# State preparation

## Table of Contents

State preparation is an essential task in quantum science; for metrology, quantum simulation, or communication. Clearly, states should be efficiently preparable – but which states fit the bill?

### MPS in log depth #

For one-dimensional systems, one of the most important classes of states are matrix-product states (MPS). All MPS can be prepared with a sequential (staircase) quantum circuit (and thus a depth linear in system size) from a product state, and all sequential quantum circuits produce MPS. But can this be done faster?

In a recent preprint [1] we have explicitly provided an algorithm (a way to construct a quantum circuit) that can be used to prepare essentially all MPS (this includes perhaps the most relevant class, unique ground states of local Hamiltonians, but not GHZ or W states) very rapidly.
Loosely speaking, given an MPS with correlation length \(\xi\) on \(N\) sites we show that it is enough to have a circuit depth of \( \xi\,\log(N) \), with a (small) constant in front.
Moreover, we can even show that this is *optimal*, i.e., that there are no algorithms that are asymptotically faster.
Another cool thing we manage to show is that if you can additional perform measurements in your quantum circuit (and gates conditional on the outcome) you can prepare the same MPS in a depth scaling with \(\mathrm{loglog}(N)\), which is exponentially faster, and you get the ability to prepare long-range correlated states such as GHZ states (or in general superpositions of short-range correlated MPS).

*Preparation of matrix product states with log-depth quantum circuits*

Malz*, Styliaris*, Wei*, Cirac, PRL (2024), arXiv

### Sequentially prepared states in higher dimension #

What about higher dimensions? Interestingly, there exist projected entangled-pair states (PEPS, the higher-dimensional generalization of MPS) that despite small bond dimension require exponential depth to prepare, so the answer is not as easy as in one dimension.

Together with Zhi-Yuan Wei and Ignacio Cirac, we introduced a natural generalization of sequentially prepared states to higher dimensions: the plaquette-PEPS [2]. These are states that can be prepared with a quantum circuit comprising a number of mutually overlapping plaquette unitaries that scales linearly in system size. We derive some of the properties of these states, and find that they can represent a wide variety of known states, including ground states of many topological models. The big advantage of defining states via quantum circuits is that this automatically provides an algorithm to prepare them on a quantum processors. In a second part, we put forward a way to produce these states as flying photons in a circuit QED platform [3]. Circuit QED is a promising platform to prepare large-scale entangled photonic states, as is demonstrated in experimental work we have been involved in [4].

*Sequential generation of projected entangled-pair states*

Wei*, Malz*, Cirac, PRL (2022), arXiv*Generation of photonic tensor network states with Circuit QED*

Wei, Cirac, Malz, PRA (2022), arXiv.*Realizing a Deterministic Source of Multipartite-Entangled Photonic Qubits*

Besse, Reuer et al, ncomms (2021), arXiv.

### Computational complexity #

If a state has a low circuit complexity, it is also easy to classically simulate it.
Conversely, if a computationally hard problem can be embedded into a state, then it cannot be *too* easy to prepare it.
Complexity is an important notion for variational ansatz states, because if the complexity of a variational family is too low, we cannot use the family to solve hard problems.
On the other hand, if it’s too high, we cannot use the family full stop.

Isometry tensor network states, introduced in (Zaletel and Pollmann, 2020) seem to inhabit a sweet spot: they can capture many states, so they aren’t trivial, but they also not too hard. But how hard are they exactly? This is what we set out to answer in Ref. [5].

We use what we have learned about sequential circuits to assess the computational complexity of isometric tensor network states (isoTNS) [2]. The key twist in this work is that properties (say, local expectation values) of 2D isoTNS can be mapped to an equivalent (1+1)D open quantum circuit, which lets us map the complexity of local expectation values to simulating quantum circuits with depolarizing noise, and sampling complexity to measurement-induced phase transitions.

*Computational complexity of isometric tensor network states*

Malz, Trivedi, arXiv

### Adiabatic state preparation #

Finally, we have also considered adiabatic preparation. This is well known to be a powerful approach, and others have proven (if you assume that there’s a gap) that you can prepare some tensor network states very rapidly. In Ref. [6] we tested how well this works in practise and we introduced some innovations on the adiabatic path that lets us propose a very efficient adiabatic algorithm to prepare the AKLT state in two dimensions.

*Efficient Adiabatic Preparation of Tensor Network States*

Wei*, Malz*, Cirac, PRR (2023), arXiv