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What is the underlying physical principle behind quantum fault tolerance in quantum computation?
I am trying to follow the mathematical steps behind quantum fault tolerance, and while I can just manage to do so, I have to admit I fail to see the underlying principle behind them, or the motivation for just why it works. I kind of understand how entanglement swapping with a heat sink lies behind quantum error correction, but what underlying principler lies behind quantum fault tolerance?
I would encourage you to think of error correction not as entanglement-swapping, per se, but the stimulated dissipation of excitations from a carefully engineered Hamiltonian — not the Hamiltonian which governs the unitary evolution, but a different Hamiltonian in whose ground-state manifold one thinks of trying to keep the state of the system, by dissipating excitations into a system of low temperature. Then we can describe fault tolerance as the threshold at which the rate of stimulated dissipation is such that the probability of an uncontrolled spontaneous dissipation can be efficiently suppressed by any desired factor.
Against the role of entanglement — per se — in error correction
The picture of 'entanglement swapping' which you describe brings to mind the question of coherence, and the issue of the state of the computation being isolated from the environment: in the quantum circuit model, we would quite like (short of what transformations are being applied) for the state of the computation to be essentially closed off from the outside world. This is the classic catch-22 behind the engineering problem of designing a quantum computer: to be open to the environment only when you want it to be (because you are part of the environment), and to control when it is open to the environment (from within the environment). It invites us to think of the problem in terms of a double-bind. Furthermore, not all errors necessarily would manifest as entanglement with the environment: an environment which deterministically performs Pauli operations on each of our qubits in some order would not be entangling and would be just as bad from an error perspective as anything which did involve entanglement.
Error correction as a form of Hamiltonian engineering
An alternative view is that the error correcting process is a sort of dissipative process.
Each detectable error corresponds to a sort of excitation of the system — an event corresponding perhaps to a projector onto a partial error syndrome — and what we are attempting to determine is whether the state of the system has become "excited" because it has a substantial overlap with the 'signature' of some error due to interaction with an environment with non-zero temperature.
In the case of stabilizer codes, for instance, you could think of the Pauli generators $S_1, S_2, \ldots$ of the code as giving rise to projectors $P_j = \tfrac12(\mathbf 1 - S_j)$. The Hamiltonian is then $H = \sum_j P_j$, and the codespace is the ground state manifold — that is, the lowest-energy eigenspace: the zero eigenspace, i.e. the joint zero eigenspace of the projectors, as opposed to the joint +1-eigenspace that one often considers. The error syndrome is then a description of a collection of commuting excitations of the projectors $P_j$ representing energy contributions — the error-free syndrome corresponding to the highly degenerate ground-state manifold, $\mathrm{null}(H)$.
What we attempt to do is to measure the energy terms in the Hamiltonian frequently enough, not to mention accurately enough, to detect the excitations (errors). Then, we attempt to dissipate away these excitations (correct the errors) the dissipation process being the efficient transmission of the excitation (error) out of the state of the computation and into some auxiliary system.
Furthermore, we would like to do it quickly enough so that the system does not accumulate too much error, and then spontaneously and uncontrollably dissipate the excitations. For the error correcting code to be useful, it has to encode a highly degenerate subspace: thus, the error-excitations which we are trying to dissipate are excitations of a highly degenerate Hamiltonian. Because of the degeneracy of the ground-state manifold, there is a risk that a spontaneous relaxation of an excited state to a ground state could give rise to a non-trivial transformation of the ground-state manifold: an uncorrectable encoded error.
In all of the above, entangled states will sometimes be involved, and one may think of codes such as the Shor 9-qubit code as exploiting entanglement (correlations in multiple bases of measurement) to represent a quantum analogue of a simple repetition code. But in all this, entanglement is an incidental feature of the more fundamental phenomenon of correlations of multi-qubit observables representing excitations (or lack of them) in a physical process. That is, the role of entanglement here is the same as in many-body systems: it just so happens that the ground states of Hamiltonians may be entangled, but the mere presence or absence of entanglement is not what makes the states ground-states.
Competing dissipation processes
Above, I talk about spontaneous versus stimulated dissipation. However, this is a distinction of engineering, not of physics: the fact that the 'stimulated' process is under our control is not relevant from a physical perspective. Physically, we merely have two interaction/dissipation processes with the computational system: an uncontrolled 'noise' process with a bath, and a controlled dissipation process which we might think of as happening with a cold reservoir (corresponding to the ancillas which we prepare to aid with the error correction process). The error rate per unit time is the interaction of the bath with the computational system; the error rate per gate is the effective three-way interaction of the bath with the reservoir and computational system. The error correction is the interaction of the computational system with the cold reservoir alone.
In this picture, what would it look like if we could make the interaction with the reservoir dominate these interactions of the computational system with the bath, or the bath-system-reservoid interaction? It would look as if any interaction of the bath with the system would be more and more efficiently transmitted into the cold reservoir, with the computational system left essentially undisturbed.
This is how we can interpret what the threshold theorem tells us — if the interaction of the system with the warm bath is sufficiently low, we can simulate a bath at arbitrarily low temperature by simulating a coupling of the computational system with a cold reservoir, where the transmission of excitations from the system to the reservoir is arbitrarily swift; and furthermore the resources to do so is $\mathrm{poly} \log(\varepsilon)$, where $\varepsilon > 0$ is the factor of suppression of noise in the computational system.
