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Given a quantum register $q_{0},\dots, q_{n-1}$ composed of $n$ qubits, I wonder if there exists a general mathematical framework to generate the quantum circuit to put this quantum register in a particular state $\lvert\psi\rangle$ from the ground initial state $\lvert 0\dots 0\rangle$ and assuming the quantum circuit can only use a given set of universal gates or native ones, which depend on the qubit type.

Also, a similar problem is how to implement an observable with this same universal gate set, or how to decompose this observable into a sequence of Pauli gates, which means which quantum circuit will simulate this observable.

I ask these questions because I used to design quantum algorithms assuming I can reach relatively easily the initial step required for this algorithm and that I can implement also easily an observable from which to retrieve meaningful classic data.

So, understanding the complexity of both initial state and observable decomposition seems to me of prior importance.

deb2014
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2 Answers2

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By a simple counting argument (see e.g Nielsen & Chuang Sec. 4.5.4), we know that almost all unitaries (or quantum states) have exponential circuit complexity. This should not come as a surprise, as this is already the case for classical Boolean functions (shown by Shannon in 49). See also https://arxiv.org/abs/2205.06977

Most of research is thus concentrated on studying poly-sized circuits and what can be approximated with those. Concerning quantum states, important efficiently preparable classes are e.g. matrix product states (with bound bond dimension).

Markus Heinrich
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Software packages like Qiskit allow for decomposition of arbitrary unitary matrices into universal gates, as detailed in this answer. Slightly tangential (and a trickier problem) is this paper discusses how to decompose $SU(N)$ operations into SNAP and displacement (universal gates in the 3D circuit QED architecture).

banercat
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