⚛️ quantum physics

Optimal two-mode bosonic loss codes from finite group symmetry

This paper leverages finite group symmetry to derive necessary-and-sufficient conditions for constructing optimal two-mode bosonic loss codes, resulting in analytic constructions that achieve maximum loss distances with minimal photon numbers and provide evidence for an infinite family of such codes.

Original authors: Argyris Giannisis Manes, Mahadevan Subramanian, Liang Jiang

Published 2026-10-02

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CC BY 4.0

Original authors: Argyris Giannisis Manes, Mahadevan Subramanian, Liang Jiang

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

In the quest to build a quantum computer, scientists face a relentless enemy: noise. Unlike classical computers, which store information as stable zeros and ones, quantum machines use delicate states of matter that can easily be disturbed by their environment. One of the most common sources of this disturbance in a specific type of quantum hardware, known as superconducting circuits, is the loss of energy. Imagine a system where information is carried by packets of light, called photons, trapped inside a cavity. If even a single photon escapes, the information it carried is corrupted or lost entirely. To protect against this, researchers use a strategy called error correction. They encode a single piece of logical information across many physical particles, creating redundancy. If one particle fails, the system can detect the error and fix it without destroying the data. A particularly clever approach involves using two separate channels to carry the light. If the total number of photons in both channels is kept constant, the system can instantly know if a photon has been lost just by checking the total count, turning a destructive error into a known missing piece that is easier to repair.

The challenge has been to find the most efficient way to arrange these photons so that the system can correct the maximum number of losses while using the fewest resources. In a new study, researchers at the University of Chicago have uncovered a hidden mathematical order that solves this problem. They began by running powerful computer simulations to search for the best possible arrangements of photons for systems containing between four and twenty-five photons. They did not impose any pre-existing rules or patterns on the search; they simply let the computer look for the configuration that offered the highest protection against errors. Surprisingly, the computer did not find random, messy solutions. Instead, every single best-performing code it discovered followed a strict, repeating pattern based on the symmetries of a finite group. In simple terms, the optimal way to arrange these photons is not a chaotic scramble, but a highly structured design that looks the same when rotated or reflected in specific ways, much like the faces of a geometric solid.

This discovery was a turning point. The researchers realized that these numerical patterns were not just coincidences but were pointing toward a deeper mathematical truth. They used this insight to derive a new set of rules for building these codes. Instead of guessing and checking, they could now construct the perfect codes by following the rules of these symmetries. This method allowed them to create exact, mathematically proven codes that matched the computer's best guesses and even found new, better solutions that the computer had missed. For example, they constructed codes that could correct for the loss of up to ten photons using fewer total photons than any previously known method. Specifically, they found codes that could handle six, eight, and ten losses using twenty-eight, forty-nine, and seventy-six photons respectively, setting new records for efficiency.

The team also developed a way to prove that their codes were truly the best possible for their size. They used a rigorous mathematical certification process to show that no other arrangement of the same number of photons could possibly correct more errors. In twenty-one out of the twenty-two cases they examined, their constructed codes reached the absolute theoretical limit of performance. In the one remaining case, the code they found was just one step below the limit, suggesting that the theoretical limit might be slightly higher than currently calculated or that an even better code exists but is extremely difficult to find. The researchers also identified a specific sequence of these codes that they believe continues infinitely, offering a blueprint for building quantum memory that scales up efficiently.

The significance of this work lies in its ability to turn a difficult, open-ended search into a systematic design process. By recognizing that the best solutions emerge from finite group symmetries, the researchers have provided a clear path forward for building more robust quantum hardware. These codes are not just theoretical curiosities; they are directly applicable to the physical systems being built today. Because the symmetries involved can be implemented using standard optical components, these codes can be realized in the lab without exotic new technology. The study suggests that the most efficient way to protect quantum information from loss is to arrange it with a specific, elegant geometric structure, a principle that could guide the development of the next generation of quantum computers.

Technical Summary: Optimal Two-Mode Bosonic Loss Codes from Finite Group Symmetry

Problem Statement
Photon loss is the dominant noise mechanism in bosonic quantum hardware, such as superconducting cavities. While dual-rail encodings ( $|0\rangle|1\rangle, |1\rangle|0\rangle$ ) allow for the detection of photon loss via total photon number measurements, they cannot correct it. Encoding logical information into higher fixed-total-photon-number subspaces ( $H_n$ ) of two bosonic modes offers the potential for both loss detection and correction. However, identifying the optimal encodings within these subspaces for arbitrary photon numbers ( $n$ ) and loss distances ( $d$ ) remains a challenging optimization problem. Previous approaches relied on mapping these codes to spin codes or permutation-invariant codes, but these mappings did not inherently identify which specific codes are optimal against loss.

Methodology
The authors employ a three-stage pipeline combining numerical optimization, representation theory, and linear programming certification:

Numerical Discovery: The authors perform unconstrained entanglement fidelity optimization for logical qubits encoded in the fixed-total-photon-number subspace $H_n$ for $4 \le n \le 25$ . They utilize a bi-convex optimization framework to find optimal encoder and decoder maps. Crucially, they decompose the recovery problem into independent semidefinite programs (SDPs) for each surviving photon number sector, significantly accelerating the search.
Symmetry Identification and Analytic Construction: Upon observing that the numerically optimal codes align closely with finite-group invariant projectors, the authors derive necessary-and-sufficient Knill–Laflamme (KL) conditions for arbitrary finite-group-invariant codes.
- They establish a direct equivalence between correcting up to $k$ photon losses and detecting irreducible spin tensors of rank up to $k$ .
- Using representation theory, they reduce the KL conditions to constraints on multiplicity spaces.
- They prove that a minimal set of "seed" operators (rotated powers of the raising operator $J_+$ ) is sufficient to enforce error correction for the entire group orbit, reducing the construction to solving equations on these seeds.
Certification: To verify optimality, the authors employ MacWilliams linear programming (LP) bounds adapted for $SU(2)$ spin codes. They generate exact Farkas certificates to prove the non-existence of codes with higher distances for specific parameters, establishing rigorous upper bounds on the achievable loss distance.

Key Contributions

Emergence of Finite Group Symmetry: The study reveals that unconstrained optimization of loss-correcting codes naturally converges to codes possessing finite-group symmetry (specifically subgroups of $SU(2)$ ). In the range $4 \le n \le 25$ , all 22 best-found codes correspond to either two-dimensional irreducible representations or reducible representations of finite groups.
General Construction Framework: The authors provide a systematic method to construct exact codes from finite group symmetry. This framework reduces the complex operator-space KL conditions to equations on multiplicity spaces and identifies the minimum number of seed operators required to span the error space for any tensor rank.
Binary-Polyhedral Sequence: Motivated by the numerical findings, the authors construct an infinite family of codes based on binary polyhedral groups ( $2O$ for even distances, $2T$ for odd distances). They propose a sequence of photon numbers $n_d = \lceil(3d^2 + 1)/4\rceil$ that achieves distance $d$ .
New Record-Breaking Codes: The authors explicitly construct codes for $(n, d) = (28, 6)$ , $(49, 8)$ , and $(76, 10)$ . To the authors' knowledge, these codes utilize the smallest reported photon numbers for their respective distances, improving upon previous records (e.g., reducing $n$ for $d=6$ from 36 to 28).

Results

Optimality in Scanned Sectors: For 21 out of the 22 scanned photon numbers ( $4 \le n \le 25$ ), the constructed codes achieve the maximum possible loss distance allowed by the MacWilliams LP bound. The sole exception is $n=18$ , where the LP permits distance 5, but the best construction found achieves distance 4.
Monotonicity: The authors prove that the LP feasibility bound is monotone in the total photon number $n$ ; if a code of distance $d$ is feasible for $n$ , it is also feasible for $n+1$ .
Exact vs. Numerical: The analytic construction recovers the numerical codes with high fidelity. In some cases (e.g., $n=15$ ), the analytic construction identifies a code structure (binary tetrahedral $2T$ ) that the initial random-start numerical search only approximated, demonstrating the predictive power of the symmetry-based approach.
Constraint Balancing: For the proposed binary-polyhedral family, the authors show that the number of symmetry-reduced KL constraints exactly matches the degrees of freedom available in the multiplicity space for rank-one projectors, providing structural evidence for the existence of this infinite family.

Significance
The paper establishes that finite-group symmetry is not merely an ansatz for constructing bosonic codes but is an emergent property of optimal loss correction. This connection bridges the gap between numerical discovery and analytic construction, turning numerically found codes into exact, provable solutions. The work provides a systematic design principle for hardware-efficient bosonic codes that retain the loss-detection advantage of dual-rail encoding while enabling correction. The authors claim that their constructions and constraint counts provide strong evidence for a conjectured infinite family of codes that attain every distance $d$ at the minimum possible photon number $n_d$ . While the existence of this family is supported by constraint counting and numerical construction up to $d=10$ , the authors note that a rigorous proof of the family's existence for all $d$ remains an open problem. Additionally, the gap between the LP bound and the constructed codes at certain parameters (like $n=18$ ) suggests that the LP relaxation may not be tight, leaving room for potentially better codes or new theoretical bounds.

Technical Summary: Optimal Two-Mode Bosonic Loss Codes from Finite Group Symmetry

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