Up to some technical subtleties, the equivalence of these two definitions follows from the quantum-circuit representation of the evolution along the adiabatic path [70].

The RG fixed points are typically defined on chains of qudits of dimension $d$. However, we can always view a local $d$-dimensional space as that of $\lceil {\log }_2(d)\rceil$ qubits. Note that any shallow circuit made of two-qudit gates on the qudit chain can always be expressed as a shallow local circuit made of two-qubit gates in the associated qubit chain, possibly up to a polylogarithmic overhead.

One could ask if $|\tilde{\varphi }[\alpha ,\beta ]⟩$ can be represented as a TI MPS (with $N$-independent tensors and without boundary tensors), for any $\alpha ,\beta \in \mathbb{C}$, or at least for a dense set of them. This is indeed the case, possibly restricting to some values of $N$. For example, by summing $n$ product states $|0{⟩}^{\otimes N}$ and $m$ product states $|1{⟩}^{\otimes N}$, we obtain a TI MPS of the form (5) with $\alpha =n/\sqrt{n^2+m^2}$ and $\beta =m/\sqrt{n^2+m^2}$, which can approximate any pair of positive real numbers. When $\alpha \in \mathbb{C}$ (we can always assume that $\beta \in {\mathbb{R}}^{+}$ up to a global phase), we can sum product states defined by local tensors with an additional phase $e^{i\phi }$, yielding the state $e^{iN\phi }|0{⟩}^{\otimes N}$. Choosing $\phi =2\pi r/s$ with $r,s\in \mathbb{N}$ and restricting to $N\equiv 1$ (mod $s$), we obtain states of the form (5) where $\alpha$ has a phase angle $\phi =2\pi r/s$.