arXiv:2405.12890 · Physical review. D/Physical review. D.

Block encodings of discrete subgroups on a quantum computer

Henry Lamm, Ying-Ying Li, Jing Shu, Yi-Lin Wang, Bin Xu

Published Sep 9, 2024• 26 citations

Abstract

We introduce a block encoding method for mapping discrete subgroups to qubits on a quantum computer. This method is applicable to general discrete groups, including crystal-like subgroups such as BI of S U ( 2 ) and V of S U ( 3 ) . We detail the construction of primitive gates—the inversion gate, the group multiplication gate, the trace gate, and the group Fourier gate—utilizing this encoding method for BT and for the first time BI group. We also provide resource estimations to extract the gluon viscosity. The inversion gates for BT and BI are benchmarked on the quantum computer with estimated fidelities of 40 − 4 + 5 % and 4 − 3 + 5 % , respectively. Published by the American Physical Society 2024

DOI: 10.1103/physrevd.110.054505

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arXiv:2405.12890 · Physical review. D/Physical review. D.

Block encodings of discrete subgroups on a quantum computer

Henry Lamm, Ying-Ying Li, Jing Shu, Yi-Lin Wang, Bin Xu

Published Sep 9, 2024• 26 citations

arXiv HTML PDF

Abstract

DOI: 10.1103/physrevd.110.054505

Related papers

Primitive quantum gates for an S U ( 3 ) discrete subgroup: Σ ( 36 × 3 )

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Bastidas et al. (2025) highlighted the importance of exploiting group symmetries to reduce computational overhead and unify simulation methods across both condensed matter physics and ab initio quantum chemistry. I discuss the discrete character transform, which is the group Fourier transform restricted to $Z(\mathbb{C}[G])$ , the center of the group algebra, where class functions form a commutative convolution algebra and irreducible characters are its spectral basis. The normalized character table of a non-abelian group can be obtained through a restriction operator, which means an efficient quantum character transform for any finite group with an efficient quantum Fourier transform. I give complementary quantum algorithms for class convolution based on direct class-sum block encoding and spectral implementation in the character basis, explaining their applications to symmetry-adapted many-body dynamics, molecular and spectroscopic symmetry analysis, and quantum walks, as well as characterizing the central information used in hidden-subgroup problems, including its limitations for graph isomorphism.Bastidas et al. (2025) highlighted the importance of exploiting group symmetries to reduce computational overhead and unify simulation methods across both condensed matter physics and ab initio quantum chemistry. I discuss the discrete character transform, which is the group Fourier transform restricted to $Z(\mathbb{C}[G])$, the center of the group algebra, where class functions form a commutative convolution algebra and irreducible characters are its spectral basis. The normalized character table of a non-abelian group can be obtained through a restriction operator, which means an efficient quantum character transform for any finite group with an efficient quantum Fourier transform. I give complementary quantum algorithms for class convolution based on direct class-sum block encoding and spectral implementation in the character basis, explaining their applications to symmetry-adapted many-body dynamics, molecular and spectroscopic symmetry analysis, and quantum walks, as well as characterizing the central information used in hidden-subgroup problems, including its limitations for graph isomorphism.

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We construct a primitive gate set for the digital quantum simulation of the binary tetrahedral (

\mathbb{BT}

) group on two quantum architectures. This non-Abelian discrete group serves as a crude approximation to

SU(2)

lattice gauge theory while requiring five qubits or one quicosotetrit per gauge link. The necessary basic primitives are the inversion gate, the group multiplication gate, the trace gate, and the

\mathbb{BT}

Fourier transform over

\mathbb{BT}

. We experimentally benchmark the inversion and trace gates on ibm_nairobi, with estimated fidelities between 14%--55%, depending on the input state.

Primitive Quantum Gates for an

SU(3)

Discrete Subgroup:

Σ(72\times3)

We construct a primitive gate set for the digital quantum simulation of a discrete subgroup of

SU(3)

: the 216-element

Σ(72\times3)

. The necessary primitives are the inversion gate, the group multiplication gate, the trace gate, and the group Fourier transform, for which we provide qubit decompositions. The resulting fault-tolerant T gate costs for a fiducial calculation of shear viscosity would require about

10^{12}

T gates which compares favorably to other modern estimates.

We describe the simulation of dihedral gauge theories on digital quantum computers. The non-Abelian discrete gauge group

{D}_{N}

---the dihedral group---serves as an approximation to $U(1)\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ lattice gauge theory. In order to carry out such a lattice simulation, we detail the construction of efficient quantum circuits to realize basic primitives including the non-Abelian Fourier transform over

{D}_{N}

, the trace operation, and the group multiplication and inversion operations. For each case the required quantum resources scale linearly or as low-degree polynomials in

n=\mathrm{log}N

. We experimentally benchmark our gates on the Rigetti Aspen-9 quantum processor for the case of

{D}_{4}

. The estimated fidelity of all

{D}_{4}

gates was found to exceed 80%.

Z(\mathbb{C}[G])

, the center of the group algebra, where class functions form a commutative convolution algebra and irreducible characters are its spectral basis. The normalized character table of a non-abelian group can be obtained through a restriction operator, which means an efficient quantum character transform for any finite group with an efficient quantum Fourier transform. I give complementary quantum algorithms for class convolution based on direct class-sum block encoding and spectral implementation in the character basis, explaining their applications to symmetry-adapted many-body dynamics, molecular and spectroscopic symmetry analysis, and quantum walks, as well as characterizing the central information used in hidden-subgroup problems, including its limitations for graph isomorphism.