arXiv:2108.13305 · Physical review. D/Physical review. D.

Primitive quantum gates for dihedral gauge theories

M. Sohaib Alam, Stuart Hadfield, Henry Lamm, Andy C. Y. Li

Published Jun 2, 2022• 49 citations

Abstract

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%.

DOI: 10.1103/physrevd.105.114501

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arXiv:2108.13305 · Physical review. D/Physical review. D.

Primitive quantum gates for dihedral gauge theories

M. Sohaib Alam, Stuart Hadfield, Henry Lamm, Andy C. Y. Li

Published Jun 2, 2022• 49 citations

arXiv HTML PDF

Abstract

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%.

DOI: 10.1103/physrevd.105.114501

Related papers

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

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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 )

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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

Dihedral non-Abelian Lattice Gauge Theories: Physics and Quantum Simulation

Edoardo Ballini

Lattice gauge theories provide a first-principles, non-perturbative formulation of the fundamental interactions of the Standard Model, and their Hamiltonian reformulation on a quantum simulator offers a promising route to regimes---real-time dynamics, finite baryon density---that lie beyond the reach of Euclidean Monte Carlo methods. With the ultimate goal of overcoming these shortcomings, over the last decade, quantum simulations of Abelian lattice gauge theories have expanded from the first proofs of principle to the experimental study of confinement and string breaking by pair production. One of the next frontiers lies in the simulation of non-Abelian theories, which are significantly more demanding both in local Hilbert space dimension and in symmetry-protection requirements, and ultimately essential for addressing open problems in high-energy physics. 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We find that both protocols significantly improve the reconstructed dynamics, with DPS tracking the exact evolution over longer times. Taken together, the results of this thesis position dihedral gauge theories as a concrete stepping stone from Abelian proofs of principle toward genuinely non-Abelian quantum simulation on near-term devices.Lattice gauge theories provide a first-principles, non-perturbative formulation of the fundamental interactions of the Standard Model, and their Hamiltonian reformulation on a quantum simulator offers a promising route to regimes---real-time dynamics, finite baryon density---that lie beyond the reach of Euclidean Monte Carlo methods. With the ultimate goal of overcoming these shortcomings, over the last decade, quantum simulations of Abelian lattice gauge theories have expanded from the first proofs of principle to the experimental study of confinement and string breaking by pair production. One of the next frontiers lies in the simulation of non-Abelian theories, which are significantly more demanding both in local Hilbert space dimension and in symmetry-protection requirements, and ultimately essential for addressing open problems in high-energy physics. In this thesis, we investigate dihedral non-Abelian lattice gauge theories $D_N$---whose small order and finite local dimension make $D_3$ and $D_4$ natural targets for near-term quantum platforms---as a minimal yet physically rich arena for the study of non-Abelian gauge dynamics. After introducing dihedral groups and the associated lattice gauge theories, we propose a gauge-fixed ladder model that reduces the effective dimensionality while preserving the essential physics. We use it to show that the $D_3$ theory on a ladder displays a smooth crossover---rather than a sharp phase transition---between electric- and magnetic-dominated regimes, with the nonstabilizerness peaking in the crossover region. Furthermore, we introduce a formulation of the model compatible with the analysis of the Eigenstate Thermalization Hypothesis. We then study string breaking in pure-gauge $D_N$ ladders in the presence of static charges through tensor-network DMRG simulations in the rishon formalism. We show that for odd $N$ static charges are screened by gluelumps---charge-glueball bound states enabled by the trivial centre of the group---while for even $N$ a fundamental flux string persists. Turning to the quantum-simulation side, we introduce a dynamical post-selection (DPS) protocol in which the local gauge charges are tested by mid-circuit measurements on auxiliary registers, and we show that gauge protection emerges through a sharp quantum Zeno transition in the spectrum of an associated Liouvillian. We verify the generality of this picture on $\mathbb{Z}_2$, $\mathbb{Z}_3$, and truncated $U(1)$ gauge groups, identifying an optimal window of measurement frequencies that survives also in the presence of noisy measurements. Finally, we benchmark DPS against an alternative post-processed symmetry verification scheme on a two-plaquette $D_3$ model designed for state-of-the-art qudit hardware. We find that both protocols significantly improve the reconstructed dynamics, with DPS tracking the exact evolution over longer times. Taken together, the results of this thesis position dihedral gauge theories as a concrete stepping stone from Abelian proofs of principle toward genuinely non-Abelian quantum simulation on near-term devices.

Jun 18, 20260 citations

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}

{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 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 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.

D_N

---whose small order and finite local dimension make

D_3

and

D_4

natural targets for near-term quantum platforms---as a minimal yet physically rich arena for the study of non-Abelian gauge dynamics. After introducing dihedral groups and the associated lattice gauge theories, we propose a gauge-fixed ladder model that reduces the effective dimensionality while preserving the essential physics. We use it to show that the

D_3

theory on a ladder displays a smooth crossover---rather than a sharp phase transition---between electric- and magnetic-dominated regimes, with the nonstabilizerness peaking in the crossover region. Furthermore, we introduce a formulation of the model compatible with the analysis of the Eigenstate Thermalization Hypothesis. We then study string breaking in pure-gauge

D_N

ladders in the presence of static charges through tensor-network DMRG simulations in the rishon formalism. We show that for odd

N

static charges are screened by gluelumps---charge-glueball bound states enabled by the trivial centre of the group---while for even

N

a fundamental flux string persists. Turning to the quantum-simulation side, we introduce a dynamical post-selection (DPS) protocol in which the local gauge charges are tested by mid-circuit measurements on auxiliary registers, and we show that gauge protection emerges through a sharp quantum Zeno transition in the spectrum of an associated Liouvillian. We verify the generality of this picture on

\mathbb{Z}_2

\mathbb{Z}_3

, and truncated

U(1)

gauge groups, identifying an optimal window of measurement frequencies that survives also in the presence of noisy measurements. Finally, we benchmark DPS against an alternative post-processed symmetry verification scheme on a two-plaquette

D_3

model designed for state-of-the-art qudit hardware. We find that both protocols significantly improve the reconstructed dynamics, with DPS tracking the exact evolution over longer times. Taken together, the results of this thesis position dihedral gauge theories as a concrete stepping stone from Abelian proofs of principle toward genuinely non-Abelian quantum simulation on near-term devices.