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arXiv:2510.08154

Classification and implementation of unitary-equivariant and permutation-invariant quantum channels

Laura Mančinska, Elias Theil

Published Oct 9, 2025• 0 citations
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Abstract

Many quantum information tasks use inputs of the form $ρ^{\otimes m}$, which naturally induce permutation and unitary symmetries. We classify all quantum channels that respect both symmetries - i.e. unitary-equivariant and permutation-invariant quantum channels from $(\mathbb{C}^{d})^{\otimes m}$ to $(\mathbb{C}^{d})^{\otimes n}$ - via their extremal points. Operationally, each extremal quantum channel factors as unitary Schur sampling $\rightarrow$ an irrep-level unitary-equivariant quantum channel $\rightarrow$ the adjoint unitary Schur sampling. We give a streaming implementation ansatz that uses an efficient streaming implementation of unitary Schur sampling together with a resource-state primitive, and we apply it to state symmetrization, symmetric cloning, and purity amplification. In these applications we obtain polynomial-time algorithms with exponential memory improvements in $m,n$. Further, for symmetric cloning we present, to our knowledge, the first efficient (polynomial-time) algorithm with explicit memory and gate bounds.

DOI
10.48550/arxiv.2510.08154

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Many quantum information tasks use inputs of the form ρ⊗mρ^{\otimes m}ρ⊗m, which naturally induce permutation and unitary symmetries. We classify all quantum channels that respect both symmetries - i.e. unitary-equivariant and permutation-invariant quantum channels from (Cd)⊗m(\mathbb{C}^{d})^{\otimes m}(Cd)⊗m to (Cd)⊗n(\mathbb{C}^{d})^{\otimes n}(Cd)⊗n - via their extremal points. Operationally, each extremal quantum channel factors as unitary Schur sampling →\rightarrow→ an irrep-level unitary-equivariant quantum channel →\rightarrow→ the adjoint unitary Schur sampling. We give a streaming implementation ansatz that uses an efficient streaming implementation of unitary Schur sampling together with a resource-state primitive, and we apply it to state symmetrization, symmetric cloning, and purity amplification. In these applications we obtain polynomial-time algorithms with exponential memory improvements in m,nm,nm,n. Further, for symmetric cloning we present, to our knowledge, the first efficient (polynomial-time) algorithm with explicit memory and gate bounds.
The Schur transform, which block-diagonalizes the tensor representation U⊗nU^{\otimes n}U⊗n of the unitary group Ud\mathbf{U}_dUd​ on nnn qudits, is an important primitive in quantum information and theoretical physics. We give a generalization of its quantum circuit implementation due to Bacon, Chuang, and Harrow (SODA 2007) to the case of mixed tensor U⊗n⊗Uˉ⊗mU^{\otimes n} \otimes \bar{U}^{\otimes m}U⊗n⊗Uˉ⊗m, where Uˉ\bar{U}Uˉ is the dual representation. This representation is the symmetry of unitary-equivariant channels, which find various applications in quantum majority vote, multiport-based teleportation, asymmetric state cloning, black-box unitary transformations, etc. The "mixed" Schur transform contains several natural extensions of the representation theory used in the Schur transform, in which the main ingredient is a duality between the mixed tensor representations and the walled Brauer algebra. Another element is an efficient implementation of a "dual" Clebsch-Gordan transform for Uˉ\bar{U}Uˉ. The overall circuit has complexity O~((n+m)d4)\widetilde{O} ((n+m)d^4)O((n+m)d4). Finally, we show how the mixed Schur transform enables efficient implementation of unitary-equivariant channels in various settings and discuss other potential applications, including an extension of permutational quantum computing that includes partial transposes.
Exploiting permutation invariance to reduce the exponential scaling of semidefinite programs in quantum information has emerged as a powerful computational technique. In this work, we develop a systematic framework for using this reduction via Schur-Weyl duality for optimization problems, and establish methods that allow one to work fully inside the permutation invariant subspace while performing operations such as (partially) applying channels and taking (partial) traces, or computing expressions like the quantum relative entropy. We then apply our techniques to the problem of computing efficient lower bounds on the channel fidelity over nnn parallel uses of a quantum channel. The algorithm, which we call symmetric seesaw method, exploits permutation-invariant codes to yield improved lower bounds on the channel fidelity over nnn uses of the depolarizing and amplitude-damping channel in the regime of tens of channel uses, and was used in [arxiv:2604.27042] to demonstrate non-asymptotic superactivation of quantum capacity for n=17n = 17n=17. An implementation of our methods, aimed at being suitable for various quantum information theoretic optimization problems, is also available as an open-source Python package.
We formalize the task of unitary Schur sampling -- an extension of weak Schur sampling -- which is the process of measuring the Young label and the unitary group register of an input mmm qudit state. Intuitively, this task is equivalent to applying the Schur transform, projecting onto the isotypic subspaces of the unitary and symmetric groups indexed by the Young labels, and discarding of the permutation register. As such unitary Schur sampling is the natural task in processes such as quantum state tomography or spectrum estimation. We generalize this task to unitary mixed Schur sampling to account for the recently introduced mixed Schur-Weyl transform. We provide a streaming algorithm which achieves an exponential reduction in the memory complexity and a polynomial reduction in the gate complexity over naïve algorithms for the task of unitary (mixed) Schur sampling. Further, we show that if the input state has limited rank, the gate and memory complexities of our streaming algorithm as well as the algorithms for the full Schur and mixed Schur transforms are further reduced. Our work generalizes and improves on the results in arXiv2309.11947.
We consider the computational task of anti-symmetrization of an unknown mixed quantum state: Given iid copies of a rank-rrr qudit state ρ=∑i=1rλi∣φi⟩ ⁣⟨φi∣ρ=\sum_{i=1}^r λ_i |\varphi_i\rangle\!\langle\varphi_i|ρ=∑i=1r​λi​∣φi​⟩⟨φi​∣, prepare a copy of the state proportional to Πantir(∣φ1⟩⊗…⊗∣φr⟩)Π_{\mathrm{anti}}^r (|\varphi_1\rangle\otimes \ldots\otimes|\varphi_r\rangle)Πantir​(∣φ1​⟩⊗…⊗∣φr​⟩), with ΠantirΠ_{\mathrm{anti}}^rΠantir​ the anti-symmetric subspace projector. We first give a simple algorithm for anti-symmetrization that uses O~(r/λmin⁡)\widetilde{O}(r / λ_{\min})O(r/λmin​) copies of ρρρ, where λmin⁡λ_{\min}λmin​ is the minimum non-zero eigenvalue of ρρρ. This algorithm works even in the streaming setting with small working memory, and it uses only simple quantum operations like controlled-SWAPs and constantly many layers of single-qubit gates. As an application of this streaming anti-symmetrization procedure, we give a reduction from gapped mixed-state one-way state generators to pure-state one-way state generators. Then, using tools from Schur-Weyl duality, we give and analyze a second (non-streaming) anti-symmetrization procedure that uses O(r/λmin⁡)O(r / λ_{\min})O(r/λmin​) copies of ρρρ, which then serves as a building block in new algorithms for weak Schur sampling and unitary Schur sampling, outperforming existing approaches in some regimes of number of copies nnn and local dimension ddd, as well as in an algorithm for optimal purity amplification for qudit states.
This thesis explores the interplay between representation theory and quantum information. Specifically, we focus on mixed Schur–Weyl duality, which considers the action of the unitary group on mixed tensors. This setting naturally arises in quantum information tasks involving unitary-equivariant channels, such as port-based teleportation, quantum majority vote, and universal transposition of unitary operators. A key contribution of this thesis is an explicit derivation of the action of the generators of the partially transposed permutation matrix algebra—the commutant of the mixed unitary action—in the Gelfand–Tsetlin basis. As another key result of this thesis, we develop efficient quantum circuits for the mixed quantum Schur transform, a novel primitive in quantum information. The key ingredient of our construction is new efficient circuits for the dual Clebsch–Gordan transform of the unitary group. A significant application of our findings is the construction of efficient quantum algorithms for port-based teleportation, a variant of quantum teleportation that eliminates the need for corrective operations. Another application is a symmetry reduction of semidefinite optimisation problems with unitary equivariance symmetry. Finally, we study the extendibility of quantum states possessing unitary, mixed unitary, or orthogonal symmetry on the complete graph. We obtain analytically the exact maximum values for projections onto the maximally entangled state and the antisymmetric state for each of the three symmetry classes. This thesis demonstrates the usefulness of mixed Schur–Weyl duality in quantum information and computing. We expect that our tools will help address other problems in other areas of quantum information processing, such as communication, cryptography, and simulation.
Understanding which subclasses of quantum circuits are efficiently classically simulable is fundamental to delineating the boundary between classical and quantum computation. In this context, it is well known that certain tasks based on permutation-equivariant unitaries-i.e., nnn-qubit circuits whose action commutes with the qubit-permuting representation of the symmetric group SnS_nSn​-can be simulated in polynomial time. However, existing approaches scale as O(n7)O(n^7)O(n7), and can rapidly become prohibitively expensive. In this work, we introduce a practical algorithm for simulating SnS_nSn​-equivariant circuits under the assumption that the gate generators are at most kkk-local, with k∈O(1)k\in O(1)k∈O(1). The resulting method runs in O(nω+1)O(n^{ω+1})O(nω+1) time for constant depth, where ωωω is the matrix multiplication exponent, significantly lowering the polynomial degree compared to existing techniques. Finally, we numerically validate this scaling by simulating the dynamical evolution of the Lipkin-Meshkov-Glick model, and show that for n=512n=512n=512 spins, a standard laptop can compute the concurrence of the evolved state in under two minutes.