Quantum Brainstorm

Notes about previous and future quantum brainstorm sessions.

FUTURE

https://arxiv.org/pdf/2310.12726.pdf

Swap test : phase problem (Lila)

24 October 2023

Chiara - QCELS:

Early fault tolerant phase estimation algorithm:

• split up the phase estimation into separate Hadamard tests, only one ancilla qubit required for each test
• Heisenberg limit (1/eps rather than 1/eps^2) scaling retained
• T_max is much lower than in conventional QPE => early Fault tolerant
• only requires a squares overlap with the GS of 0.5

Paper 1: https://journals.aps.org/prxquantum/pdf/10.1103/PRXQuantum.4.020331

Paper 2 (about one month later, they realised their analysis wasn't optimal - you only need a squared overlap with the GS of 0.5, rather than 0.71): https://arxiv.org/abs/2303.05714

Nice talk from IPAM 2023: https://www.youtube.com/watch?v=j-MaQtgzksY

7 November 2023

Choy Boy - QML, classification:

• For a variational quantum supervised classifier, data is loaded (most commonly using rotation encoding via rotation gates), and an ansatz is used to minimise the cost function associated with the data's labels
• Parameter shift rule: can use the same circuit to evaluate cost function + gradient in cost function, highly parallelisable
• Parameter shift rule holds true even under noisy conditions due to trace linearity (https://johannesjakobmeyer.com/blog/004-noisy-parameter-shift/)
• Ansatze for local cost functions do not exhibit barren plateaus (eg https://arxiv.org/pdf/2102.01828.pdf)
• MSE loss function computed classically from quantum measurements
• Quantum state preparation (loading classical data into a quantum state) is possibly the largest bottleneck? Can do this with qGANs (problem: large number of samples)
• Hypothesis that local minima could give rise to better classification rates than the global minimum under certain scenarios, in a similar vein to QAOA where certain graphs at finite number of ansatz layers have local minima with a higher probability of finding the correct max-cut solution than the global minimum

Other things

Appendix B in: https://journals.aps.org/pra/pdf/10.1103/PhysRevA.98.062324

• Looks at the effect of applying a small perturbation to one of the angles in the Ansatz (depolarising error) - due to the unitarity of the gates, the perturbation gets washed out (fidelity susceptibility is independent of the layer index in which the perturbation occurs)
• Figure 7 shows the distribution of gradients for the MMD loss function: distribution is shown to be independent of the layer; also, variance decays exponentially with the number of shots taken

Can switch loss function to get a slightly different landscape: https://www.mdpi.com/1099-4300/23/10/1281

Recent paper that compares global vs local cost functions for binary and multi-class classification: for binary classification global cost function seems to perform better, while multiclass classification local cost function seems to perform better: https://iopscience.iop.org/article/10.1088/2632-2153/acb12f/pdf

Barren plateaus

Would be nice to have an explanation of exactly what a barren plateau is... gradient and variance in gradient decays exponentially to zero with system size? Unique to quantum circuits...

• More expressive means more likely to exhibit barren plateaus: https://journals.aps.org/prxquantum/abstract/10.1103/PRXQuantum.3.010313