Simpler (classical) and faster (quantum) algorithms for Gibbs partition functions

Srinivasan Arunachalam1, Vojtech Havlicek1,2, Giacomo Nannicini1, Kristan Temme1, and Pawel Wocjan1

1IBM Quantum, IBM T.J. Watson Research Center
2School of Mathematics, University of Bristol

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Abstract

We present classical and quantum algorithms for approximating partition functions of classical Hamiltonians at a given temperature. Our work has two main contributions: first, we modify the classical algorithm of Stefankovic, Vempala and Vigoda (J. ACM, 56(3), 2009) to improve its sample complexity; second, we quantize this new algorithm, improving upon the previously fastest quantum algorithm for this problem, due to Harrow and Wei (SODA 2020).
The conventional approach to estimating partition functions requires approximating the means of Gibbs distributions at a set of inverse temperatures that form the so-called cooling schedule. The length of the cooling schedule directly affects the complexity of the algorithm. Combining our improved version of the algorithm of Stefankovic, Vempala and Vigoda with the paired-product estimator of Huber (Ann. Appl. Probab., 25(2), 2015), our new quantum algorithm uses a shorter cooling schedule than previously known. This length matches the optimal length conjectured by Stefankovic, Vempala and Vigoda. The quantum algorithm also achieves a quadratic advantage in the number of required quantum samples compared to the number of random samples drawn
by the best classical algorithm, and its computational complexity has quadratically better dependence on the spectral gap of the Markov chains used to produce the quantum samples.

Partition functions are the normalizing factors of the Gibbs distribution. Their estimation is surprisingly useful: zero temperature partition functions for a suitably chosen Hamiltonian can for example encode the number of graph colorings or the number of subgraphs of a graph. It is therefore important to understand how to improve their computation with quantum algorithms.

Here we present classical and quantum algorithms for this problem. Our work has two main contributions: first, we modify the classical algorithm of Stefankovic, Vempala and Vigoda (J. ACM, 56(3), 2009) to improve its sample complexity; second, we quantize this new algorithm, improving upon the previously fastest quantum algorithm for this problem, due to Harrow and Wei (SODA 2020).

The usual approach to estimation of partition functions requires approximating the means of Gibbs distributions at a set of inverse temperatures that form the so-called cooling schedule. The length of the cooling schedule directly affects the complexity of the algorithm. Combining our improved version of the algorithm of Stefankovic, Vempala and Vigoda with the paired-product estimator of Huber (Ann. Appl. Probab., 25(2), 2015), our new quantum algorithm uses a shorter cooling schedule than previously known. This length matches the optimal length conjectured by Stefankovic, Vempala and Vigoda. The quantum algorithm also achieves a quadratic advantage in the number of required quantum samples compared to the number of random samples drawn by the best classical algorithm, and its computational complexity has quadratically better dependence on the spectral gap of the Markov chains used to produce the quantum samples.

► BibTeX data

► References

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Cited by

[1] Patrick Rall, "Faster Coherent Quantum Algorithms for Phase, Energy, and Amplitude Estimation", arXiv:2103.09717.

[2] Chen-Fu Chiang, Anirban Chowdhury, and Pawel Wocjan, "Space-efficient Quantization Method for Reversible Markov Chains", arXiv:2206.06886.

[3] Garrett T. Floyd, David P. Landau, and Michael R. Geller, "Quantum algorithm for Wang-Landau sampling", arXiv:2208.09543.

[4] Patrick Rall and Bryce Fuller, "Amplitude Estimation from Quantum Signal Processing", arXiv:2207.08628.

[5] Pawel Wocjan and Kristan Temme, "Szegedy Walk Unitaries for Quantum Maps", arXiv:2107.07365.

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