Design Parameterized Quantum Circuit (PQC)#
When we use Variational Quantum Algorithm (VQA), especially Quantum Circuit Learning (QCL)1, we need to design quantum circuit.
As far as we know, designing good circuit is still active research area.
Expressivity and Entangle Capability#
S. Sim et al.2 proposed 2 metrics of PQC, aka. “expressivity” and “entangle capability”.
The expressivity is defined as a circuit’s ability to generate pure states that are well representative of the Hilbert space. Let’s say, if output quantum states are well distributed over all bloch spheres when moving parameters, the circuit has good expressivity. For example, arbitary unitary rotation gate has better expressivity than RX rotation gate.
The entangle capability is defined by Mayer-Wallach entanglement mesearment3. Emprically we know highly entangled circuits can capture good representation from data structure even though shallow depth.
The authors also calculated (numerically simulated) these metrics for 19 circuits which have been proposed before and they indicated good candidates;
Josephson Sampler 4 |
Circuit Centric 5 |
|
|---|---|---|
No. |
12 |
19 |
Parameters |
\((4n-4)L\) |
\(3nL\) |
2 qubit gates |
\((n-1)L\) |
\(nL\) |
Circuit depth |
\(6L\) |
\((n+2)L\) |
Saturate |
\(L \sim 3\) |
\(L \sim 3\) |
|
|
|
where \(n\) is number of qubits, \(L\) is number of layer repetition. Generally, if we repeat more layers (with independent parameters), expressivity and entangle capability become better, however, they saturate at some point. “Saturate” row at the table shows saturation number for \(n = 4, 6, 8\). (To be honest, it is not clear for large circuit like \(n = 100\).)
Another research conducted by T. Hubregtsen et al.6 showed these circuits worked well for classification of toy dataset.
Barren Plateaus#
J. R. McClean et al.7 pointed out there was “barren plateaus” at training of PQC. That means that the gradient of randomly initialized circuit with sufficient depth become exponentially small in the number of qubits.
- 1
K. Mitarai et al., “Quantum Circuit Learning”, Phys. Rev. A 98, 032309 (2018)
DOI: https://doi.org/10.1103/PhysRevA.98.032309
arXiv: https://arxiv.org/abs/1803.00745- 2
S. Sim et al., “Expressibility and entangling capability of parameterized quantum circuits for hybrid quantum-classical algorithms”, Adv. Quantum Technol. 2 (2019) 1900070
DOI: https://doi.org/10.1002/qute.201900070
arXiv: https://arxiv.org/abs/1905.10876- 3
D. A. Meyer and N. R. Wallach, “Global entanglement in multiparticle systems”, J. Math. Phys. 43, 4273 (2002)
DOI: https://doi.org/10.1063/1.1497700
arXiv: https://arxiv.org/abs/quant-ph/0108104- 4
M. Schuld et al., “Circuit-centric quantum classifiers”, Phys. Rev. A 101, 032308 (2020)
DOI: https://doi.org/10.1103/PhysRevA.101.032308
arXiv: https://arxiv.org/abs/1804.00633- 5
M. R. Geller, “Sampling and scrambling on a chain of superconducting qubits”, Phys. Rev. Applied 10, 024052 (2018)
DOI: https://doi.org/10.1103/PhysRevApplied.10.024052
arXiv: https://arxiv.org/abs/1711.11026- 6
T. Hubregtsen et al., “Evaluation of Parameterized Quantum Circuits: on the relation between classification accuracy, expressibility and entangling capability”, Quantum Machine Intelligence volume 3, Article number: 9 (2021)
DOI: https://doi.org/10.1007/s42484-021-00038-w
arXiv: https://arxiv.org/abs/2003.09887- 7
J. R. McClean et al., “Barren plateaus in quantum neural network training landscapes”, Nat Commun 9, 4812 (2018)
DOI: https://doi.org/10.1038/s41467-018-07090-4
arXiv: https://arxiv.org/abs/1803.11173