NLTS conjecture

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In quantum information theory, the no low-energy trivial state (NLTS) conjecture is a lower bound on the complexity of certain classes of quantum states which was conjectured by Michael Freedman and Matthew Hastings in 2013.^[1] It was partly intended to be a weaker consequence of a conjectural quantum PCP theorem which would be easier to prove than a full quantum PCP theorem.^[2][3][4]

A proof of the NLTS conjecture was given in 2023 by Anurag Anshu, Nikolas Breuckmann, and Chinmay Nirkhe, and was presented at STOC 2023.^[5]

NLTS is a consequence of one aspect of qPCP problems – and posits the existence of families of Hamiltonians with all low-energy states of non-trivial complexity. In other words, it claims the inability to certify an approximation of local Hamiltonians via NP completeness.^[3] In other words, it is a consequence of the QMA complexity of qPCP problems.^[6] On a high level, it is one property of the non-Newtonian complexity of quantum computation.^[6] NLTS and qPCP conjectures posit the near-infinite complexity involved in predicting the outcome of quantum systems with many interacting states.^[7] These calculations of complexity would have implications for quantum computing such as the stability of entangled states at higher temperatures, and the occurrence of entanglement in natural systems.^[8][7]

The NLTS property is the underlying set of constraints that forms the basis for the NLTS conjecture.^{[citation needed]}

A k-local Hamiltonian (quantum mechanics) ${\displaystyle H}$ is a Hermitian matrix acting on n qubits which can be represented as the sum of ${\displaystyle m}$ Hamiltonian terms acting upon at most ${\displaystyle k}$ qubits each:

${\displaystyle H=\sum _{i=1}^{m}H_{i}.}$

The general k-local Hamiltonian problem is, given a k-local Hamiltonian ${\displaystyle H}$, to find the smallest eigenvalue ${\displaystyle \lambda }$ of ${\displaystyle H}$.^[9] ${\displaystyle \lambda }$ is also called the ground-state energy of the Hamiltonian.

The family of local Hamiltonians thus arises out of the k-local problem. Kliesch states the following as a definition for local Hamiltonians in the context of NLTS:^[3]

Let I ⊂ N be an index set. A family of local Hamiltonians is a set of Hamiltonians {H⁽ⁿ⁾}, n ∈ I, where each H⁽ⁿ⁾ is defined on n finite-dimensional subsystems (in the following taken to be qubits), that are of the form

${\displaystyle H^{(n)}=\sum _{n}H_{m}^{(n)},}$

where each H_m⁽ⁿ⁾ acts non-trivially on O(1) qubits. Another constraint is the operator norm of H_m⁽ⁿ⁾ is bounded by a constant independent of n and each qubit is only involved in a constant number of terms H_m⁽ⁿ⁾.

In physics, topological order^[10] is a kind of order in the zero-temperature phase of matter (also known as quantum matter). In the context of NLTS, Kliesch states: "a family of local gapped Hamiltonians is called topologically ordered if any ground states cannot be prepared from a product state by a constant-depth circuit".^[3]

Kliesch defines the NLTS property thus:^[3]

Let I be an infinite set of system sizes. A family of local Hamiltonians {H⁽ⁿ⁾}, n ∈ I has the NLTS property if there exists ε > 0 and a function f : N → N such that

• for all n ∈ I, H⁽ⁿ⁾ has ground energy 0,
• ⟨0ⁿ|U^†H⁽ⁿ⁾U|0ⁿ⟩ > εn for any depth-d circuit U consisting of two qubit gates and for any n ∈ I with n ≥ f(d).

There exists a family of local Hamiltonians with the NLTS property.^[3]

Proving the NLTS conjecture is an obstacle for resolving the qPCP conjecture, an even harder theorem to prove.^[2] The qPCP conjecture is a quantum analogue of the classical PCP theorem. The classical PCP theorem states that satisfiability problems like 3SAT are NP-hard when estimating the maximal number of clauses that can be simultaneously satisfied in a hamiltonian system.^[8] In layman's terms, classical PCP describes the near-infinite complexity involved in predicting the outcome of a system with many resolving states, such as a water bath full of hundreds of magnets.^[7] qPCP increases the complexity by trying to solve PCP for quantum states.^[7] Though it hasn't been proven yet, a positive proof of qPCP would imply that quantum entanglement in Gibbs states could remain stable at higher-energy states above absolute zero.^[8]

NLTS on its own is difficult to prove, though a simpler no low-error trivial states (NLETS) theorem has been proven, and that proof is a precursor for NLTS.^[11]

NLETS is defined as:^[11]

Let k > 1 be some integer, and {H_n}_{n ∈ N} be a family of k-local Hamiltonians. {H_n}_{n ∈ N} is NLETS if there exists a constant ε > 0 such that any ε-impostor family F = {ρ_n}_{n ∈ N} of {H_n}_{n ∈ N} is non-trivial.