Quantum Simulation of Gauge Theories

A general scheme is presented for simulating gauge theories, with matter fields, on a digital quantum computer. A Trotterized time-evolution operator that respects gauge symmetry is constructed, and a procedure for obtaining time-separated, gauge-invariant correlators is detailed. We demonstrate the procedure on small lattices, including the simulation of a $2+1\mathrm{D}$ non-Abelian gauge theory.A general scheme is presented for simulating gauge theories, with matter fields, on a digital quantum computer. A Trotterized time-evolution operator that respects gauge symmetry is constructed, and a procedure for obtaining time-separated, gauge-invariant correlators is detailed. We demonstrate the procedure on small lattices, including the simulation of a $2+1\mathrm{D}$ non-Abelian gauge theory.

arXiv:1903.08807Physical review. D/Physical review. D.Aug 26, 2019157 citations

SU(2) non-Abelian gauge field theory in one dimension on digital quantum computers

Natalie Klco, Martin J. Savage A, Jesse R. Stryker

A dynamical quantum simulation of SU(2) non-Abelian gauge field theory on a digital quantum computer is presented. This was enabled on current quantum hardware by introducing a mapping of the field onto a register of qubits that utilizes local gauge symmetry while preserving local constraints on the fields, reducing the dimensionality of the calculation. Controlled plaquette operators and gauge-variant completions in the unphysical part of the Hilbert space were designed and used to implement time evolution. The new techniques developed in this work generalize to quantum simulations of higher dimensional gauge field theories.A dynamical quantum simulation of SU(2) non-Abelian gauge field theory on a digital quantum computer is presented. This was enabled on current quantum hardware by introducing a mapping of the field onto a register of qubits that utilizes local gauge symmetry while preserving local constraints on the fields, reducing the dimensionality of the calculation. Controlled plaquette operators and gauge-variant completions in the unphysical part of the Hilbert space were designed and used to implement time evolution. The new techniques developed in this work generalize to quantum simulations of higher dimensional gauge field theories.

arXiv:1908.06935Physical review. D/Physical review. D.Apr 21, 2020248 citations

Digital lattice gauge theories

Erez Zohar, Alessandro Farace, Benni Reznik +1 more

We propose a general scheme for a digital construction of lattice gauge theories with dynamical fermions. In this method, the four-body interactions arising in models with $2+1$ dimensions and higher are obtained stroboscopically, through a sequence of two-body interactions with ancillary degrees of freedom. This yields stronger interactions than the ones obtained through perturbative methods, as typically done in previous proposals, and removes an important bottleneck in the road towards experimental realizations. The scheme applies to generic gauge theories with Lie or finite symmetry groups, both Abelian and non-Abelian. As a concrete example, we present the construction of a digital quantum simulator for a ${\mathbb{Z}}_{3}$ lattice gauge theory with dynamical fermionic matter in $2+1$ dimensions, using ultracold atoms in optical lattices, involving three atomic species, representing the matter, gauge, and auxiliary degrees of freedom, that are separated in three different layers. By moving the ancilla atoms with a proper sequence of steps, we show how we can obtain the desired evolution in a clean, controlled way.We propose a general scheme for a digital construction of lattice gauge theories with dynamical fermions. In this method, the four-body interactions arising in models with $2+1$ dimensions and higher are obtained stroboscopically, through a sequence of two-body interactions with ancillary degrees of freedom. This yields stronger interactions than the ones obtained through perturbative methods, as typically done in previous proposals, and removes an important bottleneck in the road towards experimental realizations. The scheme applies to generic gauge theories with Lie or finite symmetry groups, both Abelian and non-Abelian. As a concrete example, we present the construction of a digital quantum simulator for a ${\mathbb{Z}}_{3}$ lattice gauge theory with dynamical fermionic matter in $2+1$ dimensions, using ultracold atoms in optical lattices, involving three atomic species, representing the matter, gauge, and auxiliary degrees of freedom, that are separated in three different layers. By moving the ancilla atoms with a proper sequence of steps, we show how we can obtain the desired evolution in a clean, controlled way.

arXiv:1607.08121Physical Review AFeb 6, 2017136 citations

Lattice Gauge Theory for a Quantum Computer

Richard C. Brower, David E Berenstein, Hiroki Kawai

The quantum link [1] Hamiltonian was introduced two decades ago a alternative to Wilson's Euclidean lattice QCD with gauge fields represented by bi-linear fermion/anti-fermion operators. When generalized this new microscopic representation of lattice field theories is referred as D-theory [2]. Recast as a Hamiltonian in Minkowski space for real time evolution, D-theory leads naturally to quantum Qubit algorithms. Here to explore digital quantum computing for gauge theories, the simplest example of U(1) compact QED on triangular lattice is defined and gauge invariant kernels for the Suzuki-Trotter expansions are expressed as Qubit circuits capable of being tested on the IMB-Q and other existing Noisy Intermediate Scale Quantum (NISQ) hardware. This is a modest step in exploring the quantum complexity of D-theory to guide future applications to high energy physics and condensed matter quantum field theories.The quantum link [1] Hamiltonian was introduced two decades ago a alternative to Wilson's Euclidean lattice QCD with gauge fields represented by bi-linear fermion/anti-fermion operators. When generalized this new microscopic representation of lattice field theories is referred as D-theory [2]. Recast as a Hamiltonian in Minkowski space for real time evolution, D-theory leads naturally to quantum Qubit algorithms. Here to explore digital quantum computing for gauge theories, the simplest example of U(1) compact QED on triangular lattice is defined and gauge invariant kernels for the Suzuki-Trotter expansions are expressed as Qubit circuits capable of being tested on the IMB-Q and other existing Noisy Intermediate Scale Quantum (NISQ) hardware. This is a modest step in exploring the quantum complexity of D-theory to guide future applications to high energy physics and condensed matter quantum field theories.

arXiv:2002.10028International Symposium on Lattice Field Theory (LATTICE)Feb 2, 202033 citations

Gauge Symmetry in Quantum Simulation

Masanori Hanada, Shunji Matsuura, Andreas Schafer +1 more

Quantum simulation of non-Abelian gauge theories requires careful handling of gauge redundancy. We address this challenge by presenting universal principles for treating gauge symmetry that apply to any quantum simulation approach, clarifying that physical states need not be represented solely by gauge singlets. Both singlet and non-singlet representations are valid, with distinct practical trade-offs, which we elucidate using analogies to BRST quantization. We demonstrate these principles within a complete quantum simulation framework based on the orbifold lattice, which enables explicit and efficient circuit constructions relevant to real-world QCD. For singlet-based approaches, we introduce a Haar-averaging projection implemented via linear combinations of unitaries, and analyze its cost and truncation errors. We also introduce an efficient simulation protocol with an additional term to the Hamiltonian that eliminates non-singlet states from the low-energy spectrum. Beyond the singlet-approach, we show how non-singlet approaches can yield gauge-invariant observables through wave packets and string excitations. This non-singlet approach is proven to be both universal and efficient. Working in temporal gauge, we provide explicit mappings of lattice Yang-Mills dynamics to Pauli-string Hamiltonians suitable for Trotterization. Classical simulations of small systems validate convergence criteria and quantify truncation and Trotter errors, showing concrete resource estimates and scalable circuit recipes for SU $(N)$ gauge theories. Our framework provides both conceptual clarity and practical tools toward quantum advantage in simulating non-Abelian gauge theories.Quantum simulation of non-Abelian gauge theories requires careful handling of gauge redundancy. We address this challenge by presenting universal principles for treating gauge symmetry that apply to any quantum simulation approach, clarifying that physical states need not be represented solely by gauge singlets. Both singlet and non-singlet representations are valid, with distinct practical trade-offs, which we elucidate using analogies to BRST quantization. We demonstrate these principles within a complete quantum simulation framework based on the orbifold lattice, which enables explicit and efficient circuit constructions relevant to real-world QCD. For singlet-based approaches, we introduce a Haar-averaging projection implemented via linear combinations of unitaries, and analyze its cost and truncation errors. We also introduce an efficient simulation protocol with an additional term to the Hamiltonian that eliminates non-singlet states from the low-energy spectrum. Beyond the singlet-approach, we show how non-singlet approaches can yield gauge-invariant observables through wave packets and string excitations. This non-singlet approach is proven to be both universal and efficient. Working in temporal gauge, we provide explicit mappings of lattice Yang-Mills dynamics to Pauli-string Hamiltonians suitable for Trotterization. Classical simulations of small systems validate convergence criteria and quantify truncation and Trotter errors, showing concrete resource estimates and scalable circuit recipes for SU$(N)$ gauge theories. Our framework provides both conceptual clarity and practical tools toward quantum advantage in simulating non-Abelian gauge theories.

quant-pharXiv:2512.22932Dec 30, 20250 citations

General quantum algorithms for Hamiltonian simulation with applications to a non-Abelian lattice gauge theory

Zohreh Davoudi, Alexander F. Shaw, Jesse R. Stryker

With a focus on universal quantum computing for quantum simulation, and through the example of lattice gauge theories, we introduce rather general quantum algorithms that can efficiently simulate certain classes of interactions consisting of correlated changes in multiple (bosonic and fermionic) quantum numbers with non-trivial functional coefficients. In particular, we analyze diagonalization of Hamiltonian terms using a singular-value decomposition technique, and discuss how the achieved diagonal unitaries in the digitized time-evolution operator can be implemented. The lattice gauge theory studied is the SU(2) gauge theory in 1+1 dimensions coupled to one flavor of staggered fermions, for which a complete quantum-resource analysis within different computational models is presented. The algorithms are shown to be applicable to higher-dimensional theories as well as to other Abelian and non-Abelian gauge theories. The example chosen further demonstrates the importance of adopting efficient theoretical formulations: it is shown that an explicitly gauge-invariant formulation using loop, string, and hadron degrees of freedom simplifies the algorithms and lowers the cost compared with the standard formulations based on angular-momentum as well as the Schwinger-boson degrees of freedom. The loop-string-hadron formulation further retains the non-Abelian gauge symmetry despite the inexactness of the digitized simulation, without the need for costly controlled operations. Such theoretical and algorithmic considerations are likely to be essential in quantumly simulating other complex theories of relevance to nature.With a focus on universal quantum computing for quantum simulation, and through the example of lattice gauge theories, we introduce rather general quantum algorithms that can efficiently simulate certain classes of interactions consisting of correlated changes in multiple (bosonic and fermionic) quantum numbers with non-trivial functional coefficients. In particular, we analyze diagonalization of Hamiltonian terms using a singular-value decomposition technique, and discuss how the achieved diagonal unitaries in the digitized time-evolution operator can be implemented. The lattice gauge theory studied is the SU(2) gauge theory in 1+1 dimensions coupled to one flavor of staggered fermions, for which a complete quantum-resource analysis within different computational models is presented. The algorithms are shown to be applicable to higher-dimensional theories as well as to other Abelian and non-Abelian gauge theories. The example chosen further demonstrates the importance of adopting efficient theoretical formulations: it is shown that an explicitly gauge-invariant formulation using loop, string, and hadron degrees of freedom simplifies the algorithms and lowers the cost compared with the standard formulations based on angular-momentum as well as the Schwinger-boson degrees of freedom. The loop-string-hadron formulation further retains the non-Abelian gauge symmetry despite the inexactness of the digitized simulation, without the need for costly controlled operations. Such theoretical and algorithmic considerations are likely to be essential in quantumly simulating other complex theories of relevance to nature.

arXiv:2212.14030QuantumDec 20, 202352 citations

2+1\mathrm{D}

non-Abelian gauge theory.

We propose a general scheme for a digital construction of lattice gauge theories with dynamical fermions. In this method, the four-body interactions arising in models with

2+1

dimensions and higher are obtained stroboscopically, through a sequence of two-body interactions with ancillary degrees of freedom. This yields stronger interactions than the ones obtained through perturbative methods, as typically done in previous proposals, and removes an important bottleneck in the road towards experimental realizations. The scheme applies to generic gauge theories with Lie or finite symmetry groups, both Abelian and non-Abelian. As a concrete example, we present the construction of a digital quantum simulator for a

{\mathbb{Z}}_{3}

lattice gauge theory with dynamical fermionic matter in

2+1

dimensions, using ultracold atoms in optical lattices, involving three atomic species, representing the matter, gauge, and auxiliary degrees of freedom, that are separated in three different layers. By moving the ancilla atoms with a proper sequence of steps, we show how we can obtain the desired evolution in a clean, controlled way.

(N)

gauge theories. Our framework provides both conceptual clarity and practical tools toward quantum advantage in simulating non-Abelian gauge theories.