What quantum volume is needed to represent a single logical qubit?

Question

The quantum volume metric $V_Q$ is a proposed metric for quantifying and comparing the performance of quantum computers¹. The quantum volume is defined as

$$V_Q = \max_{n<N} \left(\min\left[n, d(n)\right]^2\right),$$

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/average error rate of a random $SU(4)$ gate between any two qubits (including additional swap gates if the device is not fully connected)^[2].

Now setting aside the question of whether $V_Q$ is a "good" metric (asked previously here), 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?

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

^{NOTE: Cross-posted on physics.SE.}

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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).

Answer 1

Quantum volume is a bad metric for this purpose.

For example, suppose you have a ten thousand by ten thousand grid of qubits with a gate error rate of 1 in one thousand. The quantum volume of this grid is basically 0, because if you pick two qubits at random they will on average be more than one thousand steps apart. So an error will almost certainly occur as they are routed together in order to interact, and so you can't even perform 1 randomly chosen interaction.

But quantum error correction doesn't require randomly chosen interactions. The surface code only performs local interactions, and as a result the 10k x 10k with 10^-3 error grid that I described would be easily capable of large scale fault tolerant quantum computations over thousands of logical qubits.

As you make the grid bigger, the quantum volume at the physical level will continue to get worse and worse as the quantum volume at the logical level continues to get better.

Answer 2

The Quantum Volume is a benchmark for near term, noisy quantum systems. Indeed, like other random unitary benchmarks, you need to be able to sample the ideal distribution. This distribution comes from classical simulations, so your limited to about the ~40 or so qubit limit. However, the Quantum Volume itself was designed to benchmark not only the quantum device itself, but also things like control electronics, measurements, circuit compilation methods, etc, while encoding all of this in a single numerical value. Currently, IBM has shown Quantum Volume numbers for at most four qubits, 2^4=16.