Analysis and Experimental Demonstration of Amplitude Amplification for Combinatorial Optimization

TL;DR

This work extends the conventional 2-dimensional representation of Grover's (orthogonal collective states) to oracles which encode cost functions such as QUBO, and shows that linear cost functions are a special case whereby an exact formula exists for determining optimal oracle parameter settings.

Abstract

Quantum Amplitude Amplification (QAA), the generalization of Grover's algorithm, is capable of yielding optimal solutions to combinatorial optimization problems with high probabilities. In this work we extend the conventional 2-dimensional representation of Grover's (orthogonal collective states) to oracles which encode cost functions such as QUBO, and show that linear cost functions are a special case whereby an exact formula exists for determining optimal oracle parameter settings. Using simulations of problem sizes up to 40 qubits we demonstrate QAA's algorithmic performance across all possible solutions, with an emphasis on the closeness in Grover-like performance for solutions near the global optimum. We conclude with experimental demonstrations of generalized QAA on both IBMQ (superconducting) and IonQ (trapped ion) qubits, showing that the observed probabilities of each basis state match our equations as a function of varying the free parameters in the oracle and diffusion operators.

Figures (17)

• Figure 1: Quantum circuit for the diffusion operator $U_s(\theta)$ acting on $N$ qubits.
• Figure 2: Five complex plane plots of $|\Psi\rangle$ after the application of the operators shown above each panel, corresponding to the 10-qubit C$(Z)$ given in appendix \ref{['sec:prob_instance']}. The value of $p_s$ used in $U_C$ is from equation \ref{['Eqn.ssp_ps_opt3']}, for $C_i=2$. Each colored point in the plots represents the amplitudes contained within one collective state $|C_i\rangle$, with $N_i$ indicated by the accompanying color scale. The extrema states $|C_{\textrm{min}}\rangle$ and $|C_{\textrm{max}}\rangle$ are plotted as square and triangle markers respectively. In each panel the value of $\bar{\alpha}$ is depicted by the red $\times$, and the origin of the complex plane with a black +. A red-dotted line is drawn in each panel from $\bar{\alpha}$ to the collective state $|C_i=2\rangle$, illustrating their $\pi$ phase difference via the line's intersection with the origin.
• Figure 3: (solid-color lines) Combined peak probability as a function of $p_s$ for the ten highest (lowest) $|C_i\rangle$ states, for the $N=20$ linear C$(Z)$ composed of the weights $\mathbb{W}_2$ from appendix \ref{['sec:exp_states']}. (gray-dashed lines) The $p_s$ values produced from equation \ref{['Eqn.ssp_ps_opt3']} for the integers $T \in [-222,-209]$, with $N_i$ values reported atop each line.
• Figure 4: Simulations of QAA showing probability versus iterations $k$ for qubit sizes $N=10,20,30,40$, comparing the joint probability of $|C_i=2\rangle$ and its inverse (solid-black lines) to $|m\rangle$ for standard Grover's (dashed-red lines) for $N_m=2$. The $p_{\textrm{s}}$ value used for each $U_C$ is from equation \ref{['Eqn.ssp_ps_opt3']}, while the weights of each linear C($Z$) are $\mathbb{W}_1$ from appendix \ref{['sec:exp_states']}.
• Figure 5: (top) Plot of peak achievable probabilities for each $|C_i\rangle$ state (colored circles) as a function of $\sigma(p_s)$ using $p_s$ from equation \ref{['Eqn.ssp_ps_opt3']}, for the $N=40$ cases $\mathbb{W}_1$ and $\mathbb{W}_3$ given in appendix \ref{['sec:exp_states']}. (bottom) Plot of iterations $k$ corresponding to the probabilities shown in the top plot. For comparison, lines indicating the number of iterations for Grover's ($U_G(\pi)$) using various $N_m$ are plotted. Beside each line is $\Delta k$, showing a 5% increase in $k$ from Grover's.