Simulating Heat with Quantum Particles Unlocks New Materials Science Possibilities

Scientists are developing new methods to simulate the behaviour of thermal states, crucial for understanding complex quantum systems. Manuel S. Rudolph, Armando Angrisani, and Andrew Wright, alongside Iwo Sanderski, Ricard Puig, Zoë Holmes et al. from the Ecole Polytechnique Fédérale de Lausanne and Algorithmiq Ltd, present a propagation-based approach utilising Pauli and Majorana operators to model imaginary-time evolution. This research is significant because it efficiently represents high-temperature states, which are often sparse and difficult to simulate with conventional techniques, offering analytic guarantees for error control and demonstrating effectiveness through large-scale numerical simulations on established models.

Simulating Finite Temperature Quantum Systems via Pauli and Majorana Operator Propagation offers a promising avenue for exploring complex quantum phenomena

Scientists have developed a novel approach to simulating thermal states using Pauli and Majorana propagation techniques adapted for imaginary-time evolution. This work addresses a critical challenge in material science, condensed matter physics, and quantum chemistry: accurately modelling quantum systems at finite temperatures.
The research centres on the observation that high-temperature states exhibit sparsity in Pauli or Majorana bases, simplifying their representation and enabling efficient computation. By formulating imaginary-time evolution directly within these operator bases and initiating the process from a maximally mixed state, researchers have unlocked access to a range of temperatures where the quantum state remains efficiently manageable in terms of computational resources.

The study introduces a propagation-based method that begins with the maximally mixed state, represented by the identity operator, and evolves it using a sequence of imaginary-time gates. This allows for the efficient storage and manipulation of high-temperature states, as the complexity increases with decreasing temperature.
Analytic guarantees are provided for both small-coefficient truncation and Pauli-weight (Majorana-length) truncation strategies, quantifying error growth and the impact of backflow to ensure efficient operation at elevated temperatures. These strategies allow for controlled approximation of the thermal state, with demonstrable reductions in error as truncation thresholds are increased.

Large-scale numerical experiments were conducted on the 1D J1, J2 model and the triangular-lattice Hubbard model to validate the efficiency of this new method. For the 1D J1, J2 model, energy estimates were calculated across varying system sizes and temperatures, while for the triangular-lattice Hubbard model, static correlation functions were directly computed from the propagated thermal state.

Both analytical and numerical results confirm that simulating high temperatures is computationally efficient, although low-temperature simulations remain a significant challenge. This algorithm has several potential applications, including the estimation of free energies, probing finite-temperature corrections to dynamical correlation functions, and applications to Gibbs sampling and generative modelling. The method’s adaptability to various lattice topologies and its compatibility with quantum hardware further enhance its potential impact on future research and technological advancements in quantum simulation.

Imaginary-time evolution and Pauli decomposition for thermal state preparation offer a robust and efficient approach

Pauli propagation serves as the foundation for simulating thermal state preparation via imaginary-time evolution in the Schrödinger picture. The research begins by expressing the Hamiltonian as a summation of local Pauli terms, represented as H = Σm λmPm, where Pm denotes a Pauli string acting on an n-site system.

Imaginary-time evolution is then approximated using a Trotter decomposition, enabling the computation of the action of the non-unitary imaginary-time propagator e−τH on the identity operator. This process effectively “rotates” the identity operator by an imaginary angle τ, corresponding to a time step in imaginary time.

The study leverages the observation that high-temperature states exhibit sparsity in the Pauli basis, approaching the identity operator at infinite temperature. By formulating imaginary-time evolution directly within this operator basis and initiating the process from the maximally mixed state, the research accesses a continuum of temperatures where the quantum state remains efficiently representable in memory.

Two complementary truncation strategies are analysed: small-coefficient truncation and Pauli-weight truncation, both designed to manage computational complexity. Analytic guarantees are provided for these truncation strategies, quantifying error growth and the impact of backflow to ensure efficient propagation at high temperatures.

Large-scale numerical experiments were conducted on the 1D J1-J2 model to assess energy estimates across varying system sizes and temperatures. Furthermore, the Fermi-Hubbard model on a triangular lattice was used to demonstrate the direct computation of finite-temperature static correlation functions from the propagated thermal state. These numerical results confirm the efficiency of the method at high temperatures, while acknowledging the continuing challenges associated with simulating low-temperature states.

Efficient thermal state propagation using sparse operator representations and truncation strategies enables scalable quantum simulations

Thermal state preparation via propagation methods efficiently accesses a continuum of temperatures, beginning with the maximally mixed state represented by the identity operator. The research demonstrates that high-temperature states are sparse in Pauli or Majorana bases, approaching the identity at infinite temperature, enabling efficient representation and manipulation.

By formulating imaginary-time evolution directly in these operator bases, the study achieves efficient storage of the quantum state in memory across a range of temperatures. Analytic guarantees were established for both small-coefficient truncation and Pauli-weight truncation strategies, quantifying error growth and the impact of backflow during propagation.

Upper bounds derived show that the approximation error decreases as the truncation threshold increases, ensuring efficient operation at high temperatures. Specifically, the algorithm allows for propagation of thermal states while maintaining computational efficiency, even as the system cools and the state becomes less sparse.

Large-scale numerical experiments were conducted on the 1D J1, J2 model and the triangular-lattice Hubbard model to validate the efficiency of the approach. For the 1D J1, J2 model, energy estimates were computed across varying system sizes and temperatures, demonstrating the scalability of the method.

Furthermore, the Fermi-Hubbard model on a triangular lattice enabled direct computation of finite-temperature static correlation functions from the propagated thermal state, showcasing the ability to access physically relevant observables. The work provides a route to estimating free energies, as thermodynamic quantities follow with minimal additional work once a thermal state is accessible.

Combining thermal preparation with subsequent time evolution allows for probing finite-temperature corrections to infinite-temperature dynamical correlation functions. The compact representation of thermal states also suggests applications to Gibbs sampling and generative modeling, facilitating efficient sampling from thermal distributions or learning compact generative surrogates for finite-temperature data. Imaginary time propagation methods remain advantageous due to their independence from lattice topology and natural interface with quantum hardware.

Efficient Thermal State Simulation via Pauli Propagation and Controlled Truncation enables accurate and scalable results

Researchers have developed a propagation-based method for simulating thermal states, adapting Pauli and propagation techniques to imaginary-time evolution within the Schrödinger picture. The core of this approach lies in the observation that high-temperature states exhibit sparsity when expressed in Pauli or related bases, gradually approaching the identity as temperature increases.

By directly formulating imaginary-time evolution in these operator bases and initiating it from a maximally mixed state, the simulation can efficiently represent states across a range of temperatures. This method incorporates analytic guarantees for truncation strategies, specifically addressing small-coefficient and Pauli-weight limitations by quantifying error growth and the impact of backflow during the simulation.

Validation through large-scale numerical calculations on the one-dimensional J1-J2 model and the triangular-lattice Hubbard model confirms the efficiency of the technique at elevated temperatures. The authors demonstrate that the error in approximating the thermal state can be bounded by a function of the temperature, the system size, and the truncation parameters.

The study acknowledges limitations related to the bounded-degree assumption, which restricts the number of terms each qubit participates in, and the need to carefully select truncation parameters to maintain accuracy. Future research could focus on extending the approach to more complex systems or exploring alternative bases for representing thermal states. The demonstrated ability to efficiently simulate high-temperature states represents a step towards understanding the behaviour of quantum systems at thermal equilibrium and could facilitate the study of phenomena such as phase transitions and thermal transport.