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Matrix product state (MPS) is a way to write down an area-law-entangled many-body quantum state in 1D. Depending on the dimensionality and size of the system an MPS can more or less efficiently approximate the full state.

It's important to understand that while it may be possible to describe the full many-body state accurately using much less information than the full state has ("much" means exponentially less), it is not obvious that it is possible to do it within an exponentially more efficient algorithm than exact diagonalization.

Density matrix renormalization group (DMRG) is an efficient method to find the optimal MPS representation of the many-body state. For example, it can target ground states of gapped 1D systems accurately. However, that only scratches the surface of its applicability. It's important to state here that DMRG can target general states that can be well approximated by MPS and not just ground states, however, to do that you need to do Shift-Inversion with some iterative $Ax=b$ solver. Recently there has been a lot of research progress along this direction, using DMRG method to target highly-excited (finite-energy-density) states in many-body localization (MBL) systems, and the method is now called DMRG-X (with "X" stands for excited states). It is also important to state that DMRG may not be the most efficient algorithm, but it is the most common algorithm to optimize MPS.

Matrix product state (MPS) is a way to write down many-body quantum states. It's a natural representation for infinite 1D states that bipartite entanglement entropy obeys area-law ($S = constant$). This doesn't mean that it can't represent finite systems which are not 1D and $S = F(L)$, where L is some dimension of the system. Depending on the dimensionality and size of the system an MPS can more or less efficiently approximate the full state.

It's important to understand that while it may be possible to describe the full many-body state accurately using much less information than the full state has ("much" means exponentially less), it is not obvious that it is possible to do it within an exponentially more efficient algorithm than exact diagonalization.

Density matrix renormalization group (DMRG) is an efficient method to find the optimal MPS representation of the many-body state. For example, it can target ground states of gapped 1D systems accurately. However, that only scratches the surface of its applicability. It's important to state here that DMRG can target general states that can be well approximated by MPS and not just ground states, however, to do that you need to do Shift-Inversion with some iterative $Ax=b$ solver. Recently there has been a lot of research progress along this direction, using DMRG method to target highly-excited (finite-energy-density) states in many-body localization (MBL) systems, and the method is now called DMRG-X (with "X" stands for excited states). It is also important to state that DMRG may not be the most efficient algorithm, but it is the most common algorithm to optimize MPS.

Matrix product state (MPS) is a "method" to write down an area-law-entangled many-body quantum state in 1D. Depending on the dimensionality and size of the system an MPS can more or less efficiently approximate the full state.

It's important to understand that while it may be possible to describe the full many-body state accurately using much less information than the full state has ("much" means exponentially less), it is not obvious that it is possible to do it within an exponentially more efficient algorithm than exact diagonalization.

DMRG is an efficient method to perform that optimization. For example, it can target ground states of gapped 1D systems accurately. However, that only scratches the surface of its applicability. It's important to state here that DMRG can target general states that can be well approximated by MPS and not just ground states, however, to do that you need to do Shift-Inversion with some iterative $Ax=b$ solver. It is also important to state that DMRG may not be the most efficient algorithm, but it is the most common algorithm to optimize MPS.

Matrix product state (MPS) is a way to write down an area-law-entangled many-body quantum state in 1D. Depending on the dimensionality and size of the system an MPS can more or less efficiently approximate the full state.

It's important to understand that while it may be possible to describe the full many-body state accurately using much less information than the full state has ("much" means exponentially less), it is not obvious that it is possible to do it within an exponentially more efficient algorithm than exact diagonalization.

Density matrix renormalization group (DMRG) is an efficient method to find the optimal MPS representation of the many-body state. For example, it can target ground states of gapped 1D systems accurately. However, that only scratches the surface of its applicability. It's important to state here that DMRG can target general states that can be well approximated by MPS and not just ground states, however, to do that you need to do Shift-Inversion with some iterative $Ax=b$ solver. Recently there has been a lot of research progress along this direction, using DMRG method to target highly-excited (finite-energy-density) states in many-body localization (MBL) systems, and the method is now called DMRG-X (with "X" stands for excited states). It is also important to state that DMRG may not be the most efficient algorithm, but it is the most common algorithm to optimize MPS.

Matrix Product State is an "method" to write down an arbitrary Quantum State. Depending on the Dimensionality and Size of the system a MPS can more or less efficiently approximate the full state.

It's important to understand that while it may be possible to describe the Full State accurately using much less information than the full state has -much means Exponentially less-, it is not obvious that it is possible to do it within an exponentially more efficient algorithm than exact diagonalization.

DMRG is an efficient method to perform that optimization. For example it can target ground states of gapped 1D systems accurately. However that only scratches the surface of it's applicability. It's important to state here that DMRG can target general states that can be well approximated by MPS and not just ground states, however to do that you need to do Shift-Invertion with some iterative $Ax=b$ solver. It is also important to state that DMRG may not be the most efficient algorithm, but it is the most common algorithm to optimize MPS.

Matrix product state (MPS) is a "method" to write down an area-law-entangled many-body quantum state in 1D. Depending on the dimensionality and size of the system an MPS can more or less efficiently approximate the full state.

It's important to understand that while it may be possible to describe the full many-body state accurately using much less information than the full state has ("much" means exponentially less), it is not obvious that it is possible to do it within an exponentially more efficient algorithm than exact diagonalization.

DMRG is an efficient method to perform that optimization. For example, it can target ground states of gapped 1D systems accurately. However, that only scratches the surface of its applicability. It's important to state here that DMRG can target general states that can be well approximated by MPS and not just ground states, however, to do that you need to do Shift-Inversion with some iterative $Ax=b$ solver. It is also important to state that DMRG may not be the most efficient algorithm, but it is the most common algorithm to optimize MPS.