Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Average-case quantum complexity from glassiness

Alexander Zlokapa, Bobak T. Kiani, Eric R. Anschuetz

TL;DR

The paper develops a quantum glassiness framework that connects replica-symmetry-breaking clustering of Gibbs states to average-case hardness for quantum algorithms. By coupling Pauli-based overlaps with quantum optimal transport via the quantum Wasserstein distance, it proves that stable quantum procedures, including constant-time Lindbladian Gibbs samplers and shallow circuits, fail to prepare Gibbs states for glassy Hamiltonian ensembles. It delivers a concrete analysis for random $p$-local Pauli Hamiltonians: the $p=3$ ensemble exhibits a glass transition and is therefore hard for stable quantum algorithms, while sufficiently large $p$ yields non-glassy behavior, signaling a sharp phase diagram in the quantum setting. The results are underpinned by a replica-trick path integral treatment, yielding RS/RSB/FRSB saddle points, with rigorous bounds supported by sharpened log-Sobolev inequalities, and point toward a transport-theoretic avenue for future average-case quantum complexity bounds and algorithm design guidance.

Abstract

Glassiness -- a phenomenon in physics characterized by a rough free-energy landscape -- implies hardness for stable classical algorithms. For example, it can obstruct constant-time Langevin dynamics and message-passing in random $k$-SAT and max-cut instances. We provide an analogous framework for average-case quantum complexity showing that a natural family of quantum algorithms (e.g., Lindbladian evolution) fails for natural Hamiltonian ensembles (e.g., random 3-local Hamiltonians). Specifically, we prove that the standard notion of quantum glassiness based on replica symmetry breaking obstructs stable quantum algorithms for Gibbs sampling, which we define by a Lipschitz temperature dependence in quantum Wasserstein complexity. Our proof relies on showing that such algorithms fail to capture a structural phase transition in the Gibbs state, where glassiness causes the Gibbs state to decompose into clusters extensively separated in quantum Wasserstein distance. This yields average-case lower bounds for constant-time local Lindbladian evolution and shallow variational circuits. Unlike mixing time lower bounds, our results hold even when dynamics are initialized from the maximally mixed state. We apply these lower bounds to non-commuting, non-stoquastic Hamiltonians by showing a glass transition via the replica trick. We find that the ensemble of all 3-local Pauli strings with independent Gaussian coefficients is average-case hard, while providing analytical evidence that the general $p$-local Pauli ensemble is non-glassy for sufficiently large constant $p$, in contrast to its classical (Ising $p$-spin, always glassy) and fermionic (SYK, never glassy) counterparts.

Average-case quantum complexity from glassiness

TL;DR

The paper develops a quantum glassiness framework that connects replica-symmetry-breaking clustering of Gibbs states to average-case hardness for quantum algorithms. By coupling Pauli-based overlaps with quantum optimal transport via the quantum Wasserstein distance, it proves that stable quantum procedures, including constant-time Lindbladian Gibbs samplers and shallow circuits, fail to prepare Gibbs states for glassy Hamiltonian ensembles. It delivers a concrete analysis for random -local Pauli Hamiltonians: the ensemble exhibits a glass transition and is therefore hard for stable quantum algorithms, while sufficiently large yields non-glassy behavior, signaling a sharp phase diagram in the quantum setting. The results are underpinned by a replica-trick path integral treatment, yielding RS/RSB/FRSB saddle points, with rigorous bounds supported by sharpened log-Sobolev inequalities, and point toward a transport-theoretic avenue for future average-case quantum complexity bounds and algorithm design guidance.

Abstract

Glassiness -- a phenomenon in physics characterized by a rough free-energy landscape -- implies hardness for stable classical algorithms. For example, it can obstruct constant-time Langevin dynamics and message-passing in random -SAT and max-cut instances. We provide an analogous framework for average-case quantum complexity showing that a natural family of quantum algorithms (e.g., Lindbladian evolution) fails for natural Hamiltonian ensembles (e.g., random 3-local Hamiltonians). Specifically, we prove that the standard notion of quantum glassiness based on replica symmetry breaking obstructs stable quantum algorithms for Gibbs sampling, which we define by a Lipschitz temperature dependence in quantum Wasserstein complexity. Our proof relies on showing that such algorithms fail to capture a structural phase transition in the Gibbs state, where glassiness causes the Gibbs state to decompose into clusters extensively separated in quantum Wasserstein distance. This yields average-case lower bounds for constant-time local Lindbladian evolution and shallow variational circuits. Unlike mixing time lower bounds, our results hold even when dynamics are initialized from the maximally mixed state. We apply these lower bounds to non-commuting, non-stoquastic Hamiltonians by showing a glass transition via the replica trick. We find that the ensemble of all 3-local Pauli strings with independent Gaussian coefficients is average-case hard, while providing analytical evidence that the general -local Pauli ensemble is non-glassy for sufficiently large constant , in contrast to its classical (Ising -spin, always glassy) and fermionic (SYK, never glassy) counterparts.

Paper Structure

This paper contains 26 sections, 22 theorems, 277 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Glasses and quantum optimal transport
  3. Glassiness from Pauli overlaps
  4. Quantum optimal transport implications of glassiness
  5. Algorithmic obstruction
  6. Glassiness obstructs stable quantum algorithms
  7. Local continuous-time Lindbladian evolution
  8. Shallow quantum algorithms
  9. Glassiness of random -local Hamiltonians
  10. Path integral
  11. Replica trick
  12. Saddle point equations
  13. Glassiness of the 3-local Hamiltonian ensemble
  14. Non-glassiness of the q-local Hamiltonian ensemble for large q
  15. Path integrals for the -local Hamiltonian ensemble
  16. ...and 11 more sections

Key Result

Theorem 3

Let $\mathcal{H}_n$ be a Hamiltonian ensemble that is glassy at inverse temperature $\beta$ and non-glassy at $\beta'$. For $H \sim \mathcal{H}_n$, denote the corresponding Gibbs states by $\bm \rho_\beta$ and $\bm \rho_{\beta'}$. Then with probability $1-o(1)$, there exists constant $\epsilon_* > 0 must also satisfy for constant $\alpha$ independent of $\epsilon$.

Figures (2)

  • Figure 1: Cartoon of the glass decomposition $\bm \rho_\beta = \sum_i c_i \bm \rho_i$ of a Gibbs state that is non-glassy, shattered, and 1RSB. At a high temperature $\beta_0$, the Gibbs state is replica symmetric and non-glassy, and is dominated by one "cluster" as measured by $\norm{\cdot}_\mathrm{Pauli}$\ref{['eq:pauliform']}. At a cooler temperatures $\beta_1$, the Gibbs state may shatter into a glassy phase defined by exponentially many clusters $\bm \rho_i$, which we show are separated by $\Omega(n)$ in quantum Wasserstein distance. Although the state remains replica symmetric in the shattered phase (due to each cluster being exponentially small), we show that any quantum channel satisfying $\bm \varPhi(\bm \rho_{\beta_0}) \approx \bm \rho_{\beta_1}$ must have Wasserstein complexity $\Omega(n)$. At an even cooler temperature $\beta_2$, the Gibbs state is decomposed into a constant number of clusters due to (static) replica symmetry breaking. Since the transition from a non-glassy to a glassy phase occurs abruptly at some constant $\beta_*$, any algorithm that is Lipschitz in $\beta$ will fail at the glass transition.
  • Figure 2: Numerical solutions to the replica saddle point equations indicate the presence of replica symmetry breaking in the $p=3$ model. Here, values of the overlap $q_0$ in the RS solution of the $p=3$ model are plotted over various values of inverse temperature $\beta$ with $J=1$ throughout. A phase transition in the off-diagonal value indicates that the RS ansatz is incorrect at some temperature $\beta \approx 6$, demonstrating a glass transition at constant temperature. We note that because the glass transition likely occurs before the off-diagonals of the RS ansatz become nonzero, the RS solver is fairly unstable and the precise temperature of the glass transition cannot be deduced from this plot alone; $\beta \approx 6$ only provides an approximate estimate of the transition.

Theorems & Definitions (53)

  • Definition 1: Glassiness
  • Remark 2
  • Theorem 3: Extensive Wasserstein complexity across the glass transition, informal
  • Definition 4: Stable quantum algorithm, informal
  • Theorem 5: Examples of stable quantum algorithms, informal
  • Theorem 6: Glassiness obstructs stable algorithms
  • Remark 7: Topological barrier
  • Definition 8: Random $p$-local Hamiltonian ensemble
  • Corollary 9: Hardness of 3-local Hamiltonians
  • Definition 10: Replica symmetry
  • ...and 43 more