Toward Reliability in the NISQ Era: Robust Interval Guarantee for Quantum Measurements on Approximate States
This repo contains the code for the paper "Toward Reliability in the NISQ Era: Robust Interval Guarantee for Quantum Measurements on Approximate States" [1]. A tutorial that explains the implementation in Tequila is available here.
- Tequila together with psi4 as chemistry backend (see the Tequila repository for installation).
- Quantum backends: Qiskit (< 0.25) for simulations with noise and Qulacs for noiseless simulations.
To compute robustness intervals for eigenvalues of a given Hamiltonian, the first step is to run a variational quantum eigensolver. The second step is to compute required statistics, namely expectation values and fidelities and, if necessary, variances.
For example, to use a noisy VQE with an SPA Ansatz [2] to estimate bond dissociation curves for
python run_vqe.py --molecule h2 --ansatz spa --noise bitflip-depol --error-rate 0.01 --samples 8192 --backend qiskit
This runs VQE and saves the results in the subdir ./results/dir/to/results. The second step is to compute the
statistics. For example, to get the Gramian eigenvalue bound, we run the command
python compute_stats.py --results-dir ./results/dir/to/results --which hamiltonian
Finally, to compute the Gramian eigenvalue intervals, we run the command
python compute_interval.py --results-dir ./results/dir/to/results --method gramian-eigval
For the above example, this results in the following table:
| r | exact | fidelity | vqe_energy | lower_bound | upper_bound |
|---|---|---|---|---|---|
| 0.50 | -1.077562481 | 0.958205382 | -0.988021299 | -1.083851669 | -0.892573143 |
| 0.60 | -1.131093394 | 0.957995880 | -1.049706365 | -1.134667839 | -0.964082683 |
| 0.70 | -1.149766772 | 0.958045453 | -1.078714705 | -1.154566071 | -1.003039667 |
| 0.75 | -1.151648186 | 0.957997992 | -1.084870980 | -1.156489296 | -1.013617719 |
| 0.80 | -1.150463131 | 0.956361844 | -1.083803309 | -1.154374306 | -1.012676374 |
| 0.90 | -1.141818498 | 0.957287300 | -1.082813293 | -1.145940205 | -1.019863177 |
| 1.00 | -1.128477954 | 0.956343543 | -1.074174701 | -1.132620953 | -1.015699398 |
| 1.25 | -1.089243435 | 0.955575508 | -1.041665818 | -1.092590918 | -0.991141821 |
| 1.50 | -1.053578279 | 0.953605150 | -1.012057922 | -1.057875400 | -0.966444444 |
| 1.75 | -1.026226767 | 0.947645608 | -0.986626289 | -1.031472964 | -0.942610056 |
| 2.00 | -1.007246702 | 0.940898351 | -0.969707699 | -1.014640383 | -0.924539724 |
| 2.25 | -0.994987898 | 0.938526329 | -0.958604338 | -1.001223562 | -0.915913894 |
| 2.50 | -0.987402706 | 0.936911723 | -0.951663413 | -0.993183626 | -0.910199765 |
| 2.75 | -0.983085755 | 0.927166048 | -0.945083506 | -0.989858075 | -0.900383804 |
The script ./analyze/make_figure_gramian_eigenvalue_bound.py can be used to generate the corresponding figure:
[1] Maurice Weber, Abhinav Anand, Alba Cervera-Lierta, Jakob S. Kottmann, Thi Ha Kyaw, Bo Li, Alán Aspuru-Guzik, Ce Zhang and Zhikuan Zhao. "Toward Reliability in the NISQ Era: Robust Interval Guarantee for Quantum Measurements on Approximate States", arXiv:2110.09793 (2021).
[2] Jakob S. Kottmann, Alán Aspuru-Guzik. "Optimized Low-Depth Quantum Circuits for Molecular Electronic Structure using a Separable Pair Approximation", arXiv:2105.03836 (2021).
[3] Jakob S. Kottmann, Philipp Schleich, Teresa Tamayo-Mendoza, and Alán Aspuru-Guzik. "Reducing Qubit Requirements while Maintaining Numerical Precision for the Variational Quantum Eigensolver: A Basis-Set-Free Approach", J. Phys. Chem. Lett.12, 663 (2021).
[4] Jakob S. Kottmann, Florian A. Bischoff, and Edward F. Valeev. "Direct determination of optimal pair-natural orbitals in a real-space representation: The second-order Moller–Plesset energy", J. Chem. Phys. 152, 074105 (2020).