"), the bubble was very fragile. The tipping point was low.
The Solution: They realized they could rotate the noise! By applying a special "magic" twist (a coherent rotation) to the system, they changed the nature of the noise.
The Result: They rotated the noise to a diagonal direction (the "X+Z axis"). Suddenly, the bubble became much tougher. The tipping point moved way up!

The Analogy: Imagine trying to push a heavy box across a floor.

If you push it straight against a wall, it gets stuck easily (low threshold).
But if you push it at a 45-degree angle, the friction changes, and you can push it much harder before it stops moving (high threshold).
The scientists used a concept from physics called "Electric-Magnetic Duality" to find this 45-degree angle. It's like finding a secret path through a maze that avoids all the traps.

4. Why This Matters

This isn't just a cool trick; it's a huge step toward distributed quantum computing.

The Future: Imagine a future where we have many small quantum computers scattered around the world (or in different buildings). To make a super-powerful quantum computer, we need to link them all together.
The Challenge: Linking them requires sending quantum information (teleporting) over long distances, which introduces noise and errors.
The Breakthrough: This experiment proved that even with imperfect connections, we can still teleport complex, error-corrected information if we know how to "tune" the connection and use the right "magic" rotations.

Summary

The scientists built a bridge between two quantum islands. They showed that:

You can teleport a complex, protected quantum state even if the bridge is a bit wobbly.
There is a specific limit to how wobbly it can get before it breaks.
Crucially, by changing the type of wobble (using a "magic" rotation), you can make the bridge much stronger, allowing the teleportation to survive even rougher conditions.

This work is a vital step toward building a "Quantum Internet," where we can share and process quantum information across vast distances without losing the precious data inside.

1. Problem Statement

Quantum error correction (QEC), particularly using the surface code, is a leading candidate for achieving fault-tolerant quantum computation. The surface code encodes logical qubits into long-range entangled states of physical qubits, protecting them against local errors via topological order.

A critical challenge for distributed quantum computing is the teleportation of logical states between distinct modules (e.g., between two processors or distant regions of a large processor). While standard quantum teleportation is well-understood for single qubits, scaling this to many-body topological states (surface codes) under realistic conditions (imperfect entanglement, incoherent noise, and coherent errors) remains an open experimental question. Specifically, determining the entanglement threshold—the minimum quality of entanglement required to successfully teleport a logical state while preserving topological order—is crucial for designing scalable quantum networks.

2. Methodology

The researchers utilized a 125-qubit superconducting quantum processor (Zhejiang University) to experimentally demonstrate the teleportation of topological rotated surface codes.

Experimental Setup:
- Architecture: Two groups of qubits, Alice (sender) and Bob (receiver), are physically interlaced on a $d \times d$ square lattice.
- Code Distance: They prepared surface codes with distances up to $d=7$ (requiring 49 qubits per code, 98 total for the teleportation pair).
- State Preparation: Alice's qubits were initialized into a logical state $|\psi_L\rangle$ using a linear-depth unitary circuit. Bob's qubits were initialized in a simple product state $|0\dots0\rangle$ .
- Tunable Teleportation Protocol: Instead of standard Bell measurements with classical feed-forward (which is difficult to implement in real-time on current hardware), they used a transversal entangling gate followed by measurement and post-processing.
  - A parameterized unitary gate $U(\theta, \phi)$ rotates Alice's qubits.
  - A two-qubit entangling gate $R_{XX}(\pi/2 - 2t)$ $R_{X X} (π /2 - 2 t)$ connects Alice and Bob. The parameter $t$ acts as a "knob" to tune the entanglement strength.
    - $t=0$ : Maximal entanglement (ideal teleportation).
    - $t=\pi/4$ : No entanglement (teleportation blocked).
  - This setup effectively injects coherent errors controlled by $t$ , allowing the team to map the phase transition from a teleportable topological phase to a trivial phase.
Decoding:
- Measurement outcomes from Alice were processed using a Tensor Network (TN)-based Maximum Likelihood (ML) decoder to correct errors and retrieve the logical state on Bob's side.
- Alternative decoders (MWPM and TN-based Bayesian) were also tested for benchmarking.
Noise Characterization:
- The study analyzed both incoherent noise (modeled as bit-flip channels) and coherent errors (controlled by the gate parameter $t$ ).
- They defined an effective entangling parameter ( $t_{eff}$ ) to combine the effects of coherent and incoherent noise.

3. Key Contributions

Experimental Demonstration of Many-Body Teleportation: First demonstration of teleporting a logical qubit encoded in a surface code (up to $d=7$ ) from one set of physical qubits to another on a superconducting processor.
Mapping the Teleportation Phase Diagram: By continuously tuning the entanglement strength ( $t$ ), the authors experimentally charted the teleportation transition. They identified the critical threshold where the topological order is preserved (teleportation succeeds) versus where it is destroyed (teleportation fails).
Enhancement via Electric-Magnetic Duality: The paper demonstrates that the entanglement threshold can be significantly boosted by rotating the coherent error axis from the $X$ $X$ direction to the $X+Z$ direction ( $\theta = \pi/4$ $θ = π /4$ ).
- This rotation injects "magic" resources (non-Clifford operations) into the circuit.
- It restores electric-magnetic duality (self-duality) of the topological phase, making the system more robust against specific error types.
Teleportation of Magic States: The protocol was successfully applied to teleport arbitrary logical states, including H-type magic states ( $\theta_L = \pi/4$ ), which are essential for universal fault-tolerant quantum computation.

4. Key Results

High-Fidelity Teleportation:
- At maximal entanglement ( $t=0$ ), the logical state was successfully teleported.
- Logical Error Rates: For $d=7$ , the logical error rate was 0.014(1) for Alice (pre-teleportation) and 0.025(3) for Bob (post-teleportation), indicating the teleportation process adds minimal noise.
- Stabilizer Parities: Measured stabilizer parities remained high (Mean parity $\approx 0.68$ for Bob), confirming the preservation of topological order.
Teleportation Thresholds:
- Standard Case ( $\theta = \pi/2$ , X-axis errors): The experimental pseudo-threshold was found to be $t_{c, exp} \approx 0.122\pi$ , slightly higher than the ideal asymptotic threshold of $0.107\pi$ due to finite-size effects and noise.
- Self-Dual Case ( $\theta = \pi/4$ , X+Z axis errors): By exploiting electric-magnetic duality, the threshold was significantly enhanced. The experimental threshold rose to $t_{c, exp} \approx 0.175\pi$ , compared to the ideal asymptotic threshold of $0.155\pi$ . This confirms that injecting magic resources improves resilience.
Magic State Teleportation:
- The fidelity of teleported H-type magic states was 0.95(2), demonstrating that non-Clifford resources can be faithfully transferred across the topological code.
Phase Diagram:
- The study successfully mapped the transition between the "Teleportable" phase (topological order preserved) and the "Non-teleportable" phase.
- It confirmed that incoherent bit-flip noise destroys the phase at a critical probability $p_c \approx 0.109$ , consistent with theoretical predictions.

5. Significance

Foundational for Distributed Quantum Computing: This work provides a proof-of-principle for distributed fault-tolerant quantum computation. It demonstrates that logical information can be moved between modules without decoding it to the physical level, a prerequisite for scaling quantum computers beyond a single chip.
Resource Optimization: The discovery that electric-magnetic duality can boost the entanglement threshold offers a new design principle. It suggests that by carefully engineering the error types (injecting "magic"), one can tolerate lower-quality entanglement resources, potentially reducing the overhead for long-distance quantum communication.
Benchmarking Topological Matter: The experiment serves as a robust platform for simulating and probing topological quantum matter, specifically the phase transitions induced by decoherence and measurement.
Path Forward: The authors propose future improvements, such as integrating mid-circuit measurements and qubit reset to reduce circuit depth from linear to constant, and extending the protocol to truly non-local settings (e.g., between separate processors).

In summary, this paper bridges the gap between theoretical topological quantum error correction and experimental reality, demonstrating that robust logical teleportation is achievable on current noisy intermediate-scale quantum (NISQ) devices and providing a roadmap for scaling to fault-tolerant distributed networks.