  The threshold theorem is a fundamental result in the theory of fault-tolerant
quantum computation stating that arbitrarily long quantum computations can be
performed with a polylogarithmic overhead provided the noise level is below a
constant level. A recent work by Fawzi, Grospellier and Leverrier (FOCS 2018)
building on a result by Gottesman (QIC 2013) has shown that the space overhead
can be asymptotically reduced to a constant independent of the circuit provided
we only consider circuits with a length bounded by a polynomial in the width.
In this work, using a minimal model for quantum fault tolerance, we establish a
general lower bound on the space overhead required to achieve fault tolerance.
  For any non-unitary qubit channel $\mathcal{N}$ and any quantum fault
tolerance schemes against $\mathrm{i.i.d.}$ noise modeled by $\mathcal{N}$, we
prove a lower bound of
$\max\left\{\mathrm{Q}(\mathcal{N})^{-1}n,\alpha_\mathcal{N} \log T\right\}$ on
the number of physical qubits, for circuits of length $T$ and width $n$. Here,
$\mathrm{Q}(\mathcal{N})$ denotes the quantum capacity of $\mathcal{N}$ and
$\alpha_\mathcal{N}>0$ is a constant only depending on the channel
$\mathcal{N}$. In our model, we allow for qubits to be replaced by fresh ones
during the execution of the circuit and we allow classical computation to be
free and perfect. This improves upon results that assumed classical
computations to be also affected by noise, and that sometimes did not allow for
fresh qubits to be added. Along the way, we prove an exponential upper bound on
the maximal length of fault-tolerant quantum computation with amplitude damping
noise resolving a conjecture by Ben-Or, Gottesman, and Hassidim (2013).
