Thermal State Simulation with Pauli and Majorana Propagation

TL;DR

This work introduces a propagation-based method to simulate finite-temperature quantum states by performing imaginary-time evolution in the Pauli and Majorana operator bases, starting from the maximally mixed state. By truncating the operator growth via small-coefficient and weight (or length) criteria, the authors derive analytic guarantees that high-temperature regimes admit efficient, polylogarithmic-in-$1/\epsilon$ resource scaling for fixed system size and demonstrate substantial numerical validation on challenging geometries. The framework unifies Pauli/Majorana propagation with stochastic (qDRIFT) and deterministic (Trotter) schemes, offering insights into backflow limitations and the role of operator-space sparsity for thermal-state representations. The results show practical efficiency at high temperatures, with clear limitations at low temperatures, and point to promising applications in computing thermodynamic quantities and facilitating hybrid classical-quantum workflows.

Abstract

We introduce a propagation-based approach to thermal state simulation by adapting Pauli and Majorana propagation to imaginary-time evolution in the Schrödinger picture. Our key observation is that high-temperature states can be sparse in the Pauli or Majorana bases, approaching the identity at infinite temperature. By formulating imaginary-time evolution directly in these operator bases and evolving from the maximally mixed state, we access a continuum of temperatures where the state remains efficiently representable. We provide analytic guarantees for small-coefficient truncation and Pauli-weight (Majorana-length) truncation strategies by quantifying the error growth and the impact of backflow. Large-scale numerics on the 1D J1-J2 model (energies) and the triangular-lattice Hubbard model (static correlations) validate efficiency at high temperatures.

Figures (5)

• Figure 1: Schematic depiction of thermal state preparation via propagation methods. The process begins at infinite temperature ($\beta=0$) with the maximally mixed state, represented by the identity operator $\mathbb{I}$. We then apply a sequence of imaginary-time gates (propagators $e^{-\tau H}$) to evolve the operator. As the system cools (corresponding to larger $\beta$), the state becomes less sparse and the number of operators (either Paulis or Majoranas) that need to be propagated grow. This approach allows us to efficiently store and manipulate high-temperature states.
• Figure 2: Comparison between 1st-order Trotter and qDRIFT. We evolve using truncated trotterization (full) and qDRIFT (dashed) for different models and compare to the untruncated evolutions using the mean relative error of the energy as a metric. This error is plotted as a function of $\beta$ (top) and as a function of the number of Paulis it utilizes (bottom) for a 10 qubit system. The overlap of the curves between the trotter evolution and the qDRIFT suggests that weight truncation is equally justified for both types of evolution. The bottom plot gives a comparison of the effectiveness of each truncation scheme where in general the coefficient truncation leads to a better error at a given number of Paulis.
• Figure 3: Imaginary time evolution under the J1-J2 Hamiltonian. Numerical results for the 1D $J_1-J_2$ Heisenberg model across system sizes ranging from 10 to 40 qubits. The top row shows the estimated energy density $\langle H \rangle / n$ converging toward the ground state (dashed line) as the inverse temperature $\beta$ increases. The bottom row tracks the number of Pauli strings generated, showing the exponential growth in complexity as the temperature drops. Different shades of blue represent different coefficient truncation thresholds (from $2^{-9}$ to $2^{-18}$); the point where these lines diverge indicates the limit where our approximation is no longer accurate.
• Figure 4: Spin correlation in the Fermi-Hubbard model on a triangular lattice. Snapshots of spin-spin correlations ($C_{ZZ}$) emerging in a Fermi-Hubbard model on a 37 site triangular lattice (corresponding to 74 Majorana modes). As we lower the temperature from $\beta=0$ to $\beta=0.1$, we can observe magnetic order and correlations beginning to form around the central site (red). While reaching the deep frustration patterns of lower temperatures ($\beta \sim 2$ to $\beta \sim 3$) remains challenging for current propagation techniques, this demonstrates the ability to compute static correlations for complex geometries directly from the propagated state.
• Figure 5: Imaginary time evolution under the Fermi-Hubbard Hamiltonian. A look "under the hood" of the triangular lattice simulations from Figure 3. Here we track the energy (top) and the count of Majorana operators (bottom) as imaginary time progresses for 19-site and 37-site systems. The sharp rise in the bottom plots illustrates the "operator-growth barrier": even at a high temperature of $\beta \approx 0.1$, the 37-site simulation generates billions of terms, saturating 500GB of memory. As before, the diverging lines show where stricter truncation thresholds cause the simulation to lose accuracy.

Theorems & Definitions (54)

• Theorem 1: Small-angle truncation error
• Theorem 2: Weight truncation Backflow Error, Informal
• Theorem 3: Weight-truncation error for 1st-order Trotter imaginary-time evolution (informal)
• Lemma 4
• proof
• Lemma 5
• proof
• Definition 6: Max Divergence
• Lemma 7
• proof