On the universality of $S_n$-equivariant $k$-body gates

• URL: http://arxiv.org/abs/2303.00728v2
• Date: Fri, 31 May 2024 02:04:30 GMT
• Title: On the universality of $S_n$-equivariant $k$-body gates
• Authors: Sujay Kazi, Martin Larocca, M. Cerezo,
• Abstract summary: We study how the interplay between symmetry and $k$-bodyness in the QNN generators affect its expressiveness. Our results bring us a step closer to better understanding the capabilities and limitations of equivariant QNNs.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The importance of symmetries has recently been recognized in quantum machine learning from the simple motto: if a task exhibits a symmetry (given by a group $\mathfrak{G}$), the learning model should respect said symmetry. This can be instantiated via $\mathfrak{G}$-equivariant Quantum Neural Networks (QNNs), i.e., parametrized quantum circuits whose gates are generated by operators commuting with a given representation of $\mathfrak{G}$. In practice, however, there might be additional restrictions to the types of gates one can use, such as being able to act on at most $k$ qubits. In this work we study how the interplay between symmetry and $k$-bodyness in the QNN generators affect its expressiveness for the special case of $\mathfrak{G}=S_n$, the symmetric group. Our results show that if the QNN is generated by one- and two-body $S_n$-equivariant gates, the QNN is semi-universal but not universal. That is, the QNN can generate any arbitrary special unitary matrix in the invariant subspaces, but has no control over the relative phases between them. Then, we show that in order to reach universality one needs to include $n$-body generators (if $n$ is even) or $(n-1)$-body generators (if $n$ is odd). As such, our results brings us a step closer to better understanding the capabilities and limitations of equivariant QNNs.

Related papers

• Permutation-equivariant quantum convolutional neural networks [1.7034813545878589]
  We propose the architectures of equivariant quantum convolutional neural networks (EQCNNs) adherent to $S_n$ and its subgroups. For subgroups of $S_n$, our numerical results using MNIST datasets show better classification accuracy than non-equivariant QCNNs.
  arXiv  Detail & Related papers  (2024-04-28T14:34:28Z)
• A Unified Framework for Uniform Signal Recovery in Nonlinear Generative Compressed Sensing [68.80803866919123]
  Under nonlinear measurements, most prior results are non-uniform, i.e., they hold with high probability for a fixed $mathbfx*$ rather than for all $mathbfx*$ simultaneously. Our framework accommodates GCS with 1-bit/uniformly quantized observations and single index models as canonical examples. We also develop a concentration inequality that produces tighter bounds for product processes whose index sets have low metric entropy.
  arXiv  Detail & Related papers  (2023-09-25T17:54:19Z)
• Theoretical Guarantees for Permutation-Equivariant Quantum Neural Networks [0.0]
  We show how to build equivariant quantum neural networks (QNNs) We prove that they do not suffer from barren plateaus, quickly reach overparametrization, and generalize well from small amounts of data. Our work provides the first theoretical guarantees for equivariant QNNs, thus indicating the extreme power and potential of GQML.
  arXiv  Detail & Related papers  (2022-10-18T16:35:44Z)
• Monogamy of entanglement between cones [68.8204255655161]
  We show that monogamy is not only a feature of quantum theory, but that it characterizes the minimal tensor product of general pairs of convex cones. Our proof makes use of a new characterization of products of simplices up to affine equivalence.
  arXiv  Detail & Related papers  (2022-06-23T16:23:59Z)
• A lower bound on the space overhead of fault-tolerant quantum computation [51.723084600243716]
  The threshold theorem is a fundamental result in the theory of fault-tolerant quantum computation. We prove an exponential upper bound on the maximal length of fault-tolerant quantum computation with amplitude noise.
  arXiv  Detail & Related papers  (2022-01-31T22:19:49Z)
• Uncertainties in Quantum Measurements: A Quantum Tomography [52.77024349608834]
  The observables associated with a quantum system $S$ form a non-commutative algebra $mathcal A_S$. It is assumed that a density matrix $rho$ can be determined from the expectation values of observables. Abelian algebras do not have inner automorphisms, so the measurement apparatus can determine mean values of observables.
  arXiv  Detail & Related papers  (2021-12-14T16:29:53Z)
• Quantum double aspects of surface code models [77.34726150561087]
  We revisit the Kitaev model for fault tolerant quantum computing on a square lattice with underlying quantum double $D(G)$ symmetry. We show how our constructions generalise to $D(H)$ models based on a finite-dimensional Hopf algebra $H$.
  arXiv  Detail & Related papers  (2021-06-25T17:03:38Z)
• Symmetry from Entanglement Suppression [0.0]
  We show that a minimally entangling $S$-matrix would give rise to global symmetries. For $N_q$ species of qubit, the Identity gate is associated with an $[SU(2)]N_q$ symmetry.
  arXiv  Detail & Related papers  (2021-04-22T02:50:10Z)
• Epsilon-nets, unitary designs and random quantum circuits [0.11719282046304676]
  Epsilon-nets and approximate unitary $t$-designs are notions of unitary operations relevant for numerous applications in quantum information and quantum computing. We prove that for a fixed $d$ of the space, unitaries constituting $delta$-approx $t$-expanders form $epsilon$-nets for $tsimeqfracd5/2epsilon$ and $delta=left(fracepsilon3/2dright)d2$. We show that approximate tdesigns can be generated
  arXiv  Detail & Related papers  (2020-07-21T15:16:28Z)
• Quantum Algorithms for Simulating the Lattice Schwinger Model [63.18141027763459]
  We give scalable, explicit digital quantum algorithms to simulate the lattice Schwinger model in both NISQ and fault-tolerant settings. In lattice units, we find a Schwinger model on $N/2$ physical sites with coupling constant $x-1/2$ and electric field cutoff $x-1/2Lambda$. We estimate observables which we cost in both the NISQ and fault-tolerant settings by assuming a simple target observable---the mean pair density.
  arXiv  Detail & Related papers  (2020-02-25T19:18:36Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.