# What quantum volume is needed to represent a single fault-tolerant logical qubit?

The quantum volume metric $$V_Q$$ is a proposed metric for quantifying and comparing the performance of quantum computers[1]. The quantum volume is defined as

$$V_Q = \max_{n

where $$n$$ is the number of qubits used (out of the maximum $$N$$ available qubits on the device), $$d(N)\simeq 1/nε_{\text{eff}}$$ is the effective/achievable circuit depth, and $$\epsilon_{\text{eff}}$$ is the effective error rate of a random two qubit gate, averaged over all two qubit combinations of the $$n\subset N$$ considered qubits including additional swap gates if the device is not fully connected (i.e. an arbitrary SU(4) applied to two random qubits, which includes one qubit gates as a subset of possible operations)[2].

Now setting aside the question of whether $$V_Q$$ is the best metric to use, can an estimate be made for what $$V_Q$$ is needed in order to have a single logical qubit? More generally, what is the number of logical qubits a device with $$V_Q$$ can support?

Note: By logical qubit I mean in the sense of fault-tolerant operation, i.e. if I have $$m$$ logical qubits I can apply an arbitrary sequence of gates, including a sequence of indefinite lenght, to the $$m$$ qubits and end up with the ideal quantum state with bounded error (e.g. probability greater than 2/3).

References
1. Bishop, Lev S., et al. "Quantum volume." Quantum Volume. Technical Report (2017) (PDF)
2. Moll, Nikolaj, et al. "Quantum optimization using variational algorithms on near-term quantum devices." Quantum Science and Technology 3.3 (2018): 030503
3. Cross, Andrew W., et al. "Validating quantum computers using randomized model circuits." arXiv preprint arXiv:1811.12926 (2018).

• I asked this question on the quantum computing stackexchange a while ago, but due to a lack of attention (and a single incorrect answer by someone who misunderstood $V_Q$) I'm re-asking here – Punk_Physicist May 14 at 22:45