Local hamiltonians

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:^[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

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

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⁾.

Topological order

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]

NLTS property

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).

NLTS conjecture

There exists a family of local Hamiltonians with the NLTS property.^[2]