Ion-native variational ansatz for quantum approximate optimization

Variational quantum algorithms involve training parametrized quantum circuits using a classical coprocessor. An important variational algorithm, designed for combinatorial optimization, is the quantum approximate optimization algorithm. Realization of this algorithm on any modern quantum processor requires either embedding a problem instance into a Hamiltonian or emulating the corresponding propagator by a gate sequence. For a vast range of problem instances this is impossible due to current circuit depth and hardware limitations. Hence we adapt the variational approach—using ion-native Hamiltonians—to create ansatz families that can prepare the ground states of more general problem Hamiltonians. We analytically determine symmetry protected classes that make certain problem instances inaccessible unless this symmetry is broken. We exhaustively search over six qubits and consider up to 20 circuit layers, demonstrating that symmetry can be broken to solve all problem instances of the Sherrington-Kirkpatrick Hamiltonian. Going further, we numerically demonstrate training convergence and levelwise improvement for up to 20 qubits. Specifically these findings widen the class problem instances which might be solved by ion based quantum processors. Generally these results serve as a test bed for quantum approximate optimization approaches based on system native Hamiltonians and symmetry protection.

Figure 1

Energy minimization of an easy instance with respect to the QAOA ansatz (7). Here we set $A_j=1$ for all qubits $j$. The energy [see (2)] gradually decreases with increasing depth until minimization. Inset: SK instance that is being minimized; red thin edges depict $K_{jk}=+1$, while blue thick ones depict $K_{jk}=-1$.

Figure 2

Energy minimization of a hard instance with respect to the QAOA ansatz (7). Here we set $A_j=1$ for all qubits $j$. The energy [see (2)] gradually decreases with increasing depth yet stagnates at some intermediate value (here $\sim -5$). Inset: SK instance that is being minimized; red thin edges depict $K_{jk}=+1$, while blue thick ones depict $K_{jk}=-1$.

Figure 3

Energy minimization for the instance considered in Fig. 2, with respect to the QAOA ansatz (7). Here we demonstrate three random assignments of $A_j$, shown in different colors. We see that the energy monotonically decreases with depth and eventually reaches the minimum. We contrast this with the performance that was recorded with the symmetric configuration: $A_j=1$, depicted by the blue curve.

Figure 4

Fraction of SK instances with $n=6$ that could be minimized by the proposed QAOA ansatz at each depth. For the orange curve we have set $A_j=1$ for all qubits $j$. For the blue curve we use randomly assigned values of $A_j$ provided $A_j\ne A_{n-j+1}$. For the green curve an instance was classified as solved if it was minimized with at least one of the 50 randomly sampled sets of $A_j$. The red curve illustrates the fraction of SK instances that are minimized by standard QAOA circuit (1).

Figure 5

Overlap of the state prepared by the ansatz (7) with the ground space, for different number of qubits. At each $n$ the overlap is averaged over a set of fixed 20 random instances. The blue curve represents the overlap calculated for the initial QAOA state ${|+\rangle }^{\otimes n}$. The orange curve illustrates overlap for the trained $p=4$ layer circuit. The green curve demonstrates improved overlap for the $p=8$ layer circuit, obtained via repeated sampling of circuit parameters.