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Derandomised tensor product gap amplification for quantum Hamiltonians

Thiago Bergamaschi, Tony Metger, Thomas Vidick, Tina Zhang

TL;DR

This work advances the quantum PCP program by introducing a derandomised gap amplification for local Hamiltonians that mimics Dinur’s classical approach but operates within the quantum setting. The core method combines derandomised tensor-product amplification with expander-walk sampling, implemented on layered, commuting components of the Hamiltonian, and analyzed via a novel miser’s de Finetti technique to manage entanglement. The authors establish completeness and soundness guarantees for the amplified Hamiltonian H^{(2t)}, derive an auxiliary energy measurement as a diagnostic tool, and show that the procedure is iteratable to yield streaming quantum PCP and locality-gap tradeoffs. Overall, the paper provides a concrete, quasi-derandomised framework to amplify energy gaps in quantum Hamiltonians while keeping the expanded Hamiltonian tractable, with potential implications for MIP*-style protocols and sparse quantum PCP conjectures.

Abstract

The quantum PCP conjecture asks whether it is QMA-hard to distinguish between high- and low-energy Hamiltonians even when the gap between "high" and "low" energy is large (constant). A natural proof strategy is gap amplification: start from the fact that high- and low-energy Hamiltonians are hard to distinguish if the gap is small (inverse polynomial) and amplify the Hamiltonians to increase the energy gap while preserving hardness. Such a gap amplification procedure is at the heart of Dinur's proof of the classical PCP theorem. In this work, following Dinur's model, we introduce a new quantum gap amplification procedure for Hamiltonians which uses random walks on expander graphs to derandomise (subsample the terms of) the tensor product amplification of a Hamiltonian. Curiously, our analysis relies on a new technique inspired by quantum de Finetti theorems, which have previously been used to rule out certain approaches to the quantum PCP conjecture.

Derandomised tensor product gap amplification for quantum Hamiltonians

TL;DR

This work advances the quantum PCP program by introducing a derandomised gap amplification for local Hamiltonians that mimics Dinur’s classical approach but operates within the quantum setting. The core method combines derandomised tensor-product amplification with expander-walk sampling, implemented on layered, commuting components of the Hamiltonian, and analyzed via a novel miser’s de Finetti technique to manage entanglement. The authors establish completeness and soundness guarantees for the amplified Hamiltonian H^{(2t)}, derive an auxiliary energy measurement as a diagnostic tool, and show that the procedure is iteratable to yield streaming quantum PCP and locality-gap tradeoffs. Overall, the paper provides a concrete, quasi-derandomised framework to amplify energy gaps in quantum Hamiltonians while keeping the expanded Hamiltonian tractable, with potential implications for MIP*-style protocols and sparse quantum PCP conjectures.

Abstract

The quantum PCP conjecture asks whether it is QMA-hard to distinguish between high- and low-energy Hamiltonians even when the gap between "high" and "low" energy is large (constant). A natural proof strategy is gap amplification: start from the fact that high- and low-energy Hamiltonians are hard to distinguish if the gap is small (inverse polynomial) and amplify the Hamiltonians to increase the energy gap while preserving hardness. Such a gap amplification procedure is at the heart of Dinur's proof of the classical PCP theorem. In this work, following Dinur's model, we introduce a new quantum gap amplification procedure for Hamiltonians which uses random walks on expander graphs to derandomise (subsample the terms of) the tensor product amplification of a Hamiltonian. Curiously, our analysis relies on a new technique inspired by quantum de Finetti theorems, which have previously been used to rule out certain approaches to the quantum PCP conjecture.

Paper Structure

This paper contains 47 sections, 28 theorems, 126 equations.

Table of Contents

  1. Introduction
  2. Related work
  3. Quantum gap amplification.
  4. Quantum de Finetti theorems.
  5. Discussion and applications
  6. Streaming quantum PCP
  7. Locality-gap tradeoffs for local Hamiltonians.
  8. Quantum games PCP.
  9. Sparse quantum PCP.
  10. Technical overview
  11. Construction and notation
  12. Main techniques
  13. Background: tensor product amplification.
  14. Background: classical sequential repetition and its derandomisation
  15. Difficulties in quantisation
  16. ...and 32 more sections

Key Result

Theorem 1.2

Let $H$ be a $k$--local Hamiltonian with $m$ terms, satisfying the conditions in definition:layered. Then, its derandomised $2t$--fold tensor product amplification$H^{(2t)}$ (formally defined in def:amplified_ham) satisfies: where the $\Theta$ notation obscures constants dependent on the choice of expander graph family and the original Hamiltonian $H$.

Theorems & Definitions (74)

  • Definition 1.1: Dinur's template for proving the PCP theorem
  • Theorem 1.2: Gap amplification
  • Theorem 1.3: Locality-gap tradeoff; informal
  • Theorem 1.4: Streaming quantum PCP
  • Remark 1.5
  • Definition 1.6: Layered Hamiltonians
  • Definition 1.7: Paths of length $t$ in $G_\chi$
  • Remark 1.8
  • Definition 1.9: Derandomised Tensor Product Amplification
  • Remark 1.10
  • ...and 64 more