Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2603.10932 · Physical Review Research

Efficient construction of $\mathbb{Z}_2$ gauge-invariant bases for the Quantum Minimally Entangled Typical Thermal States algorithm

Reita Maeno

quant-phhep-latSubmitted Mar 11, 2026• Announced Mar 12, 2026• 0 citations
v1 · Mar 11, 2026v2 · May 18, 2026v3 · Oct 6, 2026
arXivHTMLPDF

Abstract

In quantum computations of gauge theories at finite temperature and finite density, it is challenging to enforce Gauss's law for all states contributing to the thermal ensemble. While various techniques for implementing gauge constraints have been proposed, they often involve practical trade-offs. In this work, we adopt the Quantum Minimally Entangled Typical Thermal States (QMETTS) algorithm for $\mathbb{Z}_2$ gauge-constrained systems, which allows us to capture thermal equilibrium states with chemical potential while mitigating these trade-offs. To ensure that gauge invariance is preserved throughout the procedure while maintaining computational efficiency, we derive the specific measurement bases within the algorithm. Furthermore, since the estimation of expectation values on quantum hardware is inherently noisy, we rigorously account for shot noise in estimating expectation values, and propose a sampling method that is more efficient than those in previous works. We validate our approach numerically by studying a (1+1)-dimensional $\mathbb{Z}_2$ gauge theory coupled to staggered fermions. Our proposed algorithm reproduces the correct equilibrium states at finite temperature and finite density.

Journal reference
Phys. Rev. Research 8, 033306 (2026)
DOI
10.1103/qyj2-97tw
Comments
22 pages, 12 figures

Related papers

Minimally Entangled Typical Thermal States for Classical and Quantum Simulation of 1+1-Dimensional $\mathbb Z_2$ Lattice Gauge Theory at Finite Temperature and Density

Chen, I-Chi, João C. Getelina, Klée Pollock +4 more

Simulating strongly coupled gauge theories at finite temperature and density is a longstanding challenge in nuclear and high-energy physics that also has fundamental implications for condensed matter physics. In this work, we use minimally entangled typical thermal state (METTS) approaches to facilitate both classical and quantum computational studies of such systems. METTS techniques combine classical random sampling with imaginary time evolution, which can be performed on either a classical or a quantum computer, to estimate thermal averages of observables. We study 1+1-dimensional $\mathbb{Z}_2$ gauge theory coupled to spinless fermionic matter, which maps onto a local quantum spin chain. We benchmark both a classical matrix-product-state implementation of METTS and a recently proposed adaptive variational approach that is a promising candidate for implementation on near-term quantum devices, focusing on the equation of state as well as on various measures of fermion confinement. Of particular importance is the choice of basis for obtaining new METTS samples, which impacts both the classical sampling complexity (a key factor in both classical and quantum simulation applications) and complexity of circuits used in the quantum computing approach. Our work sets the stage for future studies of strongly coupled gauge theories with both classical and quantum hardware.

arXiv:2407.11949•Jul 16, 2024•0 citations

Gauss law codes and vacuum codes from lattice gauge theories

Javier P. Lacambra, Aidan Chatwin-Davies, Masazumi Honda +1 more

We develop a comprehensive framework for constructing quantum error correcting codes (QECCs) from Abelian lattice gauge theories (LGTs) using quantum reference frames (QRFs) as a unifying formalism. We consider LGTs with arbitrary compact Abelian gauge groups supported on lattices in arbitrary numbers of spatial dimensions, and we work with both pure gauge theories and theories with couplings to bosonic and fermionic matter. The codes that we construct fall into two classes: First, Gauss law codes identify the code subspace with the full gauge-invariant sector of the theory. In models with matter coupled to gauge fields, these codes inherit a natural subsystem structure in which gauge-invariant Wilson loops and dressed matter excitations factorize the code space. Second, vacuum codes restrict the code subspace to the matter vacuum sector within the gauge-invariant subspace, yielding codes where errors correspond to gauge-invariant charge excitations rather than to violations of the Gauss law. Despite their distinct setup, we show that when the gauge group is finite, vacuum codes are unitarily equivalent to pure gauge theory Gauss law codes, and that when the group is continuous, this is only true upon a charge coarse-graining of the vacuum code. In all cases, QRFs provide a systematic apparatus for fully characterizing the codes' algebraic structures and correctable error sets. For clarity, we illustrate our general results in $\mathbb{Z}_2$-gauge theory, as well as in scalar and fermionic QED. These findings offer fundamental insights into the parallelism between quantum error correction and gauge theory and point toward practical advantages for simulating LGTs on noisy quantum devices.

quant-pharXiv:2604.06087•Apr 8, 2026•0 citations

Classical and quantum simulations of 1+1-dimensional $${{\mathbb{Z}}}_{2}$$ gauge theory at finite temperature and density

I-Chi Chen, João C. Getelina, Klée Pollock +4 more

Abstract Simulating strongly coupled gauge theories at finite temperature and density is a longstanding challenge in nuclear and high-energy physics with fundamental implications for condensed matter physics. Here, we simulate such systems using minimally entangled typical thermal state (METTS) approaches, which combine classical random sampling with imaginary-time evolution, implementable on either classical or quantum computers, to estimate thermal averages of observables. We study 1+1-dimensional $${{\mathbb{Z}}}_{2}$$ Z 2 gauge theory coupled to spinless fermionic matter, which maps onto a local quantum spin chain. We benchmark both a classical matrix-product-state implementation of METTS and a recently proposed adaptive variational approach for near-term quantum devices, focusing on the equation of state and measures of fermion confinement. Of particular importance is the choice of basis for METTS sampling, which impacts both the sampling overhead and quantum circuit complexity. Our work sets the stage for future studies of strongly coupled gauge theories using classical and quantum hardware.

Communications Physics•Sep 26, 2025•4 citations

Classical shadows for sample-efficient measurements of gauge-invariant observables

Jacob Bringewatt, Henry Froland, Andreas Elben +1 more

Classical shadows provide a versatile framework for estimating many properties of quantum states from repeated, randomly chosen measurements without requiring full quantum state tomography. When prior information is available, such as knowledge of symmetries of states and operators, this knowledge can be exploited to significantly improve sample efficiency. In this work, we develop three classical shadow protocols for $\mathbb{Z}_2$ lattice gauge theory, where a dual formulation enables a rigorous analysis of resource requirements, including both circuit depth and sample complexity. Our approaches can offer exponential improvements in sample complexity over symmetry-agnostic methods, albeit at the cost of increased circuit complexity. While our analysis is restricted to $\mathbb{Z}_2$ lattice gauge theory, our approach offers a blueprint for similar protocols for more general lattice gauge theory models which are currently at the forefront of quantum simulation efforts.

quant-pharXiv:2511.02904•Quantum•Nov 6, 2025•0 citations

Toward Exploring Phase Diagrams of Gauge Theories on Quantum Computers with Thermal Pure Quantum States

Connor Powers, Zohreh Davoudi, Niklas Mueller

Aiming at evading the notorious sign problem in classical Monte-Carlo approaches to lattice quantum chromodynamics, we present an approach for quantum computing finite-temperature lattice gauge theories at non-zero density. Based on the thermal pure-quantum-state formalism of statistical mechanics when extended to gauge-theory systems, our approach allows for sign-problem-free quantum computations of thermal expectation values and non-equal time correlation functions. By taking a simple lattice gauge theory for which classical benchmarks are possible, namely $\mathbb{Z}_2$ lattice gauge theory in 1+1 dimensions at finite chemical potential, we discuss resource requirements and robustness to algorithmic and hardware imperfections for near-term quantum-hardware realizations.

arXiv:2212.11388•International Symposium on Lattice Field Theory (LATTICE)•Jan 29, 2023•0 citations

Arbitrary-Distance Quantum Error Correction with Gauss's Law for $\mathbb Z_2$ Lattice Gauge Theory

Neel S. Modi, Lento Nagano, Masazumi Honda +2 more

It has previously been shown by Rajput, Roggero, and Wiebe that $\mathbb Z_2$ Gauss's law constraints can be used to build efficient quantum error-correcting codes (QECCs) that are robust against arbitrary single-qubit errors. In this work, we generalize the construction to be robust against arbitrary $t$-qubit errors, where $t$ is any positive integer. This includes a derivation of the optimal Gauss's law code within the considered family by minimizing the number of physical qubits required for a given code distance. Finally, we compare our codes against other efficient QECCs on metrics such as the number of physical qubits, the locality of the encoded Hamiltonian, and the logical error rate in the code capacity setting. Compared to using a domain-agnostic code for every lattice degree of freedom, we find that the Gauss's law code primarily excels at reducing the locality of the encoded Hamiltonian. Moreover, the physical qubit overhead is also reduced for $t \le 3$ (distance $d \le 7$).

hep-latCross-listedarXiv:2608.00165•Aug 4, 2026•0 citations
Efficient construction of Z2\mathbb{Z}_2Z2​ gauge-invariant bases for the Quantum Minimally Entangled Typical Thermal States algorithm
In quantum computations of gauge theories at finite temperature and finite density, it is challenging to enforce Gauss's law for all states contributing to the thermal ensemble. While various techniques for implementing gauge constraints have been proposed, they often involve practical trade-offs. In this work, we adopt the Quantum Minimally Entangled Typical Thermal States (QMETTS) algorithm for Z2\mathbb{Z}_2Z2​ gauge-constrained systems, which allows us to capture thermal equilibrium states with chemical potential while mitigating these trade-offs. To ensure that gauge invariance is preserved throughout the procedure while maintaining computational efficiency, we derive the specific measurement bases within the algorithm. Furthermore, since the estimation of expectation values on quantum hardware is inherently noisy, we rigorously account for shot noise in estimating expectation values, and propose a sampling method that is more efficient than those in previous works. We validate our approach numerically by studying a (1+1)-dimensional Z2\mathbb{Z}_2Z2​ gauge theory coupled to staggered fermions. Our proposed algorithm reproduces the correct equilibrium states at finite temperature and finite density.
Minimally Entangled Typical Thermal States for Classical and Quantum Simulation of 1+1-Dimensional Z2\mathbb Z_2Z2​ Lattice Gauge Theory at Finite Temperature and Density
Simulating strongly coupled gauge theories at finite temperature and density is a longstanding challenge in nuclear and high-energy physics that also has fundamental implications for condensed matter physics. In this work, we use minimally entangled typical thermal state (METTS) approaches to facilitate both classical and quantum computational studies of such systems. METTS techniques combine classical random sampling with imaginary time evolution, which can be performed on either a classical or a quantum computer, to estimate thermal averages of observables. We study 1+1-dimensional Z2\mathbb{Z}_2Z2​ gauge theory coupled to spinless fermionic matter, which maps onto a local quantum spin chain. We benchmark both a classical matrix-product-state implementation of METTS and a recently proposed adaptive variational approach that is a promising candidate for implementation on near-term quantum devices, focusing on the equation of state as well as on various measures of fermion confinement. Of particular importance is the choice of basis for obtaining new METTS samples, which impacts both the classical sampling complexity (a key factor in both classical and quantum simulation applications) and complexity of circuits used in the quantum computing approach. Our work sets the stage for future studies of strongly coupled gauge theories with both classical and quantum hardware.
We develop a comprehensive framework for constructing quantum error correcting codes (QECCs) from Abelian lattice gauge theories (LGTs) using quantum reference frames (QRFs) as a unifying formalism. We consider LGTs with arbitrary compact Abelian gauge groups supported on lattices in arbitrary numbers of spatial dimensions, and we work with both pure gauge theories and theories with couplings to bosonic and fermionic matter. The codes that we construct fall into two classes: First, Gauss law codes identify the code subspace with the full gauge-invariant sector of the theory. In models with matter coupled to gauge fields, these codes inherit a natural subsystem structure in which gauge-invariant Wilson loops and dressed matter excitations factorize the code space. Second, vacuum codes restrict the code subspace to the matter vacuum sector within the gauge-invariant subspace, yielding codes where errors correspond to gauge-invariant charge excitations rather than to violations of the Gauss law. Despite their distinct setup, we show that when the gauge group is finite, vacuum codes are unitarily equivalent to pure gauge theory Gauss law codes, and that when the group is continuous, this is only true upon a charge coarse-graining of the vacuum code. In all cases, QRFs provide a systematic apparatus for fully characterizing the codes' algebraic structures and correctable error sets. For clarity, we illustrate our general results in Z2\mathbb{Z}_2Z2​-gauge theory, as well as in scalar and fermionic QED. These findings offer fundamental insights into the parallelism between quantum error correction and gauge theory and point toward practical advantages for simulating LGTs on noisy quantum devices.
Classical and quantum simulations of 1+1-dimensional Z2{{\mathbb{Z}}}_{2}Z2​ gauge theory at finite temperature and density
Abstract Simulating strongly coupled gauge theories at finite temperature and density is a longstanding challenge in nuclear and high-energy physics with fundamental implications for condensed matter physics. Here, we simulate such systems using minimally entangled typical thermal state (METTS) approaches, which combine classical random sampling with imaginary-time evolution, implementable on either classical or quantum computers, to estimate thermal averages of observables. We study 1+1-dimensional Z2{{\mathbb{Z}}}_{2}Z2​ Z 2 gauge theory coupled to spinless fermionic matter, which maps onto a local quantum spin chain. We benchmark both a classical matrix-product-state implementation of METTS and a recently proposed adaptive variational approach for near-term quantum devices, focusing on the equation of state and measures of fermion confinement. Of particular importance is the choice of basis for METTS sampling, which impacts both the sampling overhead and quantum circuit complexity. Our work sets the stage for future studies of strongly coupled gauge theories using classical and quantum hardware.
Classical shadows provide a versatile framework for estimating many properties of quantum states from repeated, randomly chosen measurements without requiring full quantum state tomography. When prior information is available, such as knowledge of symmetries of states and operators, this knowledge can be exploited to significantly improve sample efficiency. In this work, we develop three classical shadow protocols for Z2\mathbb{Z}_2Z2​ lattice gauge theory, where a dual formulation enables a rigorous analysis of resource requirements, including both circuit depth and sample complexity. Our approaches can offer exponential improvements in sample complexity over symmetry-agnostic methods, albeit at the cost of increased circuit complexity. While our analysis is restricted to Z2\mathbb{Z}_2Z2​ lattice gauge theory, our approach offers a blueprint for similar protocols for more general lattice gauge theory models which are currently at the forefront of quantum simulation efforts.
Aiming at evading the notorious sign problem in classical Monte-Carlo approaches to lattice quantum chromodynamics, we present an approach for quantum computing finite-temperature lattice gauge theories at non-zero density. Based on the thermal pure-quantum-state formalism of statistical mechanics when extended to gauge-theory systems, our approach allows for sign-problem-free quantum computations of thermal expectation values and non-equal time correlation functions. By taking a simple lattice gauge theory for which classical benchmarks are possible, namely Z2\mathbb{Z}_2Z2​ lattice gauge theory in 1+1 dimensions at finite chemical potential, we discuss resource requirements and robustness to algorithmic and hardware imperfections for near-term quantum-hardware realizations.
Arbitrary-Distance Quantum Error Correction with Gauss's Law for Z2\mathbb Z_2Z2​ Lattice Gauge Theory
It has previously been shown by Rajput, Roggero, and Wiebe that Z2\mathbb Z_2Z2​ Gauss's law constraints can be used to build efficient quantum error-correcting codes (QECCs) that are robust against arbitrary single-qubit errors. In this work, we generalize the construction to be robust against arbitrary ttt-qubit errors, where ttt is any positive integer. This includes a derivation of the optimal Gauss's law code within the considered family by minimizing the number of physical qubits required for a given code distance. Finally, we compare our codes against other efficient QECCs on metrics such as the number of physical qubits, the locality of the encoded Hamiltonian, and the logical error rate in the code capacity setting. Compared to using a domain-agnostic code for every lattice degree of freedom, we find that the Gauss's law code primarily excels at reducing the locality of the encoded Hamiltonian. Moreover, the physical qubit overhead is also reduced for t≤3t \le 3t≤3 (distance d≤7d \le 7d≤7).