MORE Thermal Gauge Theories at Finite theta and mu from Real-Time Quantum Simulation

Imaginary-time evolution can be reconstructed from real-time quantum simulations using exact integral transforms, enabling the study of finite-(T, θ, µ) physics without needing to prepare thermal states or sample complex weights. The paper introduces a method called More for measure once that reconstructs thermal traces and Euclidean correlators from a single real-time dataset.

The gist

One real-time dataset reconstructs targeted inverse temperatures, Euclidean separations, and chemical potentials through classical post-processing, bringing finite-(T, θ, µ) physics within reach of real-time quantum simulation without thermal-state preparation.

How it works: The Core Identity and Kernel Decomposition

The method is based on the exact operator identity connecting imaginary-time evolution to real-time measurements:

e−Hτ = i2π ∫∫ dt e−i(t+iτ)t + iτ e−iHt, where τ ≥ 0 is the imaginary-time interval and c is a lower bound on the spectrum. Hadamard tests measure real-time amplitudes, which are combined classically using a kernel in Eq. (1). The paper identifies this construction as a continuous linear combination of Hamiltonian simulations (LCU) and shows that its slow convergence arises from a redundant kernel component. Removing this component gives an exact finite-l1 representation, while near-optimal kernels reduce the required real-time extent exponentially.

How it works: The More Pipeline

The reconstruction process is summarized in three stages:

1. Quantum stage: Real-time amplitudes are measured on a fixed grid and combined with weights wj that carry the entire (β, τ, µ) dependence.

2. Measure once: The dataset consists of one row per computational-basis trace state and one column per node of the real-time grid, where weights wj (β) are drawn for different βJ values.

3. Reconstruct everywhere: The parameters β, τ, and µ are reached by classical reweighting. This allows the method to reuse one real-time dataset across all targeted values of β, τ, and µ.

How it works: Kernel Optimization and Error Quantification

The paper compares different kernel constructions. The original kernel has a redundant odd component with a 1/t tail causing logarithmic divergence. The near-optimal kernels, such as the one from An, Childs, and Lin [20], converge near-exponentially. Kernel error is quantified as the deviation of the reconstructed filter from the target filter. The even kernel has a finite l1 norm and truncation error O(τ /[(E − c)tcut]), whereas Eq. (5) converges near-exponentially.

How it works: Handling Finite (T, θ, µ) Physics

The framework extends to thermal traces and Euclidean correlators within the EρOQ framework. Thermal expectation values are reconstructed via:

⟨ne−βHOm⟩ = Z ∫dt Kβ(t)⟨ne−iHt Om⟩, where Kβ is any kernel of Section II. Euclidean correlators follow from two insertions, involving the product of kernels Kβ-τ (t1) and Kτ (t2).

How it works: Managing θ and µ Dependence

The paper addresses the different costs associated with modifying the Hamiltonian versus coupling to conserved charges. The coupling to a conserved charge like baryon number commutes out of the Boltzmann factor, allowing the whole µ axis can in favorable cases be recovered from a single µ = 0 dataset by classical reweighting. In contrast, θ modifies H and requires a separate dataset because [H(θ), H(θ′)] ≠ 0 for θ ≠ θ′.

How it works: Benchmarking and Costs

The method is benchmarked on one- and two-flavor lattice Schwinger models. The total T-gate cost is quantified by Eq. (25), combining rotation synthesis, circuit executions, and shot noise. Five reductions are proposed to improve resource usage, including allocating shots as Mj ∝ wj to minimize variance at fixed budget. The analysis shows that for the one-flavor model, deviation from exact diagonalization never exceeds 0.49 times the statistical spread across all reconstructed points.

How it works: Noise and Hardware Considerations

The paper models hardware noise as depolarizing noise parameterized by circuit volume, A(t) → e−γnCX(t)A(t). It demonstrates that reweighting by the fitted gate-count dependence reduces the error relative to the Trotterized noiseless reconstruction by a factor of 11–13 and keeps mean error below 1% for γ ≤ 10−4.

Here are the specific improvements to AI systems that can be derived from this scientific paper, and what those improved systems could achieve:

)1. Improved Thermal State Preparation (Replacing Classical Emulation):

The system can move beyond preparing thermal states via classical emulation (like Variational Free Energies or thermal pure quantum states).

• This is achieved by using the More pipeline to reconstruct thermal traces and correlators directly from a single real-time quantum simulation dataset.

• By utilizing the kernel expansion (Eq. 10) and the measure once approach, the system can reconstruct the expectation value of any observable, including thermal averages, without needing to explicitly prepare or sample complex weights associated with a specific thermal state.

• It can achieve this for finite temperature and finite chemical potential physics (finite T, θ, µ).

)2. Finite (T, θ, µ) Physics Prediction and Characterization:

The system can model and predict the phase structure of lattice field theories under non-zero topological angle θ and quark chemical potential µ.

• It can study the transition lines (e.g., the first-order line at θ = π terminating at an Ising endpoint) on quantum simulators, providing data that probes physics beyond zero temperature or zero chemical potential limitations.

• It can characterize physical observables like the topological charge and chiral condensate as functions of T, θ, and µ by reconstructing them from real-time evolution data.

)3. Advanced Quantum Simulation for Gauge Theories (QCD/Yang-Mills):

The system can be applied to complex non-Abelian gauge theories (like QCD or Yang-Mills theory) that are currently computationally expensive or ill-defined due to sign problems.

• It can handle the modification of the Hamiltonian by topological terms (θ) and chemical potential terms (µ).

• It enables simulations on systems like the lattice Schwinger model and potentially larger gauge theories using techniques like loop-string hadron digitization, maximal tree gauge fixing, or orbifold formulations.

)4. Robust Quantum Algorithm Design for Time Evolution:

The system can optimize the construction of quantum algorithms for imaginary-time evolution using exact integral transforms (the LCU construction).

• It provides a formal framework to identify and remove redundant kernel components (the odd part), leading to near-optimal kernels that reduce the required real-time simulation extent exponentially.

• It allows for the comparison and selection of optimal kernels based on error metrics like truncation error versus shot noise.

)5. Resource Optimization and Cost Modeling:

The system can provide a rigorous methodology for estimating hardware requirements (gate counts) for complex simulations.

• It can quantify the total resource cost by combining state preparation, trace sampling, and measurement costs (Eq. 25).

• It allows researchers to compare different algorithmic implementations (e.g., separate circuits vs. block encoding) and determine when specific hardware constraints (like Trotter error or noise damping) become dominant over other sources of error.

)6. Precision Error Budgeting for Quantum Simulations:

The system can provide a detailed, quantifiable budget for reconstruction errors in quantum simulation results.

• It decomposes the total reconstruction error into components: truncation error, discretization error, Trotter error, hardware noise (depolarizing noise), statistical shot noise, and synchronization errors.

• This allows researchers to systematically tune simulation parameters (like increasing grid size or Trotter steps) to minimize specific types of errors for a target precision.

Imaginary-time evolution can be reconstructed from real-time quantum simulations using exact integral transforms. We identify the construction of Guo, Shibu, Lin, and Zhao as a continuous linear combination of Hamiltonian simulations and show that its slow 1/t cut convergence arises from a redundant kernel component. We extend the construction from pure-state matrix elements to thermal traces and correlators. One real-time dataset then reconstructs targeted inverse temperatures, Euclidean separations, and chemical potentials through classical post-processing, bringing finite- (T,θ,μ) physics within reach of real-time quantum simulation without thermal-state preparation. We benchmark the method on one- and two-flavor lattice Schwinger models, including circuit-level simulations with depolarizing noise.