Super happy to share our new preprint today on ArXiv: 🥳
arxiv.org/pdf/2510.05531

It’s about efficient learning of bosonic Gaussian unitaries with provable recovery guarantees in a physically motivated accuracy metric: the "energy-constrained diamond-norm".

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📐 We also introduce a graphical calculus tool to diagrammatically manipulate and visualize the elements of this new basis 🎨.

Perfect for hands-on calculations: It allowed us to save lots of writing 📚, as many pages of analytical calculations became just a few diagrams 😉.

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This new "Pauli-sum" basis is intuitive and computationally friendly 💻.

It’s (gracefully) generated by products of:
• Permutation operators (which generate the commutant of the unitary group) 🔄
• Just three additional operators 🔑

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🔧 In our work, we give a full description of the commutant for arbitrary n (qubits) and k (tensor powers):

- An explicit orthogonal basis 🧮

- The exact dimension of the commutant 📏

- A new, compact, and easy-to-manipulate basis formed by isotropic sums of Pauli operators 🔀

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At the heart of understanding many properties of the Clifford group 💡 — and unlocking its broad range of applications 🚀 — lies its commutant: the set of operators that commute with the k-fold tensor powers of all Clifford unitaries.

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The Clifford group is ubiquitous in quantum information 🌐.
It lies at the core of many key applications 🔑, including error correction, tomography, benchmarking, and more.

It consists of unitaries that map Pauli operators to Pauli operators under conjugation 🔄.