Quantum Metropolis Sampling via Weak Measurement

TL;DR

The paper presents a Metropolis-style quantum Gibbs sampler that uses weak measurement and Boosted Quantum Phase Estimation to approximate Gibbs-state preparation for general quantum Hamiltonians. By decomposing each iteration into Accept, AltAccept, and Reject branches and constructing an effective Lindbladian $oldsymbol{ m{L}}$ with a fast-mixing fixed point, the authors show that repeated iterations converge toward a state close to the Gibbs state $ ho_eta$. The analysis bounds the impact of imperfect energy estimation and finite-precision QPE, and demonstrates polynomial-time scalability for the mixing with appropriate parameter choices, while avoiding Marriott–Watrous rewinding and shift-invariance requirements. This approach provides a conceptually simple, rewind-free quantum Metropolis sampler that connects directly to Lindbladian dynamics and offers a platform for comparing quantum Gibbs samplers across different formulations. The work situates itself among Davies-generator-based methods and contemporaneous quantum Gibbs samplers, highlighting the tradeoffs between energy-estimation precision, circuit depth, and mixing behavior with potential improvements via advanced Lindbladian simulation techniques.

Abstract

Gibbs sampling is a crucial computational technique used in physics, statistics, and many other scientific fields. For classical Hamiltonians, the most commonly used Gibbs sampler is the Metropolis algorithm, known for having the Gibbs state as its unique fixed point. For quantum Hamiltonians, designing provably correct Gibbs samplers has been more challenging. [TOV+11] introduced a novel method that uses quantum phase estimation (QPE) and the Marriot-Watrous rewinding technique to mimic the classical Metropolis algorithm for quantum Hamiltonians. The analysis of their algorithm relies upon the use of a boosted and shift-invariant version of QPE which may not exist [CKBG23]. Recent efforts to design quantum Gibbs samplers take a very different approach and are based on simulating Davies generators [CKBG23,CKG23,RWW23,DLL24]. Currently, these are the only provably correct Gibbs samplers for quantum Hamiltonians. We revisit the inspiration for the Metropolis-style algorithm of [TOV+11] and incorporate weak measurement to design a conceptually simple and provably correct quantum Gibbs sampler, with the Gibbs state as its approximate unique fixed point. Our method uses a Boosted QPE which takes the median of multiple runs of QPE, but we do not require the shift-invariant property. In addition, we do not use the Marriott-Watrous rewinding technique which simplifies the algorithm significantly.

Figures (2)

• Figure 1: One iteration of the algorithm. The operation $U$ is $\rm{QPE}_{1,3} \circ C \circ \rm{QPE}_{1,2}$. The two measurements are performed on the last qubit only. The $\blacksquare$ symbol indicates that the last three registers are traced out and replaced by fresh qubits in the $\left\vert 0 \right\rangle$ state.
• Figure 2: The function shown in Equation (\ref{['eq:function']}) is minimized for $\theta = 1/2$.

Theorems & Definitions (41)

• Theorem 1: Informal version of theorem \ref{['thm:main']}
• Theorem 2: Informal version of Theorem \ref{['thm:mix']}
• Lemma 3
• Theorem 5
• Definition 6: Mixing time
• Theorem 7: Error bounds w.r.t Mixing time
• Theorem 8: Bounding mixing time w.r.t spectral gap
• Corollary 9
• Corollary 10: Error bounds w.r.t Spectral gap
• Definition 11