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Spectral Certificates and Sum-of-Squares Lower Bounds for Semirandom Hamiltonians

Nicholas Kocurek

TL;DR

This work studies a quantum analogue of k-XOR by formulating Hamiltonian k-XOR instances as sums of signed k-local Pauli operators and analyzes their ground-state energies under semirandom and Gaussian-signed randomness. The authors develop a classical spectral refutation approach based on a quantum variant of the Kikuchi matrix, yielding an $n^{O}(\ell)$-time algorithm that certifies ground energy no greater than $\tfrac{1}{2}+\varepsilon$ for dense semirandom or Gaussian-signed instances, with a matching refutation threshold depending on the number of terms $|\mathcal{H}|$. They provide detailed constructions for even and odd arity using degree-regularized Kikuchi matrices and, in the odd case, a bipartite decomposition plus edge-deletion to control local degrees. Additionally, they prove near-tight non-commutative SOS lower bounds by lifting classical hard $k$-XOR instances to one-basis Hamiltonians, showing that in the semirandom regime the spectral refutation is essentially optimal within ncSoS. Together, these results clarify the interplay between randomness, non-commutativity, and certifiable ground-state energy in quantum many-body systems and connect quantum refutation thresholds to classical CSP refutation theory.

Abstract

The $k$-$\mathsf{XOR}$ problem is one of the most well-studied problems in classical complexity. We study a natural quantum analogue of $k$-$\mathsf{XOR}$, the problem of computing the ground energy of a certain subclass of structured local Hamiltonians, signed sums of $k$-local Pauli operators, which we refer to as $k$-$\mathsf{XOR}$ Hamiltonians. As an exhibition of the connection between this model and classical $k$-$\mathsf{XOR}$, we extend results on refuting $k$-$\mathsf{XOR}$ instances to the Hamiltonian setting by crafting a quantum variant of the Kikuchi matrix for CSP refutation, instead capturing ground energy optimization. As our main result, we show an $n^{O(\ell)}$-time classical spectral algorithm certifying ground energy at most $\frac{1}{2} + \varepsilon$ in (1) semirandom Hamiltonian $k$-$\mathsf{XOR}$ instances or (2) sums of Gaussian-signed $k$-local Paulis both with $O(n) \cdot \left(\frac{n}{\ell}\right)^{k/2-1} \log n /\varepsilon^4$ local terms, a tradeoff known as the refutation threshold. Additionally, we give evidence this tradeoff is tight in the semirandom regime via non-commutative Sum-of-Squares lower bounds embedding classical $k$-$\mathsf{XOR}$ instances as entirely classical Hamiltonians.

Spectral Certificates and Sum-of-Squares Lower Bounds for Semirandom Hamiltonians

TL;DR

This work studies a quantum analogue of k-XOR by formulating Hamiltonian k-XOR instances as sums of signed k-local Pauli operators and analyzes their ground-state energies under semirandom and Gaussian-signed randomness. The authors develop a classical spectral refutation approach based on a quantum variant of the Kikuchi matrix, yielding an -time algorithm that certifies ground energy no greater than for dense semirandom or Gaussian-signed instances, with a matching refutation threshold depending on the number of terms . They provide detailed constructions for even and odd arity using degree-regularized Kikuchi matrices and, in the odd case, a bipartite decomposition plus edge-deletion to control local degrees. Additionally, they prove near-tight non-commutative SOS lower bounds by lifting classical hard -XOR instances to one-basis Hamiltonians, showing that in the semirandom regime the spectral refutation is essentially optimal within ncSoS. Together, these results clarify the interplay between randomness, non-commutativity, and certifiable ground-state energy in quantum many-body systems and connect quantum refutation thresholds to classical CSP refutation theory.

Abstract

The - problem is one of the most well-studied problems in classical complexity. We study a natural quantum analogue of -, the problem of computing the ground energy of a certain subclass of structured local Hamiltonians, signed sums of -local Pauli operators, which we refer to as - Hamiltonians. As an exhibition of the connection between this model and classical -, we extend results on refuting - instances to the Hamiltonian setting by crafting a quantum variant of the Kikuchi matrix for CSP refutation, instead capturing ground energy optimization. As our main result, we show an -time classical spectral algorithm certifying ground energy at most in (1) semirandom Hamiltonian - instances or (2) sums of Gaussian-signed -local Paulis both with local terms, a tradeoff known as the refutation threshold. Additionally, we give evidence this tradeoff is tight in the semirandom regime via non-commutative Sum-of-Squares lower bounds embedding classical - instances as entirely classical Hamiltonians.

Paper Structure

This paper contains 23 sections, 25 theorems, 95 equations.

Table of Contents

  1. Introduction
  2. Our results
  3. Related work
  4. Structure of the paper
  5. Preliminaries
  6. Quantum computation
  7. Pauli operators
  8. Sum-of-Squares hierarchy
  9. Non-commutative Sum-of-Squares
  10. Hamiltonian $k$-$\mathsf{XOR}$
  11. Binomial coefficient inequalities
  12. Certifying Semirandom $k$-$\mathsf{XOR}$ Hamiltonians for even $k$
  13. Step 1: Expressing $\bra{\psi}\mathbf{H}_\mathcal{I}\ket{\psi}$ as a quadratic form of a Kikuchi matrix
  14. Step 2: Bounding the spectral norm of $\tilde{K}$ via the trace moment method
  15. Certifying Semirandom $k$-$\mathsf{XOR}$ Hamiltonians for odd $k$
  16. ...and 8 more sections

Key Result

Theorem 1.2

Fix $n/2 \geq \ell \geq k/2$. There is a classical algorithm taking as input a Hamiltonian $k$-$\mathsf{XOR}$ instance $\mathcal{I} = (\mathcal{H}, \{(P_C, b_C)\}_{C \in \mathcal{H}})$ describing an $n$-qubit Hamiltonian $\mathbf{H}_{\mathcal{I}}$ and outputs a number $\mathrm{algval}(\mathbf{H}_{\m

Theorems & Definitions (83)

  • Definition 1.1: Hamiltonian $k$-$\mathsf{XOR}$
  • Theorem 1.2: Spectral refutation of semirandom $k$-$\mathsf{XOR}$ Hamiltonians
  • Corollary 1.3: Semirandom refutation of Gaussian polynomials
  • Theorem 1.4: Non-commutative Sum-of-Squares lower bounds for certifying random one-basis $k$-$\mathsf{XOR}$ Hamiltonians
  • Theorem 1.5: SoS-hardness of $k$-$\mathsf{XOR}$ $\implies$ ncSoS-hardness of Hamiltonian $k$-$\mathsf{XOR}$
  • Corollary 1.6: Theorem 1.1 HopkinsL22 + \ref{['thm:lifting']}
  • Definition 2.1: Density operator
  • Definition 2.2: Local Hamiltonians
  • Definition 2.3: Pauli operators
  • Definition 2.5: Pauli basis
  • ...and 73 more