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

Laura Mančinska, Elias Theil

TL;DR

This work classifies all quantum channels that are both unitary-equivariant and permutation-invariant from $\mathbb{C}^{d}$-tensor powers, revealing that extremal channels factor into unitary Schur sampling, irrep-level unitary-equivariant channels, and the adjoint Schur sampling. It then provides a streaming implementation ansatz that reduces the problem to implementing unitary Schur sampling, its adjoint, and unitary-equivariant channels on irreps via a resource-state, with explicit memory and gate complexity bounds. The authors apply this general framework to state symmetrization, symmetric cloning, and purity amplification, achieving polynomial-time streaming algorithms with exponential memory improvements in $m$ and $n$, and the first efficient algorithm for symmetric cloning at general dimension $d$. The results leverage mixed Schur–Weyl duality, Clebsch–Gordan transforms, and a structured decomposition over $SU(d)$ irreps, yielding practical, memory-efficient quantum circuits for symmetry-laden tasks. Overall, the paper provides a principled, representation-theoretic foundation for constructing and implementing symmetry-respecting quantum channels and demonstrates substantial practical gains in three emblematic tasks.

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.

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

TL;DR

This work classifies all quantum channels that are both unitary-equivariant and permutation-invariant from -tensor powers, revealing that extremal channels factor into unitary Schur sampling, irrep-level unitary-equivariant channels, and the adjoint Schur sampling. It then provides a streaming implementation ansatz that reduces the problem to implementing unitary Schur sampling, its adjoint, and unitary-equivariant channels on irreps via a resource-state, with explicit memory and gate complexity bounds. The authors apply this general framework to state symmetrization, symmetric cloning, and purity amplification, achieving polynomial-time streaming algorithms with exponential memory improvements in and , and the first efficient algorithm for symmetric cloning at general dimension . The results leverage mixed Schur–Weyl duality, Clebsch–Gordan transforms, and a structured decomposition over irreps, yielding practical, memory-efficient quantum circuits for symmetry-laden tasks. Overall, the paper provides a principled, representation-theoretic foundation for constructing and implementing symmetry-respecting quantum channels and demonstrates substantial practical gains in three emblematic tasks.

Abstract

Many quantum information tasks use inputs of the form , 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 to - via their extremal points. Operationally, each extremal quantum channel factors as unitary Schur sampling an irrep-level unitary-equivariant quantum channel 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 . Further, for symmetric cloning we present, to our knowledge, the first efficient (polynomial-time) algorithm with explicit memory and gate bounds.

Paper Structure

This paper contains 25 sections, 21 theorems, 133 equations, 13 figures, 3 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Technical results
  3. Classification and operational interpretation
  4. General ansatz
  5. Application to three tasks
  6. Discussion
  7. Mathematical preliminaries and notation
  8. Linear algebra and quantum channels
  9. Representations and dual representations
  10. Partitions and staircases
  11. Representation theory of $SU(d)$
  12. Representation theory of $\mathcal{A}_{m,n}^{d}$
  13. Mixed Schur-Weyl duality
  14. Classification and operational interpretation
  15. Classification of all unitary-equivariant and permutation-invariant quantum channels
  16. ...and 10 more sections

Key Result

Theorem 1

Each extremal point $\Phi \in \mathcal{C}_{up}^{d}(m,n)$ corresponds to a collection of touples $(\mu_{\lambda},\gamma_{\lambda},\ket{\psi_{\lambda}})$ for each $\lambda\vdash_{d} m$, so that For a given collection of touples, the Choi matrix $C_{\Phi}$ is Such a quantum channel $\Phi$ is equivalent to Here, $\Phi_{\rm USS}^{m}$ is the unitary Schur sampling quantum channel on $(\mathbb{C}^{d})

Figures (13)

  • Figure 1: Circuit diagram showing the factoring of quantum channels in Theorem \ref{['thm:operational_interpretation_extremal_quantum channels']}. Unitary Schur sampling $\Phi_{\rm USS}^{m}$ provides an output irrep label $\lambda$, together with a state on that output irrep $\rho_{\mathcal{Q}_{\lambda}^{d}}$. For each label $\lambda$, a unitary-equivariant quantum channel $\Phi_{\lambda,\mu_{\lambda}}^{\gamma_{\lambda},\psi_{\lambda}}$ maps the unitary irrep $\mathcal{Q}_{\lambda}^{d}$ into the unitary irrep $\mathcal{Q}_{\mu_{\lambda}}^{d}$, which is then transformed back via dual unitary Schur sampling. The superscripts $\gamma_{\lambda},\psi_{\lambda}$ denote a particular choice of quantum channel, explained in detail in Definition \ref{['def:unitary_equivariant_quantum channel_irreps']}.
  • Figure 2: Implementing the mixed Schur transform Nguyen_2023Grinko_2024.
  • Figure 3: Implementing unitary mixed Schur sampling CerveroMartin_2024.
  • Figure 4: Implementing the dual quantum channel to unitary mixed Schur sampling.
  • Figure 5: Implementing the embedding $\iota_{\mu,\gamma}^{\lambda,\psi}$.
  • ...and 8 more figures

Theorems & Definitions (62)

  • Definition 1
  • Theorem 1: Corollary \ref{['cor:classification_extremal_points_Cusd']} and Theorem \ref{['thm:operational_interpretation_extremal_quantum channels']}
  • Theorem 2: Propositions \ref{['prop:paths_embedding']} and \ref{['prop:implement_paths_embedding']}, and Theorem \ref{['thm:complexity_unitary_equivariant_permutation_invariant_quantum channels']}
  • Theorem 3: Corollaries \ref{['cor:result_state_symmetrization']}, \ref{['cor:result_symmetric_cloning']} and \ref{['cor:result_purity_amplification']}
  • Example 1
  • Example 2
  • Example 3
  • Proposition 4
  • Remark 1
  • proof
  • ...and 52 more