  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.
