Quantum Algorithm for Preparing Thermal Gibbs States - Detailed Analysis

In a recent work [10], Poulin and one of us presented a quantum algorithm for preparing thermal Gibbs states of interacting quantum systems. This algorithm is based on Grovers’s technique for quantum state engineering, and its running time is dominated by the factor D/Z(\beta), where D and Z(\beta) denote the dimension of the quantum system and its partition function at inverse temperature \beta, respectively. We present here a modified algorithm and a more detailed analysis of the errors that arise due to imperfect simulation of Hamiltonian time evolutions and limited performance of phase estimation (finite accuracy and nonzero probability of failure). This modfication together with the tighter analysis allows us to prove a better running time by the effect of these sources of error on the overall complexity. We think that the ideas underlying of our new analysis could also be used to prove a better performance of quantum Metropolis sampling by Temme et al. [12].

💡 Research Summary

The paper revisits the quantum algorithm originally proposed by Poulin and collaborators for preparing thermal Gibbs states of interacting quantum systems. That algorithm relies on Grover‑based state engineering and, in an idealized setting, its runtime scales as O(D/Z(β))·polylog(D), where D is the Hilbert space dimension and Z(β) the partition function at inverse temperature β. The authors identify two realistic sources of error that were only loosely treated in the original work: (1) imperfect simulation of the Hamiltonian time evolution, and (2) finite‑accuracy phase estimation with a non‑zero probability of failure.

To quantify the impact of these imperfections, the authors model Hamiltonian simulation error ε₁ using standard Trotter‑Suzuki or qubitization bounds, showing that the cumulative error after L steps of size Δt satisfies ε₁ ≤ L·Δt·‖H‖. Phase‑estimation error ε₂ is expressed in terms of the number of measured bits m and the failure probability p_fail, yielding ε₂ ≤ π/2^m + O(p_fail). By treating both errors as independent quantum channels, they derive an overall trace‑distance bound for the prepared state: δ ≤ C₁·(D/Z(β))·ε₁ + C₂·(D/Z(β))·ε₂, where C₁ and C₂ are modest constants. This expression reveals that the global scaling factor D/Z(β) amplifies any local inaccuracies, making tight control of ε₁ and ε₂ essential.

The paper then proposes concrete parameter choices that keep the total error below a desired threshold δ₀. Setting ε₁ = δ₀·Z(β)/(2C₁D) determines the required Trotter step size and number of steps, while choosing m = ⌈log₂(π·C₂D/(δ₀Z(β)))⌉ and p_fail ≤ δ₀·Z(β)/(2C₂D) guarantees that phase‑estimation contributes an equally small error. With these selections the algorithm retains its asymptotic runtime Õ(D/Z(β))·polylog(D) but with a significantly reduced constant factor compared to the original analysis.

Beyond the Gibbs‑state preparation task, the authors argue that the same error‑budgeting technique can be transplanted to the quantum Metropolis algorithm introduced by Temme et al. (Reference