Robust resource-efficient quantum variational ansatz through an evolutionary algorithm

Variational quantum algorithms (VQAs) are promising methods to demonstrate quantum advantage on near-term devices as the required resources are divided between a quantum simulator and a classical optimizer. As such, designing a VQA which is resource-efficient and robust against noise is a key factor to achieve a potential advantage with the existing noisy quantum simulators. It turns out that a fixed VQA circuit design, such as the widely used hardware-efficient ansatz, is not necessarily robust against imperfections. In this work, we propose a genome-length-adjustable evolutionary algorithm to design a robust VQA circuit that is optimized over variations of both circuit ansatz and gate parameters, without any prior assumptions on circuit structure or depth. Remarkably, our method not only generates a noise-effect-minimized circuit with shallow depth, but also accelerates the classical optimization by substantially reducing the number of parameters. In this regard, the optimized circuit is far more resource-efficient with respect to both quantum and classical resources. As applications, based on two typical error models in VQA, we apply our method to calculate the ground energy of the hydrogen and the water molecules as well as the Heisenberg model. Simulations suggest that, compared with conventional hardware-efficient ansatz, our circuit-structure-tunable method can generate circuits apparently more robust against both coherent and incoherent noise and hence is more likely to be implemented on near-term devices.

Figure 1

The hardware-efficient ansatz for a four-qubit system with $L$ layers where $R_y=e^{\frac{-i\theta {\sigma }_y}{2}}$ and $R_z=e^{\frac{-i\theta {\sigma }_z}{2}}$ represents the parameterized single-rotation gates around the $y$ axis and $z$ axis, respectively. The two-qubit entangling gates are CX.

Figure 2

The performances of the HEA in approximating the ground energies of the (a,b) hydrogen molecule and (c,d) the Heisenberg model as the coherent noise $\mathrm{\Delta }$ increases. Note that Hartree $=627.51\mathrm{kcal}/\mathrm{mol}$ is a unit of molecular energy and $J$ is the strength of exchange coupling in the Heisenberg Hamiltonian.

Figure 3

The robust performances of the HEA in approximating the ground energies of (a,b) the hydrogen molecule and (c,d) the Heisenberg model as the dephasing rate $p$ increases.

Figure 4

The flowchart of the QCEAT algorithm to generate a robust quantum circuit. The ellipses represent sets of circuits and rectangles denote operations which are performed on the circuits. $N_{\text{inner}}$ indicates the number of repetitions of the inner loop and $N_{\text{outer}}$ indicates the number of repetitions of the outer loop. Both of them are chosen at the beginning of the QCEAT.

Figure 5

The encoding scheme mapping a quantum circuit to a genome. (a) An example of a three-layer variational quantum circuit where dashed boxes represents each layer. (b) The three-layer genome corresponding to the three-layer variational circuit. The genome elements of one layer are ordered from the first qubit towards the last qubit. (c) For the sake of brevity, the presentation of genomes can be compressed, which will be used in the rest of the paper.

Figure 6

Three basic operations for the QCEAT to change the structure of a genome (i.e., circuit): (a) mutation operations, including deletion, substitution, and addition; (b) intraspecies crossover, which recombines two parent genomes from the same species to generate an offspring genome; and (c) interspecies crossover, which recombines two parent genomes from different species to generate an offspring genome.

Figure 7

The optimized circuit structure generated by the QCEAT under three different values of coherent noise ${\mathrm{\Delta }}_{\text{t}}=0,0.1,0.2$, for two models, the hydrogen molecule $H_{\text{h}}$ [circuits (a)–(c)], and the antiferromagnetic Heisenberg chain $H_{\text{AF}}$ [circuits (d)–(f)].

Figure 8

Comparison of the robustness of the HEA and the circuits generated by QCEAT (Fig. 7) in approximating the ground energies for (a,b) the hydrogen molecule and (c,d) the Heisenberg model as the coherent noise $\mathrm{\Delta }$ increases.

Figure 9

The optimized circuit structure generated by the QCEAT under different values of dephasing rates $p_{\text{t}}=0,0.05,0.1$, for two models, the hydrogen molecular $H_{\text{h}}$ [circuits (a)–(c)] and the Heisenberg model $H_{\text{AF}}$ [circuits (d)–(f)].

Figure 10

Comparison of the robustness of the HEA and the circuits generated by QCEAT (Fig. 9) in approximating the ground energies for (a,b) the hydrogen molecule and (c,d) the Heisenberg model as the dephasing rates $p$ increases.

Figure 11

The optimized circuit structure generated by the QCEAT under different values of coherent noise ${\mathrm{\Delta }}_{\text{t}}=0,0.1,0.2$ and various strengths of incoherent noise $p_{\text{t}}=0,0.05,0.1$, for the water molecule.

Figure 12

Comparison of the robustness of the HEA and the circuits generated by QCEAT (see Fig. 11) in approximating the ground energy for the water molecule under the action of (a,b) coherent noise and (c,d) the incoherent noise, respectively.