Quality measures

• number of physical qubits
• effective space size
• number of gates
• number of parameterized gates
• depth
• depth on LNN
• energy span

No. qubits and effective space size

XY-QAOA for TSP

mixer: \(X_iX_j + Y_iY_j\)

• \(n^2\) qubits are needed
• with good mixer, only one hot states are present (for example \(|001\rangle |010\rangle |001\rangle\))
• There is only \(n^n\) of them
• effective space space size is \(\log(n^n) = n\log n\)
• both lower bouned by log of the number of solutions

No. (parameterized) gates

• The larger number of gates - the more destructive noise
• number of parameterized gates influences the cost of estimating analytical gradient.
• both lower bounded by the number of degrees of freedom in the problem (for example number of possible edges in Max-K-Cut)

Depth (on LNN)

Energy span

From Hoeffding Theorem

Better than the state-of-the-art way for VQE

Max-K-Cut

• X-QAOA - X mixer, standard QUBO,
• XY-QAOA - XY mixer, standard QUBO
• HOBO - binary encoding, X-mixer
• SIM-QAOA - as in Fuchs, Franz G., et al. "Efficient Encoding of the Weighted MAX-k-CUT on a Quantum Computer Using QAOA." SN Computer Science 2.2 (2021): 1-14.

Very bad when \(K=2^k+1\)

• All colors have meaning - last color is multiplied
• Fix incorrectly assumed different colors

• we start in product of W-like state in a binary encoding
• quantum version of the classical pseudocode,
• for double loop: use swap network
• Dependency on K basically disappeared

TSP

number of qubits times depth at least \(O(n^3)\) for

TSP - SIM-QAOA

• We start in binary encoding like in HOBO (product of \(n\) W-state-like states in binary encoding
• we choose Grover mixer for our purpose (different ones can be used)
• Hamiltonian implemented through a quantum version of the classical pseudocode (with some parallelization + swapping strategy)

TSP - SIM-QAOA

TSP - SIM-QAOA