Classifying complete orthogonal product bases now reduces to solving graph isomorphism problems, previously achieved using more complex methods. Yvkai Zhao and Lin Chen from Beihang University accomplished this by associating each basis with a specific type of coloured network termed an edge-coloured complete multigraph. This approach enables structural analysis and provides an upper bound of 2^n-1 variables, while showing that the number of possible groupings grows as 2^2^n+o(n). The classification of quantum systems, specifically, complete orthogonal product bases used in quantum information processing, is linked to graph theory, a mathematical study of networks.

This connection reframes identifying equivalent quantum states as determining whether two corresponding network diagrams are identical; this is known as ‘graph isomorphism’. The team showed that the number of possible groupings for these states increases at a doubly exponential rate with each additional qubit added to the system. These bases can be understood as different ways to encode information using multiple qubits, much like choosing various combinations of switches within an electrical circuit.

Determining whether two such quantum states are equivalent is akin to recognising if two maps depict the same city layout despite differing colours or symbols. This utilises ‘graph isomorphism’. By associating each basis with a specific type of network called an edge-coloured complete multigraph, they’ve shown the number of possible groupings for these states increases at a doubly exponential rate with added qubits.

Graph theory reduces complexity in classifying complete orthogonal product bases

A novel graph-theoretic approach originating has reduced the number of variables needed for classification of complete orthogonal product bases from an initial upper bound of 2ⁿ down to just 2^n-1. This breakthrough addresses limitations imposed by previous methods which struggled with systems exceeding this threshold; earlier techniques were unable to efficiently handle such complex calculations without becoming computationally intractable. The method links these quantum states to edge-coloured networks, enabling structural analysis and bounding using combinatorial principles, offering important advantages when understanding multiqubit OPBs.

A variable pairing limit of 2^n-1 now successfully bounds possible configurations within these quantum systems. Further refinement revealed that equivalence classes, representing fundamentally different arrangements of qubits, grow with a factor of two raised to the power of 2ⁿ, demonstrating rapid complexity as system size increases. Recursive construction proved that the quantity of OPB classes is at least one plus half the previous value; however determining whether these theoretical bounds translate into practical scalability remains challenging given the exponential worst-case complexity inherent in their equivalence testing algorithm.

The team employed a formal matrix formalism, a mathematical framework for representing and manipulating complete orthogonal product bases, translating the complex problem of quantum state classification into more manageable terms. This approach allowed representation of each basis as structured data akin to a spreadsheet. They then associated these matrices with edge-coloured complete multigraphs, networks where every node connects to every other node and edges have assigned colours.

Quantum state classification via edge-coloured graph structural comparison

The researchers cleverly recast classifying these quantum building blocks, complete orthogonal product bases, as assessing network similarity; specifically, determining if two edge-coloured graphs share identical structure. While offering improvements over previous methods for bounding system variables, the technique ultimately relies on solving ‘graph isomorphism’, a notoriously difficult computational challenge. This reframing of a physics problem into the language of graph theory offers new analytical tools and potentially faster ways to estimate key properties like system variables.

Complete orthogonal product bases, essential tools for encoding information in qubits, can now be understood through the perspective of network structures called edge-coloured complete multigraphs, establishing a strong connection between quantum states and graph theory. Successfully linking these abstract mathematical concepts provides novel analytical pathways previously unavailable within quantum information science.

Researchers demonstrated that classifying complex quantum systems, specifically, complete orthogonal product bases, can be reframed as comparing edge-coloured complete multigraphs. This approach allows estimation of system variables with new bounds, although determining if two quantum states are equivalent still requires solving a computationally difficult graph comparison task. The team showed that the number of possible classifications grows at a rate of $2^{2}$ ^n+o(n), where n represents the size of the qubit system. By connecting quantum physics to graph theory, they provide alternative mathematical tools for analysing these systems and their properties.

👉 More information
🗞 Multiqubit orthogonal product bases
✍️ Yvkai Zhao and Lin Chen
🧠 ArXiv: https://arxiv.org/abs/2608.18421

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