Representation Theory for Geometric Quantum Machine Learning

TL;DR

This work argues that incorporating symmetries into quantum learning via representation theory provides a principled path to geometry-aware QML. It develops a formal framework around label invariance under group representations, and bridges discrete and continuous groups through Lie algebras, the exponential map, and the Weyl unitary trick. Key tools—Haar integration, twirling, commutants, and Schur-Weyl duality—are shown to yield block-diagonal structures that constrain accessible information and guide the design of equivariant quantum networks and measurements. Through a rich array of examples and a field-guide style presentation, the paper equips researchers to detect symmetries and construct symmetry-respecting QML models with potential for robust quantum advantages.

Abstract

Recent advances in classical machine learning have shown that creating models with inductive biases encoding the symmetries of a problem can greatly improve performance. Importation of these ideas, combined with an existing rich body of work at the nexus of quantum theory and symmetry, has given rise to the field of Geometric Quantum Machine Learning (GQML). Following the success of its classical counterpart, it is reasonable to expect that GQML will play a crucial role in developing problem-specific and quantum-aware models capable of achieving a computational advantage. Despite the simplicity of the main idea of GQML -- create architectures respecting the symmetries of the data -- its practical implementation requires a significant amount of knowledge of group representation theory. We present an introduction to representation theory tools from the optics of quantum learning, driven by key examples involving discrete and continuous groups. These examples are sewn together by an exposition outlining the formal capture of GQML symmetries via "label invariance under the action of a group representation", a brief (but rigorous) tour through finite and compact Lie group representation theory, a reexamination of ubiquitous tools like Haar integration and twirling, and an overview of some successful strategies for detecting symmetries.

Figures (10)

• Figure 1: Symmetries in QML. Consider the QML task of classifying between spherical cows and cubic tardigrades lee2021entanglementvedral2021microscopic whose information has been encoded into quantum states havlivcek2019supervised. Note that a rotated spherical cow is still a spherical cow, and as such the QML model should be able to accurately classify it regardless of how the cow was rotated. In this case a rotation in three-dimensions corresponds to a symmetry of the dataset, as the group of rotations do not change the labels.
• Figure 2: Examples of discrete and continuous symmetry groups. a) In this QML task the goal is to classify real valued data $x$. The data $x_i$ with label $y_i=0$ (blue circles) is sampled from the interval $[-\frac{\pi}{4},\frac{\pi}{4}]$, while the data $x_i$ with label $y_i=1$ (orange triangles) is sampled from $(-\frac{\pi}{2},-\frac{\pi}{4}]\cup[\frac{\pi}{4},\frac{\pi}{2})$. As indicated by the pink dashed arrow, the symmetry operation $x\rightarrow -x$ preserves the label. We encode this classical data in a quantum state by initializing a single qubit to the state $\ket{+}$ and performing a rotation about the $y$-axis with an angle $x_i$. As schematically depicted, the quantum states $\rho_i$ in the dataset are pure states living on the surface of the Bloch sphere. Here, the symmetry group that preserves the labels of the quantum states is $G_{\rm{bflip}}=\{\openone,X\}$ of Eq. \ref{['eq:par_group']} (see pink dashed arrow over the Bloch sphere). b) In this QML task the goal is to classify single-qubit pure states from single-qubit mixed states. The data $\rho_i$ with label $y_i=0$ (blue circles) correspond to pure states living on the surface of the Bloch sphere, while the data $x_i$ with label $y_i=1$ (orange triangles) are mixed states living in a shell inside of the Bloch sphere. Since the purity is a spectral property, it gets preserved by the action of any unitary. As such, the symmetry group preserving the labels is $G_{\rm{uni}}=\{U\in SU(2)\}$ of Eq. \ref{['eq:uni_group']} (see pink dashed arrow over the Bloch sphere). Note that here the data is quantum mechanical in nature, as it does not correspond to classical data encoded in quantum states.
• Figure 3: Example of another discrete symmetry group. In this QML task the goal is to classify real valued data $x=(x^1,x^2)$ living in a two-dimensional plane. The data $x_i$ with label $y_i=0$ (blue circles) is sampled from the green region of the plane, while the data $x_i$ with label $y_i=1$ (orange triangles) is sampled from the yellow region of the plane. As indicated by the pink dashed arrow, the symmetry operation $(x^1,x^2)\rightarrow (x^2,x^1)$ preserves the label. We encode this classical data in a two-qubit quantum state by initializing the qubits to the state $\ket{+}\otimes \ket{+}$ and performing a rotation about the $y$-axis with a angle $x_i^1$ for the first qubit, and $x_i^2$ for the first qubit. Now, the symmetry group preserving the labels of the quantum states is $G_{\mathrm{SWAP}}=\{\openone,\mathrm{SWAP}\}$ of Eq. \ref{['eq:swap-group']}.
• Figure 4: Example of another continuous symmetry group. In this QML task the goal is to distinguish ferromagnetic from antiferromagnetic two-qubit states. The data $\rho_i$ with label $y_i=0$ is given by states where the single-qubit reduced states $\rho_A$ and $\rho_B$ are aligned (i.e., $\Tr[\rho_A\sigma]=\Tr[\rho_B\sigma]$ for any $\sigma=X,Y,Z$). The data $\rho_i$ with label $y_i=1$ is given by states where the single-qubit reduced states $\rho_A$ and $\rho_B$ are anti-aligned (i.e., $\Tr[\rho_A\sigma]=-\Tr[\rho_B\sigma]$). Here the symmetry group preserving the labels is $G_{\rm{prod}}=\{U^{\otimes 2}\,|\, U\in SU(2)\}$ of Eq. \ref{['eq:loc_group']}. Note that here the data is quantum mechanical in nature, as it does not correspond to classical data encoded in quantum states.
• Figure 5: The symmetric group $S_3$ and one of its representations. a) The elements of the symmetric group $S_n$ correspond to the permutations of a set of size $n$. Here we depict the case for $S_3$ where its elements are denoted as $P_{\pi}$. For instance taking $\pi = (12)$ (using cycle notation), the element $P_{\pi}$ maps $123$ to $213$. Here we can see that a generating set for $S_3$ is $\{(1,2),(2,3),(1,3)\}$ as any permutation can be obtained via transpositions of two elements. b) Consider a QML task where the states $\rho_i$ can be thought of as representing an $n$-qubit quantum system whose interaction topology follows that of a graph verdon2019quantumgraphlarocca2022group. Since the way one labels the vertices and assigns them to qubits is completely arbitrary, the problem should be invariant under the action of $S_n$. By conjugating the quantum states $\rho_i$ with elements $P_\sigma\in S_n$ one obtains a new quantum state $P_\sigma \rho_i P_\sigma^{\dagger}$ whose underlying graph vertices are permuted according to $P_\sigma$.

Theorems & Definitions (32)

• Proposition 1
• Definition 1: Label invariance
• Definition 2
• Definition 3
• Definition 4
• Definition 5
• Definition 6
• Theorem 1: Exponential map is a local diffeomorphism
• Proposition 2
• Proposition 3