Figure 1.
The Swap (left) and the Hadamard (right) tests.
Figure 2.
Implementation of the Quantum Fourier Transform. (Reproduced from https://commons.wikimedia.org/wiki/File:Q_fourier_nqubits.pngviaWikimediaCommons).
Figure 3.
Phase Estimation Algorithm.
Figure 4.
Schematic representation of the HHL algorithm ([16]).
Figure 5.
Diagram of the VQLS algorithm ([24]).
Table 1.
Comparison of the quantum solvers.
| Algorithm | Complexity | Ref. | Type | Limitations |
|---|---|---|---|---|
| HHL | O(s2κ2 log n/ϵ) | [13] | eigendecomposition | phase estimation, matrix density, condition number, Hamiltonians |
| WZP | [18] | eigendecomposition | phase estimation, condition number | |
| Row and column iteration | [20] | iterative | ancilla qubits, condition number | |
| VQLS | [24] | hybrid | ansatz option, condition number | |
| Random walks | ≥ O(log n) per xi | [27] | hybrid, Monte Carlo | scalability |
Figure 6.
Inner product circuits for 〈Φ|Φ〉 (left) and |〈b|Φ〉|2 (right).
Figure 7.
The convergence of the optimizers.
Figure 8.
The cost function of the optimizers (left) and the corresponding residuals |b−Ax| (right).
Figure 9.
Density matrix of the COBYLA quantum solution vector.
Figure 10.
The initial ansatz structure (Ry, CZ gates) on the left and the proposed ansatz structure (U gates) on the right.
Figure 11.
Density matrix of the quantum solution vector produced by the proposed ansatz.
Figure 12.
Comparison of the ansatz structures A = 0.55𝕀 + 0.45Z3 (left) and A = 0.3Z1 + 0.4Z2 (right). The metrics shown are 〈Φ|Φ〉, |〈b|Φ〉|, and the cost . Here 〈Φ|Φ〉 indicates the norm of A|x(a)〉, while |〈b|Φ〉| measures the raw overlap with |b〉. Since these two quantities are not meaningful in isolation, the cost is the definitive convergence criterion. The additional panel reports the cost reduction Δ = CostRy-CZ − CostU, which is the quantity used to evaluate the improvement achieved by the U-gates ansatz. The trends are averaged over 10 runs. While statistical uncertainties are not quantified due to resource limitations, the consistent reductions suggest improved convergence for the U-gates ansatz.