DWave Systems just released its last iteration of its quantum chip, moving from an architecture based on the Chimera graph with ~2000 Qubits to a more connected architecture with the Pegasus graph and ~5400 Qubits.

Quantum volume is a new metric proposed by IBM to compare different quantum computers and their reliability [1].

We allow us to reinterpret this metric as follows: what is the volume span by the maximal number of qubits in a chip operating which do operate completely error-free and are fully interconnected? So the logarithm in base 2 of the quantum volume can be interpreted as the number of qubits fully interconnected, operating error-free.

With this reintepretation of quantum volume, we can run the Travelling Salesman Problem (TSP) on DWave's chips and get an estimation for the quantum volume. TSP requires for N-cities a complete graph K_N^2 as each city is hotcoded with N-Qubits (city 1: 0....1, city 2: 0...10, city N: 1...0). So N cities where each city is hotcoded with N qubits requires N^2 qubits and this N^2 qubits need to be fully interconnected with each other in a complete graph K_N^2. On the interconnections the distance matrix (NxN) is applied. In other words each interconnection gets a weight attached to it, corresponding to the distance between two nodes. DWave's Ocean suite then remaps the fully interconnected graph to the less connected Pegasus graph of Advantage (or to the even less connected Chimera graph) using way more qubits.

If we launch the TSP problem with the source code available here by Bohr Techonlogy on DWave's Leap platform and we add just some tweaks to punish incorrect solutions (too short by not visiting all cities or too long when visiting cities twice),

and then run it on the bare quantum chip (no hybrid things) we receive the following results:

The Dwave chip DW_2000_Q6 calculates the exact solution for 4 cities, but fails with 5 [3]. So number of qubits working precisely is for sure greater than 4^2=16 and for sure lower than 5^2 (25). The quantum volume itself is for sure at least 2^16=65'536 but below 2^25.

The Dwave chip Advantage_system1.1 calculates the exact solution for 6 cities but fails with 7 [4]. The number of qubits working precisely and fully interconnected is 6^2=36 but for 7^2=49 we get an incorrect solution. The quantum volume for Advantage is at least 2^36= but below 2^49.

IonQ currently claims a 2^22 quantum volume [2], so at least in our intepretation Advantage is far better while DW_2000_Q6 might lay behind.

One difference with our approach and the IBM one [1] which makes our lower bound more optimistic is that we choose the best proposed solution from Advantage while in [1] it is looked at the percentage of correct solutions over a certain threshold.

We might change the quantum volume reinterpretation sentence by adding "for which at least one solution in the sample is optimal", but this would invalidate IonQ and IBM calculations so far and make the quantum volume for them greater. If we leave the reinterpretation as it is, Advantage Quantum Volume is lower than 2^36...

As a complete graph with 180 nodes is embeddable directly in the Pegasus graph of the QPU, we can see them as 180 logical qubits fully interconnected. If they would operate error free, the theoretical quantum volume span by Advantage would be 2^180.

One last note, theoretically Advantage can solve a TSP up to about 13 cities in bare QPU mode, however the amount of noise is too high for the correct solution to appear. Currently, the biggest TSP solved by classical computers has N=85'900 cities, so there is still some way to go, unless the hybrid approach hides some surprises ...

[2] IonQ Releases A New 32-Qubit Trapped-Ion Quantum Computer With Massive Quantum Volume Claims

[3] Below the log of the achieved results on DW_2000_Q6 (2041 Qubits, Chimera graph):

N=4 QPU DW_2000Q_6

Brute force: [0, 1, 2, 3] 19.25965669651482

DWave: [1, 0, 3, 2] 19.25965669651482

N=5 QPU DW_2000Q_6

Brute force: [0, 3, 4, 1, 2] 24.883972062212624

DWave: [2, 4, 3, 1, 0] 32.74600931766605

[4] Below the log of the achieved results on Advantage_system1.1 (5436 Qubits, Pegasus graph):

N=5 QPU Advantage_system1.1

Brute force: [0, 3, 1, 2, 4] 18.084364670329673

DWave: [1, 3, 0, 4, 2] 18.084364670329673

N=6 QPU Advantage_system1.1

Brute force: [0, 3, 4, 2, 1, 5] 20.25544329378932

DWave: [1, 2, 4, 3, 0, 5] 20.255443293789323

N=7 QPU Advantage_system1.1

Brute force: [0, 1, 3, 6, 5, 4, 2] 23.21220870271781

DWave: [5, 1, 0, 3, 2, 4, 6] 32.50901939330402