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There are many possible ways to compactly represent a state, the usefulness of which strongly depend on the context.

First of all, it is important to notice that it is not possible to have a procedure that can map any state into a more efficient representation of the same state (for the same reason why it is obviously not possible to faithfully compress any 2-bit string as a 1-bit string, with a mapping that does not depend on the string).

However, as soon as you start making some assumptions, you can find more efficient ways to represent a state in a given context. There is a multitude of possible ways to do this, so I'll just mention a few that come to mind:

1. Already the standard vector representation of a ket state can be thought of as a "compressed representation", that works under the assumption of the state being pure. Indeed, you need $4^n-1$ real degrees of freedom to represent an arbitrary (generally mixed) $n$-qubit state, but only $2^{n+1}-2$ to represent a pure one.

2. If you assume a state $\rho$ to be almost pure, that is, such that $\rho$ is sparse in some representation (equivalently, $\rho$ is low rank), then again the state can be efficiently characterised. For a $d$-dimensional system (so $d=2^n$ for an $n$-qubit system), instead of using ~$d^2$ parameters, you can have a faithful representation using only $\mathcal O(r d \log^2 d)$, where $r$ is the sparsity of the state (see 0909.3304 and the works that came after that).

3. If you are only interested in a limited number $|S|$ of expectation values, you can find a compressed representation of an $n$-qubit state of size $\mathcal O(n\log(n)\log(|S|))$. Note that this amounts to an exponential reduction. This was shown (I think) in quant-ph/0402095, but the introduction given in 1801.05721 may be more accessible for a physicist (as well as presenting improvements in the optimisation method). See references in this last paper for a number of similar results.

4. If you know that the entanglement of the state is limited (in a sense that can be precisely defined), then again efficient representations can be found, in terms of tensor networks (an introduction is found e.g. in 1708.00006). More recently, it was also shown that ground states of some notable Hamiltonians can be represented using machine-learning-inspired ansatze ( (1606.02318 and many following works). This was also recently shown/claimed to be equivalent to a specific Tensor Network representation however (1710.04045) so I'm not sure whether it should go to a category of its own.

Note that in all of the above you can more efficiently represent a given state, but to then simulate the evolution of the system you generally need do go back to the original inefficient representation. If you want to efficiently represent the dynamics of a state through a given evolution, you again need assumptions on the evolution for this to be possible. The only result that comes to mind on this regard is the classical (as in enstablished, not as in "non quantum") Gottesman-Knill theorem, which allows to efficiently simulate any Clifford quantum circuit.

There are many possible ways to compactly represent a state, the usefulness of which strongly depends on the context.

First of all, it is important to notice that it is not possible to have a procedure that can map any state into a more efficient representation of the same state (for the same reason why it is obviously not possible to faithfully compress any 2-bit string as a 1-bit string, with a mapping that does not depend on the string).

However, as soon as you start making some assumptions, you can find more efficient ways to represent a state in a given context. There is a multitude of possible ways to do this, so I'll just mention a few that come to mind:

1. Already the standard vector representation of a ket state can be thought of as a "compressed representation", that works under the assumption of the state being pure. Indeed, you need $4^n-1$ real degrees of freedom to represent an arbitrary (generally mixed) $n$-qubit state, but only $2^{n+1}-2$ to represent a pure one.

2. If you assume a state $\rho$ to be almost pure, that is, such that $\rho$ is sparse in some representation (equivalently, $\rho$ is low rank), then again the state can be efficiently characterised. For a $d$-dimensional system (so $d=2^n$ for an $n$-qubit system), instead of using ~$d^2$ parameters, you can have a faithful representation using only $\mathcal O(r d \log^2 d)$, where $r$ is the sparsity of the state (see 0909.3304 and the works that came after that).

3. If you are only interested in a limited number $|S|$ of expectation values, you can find a compressed representation of an $n$-qubit state of size $\mathcal O(n\log(n)\log(|S|))$. Note that this amounts to an exponential reduction. This was shown (I think) in quant-ph/0402095, but the introduction given in 1801.05721 may be more accessible for a physicist (as well as presenting improvements in the optimisation method). See references in this last paper for a number of similar results.

4. If you know that the entanglement of the state is limited (in a sense that can be precisely defined), then again efficient representations can be found, in terms of tensor networks (an introduction is found e.g. in 1708.00006). More recently, it was also shown that ground states of some notable Hamiltonians can be represented using machine-learning-inspired ansatze ( (1606.02318 and many following works). This was also recently shown/claimed to be equivalent to a specific Tensor Network representation however (1710.04045) so I'm not sure whether it should go to a category of its own.

Note that in all of the above, you can more efficiently represent a given state, but to then simulate the evolution of the system you generally need to go back to the original inefficient representation. If you want to efficiently represent the dynamics of a state through a given evolution, you again need assumptions on the evolution for this to be possible. The only result that comes to mind in this regard is the classical (as in established, not as in "non-quantum") Gottesman-Knill theorem, which allows to efficiently simulate any Clifford quantum circuit.

There are many possible ways to compactly represent a state, the usefulness of which strongly depend on the context.

First of all, it is important to notice that it is clearly not possible to have a procedure that can map any state into a more efficient representation of the same state (for the same reason why it is obviously not possible to faithfully compress any 2-bit string as a 1-bit string, with a mapping that does not depend on the string).

However, as soon as you start making some assumptions, you can find more efficient ways to represent a state in a given context. There is a multitude of possible ways to do this, so I'll just mention a few that comes to mind:

1. Already the standard vector representation of a ket state can be thought of as a "compressed representation", that works under the assumption of the state being pure. Indeed, you need $4^n-1$ real degrees of freedom to represent an arbitrary (generally mixed) $n$-qubit state, but only $2^{n+1}-2$ to represent a pure one.

2. If you assume a state $\rho$ to be almost pure, that is, such that $\rho$ is sparse in some representation (equivalently, $\rho$ is low rank), then again the state can be efficiently characterised. For a $d$-dimensional system (so $d=2^n$ for an $n$-qubit system), instead of using ~$d^2$ parameters, you can have a faithful representation using only $\mathcal O(r d \log^2 d)$, where $r$ is the sparsity of the state (see 0909.3304 and the works that came after that).

3. If you are only interested in a limited number $|S|$ of expectation values, you can find a compressed representation of an $n$-qubit state of size $\mathcal O(n\log(n)\log(|S|))$. Note that this amounts to an exponential reduction. This was shown (I think) in quant-ph/0402095, but the introduction given in 1801.05721 may be more accessible for a physicist (as well as presenting improvements in the optimisation method). See references in this last paper for a number of similar results.

4. If you know that the entanglement of the state is limited (in a sense that can be precisely defined), then again efficient representations can be found, in terms of tensor networks (an introduction is found e.g. in 1708.00006). More recently, it was also shown that ground states of some notable Hamiltonians can be represented using machine-learning-inspired ansatze ( (1606.02318 and many following works). This was also recently shown/claimed to be equivalent to a specific Tensor Network representation however (1710.04045) so I'm not sure whether it should go to a category of its own.

Note that in all of the above you can more efficiently represent a given state, but to then simulate the evolution of the system you generally need do go back to the original inefficient representation. If you want to efficiently represent the dynamics of a state through a given evolution, you again need assumptions on the evolution for this to be possible. The only result that comes to mind on this regard is the classical (as in enstablished, not as in "non quantum") Gottesman-Knill theorem, which allows to efficiently simulate any Clifford quantum circuit.

There are many possible ways to compactly represent a state, the usefulness of which strongly depend on the context.

First of all, it is important to notice that it is not possible to have a procedure that can map any state into a more efficient representation of the same state (for the same reason why it is obviously not possible to faithfully compress any 2-bit string as a 1-bit string, with a mapping that does not depend on the string).

However, as soon as you start making some assumptions, you can find more efficient ways to represent a state in a given context. There is a multitude of possible ways to do this, so I'll just mention a few that come to mind:

1. Already the standard vector representation of a ket state can be thought of as a "compressed representation", that works under the assumption of the state being pure. Indeed, you need $4^n-1$ real degrees of freedom to represent an arbitrary (generally mixed) $n$-qubit state, but only $2^{n+1}-2$ to represent a pure one.

2. If you assume a state $\rho$ to be almost pure, that is, such that $\rho$ is sparse in some representation (equivalently, $\rho$ is low rank), then again the state can be efficiently characterised. For a $d$-dimensional system (so $d=2^n$ for an $n$-qubit system), instead of using ~$d^2$ parameters, you can have a faithful representation using only $\mathcal O(r d \log^2 d)$, where $r$ is the sparsity of the state (see 0909.3304 and the works that came after that).

3. If you are only interested in a limited number $|S|$ of expectation values, you can find a compressed representation of an $n$-qubit state of size $\mathcal O(n\log(n)\log(|S|))$. Note that this amounts to an exponential reduction. This was shown (I think) in quant-ph/0402095, but the introduction given in 1801.05721 may be more accessible for a physicist (as well as presenting improvements in the optimisation method). See references in this last paper for a number of similar results.

4. If you know that the entanglement of the state is limited (in a sense that can be precisely defined), then again efficient representations can be found, in terms of tensor networks (an introduction is found e.g. in 1708.00006). More recently, it was also shown that ground states of some notable Hamiltonians can be represented using machine-learning-inspired ansatze ( (1606.02318 and many following works). This was also recently shown/claimed to be equivalent to a specific Tensor Network representation however (1710.04045) so I'm not sure whether it should go to a category of its own.

Note that in all of the above you can more efficiently represent a given state, but to then simulate the evolution of the system you generally need do go back to the original inefficient representation. If you want to efficiently represent the dynamics of a state through a given evolution, you again need assumptions on the evolution for this to be possible. The only result that comes to mind on this regard is the classical (as in enstablished, not as in "non quantum") Gottesman-Knill theorem, which allows to efficiently simulate any Clifford quantum circuit.

There are many possible ways to compactly represent a state, the usefulness of which strongly depend on the context.

First of all, it is important to notice that it is clearly not possible to have a procedure that can map any state into a more efficient representation of the same state (for the same reason why it is obviously not possible to faithfully compress any 2-bit string as a 1-bit string, with a mapping that does not depend on the string).

However, as soon as you start making some assumptions, you can find more efficient ways to represent a state in a given context. There is a multitude of possible ways to do this, so I'll just mention a few that come to mind:

1. Already the standard vector representation of a ket state can be thought of as a "compressed representation", that works under the assumption of the state being pure. Indeed, you need $4^n-1$ real degrees of freedom to represent an arbitrary (generally mixed) $n$-qubit state, but only $2^{n+1}-2$ to represent a pure one.

2. If you assume a state $\rho$ to be almost pure, that is, such that $\rho$ is sparse in some representation (equivalently, $\rho$ is low rank), then again the state can be efficiently characterised. For a $d$-dimensional system (so $d=2^n$ for an $n$-qubit system), instead of using ~$d^2$ parameters, you can have a faithful representation using only $\mathcal O(r d \log^2 d)$, where $r$ is the sparsity of the state (see 0909.3304 and the works that came after that).

3. If you are only interested in a limited number $|S|$ of expectation values, you can find a compressed representation of an $n$-qubit state of size $\mathcal O(n\log(n)\log(|S|))$. Note that this amounts to an exponential reduction. This was shown (I think) in quant-ph/0402095, but the introduction given in 1801.05721 may be more accessible for a physicist (as well as presenting improvements in the optimisation method). See references in this last paper for a number of similar results.

4. If you know that the entanglement of the state is limited (in a sense that can be precisely defined), then again efficient representations can be found, in terms of tensor networks (an introduction is found e.g. in 1708.00006). More recently, it was also shown that ground states of some notable Hamiltonians can be represented using machine-learning-inspired ansatze ( (1606.02318 and many following works). This was also recently shown/claimed to be equivalent to a specific Tensor Network representation however (1710.04045) so I'm not sure whether it should go to a category of its own.

There are many possible ways to compactly represent a state, the usefulness of which strongly depend on the context.

First of all, it is important to notice that it is clearly not possible to have a procedure that can map any state into a more efficient representation of the same state (for the same reason why it is obviously not possible to faithfully compress any 2-bit string as a 1-bit string, with a mapping that does not depend on the string).

However, as soon as you start making some assumptions, you can find more efficient ways to represent a state in a given context. There is a multitude of possible ways to do this, so I'll just mention a few that comes to mind:

1. Already the standard vector representation of a ket state can be thought of as a "compressed representation", that works under the assumption of the state being pure. Indeed, you need $4^n-1$ real degrees of freedom to represent an arbitrary (generally mixed) $n$-qubit state, but only $2^{n+1}-2$ to represent a pure one.

2. If you assume a state $\rho$ to be almost pure, that is, such that $\rho$ is sparse in some representation (equivalently, $\rho$ is low rank), then again the state can be efficiently characterised. For a $d$-dimensional system (so $d=2^n$ for an $n$-qubit system), instead of using ~$d^2$ parameters, you can have a faithful representation using only $\mathcal O(r d \log^2 d)$, where $r$ is the sparsity of the state (see 0909.3304 and the works that came after that).

3. If you are only interested in a limited number $|S|$ of expectation values, you can find a compressed representation of an $n$-qubit state of size $\mathcal O(n\log(n)\log(|S|))$. Note that this amounts to an exponential reduction. This was shown (I think) in quant-ph/0402095, but the introduction given in 1801.05721 may be more accessible for a physicist (as well as presenting improvements in the optimisation method). See references in this last paper for a number of similar results.

4. If you know that the entanglement of the state is limited (in a sense that can be precisely defined), then again efficient representations can be found, in terms of tensor networks (an introduction is found e.g. in 1708.00006). More recently, it was also shown that ground states of some notable Hamiltonians can be represented using machine-learning-inspired ansatze ( (1606.02318 and many following works). This was also recently shown/claimed to be equivalent to a specific Tensor Network representation however (1710.04045) so I'm not sure whether it should go to a category of its own.

Note that in all of the above you can more efficiently represent a given state, but to then simulate the evolution of the system you generally need do go back to the original inefficient representation. If you want to efficiently represent the dynamics of a state through a given evolution, you again need assumptions on the evolution for this to be possible. The only result that comes to mind on this regard is the classical (as in enstablished, not as in "non quantum") Gottesman-Knill theorem, which allows to efficiently simulate any Clifford quantum circuit.