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Quantum Algorithm for Estimating Gibbs Free Energy and Entropy via Energy Derivatives

Shangjie Guo, Corneliu Buda, Nathan Wiebe

TL;DR

The paper develops a quantum algorithm to estimate vibrational contributions to Gibbs free energy and entropy by exploiting energy derivatives. It builds a block-encoded second-derivative operator $K=V^ op (E_n I - H_0)^+ V$, uses LCU and quantum singular value transformation to realize $K$, and applies the Hadamard test with amplitude estimation to extract $\langle E_0|K|E_0\rangle$, subsequently yielding $S_{ m vib}$ via a mode-sum expression involving $\theta_i$. The authors provide detailed resource estimates, show a dependence on the partition-function-like quantity $\mathcal{Z}$ and condition number $\kappa$, and prove that the problem is $\,\text{BQP}$-hard/complete, underscoring a genuine quantum advantage over classical approaches under reasonable assumptions. They discuss practical implications for quantum chemistry and materials science, and outline future directions toward optimization, near-term implementations, and extensions to other observables.

Abstract

Estimating vibrational entropy is a significant challenge in thermodynamics and statistical mechanics due to its reliance on quantum mechanical properties. This paper introduces a quantum algorithm designed to estimate vibrational entropy via energy derivatives. Our approach block encodes the exact expression for the second derivative of the energy and uses quantum linear systems algorithms to deal with the reciprocal powers of the gaps that appear in the expression. We further show that if prior knowledge about the values of the second derivative is used then our algorithm can $ε$-approximate the entropy using a number of queries that scales with the condition number $κ$, the temperature $T$, error tolerance $ε$ and an analogue of the partition function $\mathcal{Z}$, as $\widetilde{O}\left(\frac{\mathcal{Z}κ^2 }{εT}\right)$. We show that if sufficient prior knowledge is given about the second derivative then the query scales quadratically better than these results. This shows that, under reasonable assumptions of the temperature and a quantum computer can be used to compute the vibrational contributions to the entropy faster than analogous classical algorithms would be capable of. Our findings highlight the potential of quantum algorithms to enhance the prediction of thermodynamic properties, paving the way for advancements in fields such as material science, molecular biology, and chemical engineering.

Quantum Algorithm for Estimating Gibbs Free Energy and Entropy via Energy Derivatives

TL;DR

The paper develops a quantum algorithm to estimate vibrational contributions to Gibbs free energy and entropy by exploiting energy derivatives. It builds a block-encoded second-derivative operator , uses LCU and quantum singular value transformation to realize , and applies the Hadamard test with amplitude estimation to extract , subsequently yielding via a mode-sum expression involving . The authors provide detailed resource estimates, show a dependence on the partition-function-like quantity and condition number , and prove that the problem is -hard/complete, underscoring a genuine quantum advantage over classical approaches under reasonable assumptions. They discuss practical implications for quantum chemistry and materials science, and outline future directions toward optimization, near-term implementations, and extensions to other observables.

Abstract

Estimating vibrational entropy is a significant challenge in thermodynamics and statistical mechanics due to its reliance on quantum mechanical properties. This paper introduces a quantum algorithm designed to estimate vibrational entropy via energy derivatives. Our approach block encodes the exact expression for the second derivative of the energy and uses quantum linear systems algorithms to deal with the reciprocal powers of the gaps that appear in the expression. We further show that if prior knowledge about the values of the second derivative is used then our algorithm can -approximate the entropy using a number of queries that scales with the condition number , the temperature , error tolerance and an analogue of the partition function , as . We show that if sufficient prior knowledge is given about the second derivative then the query scales quadratically better than these results. This shows that, under reasonable assumptions of the temperature and a quantum computer can be used to compute the vibrational contributions to the entropy faster than analogous classical algorithms would be capable of. Our findings highlight the potential of quantum algorithms to enhance the prediction of thermodynamic properties, paving the way for advancements in fields such as material science, molecular biology, and chemical engineering.

Paper Structure

This paper contains 8 sections, 11 theorems, 79 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Entropy Approximation
  3. Vibrational Entropy as a Function of Energy Derivative
  4. Energy Derivative as Block Encoding
  5. Evaluation of operator $K$
  6. Computational Hardness of Second Derivative
  7. Discussion
  8. Derivation of the Coefficients $c_n$

Key Result

Lemma 1

Let $H(s) = H_0 + sV$ where $\bra{E_k} V \ket{E_j} = 0$ if $E_j = E_k$ and $j\ne k$ for $V$ a bounded operator acting on the same Hilbert space and $K = V^\dagger (E_0 I - H_0)^{+} V$. then we have that for any eigenstate of $H_0$, $\ket{E_0}$, with known eigenvalue $E_0$ we have that

Figures (2)

  • Figure 1: Recursive block encodings used in the block-encoding of $VAV$.
  • Figure 2: Quantum circuit representation for amplitude estimation for $M=2^{k+1}$

Theorems & Definitions (19)

  • Lemma 1: Second Derivative Lemma
  • proof
  • Theorem 2
  • Lemma 3
  • proof
  • Lemma 4
  • Theorem 5
  • proof : Proof of Theorem \ref{['thm:blockEnc']}
  • Lemma 6: Amplitude Estimation
  • Lemma 7
  • ...and 9 more