Table 2 VQA and settings used for the hardware experiments in Figs. 3–5. An initial parameter \({| +\rangle}^{\otimes N}\) means that the initial angles of all \(R_{y}\) in the first (second) layer of the ansatz are set to 0 (\(\pi /2\)). The last line highlights some key findings from our experiments

Ansatz

5

>5

Fig. 2 (p = 2)

–

Eq. (7) (p = 2)

–

Fig. 2 (p = 2)

–

Fig. 2 (p = 2)

Fig. 2 (p = 1)

Initial param.

\({|+\rangle}^{\otimes N}\)

uniform in [0,π]

\({|+\rangle}^{\otimes N}\)

\({|+\rangle}^{\otimes N}\)

Objective

CVaR Eq. (6)

(α = 0.5)

CVaR Eq. (8)

(α = 0.5)

Mean energy Eq. (9)

Custom Eq. (13)

Optimizer

COBYLA

COBYLA

Eq. (12)

Eq. (14)

No. shots

5

10

12

16

23

1000

–

–

–

–

1000

–

–

–

–

1000

–

–

–

–

1000

500

550

650

450

Quantum chip

5

10

12

16

23

multiple

–

–

–

–

multiple

–

–

–

–

multiple

–

–

–

–

multiple

ibmq_toronto

ibmq_guadalupe

ibmq_manhattan

ibmq_manhattan

Key findings

Flexible ansatz;

converges slower than F-VQE

Ansatz fixed by problem topology;

poor convergence likely due to noise

Flexible ansatz;

strongly varying performance across runs; converges slower than F-VQE

Flexible ansatz;

fastest, most consistent convergence