arXiv:2208.12309 · Physical review. D/Physical review. D.

Primitive quantum gates for an SU(2) discrete subgroup: Binary tetrahedral

Erik Gustafson, Henry Lamm, Felicity Lovelace, Damian Musk

Published Dec 8, 2022• 54 citations

Abstract

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.

DOI: 10.1103/physrevd.106.114501

Related papers

Primitive quantum gates for an S U ( 2 ) discrete subgroup: Binary octahedral

Erik Gustafson, Henry Lamm, Felicity Lovelace

arXiv:2312.10285Physical review. D/Physical review. D.Mar 13, 202425 citations

Primitive quantum gates for dihedral gauge theories

M. Sohaib Alam, Stuart Hadfield, Henry Lamm +1 more

arXiv:2108.13305Physical review. D/Physical review. D.Jun 2, 202249 citations

Primitive Quantum Gates for an $SU(3)$ Discrete Subgroup: $Σ(72\times3)$

Sebastian Osorio Perez, Edison M. Murairi, Erik J. Gustafson +1 more

hep-latCross-listedarXiv:2511.17437Nov 24, 20250 citations

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

Erik Gustafson, Yao Ji, Henry Lamm +3 more

arXiv:2405.05973Physical review. D/Physical review. D.Aug 23, 202425 citations

Elementary Quantum Gates from Lie Group Embeddings in $U(2^n)$: Geometry, Universality, and Discretization

Antonio Falco, Daniela Falco-Pomares, Hermann G. Matthies

quant-pharXiv:2601.17936Jan 27, 20260 citations

Quantum Simulation of Dihedral Gauge Theories

M. Sohaib Alam, Stuart Hadfield, Henry Lamm +1 more

Aug 30, 20211 citation

arXiv:2208.12309 · Physical review. D/Physical review. D.

Primitive quantum gates for an SU(2) discrete subgroup: Binary tetrahedral

Erik Gustafson, Henry Lamm, Felicity Lovelace, Damian Musk

Published Dec 8, 2022• 54 citations

arXiv HTML PDF

Abstract

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.

DOI: 10.1103/physrevd.106.114501

Related papers

Primitive quantum gates for an S U ( 2 ) discrete subgroup: Binary octahedral

Erik Gustafson, Henry Lamm, Felicity Lovelace

We construct a primitive gate set for the digital quantum simulation of the 48-element binary octahedral ( BO ) group. This non-Abelian discrete group better approximates S U ( 2 ) lattice gauge theory than previous work on the binary tetrahedral group at the cost of one additional qubit—for a total of six—per gauge link. The necessary primitives are the inversion gate, the group multiplication gate, the trace gate, and the BO Fourier transform. Published by the American Physical Society 2024We construct a primitive gate set for the digital quantum simulation of the 48-element binary octahedral ( BO ) group. This non-Abelian discrete group better approximates S U ( 2 ) lattice gauge theory than previous work on the binary tetrahedral group at the cost of one additional qubit—for a total of six—per gauge link. The necessary primitives are the inversion gate, the group multiplication gate, the trace gate, and the BO Fourier transform. Published by the American Physical Society 2024

arXiv:2312.10285Physical review. D/Physical review. D.Mar 13, 202425 citations

Primitive quantum gates for dihedral gauge theories

M. Sohaib Alam, Stuart Hadfield, Henry Lamm +1 more

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%.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%.

arXiv:2108.13305Physical review. D/Physical review. D.Jun 2, 202249 citations

Primitive Quantum Gates for an $SU(3)$ Discrete Subgroup: $Σ(72\times3)$ Primitive Quantum Gates for an $SU(3)$ Discrete Subgroup: $Σ(72\times3)$

Sebastian Osorio Perez, Edison M. Murairi, Erik J. Gustafson +1 more

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 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.

hep-latCross-listedarXiv:2511.17437Nov 24, 20250 citations

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

Erik Gustafson, Yao Ji, Henry Lamm +3 more

We construct the primitive gate set for the digital quantum simulation of the 108-element Σ ( 36 × 3 ) group. This is the first time a non-Abelian crystal-like subgroup of S U ( 3 ) has been constructed for quantum simulation. The gauge link registers and necessary primitives—the inversion gate, the group multiplication gate, the trace gate, and the Σ ( 36 × 3 ) Fourier transform—are presented for both an eight-qubit encoding and a heterogeneous three-qutrit plus two-qubit register. For the latter, a specialized compiler was developed for decomposing arbitrary unitaries onto this architecture. Published by the American Physical Society 2024We construct the primitive gate set for the digital quantum simulation of the 108-element Σ ( 36 × 3 ) group. This is the first time a non-Abelian crystal-like subgroup of S U ( 3 ) has been constructed for quantum simulation. The gauge link registers and necessary primitives—the inversion gate, the group multiplication gate, the trace gate, and the Σ ( 36 × 3 ) Fourier transform—are presented for both an eight-qubit encoding and a heterogeneous three-qutrit plus two-qubit register. For the latter, a specialized compiler was developed for decomposing arbitrary unitaries onto this architecture. Published by the American Physical Society 2024

arXiv:2405.05973Physical review. D/Physical review. D.Aug 23, 202425 citations

Elementary Quantum Gates from Lie Group Embeddings in $U(2^n)$ : Geometry, Universality, and DiscretizationElementary Quantum Gates from Lie Group Embeddings in $U(2^n)$: Geometry, Universality, and Discretization

Antonio Falco, Daniela Falco-Pomares, Hermann G. Matthies

In the standard circuit model, elementary gates are defined relative to a chosen tensor factorization and are therefore extrinsic to the ambient group $U(2^n)$ . Writing $N=2^n$ , we introduce an \emph{intrinsic descriptor layer} in $U(N)$ by declaring as primitive the motions inside faithful embedded copies of $SU(2)$ (phase-free), together with a phase-inclusive $U(2)$ variant. We describe the embedding landscape $\Emb(SU(2),U(N))$ as a finite union of $U(N)$ -homogeneous strata indexed by isotypic multiplicities, with stabilizers given by centralizers, and we isolate a canonical \emph{two-level sector} parameterized by $\Gr_2(\C^N)$ up to a $PSU(2)$ gauge. Equipping $U(N)$ with the Hilbert--Schmidt bi-invariant metric, each embedded subgroup is totally geodesic, yielding a variational characterization of elementary motions via minimal-norm logarithms. On the constructive side, we prove phase-free universality in $SU(N)$ from two-level primitives using QR/Givens factorizations together with explicit diagonal generation, and we obtain full universality in $U(N)$ by explicit abelian phase bookkeeping (equivalently, via the $U(2)$ two-level dictionary). Finally, we formalize a modular finite-alphabet compilation interface: any approximation routine in $SU(2)$ (e.g.\ Solovay--Kitaev) can be lifted through two-level embeddings to yield $U(N)$ -level synthesis with global operator-norm error control.In the standard circuit model, elementary gates are defined relative to a chosen tensor factorization and are therefore extrinsic to the ambient group $U(2^n)$. Writing $N=2^n$, we introduce an \emph{intrinsic descriptor layer} in $U(N)$ by declaring as primitive the motions inside faithful embedded copies of $SU(2)$ (phase-free), together with a phase-inclusive $U(2)$ variant. We describe the embedding landscape $\Emb(SU(2),U(N))$ as a finite union of $U(N)$-homogeneous strata indexed by isotypic multiplicities, with stabilizers given by centralizers, and we isolate a canonical \emph{two-level sector} parameterized by $\Gr_2(\C^N)$ up to a $PSU(2)$ gauge. Equipping $U(N)$ with the Hilbert--Schmidt bi-invariant metric, each embedded subgroup is totally geodesic, yielding a variational characterization of elementary motions via minimal-norm logarithms. On the constructive side, we prove phase-free universality in $SU(N)$ from two-level primitives using QR/Givens factorizations together with explicit diagonal generation, and we obtain full universality in $U(N)$ by explicit abelian phase bookkeeping (equivalently, via the $U(2)$ two-level dictionary). Finally, we formalize a modular finite-alphabet compilation interface: any approximation routine in $SU(2)$ (e.g.\ Solovay--Kitaev) can be lifted through two-level embeddings to yield $U(N)$-level synthesis with global operator-norm error control.

quant-pharXiv:2601.17936Jan 27, 20260 citations

Quantum Simulation of Dihedral Gauge Theories

M. Sohaib Alam, Stuart Hadfield, Henry Lamm +1 more

We describe the simulation of dihedral gauge theories on digital quantum computers. The nonabelian discrete gauge group $D_N$ -- the dihedral group -- serves as an approximation to $U(1)\times\mathbb{Z}_2$ 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 nonabelian 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 $n=\log N$ . We experimentally benchmark our gates on the Rigetti Aspen-9 quantum processor for the case of $D_4$ . The fidelity of all $D_4$ gates was found to exceed $80\%$ .We describe the simulation of dihedral gauge theories on digital quantum computers. The nonabelian discrete gauge group $D_N$ -- the dihedral group -- serves as an approximation to $U(1)\times\mathbb{Z}_2$ 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 nonabelian 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 $n=\log N$. We experimentally benchmark our gates on the Rigetti Aspen-9 quantum processor for the case of $D_4$. The fidelity of all $D_4$ gates was found to exceed $80\%$.

Aug 30, 20211 citation

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.

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%.

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.

Elementary Quantum Gates from Lie Group Embeddings in

U(2^n)

: Geometry, Universality, and Discretization

In the standard circuit model, elementary gates are defined relative to a chosen tensor factorization and are therefore extrinsic to the ambient group

U(2^n)

. Writing

N=2^n

, we introduce an \emph{intrinsic descriptor layer} in

U(N)

by declaring as primitive the motions inside faithful embedded copies of

SU(2)

(phase-free), together with a phase-inclusive

U(2)

variant. We describe the embedding landscape $\Emb(SU(2),U(N))$ as a finite union of

U(N)

-homogeneous strata indexed by isotypic multiplicities, with stabilizers given by centralizers, and we isolate a canonical \emph{two-level sector} parameterized by $\Gr_2(\C^N)$ up to a

PSU(2)

gauge. Equipping

U(N)

with the Hilbert--Schmidt bi-invariant metric, each embedded subgroup is totally geodesic, yielding a variational characterization of elementary motions via minimal-norm logarithms. On the constructive side, we prove phase-free universality in

SU(N)

from two-level primitives using QR/Givens factorizations together with explicit diagonal generation, and we obtain full universality in

U(N)

by explicit abelian phase bookkeeping (equivalently, via the

U(2)

two-level dictionary). Finally, we formalize a modular finite-alphabet compilation interface: any approximation routine in

SU(2)

(e.g.\ Solovay--Kitaev) can be lifted through two-level embeddings to yield

U(N)

-level synthesis with global operator-norm error control.

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

D_N

-- the dihedral group -- serves as an approximation to

U(1)\times\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 nonabelian 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=\log N

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

D_4

. The fidelity of all

D_4

gates was found to exceed

80\%