Optimizing Variational Quantum Algorithms Using Pontryagin’s Minimum Principle

We use Pontryagin’s minimum principle to optimize variational quantum algorithms. We show that for a fixed computation time, the optimal evolution has a bang-bang (square pulse) form, both for closed and open quantum systems with Markovian decoherence. Our findings support the choice of evolution ansatz in the recently proposed quantum approximate optimization algorithm. Focusing on the Sherrington-Kirkpatrick spin glass as an example, we find a system-size independent distribution of the duration of pulses, with characteristic time scale set by the inverse of the coupling constants in the Hamiltonian. The optimality of the bang-bang protocols and the characteristic time scale of the pulses provide an efficient parametrization of the protocol and inform the search for effective hybrid (classical and quantum) schemes for tackling combinatorial optimization problems. Furthermore, we find that the success rates of our optimal bang-bang protocols remain high even in the presence of weak external noise and coupling to a thermal bath.

Popular Summary

Quantum information processing promises solutions to computational problems that are beyond the reach of even the most powerful traditional computers. Whereas digital computers encode information in discrete binary numbers (1s and 0s), quantum computers leverage the ability of particles to be in many states at once. By manipulating and evolving those states over time, the solution to a computation problem ends up encoded in the final state of a quantum machine. One approach to that evolution, known as a variational quantum algorithm (VQA), is a hybrid between a quantum machine, which performs the time evolution, and a classical optimizer, which iteratively improves the controls based on feedback from the quantum part. In our work, we investigate optimal methods for implementing a VQA.

Using a principle from control theory known as Pontryagin’s minimum principle, we find that the optimal protocol for generic VQA is what is known as “bang-bang”—a type of feedback control that abruptly switches between two states (much like how a thermostat is typically either fully on or off). Focusing on the problem of finding the minimum energy of a spin glass, we also uncover key characteristics of this control, such as pulse durations of the bang-bang controller that are independent of the size of the quantum system. Identifying this time scale significantly reduces the number of variational parameters in VQA, decreasing the computational cost of the algorithm.

Our results could foster the development of new high-performance quantum-assisted computation methods. Further progress depends on hardware development and improvements to supporting algorithms.

Figure 1

(a) Variational quantum algorithm as a closed-loop learning control problem. (b) Increasing the total time expands the set of final states that one can reach with the variational protocols. The optimal protocol for a given time generates the closest state to a low energy target state within this reachable set.

Figure 2

(a) The optimal protocol obtained from MC simulations for a fixed instance of Hamiltonian Eq. (9) with $n=5$ spins and total annealing time $T=0.8$. Different colors represent different initial protocols. The plots are for $S=40$, but the optimal protocol does not change upon increasing $S$. (b) A typical protocol obtained for a given instance of Hamiltonian Eq. (9) with $n=5$ spins and $T=2$, using a classical optimization solver. We start from a uniform initial protocol with $S$ slices such that $\delta t=T/S=0.1$.

Figure 3

The average probability distribution of the time scales of bangs for different system sizes: $n=6$, 7, 8, 9, and 10. The total annealing time is fixed to be $T=2$, leading to an average success rate around 0.33–0.47, depending on the system sizes. Each curve is averaged over 50 instances of Hamiltonian Eq. (9).

Figure 4

Errors in the fidelity (upper panel) and final energies (lower panel) evolved with the bang-bang and QAA protocols in the presence of noise with different strengths for $n=5$.

Figure 5

Errors in the fidelity of final states evolved with the bang-bang and QAA protocols in the presence of different strengths of coupling to the environment for $n=5$. The inverse temperature is chosen to be $\beta =2/J$.