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In quantum information theory, the no low-energy trivial state (NLTS) conjecture is a precursor to a quantum PCP theorem (qPCP) and posits the existence of families of Hamiltonians with all low-energy states of non-trivial complexity. [1] [2] [3] [4] It was formulated by Michael Freedman and Matthew Hastings in 2013. An NLTS proof would be a consequence of one aspect of qPCP problems – the inability to certify an approximation of local Hamiltonians via NP completeness. [2] In other words, an NLTS proof would be one consequence of the QMA complexity of qPCP problems. [5] On a high level, if proved, NLTS would be one property of the non- Newtonian complexity of quantum computation. [5] NLTS and qPCP conjectures posit the near-infinite complexity involved in predicting the outcome of quantum systems with many interacting states. [6] 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. [7] [6] There is currently a proof of NLTS conjecture published in preprint. [8]
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) is a Hermitian matrix acting on n qubits which can be represented as the sum of Hamiltonian terms acting upon at most qubits each:
The general k-local Hamiltonian problem is, given a k-local Hamiltonian , to find the smallest eigenvalue of . [9] 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: [2]
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
where each Hm(n) acts non-trivially on O(1) qubits. Another constraint is the operator norm of Hm(n) is bounded by a constant independent of n and each qubit is only involved in a constant number of terms Hm(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". [2]
Kliesch defines the NLTS property thus: [2]
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. [2]
Proving the NLTS conjecture is an obstacle for resolving the qPCP conjecture, an even harder theorem to prove. [1] 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. [7] 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. [6] qPCP increases the complexity by trying to solve PCP for quantum states. [6] 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. [7]
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]