⚛️ quantum physics

Enumerating all bilocal Clifford distillation protocols through symmetry reduction

This paper introduces a symmetry-based enumeration method using double cosets of the symplectic group to identify and optimize all bilocal Clifford entanglement distillation protocols, successfully finding high-fidelity circuits for up to five Bell-diagonal states and eight Werner state copies on a standard desktop computer.

Original authors: Sarah Jansen, Kenneth Goodenough, Sébastian de Bone, Dion Gijswijt, David Elkouss

Published 2026-09-02

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CC BY 4.0

Original authors: Sarah Jansen, Kenneth Goodenough, Sébastian de Bone, Dion Gijswijt, David Elkouss

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

Quantum communication promises a future where information can be sent with perfect security and computers can solve problems that are currently impossible. At the heart of this technology lies a strange connection between particles called entanglement, where two objects remain linked regardless of the distance separating them. However, in the real world, this connection is fragile. As particles travel or sit in a machine, they interact with their environment, causing the link to become noisy and imperfect. This noise reduces the quality of the connection, making it difficult to use for tasks like secret messaging or distributed computing. To fix this, scientists use a process called distillation. Imagine taking many weak, noisy connections and combining them to produce a single, much stronger, and cleaner connection. This is the essential repair work required to build a functional quantum internet.

The challenge has always been figuring out exactly how to combine these noisy pairs most effectively. While theoretical methods exist, they often require an infinite number of steps or complex operations that are impossible to build in a lab. Researchers have been looking for practical recipes that use a small number of noisy pairs and simple, standard operations to create a high-quality result. A team of scientists has now mapped out every possible way to do this for a specific, important class of operations. They focused on a method where two people, traditionally named Alice and Bob, share several pairs of noisy particles. They each perform a specific set of standard quantum logic gates on their own particles, measure most of them, and then compare the results over a classical channel. If the results match in a certain way, they keep the one remaining pair, which is now much cleaner than the ones they started with.

The researchers treated this problem like a massive search for the best path through a complex maze. They realized that all the possible ways Alice and Bob could arrange their logic gates could be grouped into distinct categories based on symmetry. Instead of checking every single possibility one by one, which would take longer than the age of the universe, they used mathematical tools to find a representative for each category. This allowed them to systematically test every unique strategy without missing any or wasting time on duplicates. They ran these tests on a standard desktop computer, a tool available to most researchers, rather than needing a supercomputer.

Their investigation covered two main scenarios. First, they looked at cases where the noisy pairs were all different from one another, representing a general situation. For this, they found and evaluated every possible protocol using up to five pairs of particles. Second, they looked at a more specific but common situation where all the noisy pairs were identical, known as Werner states. Because these pairs are identical, the symmetry of the problem is even higher, allowing the team to extend their search to up to eight pairs. In both cases, they identified which specific arrangement of gates produced the highest quality output.

The results showed that for small numbers of pairs, the best methods were already known to the scientific community. However, when they increased the number of pairs to four or more, they discovered new protocols that outperformed everything previously known. These new methods could turn a set of noisy pairs into a cleaner one with significantly higher fidelity, or quality, than older techniques. For instance, with five pairs, the new optimal method improved the quality of the connection much more effectively than the standard approach, which had been the best option for years. The team also provided the exact blueprints, or circuits, for how to build these new protocols in a lab. These circuits are surprisingly simple, requiring only a modest number of steps and a small count of two-particle logic gates, making them feasible to implement with current technology.

One of the most significant findings was that these new protocols do not just work better in theory; they work better in practice. The researchers compared their optimal methods against a popular existing strategy called the DEJMPS protocol. While the old method worked well for very high-quality inputs, the new protocols showed a massive advantage when the starting noise was higher. In some cases, the new method produced a successful result at a rate three times higher than the old method. This is crucial because real-world quantum networks will likely start with very noisy connections. The study also confirmed that these improvements come without a heavy cost in complexity; the new circuits are not significantly deeper or more complicated than the ones already in use.

By mapping out the entire landscape of these distillation protocols, the researchers have given engineers a complete toolkit. Instead of guessing or using heuristics to find a good method, they can now choose the exact protocol that maximizes performance for their specific situation. Whether the goal is to get the highest possible quality from a few pairs or to get a usable connection from a very noisy batch, the best path is now known. This work removes the uncertainty from the design phase of future quantum networks, ensuring that the first generation of quantum repeaters and communication devices can be built with the most efficient methods available. The findings suggest that near-term quantum networks could perform much better than previously forecasted, bringing the dream of a global quantum internet closer to reality.

Technical Summary: Enumerating all bilocal Clifford distillation protocols through symmetry reduction

Problem Statement
Entanglement distillation is a critical procedure for transforming multiple copies of noisy entangled states into fewer copies of higher fidelity, serving as a foundational resource for quantum communication tasks such as secret-key generation and distributed quantum computation. While asymptotic protocols (e.g., the hashing protocol) exist, they often require an unbounded number of communication rounds, rendering them impractical for near-term implementations.

This paper focuses on a specific class of near-term implementable protocols: bilocal Clifford protocols. These protocols involve Alice and Bob sharing $n$ entangled pairs, performing local Clifford operations (where Bob applies the complex conjugate of Alice's operation), measuring $n-1$ pairs in the computational basis, and communicating results to post-select on successful outcomes. The goal is to systematically enumerate and optimize over all such protocols to find those that maximize output fidelity and success probability for Bell-diagonal input states, specifically addressing the computational intractability of searching the full space of Clifford operations as $n$ increases.

Methodology
The authors develop a mathematical framework that maps the search for optimal distillation protocols to a problem in group theory, specifically involving the symplectic group $Sp(2n, \mathbb{F}_2)$ over the field with two elements.

Symplectic Representation: The paper establishes a correspondence between bilocal Clifford circuits and elements of the symplectic group. The action of a Clifford circuit on the probabilities of Bell-diagonal states is represented by a symplectic matrix $M$ .
Reduction via Cosets:
- General Input States: For general Bell-diagonal states, the authors identify that distinct distillation statistics correspond to the right cosets of a specific "distillation subgroup" $D_n$ within $Sp(2n, \mathbb{F}_2)$ . The subgroup $D_n$ consists of operations that leave the distillation statistics (success probability and fidelity) invariant.
- Werner States: For the specific case of an $n$ -fold tensor product of Werner states, the input state possesses additional symmetries (invariance under SWAP, Hadamard, and Phase gates). This allows for a further reduction of the search space to double cosets of the form $D_n \setminus Sp(2n, \mathbb{F}_2) / K_n$ , where $K_n$ is the symmetry group of the Werner state.
Characterization of Subgroups: The paper provides a rigorous characterization of the distillation subgroup $D_n$ , including its generators and a closed-form formula for its order. This allows the authors to calculate the index of $D_n$ in $Sp(2n, \mathbb{F}_2)$ , quantifying the reduction in the search space.
Enumeration Algorithm: Using these reductions, the authors implement an algorithm to enumerate all unique protocols (coset representatives) for $n$ up to 5 for general states and up to $n=8$ for Werner states. This enumeration is performed on a commodity desktop computer, a feat made possible by the symmetry reduction which reduces the search space by factors as large as $2 \cdot 10^{34}$ for $n=8$ .

Key Contributions

Systematic Enumeration: The paper introduces a systematic procedure to find all bilocal Clifford protocols, moving beyond heuristic methods like the "seesaw" method or genetic optimization.
Group-Theoretic Reduction: It establishes a correspondence between distillation protocols and double cosets, providing a rigorous method to reduce the optimization problem size.
Subgroup Characterization: The authors characterize the distillation subgroup $D_n$ in terms of its generators and derive the formula for its order, enabling the calculation of the exact number of distinct protocols to check.
Explicit Circuit Construction: For the optimal protocols found, the authors provide explicit quantum circuits (in terms of Clifford gates) that achieve the highest fidelity.

Results

General States ( $n \leq 5$ ): The authors enumerated all protocols for up to 5 copies of general Bell-diagonal states. They confirmed that for $n=2$ and $n=3$ with equal input states, the optimal protocols correspond to previously known schemes (DEJMPS and double-selection).
Werner States ( $n \leq 8$ ):
- For $n=2$ and $n=3$ , the optimal protocols match existing literature.
- For $n=4$ to $n=8$ , the authors discovered protocols that achieve higher fidelities than previously considered schemes, including concatenated DEJMPS protocols.
- Specifically, for $n=5$ , the optimal protocol exhibits a cubic scaling in infidelity ( $1-F$ ) for high input fidelities, whereas previous protocols (including concatenated DEJMPS) were limited to quadratic scaling.
- The optimal circuits for $n=4$ to $8$ have modest depth (3 to 7) and a low number of two-qubit gates (4 to 13), comparable to or better than existing protocols.
Performance Metrics:
- Fidelity: The new protocols outperform concatenated DEJMPS protocols, particularly for $n \geq 4$ .
- Rate: When combined with a hashing protocol, the optimized bilocal Clifford protocols achieve rates up to three times higher than concatenated DEJMPS for $n > 3$ and input fidelities $\lesssim 0.76$ .
- Device-Independent QKD: The optimized protocols allow for a higher rate of distilling entanglement to the threshold fidelity required for device-independent quantum key distribution ( $F_{tar} \approx 0.93$ ) at lower input fidelities.
- Success Probability vs. Entanglement: While the optimized protocols sometimes have lower success probabilities, the product of success probability and relative entropy of entanglement is maximized by the full optimization for $n > 3$ , particularly at higher fidelities.

Significance
The paper claims that its primary significance lies in providing a complete and systematic enumeration of bilocal Clifford protocols, which allows for the identification of globally optimal protocols rather than relying on heuristics. By exploiting symmetries, the authors make the exhaustive search feasible for $n=8$ Werner states, a scale previously inaccessible.

The results demonstrate that higher-fidelity distillation is achievable with modest circuit resources (low depth and gate count) by moving beyond standard concatenated protocols. The authors note that if these performance gains hold under noisy operations, they could translate to improved forecasts for the performance of near-term quantum networks and distributed quantum computation. The work also provides a software tool to select the protocol that maximizes any specific figure of merit for a given set of input states. The paper concludes modestly, noting that full enumeration becomes infeasible for $n \gtrsim 11$ with current methods, but the framework provides a foundation for future extensions to $n$ -to- $m$ distillation or higher-dimensional states.

Technical Summary: Enumerating all bilocal Clifford distillation protocols through symmetry reduction

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