> For the complete documentation index, see [llms.txt](https://docs.quantaria.xyz/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://docs.quantaria.xyz/deep-dives/staircase-circuits.md).

# Staircase circuits: preparing an MPS on qubits

A matrix product state can be prepared on a quantum computer by a circuit whose gates follow the shape of the chain. This page explains the staircase construction that Quantaria's quantum path cites.

## The construction

An MPS with bond dimension 2ᵏ can be prepared sequentially. Start with qubits in the zero state. Apply gates in a staircase, each acting on k + 1 qubits: the new qubit and the k qubits that carry the bond. Each gate produces one core of the MPS.

The reference is Schön et al., sequential generation of entangled states ([arXiv:quant-ph/0501096](https://arxiv.org/abs/quant-ph/0501096)).

## Why it matters

The circuit's structure mirrors the tensor network. A Quantaria core maps to one gate, so a layer already in MPS form does not need to be re-derived to become a circuit.

## Limits

* The gates act on k + 1 qubits, which must be implemented on the hardware. Gates on several qubits are harder to run than single or two-qubit gates.
* The circuit depth grows with the number of cores. Noise accumulates across the staircase.
* A bond of 2ᵏ needs k bits of entanglement. Larger bonds need more qubits per gate.

## What the docs show

The [Quantum Path](https://github.com/bhavy2908/dataai_docs/tree/main/architecture/quantum-path.md) page includes a staircase diagram for a bond dimension of 4 (k = 2), with three gates on five qubit wires. The figure is illustrative of the construction, not a Quantaria circuit.

## Quantaria's use

Staircase preparation is a research step in the quantum path. Hilbert-1 does not run on quantum hardware. See [Quantum Path](https://github.com/bhavy2908/dataai_docs/tree/main/architecture/quantum-path.md).

## Related

* [MERA and disentanglers](/deep-dives/mera-and-disentanglers.md)
* [Bond dimension and entanglement entropy](/deep-dives/bond-dimension-and-entanglement.md)


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