Problem specific ion native ansatz for combinatorial optimization

Abstract

Variational quantum algorithms have become a standard approach for solving a wide range of problems on near-term quantum computers. Identifying an appropriate ansatz configuration for variational algorithms, however, remains a challenging task, especially when taking into account restrictions imposed by real quantum platforms. This motivated the development of digital-analog quantum circuits, where sequences of quantum gates are alternated with natural Hamiltonian evolutions. A prominent example is the use of the controllable long-range Ising interaction induced in ion-based quantum computers. This interaction has recently been applied to develop an algorithm similar to the quantum approximate optimization algorithm (QAOA), but native to the ion hardware. The performance of this algorithm has demonstrated a strong dependence on the strengths of the individual ion-ion interactions, which serve as ansatz hyperparameters. In this work, we propose a heuristic for identifying a problem-specific ansatz configuration, which enhances the trainability of the ion native digital-analog circuit. The proposed approach is systematically applied to random instances of the Sherrington-Kirkpatrick Hamiltonian for up to 15 qubits, providing favorable cost landscapes. As the result, the developed approach identifies a well-trainable ion native ansatz, which requires a lower circuit depth to solve specific problems as compared to standard QAOA. This brings the algorithm one step closer to its large scale practical implementation.

Figures (7)

• Figure 1: Typical cost landscape for the energy \ref{['cost-energy-single-layer']} of a single-layered ion native ansatz as a function of variational parameters $(\beta,\gamma) \in \Theta$ calculated using different configurations of hyperparameters: a) asymmetric ($A_i = 1$, $\forall i \neq 1$; $A_1 = -0.3$), b) $\bm{A}^*$ trained by our heuristic and c) $\alpha \bm{A}^*$ rescaled by the factor $\alpha = 0.55$ after training. Green points depict the global minima.
• Figure 2: Performance of the ion native QAOA in terms of the fractional error $1-r$ as a function of the circuit depth $p$. The results are obtained for (i) the asymmetric configuration ($A_i = 1$, $\forall i \neq 1$; $A_1 = -0.3$), shown in green, and (ii) found by the heuristic, shown in blue. Red dashed line depicts the threshold, which guarantees the ground state overlap $g(\psi) = 0.5$ from the stability lemma biamonte2021universal. The inset depicts the coefficients $K_{ij}$ for the considered SK problem instance \ref{['H-SK']}.
• Figure 3: Fraction of SK instances \ref{['H-SK']} solved with the ion native QAOA for different system sizes $n$ and plotted a) after each cycle of training $p=n$ layered QAOA circuit and b) after the 4-th cycle of training as a function of the QAOA circuit depth $p$.
• Figure 4: Fraction of SK instances \ref{['H-SK']} for $n=6$ and 8 qubits solved by the ion native and standard QAOA as a function of circuit depth $p$. The results for the ion-based QAOA are obtained using problem-specific hyperparameters found by the proposed heuristic.
• Figure 5: KL divergence \ref{['DKL']} as a function of the circuit depth $p$ for $n=6$ qubits. The results are obtained for the ion native ansatz \ref{['ion-native-ansatz']} using the asymmetric configuration $\bm{A}$ (green), using $\bm{A}^*$ found by the heuristic (blue), and for standard QAOA (red). The considered SK instance is shown in the inset of Fig. \ref{['fig:single_instance']}a. Note that lower values of $D_{\rm KL}$ correspond to a more expressible circuit.