Spectral Certificates and Non-commutative Sum-of-Squares Lower Bounds for Hamiltonians

Spectral certificates and non-commutative Sum-of-Squares lower bounds for Hamiltonians establish an efficient classical spectral algorithm to certify ground energy at most 1/2 + ε in semirandom Hamiltonian k-XOR instances, providing a tight tradeoff between time complexity and the number of local terms. This result connects the problem to classical k-XOR refutation models while demonstrating that entanglement structure is not robust to internal noise in this model.

The Gist

There is an n O(l)-time classical spectral algorithm certifying ground energy at most 1/2 + ε in (1) semirandom Hamiltonian k-XOR instances or (2) sums of Gaussian-signed k-local Paulis both with O(n·nlk/2−1 log n/ε4 local terms, a tradeoff known as the refutation threshold.

Certifying Semirandom Hamiltonians for even k

The core of the certification method relies on a novel quantum variant of the Kikuchi hierarchy, built as the signed adjacency matrix of a graph on (degree-l) Pauli operators. The algorithm expresses the ground energy, ⟨ψHIψ⟩, as a quadratic form involving this matrix: ⟨ψ HI ψ⟩ = 1/∆(ψ⊙l)†KHI ψ⊙l. To bound this value, the paper introduces a degree-regularized Kikuchi matrix K˜ = Γ−1/2A∗Γ−1/2 ⊗ I2n. The main technical component is Lemma 3.6, which provides a spectral bound on the underlying Kikuchi adjacency matrix: A˜2 ≤ O(rl log n d!) with high probability, where r is chosen as O(log N). This leads to the final bound: λmax(HI) ≤ 1/2 + A˜2, which simplifies to 1/2 + O(rl log n d!), satisfying the desired tradeoff when H is sufficiently large.

Certifying Semirandom Hamiltonians for odd k

For odd clarity, the proof extends to odd parity using a regularity decomposition algorithm (Lemma 4.4). This involves decomposing an instance I into subinstances I(t) based on a t-sparse U-bipartite decomposition, where the regularity condition ensures that any non-trivial subset H' is sufficiently large relative to the size of the set U. The certification then proceeds by applying Lemma 4.5, which uses an odd-arity Kikuchi matrix KUt to output a certificate algval(Ut) in time n O(l). This approach leverages the Cauchy-Schwarz trick (Lemma 4.3) and bounds the resulting quadratic form using Lemma 4.11, which establishes a bound on the spectral norm of the degree-regularized matrix K˜Ut, ultimately yielding λmax(HI) ≤ 1/2 + ε as desired for odd k instances.

Non-commutative Sum-of-Squares Lower Bounds

The paper proves that Theorem 1.2 is tight by showing a near-matching non-commutative Sum-of-Squares lower bound for random one-basis k-XOR Hamiltonians (Theorem 1.4). This involves lifting classical kXOR instances to non-commutative Sum-of-Squares lower bounds using Theorem 1.5, which computes a description of a Hamiltonian kXOR instance J such that val(I) = λmax(HJ) and the degree-d non-commutative Sum-of-Squares value of HJ is the degree-d Sum-of-Squares value of I. This demonstrates that classical hard XOR instances make classically hard Hamiltonians for it, which in turn makes them difficult to refute using non-commutative Sum-of-Squares for worst case.

Refutation Threshold and Tightness

The analysis shows that semirandomness allows the algorithm to beat NP in the ground state, but the one-basis assumption limits its effectiveness against non-commutative Sum-of-Squares for worst case. The paper introduces an edge deletion algorithm (Lemma 4.13) to prune the Kikuchi graph, ensuring that even when t < k/2, a subgraph AbUt with O(1)k·ε−2-bounded local degree exists. This edge deletion allows the final bound to be refined from algval∗(Ut) ≤ ε2 to algval(Ut) ≤ ε2·1/Dk s ηt τt, which yields λmax(Ut) ≤ ε2, confirming the refutation threshold observed in Theorem 1.2 is tight.

Lifting Sum-of-Squares Lower Bounds

The method for lifting classical lower bounds to non-commutative ones (Theorem 5.11) involves constructing a pseudo-expectation E˜ρ from a degree-d Boolean pseudo-expectation E˜µ for the classical instance I.

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1. Improvement: Development of a Polynomial-Time (n O(l)) Spectral Certificate for Semirandom Hamiltonian k-XOR Ground Energy.

2. Improvement: Creation of a Classically Verifiable Refutation Threshold Tradeoff for High-Dimensional Boolean Constraint Satisfaction Problems (CSP).

3. Improvement: Application of Non-commutative Sum-of-Squares (ncSoS) Lower Bounds to Certify Quantum Hardness in Hamiltonian Models.

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The improved AI system, leveraging these advancements, can perform the following specific tasks:

The improved AI system, leveraging these advancements, can perform the following specific tasks with high precision and efficiency:

A central question in quantum many-body physics is estimating the ground energy of a k-local Hamiltonian system. In this work, we present a spectral technique for certifying a lower bound on the ground energy of a random n-qubit Hamiltonian system defined as the sum of signed k-local Pauli operators. In particular, we prove that for any constant, there exists an efficiently computable length n O certificate that is always a lower bound on the ground energy with the promise that, with high probability over the random Hamiltonian distribution, the certificate value is an epsilon-good approximation of the true ground energy when the number of terms is sufficiently large. Second, we show by construction that this technique, while successful on average over random Hamiltonian systems, can fail to produce good certificates on worst-case instances. Our spectral technique for producing these certificates comes from extending classical results on k-XOR refutations to k-local Pauli Hamiltonians by crafting a quantum variant of the Kikuchi matrix for CSP refutations. To show the limitations of this technique, we prove non-commutative Sum-of-Squares lower bounds for worst-case signed k-local Pauli operators. More generally, we explore how the non-commutative Sum-of-Squares relaxation can be understood as augmenting the standard Sum-of-Squares relaxation with the commutation relations between the Pauli operators. We instantiate the resulting framework with a modification to prior quantum code-based NLTS Hamiltonians that yields stronger complexity guarantees for the low-energy space; our Hamiltonian family satisfies simultaneously (1) Ω(n) -circuit depth lower bounds for all low-energy states, (2) constant-gap NP-hardness to approximate the ground energy, and (3) a constant-gap non-commutative Sum-of-Squares integrality gap up to Ω(n) -levels.