Genetic Algorithms for Digital Quantum Simulations

We propose genetic algorithms, which are robust optimization techniques inspired by natural selection, to enhance the versatility of digital quantum simulations. In this sense, we show that genetic algorithms can be employed to increase the fidelity and optimize the resource requirements of digital quantum simulation protocols while adapting naturally to the experimental constraints. Furthermore, this method allows us to reduce not only digital errors but also experimental errors in quantum gates. Indeed, by adding ancillary qubits, we design a modular gate made out of imperfect gates, whose fidelity is larger than the fidelity of any of the constituent gates. Finally, we prove that the proposed modular gates are resilient against different gate errors.

Figure 1

Scheme of the GA-based protocol for digital quantum simulations. First, the simulated Hamiltonian is decomposed in local interaction blocks, separately implemented in different unitary evolutions $U_j$ which act on a subset of $k$ particles of the system. Second, the set of gates is selected according to the constraints of the simulating quantum technology: total number of gates to avoid an experimental gate error, interactions restricted to adjacent physical qubits, and implementable phases of the Hamiltonian, among others. Once the set of gates is determined, GAs provide a constraint-fulfilling sequence of gates, which effectively perform the resulting dynamics $U_{\mathrm{GA}}$ similar to $U_T$.

Figure 2

Logarithmic plot of the error $E=1-|⟨\mathrm{\Psi }|U_I^{†}\tilde{U}|\mathrm{\Psi }⟩{|}^2$ in the evolution of (a) Ising and (b) Heisenberg spin models for $N=5$ qubits, $J=2$, $B=1$, and $|\mathrm{\Psi }⟩=|0{⟩}^{\otimes 5}$. Here, $U_I$ is the ideal unitary evolution, while $\tilde{U}$ refers to the unitary evolution using either a digital expansion in 1 (blue line) and 2 (red line) Trotter steps or GAs (dashed green line). The GA protocol requires fewer gates than the digital method for a single Trotter step achieving similar fidelities to two Trotter steps.

Figure 3

Error resilience for architectures with $n=3$, 5, 7 imperfect cnot gates using 1000 runs. Pie charts show the percentage of cases in which the fidelity of the effective cnot gate overmatches the best cnot gate employed in the architecture. Bar charts show the distribution of cases according to the relative improvement in the error, again when compared with the best cnot gate.

Figure 4

Scheme of the optimal architecture for constructing a cnot gate with five imperfect gates, by using two ancillary qubits initialized in state $|0⟩$. Here, $C$ is the control, $T$ is the target, and $A_1$ and $A_2$ are the ancillary qubits.