The short story, including the question:
Cluster states are used for measurement-based quantum computation in which a circuit is simulated by way of single-site measurements of the bulk of a quantum state. I have been told repeatedly that 2d cluster states cannot simulate universal fault-tolerant quantum computation (UFTQC). However, Raussendorf's thesis explicitly argues that the 2d cluster state can perform UFTQC even in the presence of non-symmetric noise channels before every measurement (he does depolarizing channel specifically). Can $2$d cluster states perform UFTQC or not?
Important clarification:
I believe the paper Fault-tolerant quantum computation with high threshold in two dimensions does not answer my question. That paper is actually constructing a 3d cluster in time from 2d cluster state fragments. I quote:
Mapping to the 2D lattice. The dimensionality of the spatial layout can be reduced by one if the cluster is created slice by slice. That is, we convert the axis of "simulated time"—introduced as a means to explain the connection with surface codes—into real time.
Please note that Raussendorf's claim in his thesis is different. His claim is for a purely 2d cluster state with depolarizing noise immediately before measurements.
To put this one more way, if I allow building the state in time (say to prevent idling), then I would be building a 2d cluster state from 1d fragments, not a 3d cluster state from 2d fragments.
The long story:
Variants of the following claim are often stated:
While a cluster state with two-dimensional (2D) connectivity is required for universality, a three-dimensional (3D) cluster state is necessary for additionally achieving fault tolerance.
and
However, to advance toward fault-tolerant quantum computing, three-dimensional (3D) cluster states are ultimately required [16, 17]. In particular, a 3D cluster state of the Raussendorf-Harrington-Goyal (RHG) lattice— which is a foliation of the surface code for topological quantum computing [18–21]—introduces fault tolerance to the MBQC paradigm [16, 17].
Both of these claims are stated directly in the paper Generation of three-dimensional cluster entangled state.
These sorts of claims are additionally implied in other parts of the literature.
In particular, in the SPT literature, the 3d RBH state is privileged because it is a so-called $\mathbb{Z}^{(1)}_2 \times \mathbb{Z}^{(1)}_2$-protected SPT with "one-form" symmetries, and one-form symmetries (unlike zero-form symmetries) are viewed as resilient to weak amounts of explicit symmetry breaking through the formulation of emergent symmetries at low energy scales. As there are no such one-form SPTs in 2d (just $\mathbb{Z}^{(1)}_2 \times \mathbb{Z}^{(0)}_2$ or purely zero-form $\mathbb{Z}^{(0)}_2$), the 2d SPT phases are expected to be destroyed on adding small amounts of explicit symmetry breaking. It is often viewed that the MBQC and SPT properties are tied together, and so explicit symmetry breaking will destroy computational properties in 2d. This is implied in e.g. Two-dimensional symmetry-protected topological phases and transitions in open quantum systems
Given that the 2D cluster state possesses universal computational capabilities in the context of measurement-based quantum computation, the topological phase transition found here can also be interpreted as a transition in the computational power.
They find an immediate transition out of the 2d $\mathbb{Z}^{(1)}_2 \times \mathbb{Z}^{(0)}_2$ phase of the cluster state on the Lieb lattice for noise that explicitly breaks the zero-form symmetry that is identified with a loss of computational power.
I have also heard that the 2d cluster state on a Lieb lattice has repetition codes on its boundaries, and so can again only handle appropriately-symmetric noise.
Putting aside the Lieb lattice, I have heard further claims that the cluster state on the square lattice, which is merely a subsystem symmetry SPT, also cannot perform fault tolerant universal computation because generic errors will destroy the subsystem symmetry. In fact, I have heard the square lattice cluster state presented as even less fault-tolerant than the Lieb lattice cluster state (although, given that one can construct the Lieb lattice cluster state from a square lattice cluster state, this does not seem like a reasonable statement).
To reiterate, I have both read and heard from multiple people that 2d cluster states generically cannot perform fault tolerant computation.
However, Raussendorf's thesis argues that the 2d cluster state can indeed perform UFTQC. The main discussion is in chapter 4 including and after page 123. The idea, if I understand it correctly, is that the 2d cluster state can simulate 1d circuits. One might worry that errors on the cluster state might become terrible errors on the simulated circuit, but this isn't the case. Errors like depolarizing noise on the 2d cluster state can be mapped to relatively benign Markovian noise on the simulated 1d circuit. Fault tolerant universal quantum computation can be performed with only local gates in 1d (a result I still find somewhat surprising) by Aharonov and Ben-Or as well as Gottesman. To summarize, we can choose to simulate these 1d fault tolerant circuits with the 2d cluster state, and then errors on the cluster state will map to errors in the simulated circuit, but the fault tolerance and threshold theorem for the simulated circuit takes care of these errors, giving a threshold theorem for the cluster state.
I should emphasize that when I say fault tolerant, I mean definitions like in Nielsen and Chuang page 481 with threshold theorems, where there is a threshold under which the error of the output can be made arbitrarily small with at most polylog multiplicative increase in cost. This is a strong statement.
Putting everything together, it naively seems that there is a direct contradiction in the literature about the computational properties of 2d cluster states. What's going on?
