Single-Trajectory Gibbs Sampling for Non-Commuting Observables

Abstract: Estimating thermal expectation values of quantum many-body systems is a central challenge in physics, chemistry, and materials science. Standard quantum Gibbs sampling protocols address this task by preparing the Gibbs state from scratch after every measurement, incurring a full mixing-time cost at each step. Recent advances in single-trajectory Gibbs sampling \cite{jiang2026} substantially reduce this overhead: once stationarity is reached, measurements can be collected along a single trajectory without re-thermalizing, provided the measurement channel preserves the Gibbs ensemble. However, explicit constructions of such non-destructive measurements have been limited primarily to observables that commute with the Hamiltonian. In this work, we fundamentally extend the single-trajectory framework to arbitrary, non-commuting observables. We provide two measurement constructions that extract measurement information without fully destroying the Gibbs state, thereby eliminating the need for full re-mixing between samples. First, we construct a measurement that satisfies exact detailed balance. This ensures the system remains in equilibrium throughout the trajectory, allowing measurement outcomes to decorrelate in an autocorrelation time that could be significantly shorter than the global mixing time. Second, assuming the underlying quantum Gibbs sampler has a positive spectral gap, we design a simplified measurement scheme that ensures the post-selected state serves as a warm start for rapid re-mixing. This approach successfully decouples the resampling cost from the global mixing time. Both measurement schemes admit efficient quantum circuit implementations, requiring only polylogarithmic Hamiltonian simulation time.

• The paper presents a detailed-balance measurement channel that preserves the Gibbs state, enabling unbiased estimation of thermal expectation values for non-commuting observables.
• It introduces a warm-start measurement strategy that decouples resampling cost from mixing time, significantly reducing sample complexity.
• The study demonstrates efficient circuit implementations with polylogarithmic overhead, advancing scalable quantum simulation of many-body thermal systems.

Single-Trajectory Gibbs Sampling for Non-Commuting Observables: An Expert Summary

Background and Motivation

The estimation of thermal expectation values in quantum many-body systems is fundamental for quantum physics, chemistry, and materials science. The thermal (Gibbs) state $\sigma_\beta = e^{-\beta H}/\mathcal{Z}$ characterizes equilibrium properties at inverse temperature $\beta$, and expectation values $$\langle A \rangle_{\sigma_\beta} = \mathrm{Tr}(\sigma_\beta A)$$ encode various physical observables. Classical computational approaches are generally intractable due to the sign problem and exponential Hilbert space growth, motivating quantum algorithms for Gibbs state preparation and measurement.

Quantum Gibbs sampling algorithms [Chen2025efficient, Ding_2025, chen2025efficientexactnoncommutativequantum, gilyen2026] achieve preparation and measurement by driving an initial state to equilibrium within a mixing time $t_{\mathrm{mix}}$. For precision $\epsilon$, the standard protocol incurs a total cost $\mathcal{O}(t_{\mathrm{mix}}/\epsilon^2)$, with substantial algorithmic overhead due to re-thermalization after each destructive measurement. The paradigm of single-trajectory Gibbs sampling [jiang2026] addresses this by leveraging non-destructive measurement channels that preserve the stationary Gibbs state, so independent samples can be obtained at intervals determined by autocorrelation time $t_{\mathrm{aut}}$ (typically $t_{\mathrm{aut}} \ll t_{\mathrm{mix}}$). Previously, explicit non-destructive measurements were limited to observables commuting with the Hamiltonian.

Main Contributions

This paper provides a comprehensive extension of single-trajectory Gibbs sampling to arbitrary non-commuting observables. The following are the principal technical innovations:

• Detailed-Balance Measurement Channel Construction:

The authors construct a quantum measurement channel $\mathcal{M}$ for general Hermitian observables $A$ ($\|A\|\le 1$) that satisfies exact Kubo–Martin–Schwinger (KMS) detailed balance relative to the Gibbs state. This ensures $\mathcal{M}(\sigma_\beta) = \sigma_\beta$ and preserves stationarity along the measurement trajectory, allowing samples to decorrelate with autocorrelation time. The construction employs operator Fourier transform (Gaussian filtering) and a nontrivial combination of time-shifted filters, ensuring estimation of $\mathrm{Tr}(\sigma_\beta A)$ with variance $\widetilde{\mathcal{O}}(1)$ and efficient quantum circuit implementation (polylogarithmic in parameters, $\mathcal{O}(\beta)$ Hamiltonian simulation time).

• Warm-Start Measurement Channel Construction:

Under spectral gap assumption ($\lambda > 0$) for the underlying sampler, a simplified measurement channel is devised that forgoes exact detailed balance but guarantees post-selected states are a warm start, i.e., have bounded $\chi^2$-divergence from $\sigma_\beta$. This enables rapid remixing (cost $\mathcal{O}(\lambda^{-1}\log(1/\epsilon))$ per sample), decoupling resampling cost from mixing time.

• Sample Complexity Analysis:

Rigorous sample complexity estimates are derived, demonstrating that for general observables, the total Gibbs sampling time for estimating $\mathrm{Tr}(\sigma_\beta A)$ to precision $\epsilon$ (failure probability $\eta$) is $$t_{\mathrm{mix}}+\mathcal{O}(t_{\mathrm{aut}}/(\epsilon^2\eta))$$, with $t_{\mathrm{aut}}$ upper bounded by the inverse spectral gap (up to polylogarithmic factors). For warm-start measurements, the complexity is $$t_{\mathrm{mix}}+\mathcal{O}(\lambda^{-1}\log(1/\eta)/\epsilon^2)$$.

• Efficient Circuit Realization:

Both measurement schemes facilitate efficient implementation via block-encoding and quantum singular value transformation (QSVT), requiring only polylogarithmic overhead in Hamiltonian simulation and ancilla qubits.

Technical Approach

The detailed-balance measurement channel design leverages smoothed observables via operator Fourier transform,

$$A_f = \int_{-\infty}^{+\infty} f(t) e^{iHt} A e^{-iHt} \,dt,$$

with Gaussian filter $f(t)$ (width $\tau$) and controlled imaginary-time shifts. The construction ensures that the measurement preserves the Gibbs ensemble for non-commuting $A$, utilizing Kraus operators $K_1, K_2$ corresponding to filters $f$ and $f(t-i\beta/2)$, and a rejection channel for trace preservation modeled after quantum Metropolis–Hastings [gilyen2026].

For warm-start measurement, a two-outcome POVM with Kraus operators $K_\pm = \sqrt{\frac{1}{2}(I \pm uA_f)}$ is used. The post-selected states have controlled proximity to $\sigma_\beta$, facilitating rapid convergence under the sampler's spectral gap.

The sample complexity and variance analyses capitalize on KMS detailed balance properties and autocorrelation time bounds, connecting variance reduction in trajectory-based estimation to spectral gap analysis [temme2010chi, jiang2026].

Strong Numerical Results and Claims

• Exact unbiased estimation:

The constructed channels yield unbiased estimators for $\mathrm{Tr}(\sigma_\beta A)$ with variance $\mathcal{O}(\log^2(\beta\|H\|))$ for detailed-balance and $\mathcal{O}(1)$ for warm-start measurement.

• Decorrelation scaling:

The per-sample cost is governed by $t_{\mathrm{aut}}$ (for detailed-balance), which can be substantially less than $t_{\mathrm{mix}}$. Therefore, significantly fewer Gibbs sampling steps are needed for high-precision estimation.

• Implementation complexity:

Both constructions require only $\widetilde{\mathcal{O}}(1)$ queries to block-encoding of $A$, $\mathcal{O}(\beta)$ controlled Hamiltonian simulation time, and $\widetilde{\mathcal{O}}(1)$ ancilla qubits. Warm-start implementation is especially efficient in both query and ancilla requirements.

• Failure probability dependence:

The warm-start strategy achieves logarithmic dependence of complexity on the failure probability $\eta$ ($\mathcal{O}(\epsilon^{-2}\log(1/\eta))$), which is optimal with respect to empirical outcome concentration.

Theoretical and Practical Implications

The extension to non-commuting observables closes a substantial gap in quantum measurement theory for thermal systems, enabling efficient, high-fidelity estimation of a broader class of observables. The explicit detailed-balance construction is foundational for quantum statistical mechanics and quantum simulation algorithms, as it ensures the stationary ensemble is preserved for arbitrary many-body measurements.

Practically, the warm-start and detailed-balance strategies advance quantum algorithms toward more efficient resource usage—decoupling sample cost from mixing time, minimizing circuit overhead, and optimizing error scaling. This development is critical for scalable quantum simulation of strongly correlated systems, thermodynamic properties, and materials modeling.

Theoretically, the connection between measurement-induced disturbance, spectral gap, and autocorrelation time motivates further exploration of quantum Markov dynamics, metastability, and mixing properties, with implications for quantum complexity theory and quantum channel design.

Future Directions

This work lays the foundation for improved quantum measurement protocols in thermal and ground-state contexts. Potential future avenues include:

• Optimization and simplification of measurement channels for even higher efficiency
• Adaptation to local observables and quasi-local recovery protocols via Markov properties
• Integration with cluster expansion and weak-coupling methods for classical simulability of quantum systems
• Application in quantum shadow tomography and error mitigation for realistic quantum hardware
• Extension of autocorrelation time analysis to systems with polynomial spectral gap scaling or to Lindbladian-driven open systems

Conclusion

This paper rigorously extends single-trajectory quantum Gibbs sampling to non-commuting observables by constructing measurement channels with exact detailed balance or warm-start properties. The reduction in sampling cost, efficient circuit realization, and precise sample complexity bounds represent a substantive advance in quantum algorithm design for many-body thermal systems. The methodology enables practical, scalable computation of thermal expectation values across a wide range of physical models, deepening the theoretical understanding of non-destructive measurement and stationary quantum dynamics (2603.21595).