In
quantum information theory
Quantum information is the information of the quantum state, state of a quantum system. It is the basic entity of study in quantum information theory, and can be manipulated using quantum information processing techniques. Quantum information re ...
, 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.
An NLTS proof would be a consequence of one aspect of qPCP problems – the inability to
certify
Certification is the provision by an independent body of written assurance (a certificate) that the product, service or system in question meets specific requirements. It is the formal attestation or confirmation of certain characteristics of a ...
an approximation of local Hamiltonians via
NP-completeness
In computational complexity theory, a problem is NP-complete when:
# it is a problem for which the correctness of each solution can be verified quickly (namely, in polynomial time) and a brute-force search algorithm can find a solution by trying ...
.
In other words, an NLTS proof would be one consequence of the
QMA complexity of qPCP problems.
On a high level, if proved, NLTS would be one property of the non-
Newtonian complexity of
quantum computation
Quantum computing is a type of computation whose operations can harness the phenomena of quantum mechanics, such as superposition, interference, and entanglement. Devices that perform quantum computations are known as quantum computers. Though ...
.
NLTS and qPCP conjectures posit the near-infinite complexity involved in predicting the outcome of quantum systems with many interacting states.
These calculations of complexity would have implications for quantum computing such as the stability of
entangled states at higher
temperatures
Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer.
Thermometers are calibrated in various temperature scales that historically have relied o ...
, and the occurrence of entanglement in natural systems.
There is currently a proof of NLTS conjecture published in preprint.
NLTS property
The NLTS property is the underlying set of constraints that forms the basis for the NLTS conjecture.
Definitions
Local hamiltonians
A k-local
Hamiltonian (quantum mechanics)
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified Hamiltonian ...
is a
Hermitian matrix
In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the -th row and -th column is equal to the complex conjugate of the element in the -th ...
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
.
is 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:
''Let I ⊂ N be an index set. A family of local Hamiltonians is a set of Hamiltonians 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 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
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
,
topological
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing h ...
order
is a kind of order in the zero-temperature
phase of matter
In the physical sciences, a phase is a region of space (a thermodynamic system), throughout which all physical properties of a material are essentially uniform. Examples of physical properties include density, index of refraction, magnetizati ...
(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.''"
NLTS property
Kliesch defines the NLTS property thus:
''Let I be an infinite set of system sizes. A family of local Hamiltonians 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''
* '' > 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.
Quantum PCP conjecture
Proving the NLTS conjecture is an
obstacle
An obstacle (also called a barrier, impediment, or stumbling block) is an object, thing, action or situation that causes an obstruction. Different types of obstacles include physical, economic, biopsychosocial, cultural, political, technological ...
for resolving the qPCP conjecture, an even harder theorem to prove.
The qPCP conjecture is a quantum analogue of the classical PCP theorem. The classical PCP theorem states that
satisfiability
In mathematical logic, a formula is ''satisfiable'' if it is true under some assignment of values to its variables. For example, the formula x+3=y is satisfiable because it is true when x=3 and y=6, while the formula x+1=x is not satisfiable over ...
problems like
3SAT
In logic and computer science, the Boolean satisfiability problem (sometimes called propositional satisfiability problem and abbreviated SATISFIABILITY, SAT or B-SAT) is the problem of determining if there exists an interpretation that satisfie ...
are
NP-hard
In computational complexity theory, NP-hardness ( non-deterministic polynomial-time hardness) is the defining property of a class of problems that are informally "at least as hard as the hardest problems in NP". A simple example of an NP-hard pr ...
when estimating the max number of clauses that can be simultaneously satisfied in a hamiltonian system.
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
A magnet is a material or object that produces a magnetic field. This magnetic field is invisible but is responsible for the most notable property of a magnet: a force that pulls on other ferromagnetic materials, such as iron, steel, nickel, ...
.
qPCP increases the complexity by trying to solve PCP for
quantum states
In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement in quantum mechanics, measurement on a system. Knowledge of the quantum state together with the rul ...
.
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
Absolute zero is the lowest limit of the thermodynamic temperature scale, a state at which the enthalpy and entropy of a cooled ideal gas reach their minimum value, taken as zero kelvin. The fundamental particles of nature have minimum vibration ...
.
NLETS proof
NLTS on its own is difficult to prove, though a simpler No Low-Error Trivial States (NLETS) has been proven, and that proof is a precursor for NLTS.
NLETS is defined as: ''Let k > 1 be some integer and n∈N be a family of k-local Hamiltonians. n∈N is NLETS if there exists a constant ε > 0 such that any ε-impostor family F = n∈N of n∈N is non-trivial.''
References
{{Authority control
Quantum information theory
Conjectures