Variational quantum eigensolvers in the era of distributed quantum computers

The computational power of a quantum computer is limited by the number of qubits available for information processing. Increasing this number within a single device is difficult; it is widely accepted that distributed modular architectures are the solution to large-scale quantum computing. The major challenge in implementing such architectures is the need to exchange quantum information between modules. In this work, we show that a distributed quantum computing architecture with limited capacity to exchange information between modules can accurately solve quantum computational problems. Using the example of a variational quantum eigensolver with an ansatz designed for a two-module (dual-core) architecture, we show that three intermodule operations provide a significant advantage over no intermodule (or serially executed) operations. These results provide a strong indication that near-term modular quantum processors can be an effective alternative to their monolithic counterparts.

Figure 1

A VQE ansatz circuit for a dual-core quantum architecture. (a) VQE ansatz's structure: Each QPU contains half of the available qubits, $\frac{N}{2}$. The unitary $U\left({\vec{\theta }}_{i,j}\right)$ (yellow block) acts on the qubits of the $j\mathrm{th}$ QPU at the $i\mathrm{th}$ stage, followed by a parametric remote gate operation, $ZZ$ (blue block), allowing entanglement to be shared between QPUs. Throughout this Letter, the number of intercore operations is $n_i=3$. (b) The unitary $U\left({\vec{\theta }}_{i,j}\right)$ contains $m$ layers; gates ($R$) are implicitly parametrized by elements of ${\vec{\theta }}_{i,j}$. Throughout this Letter, $m=3$ for a total 12 layers executed on each QPU.

Figure 2

TFIM model with 12 spins at $J=1$, Eq. (2), studied as a function of the transverse magnetic field $h_x$. We depict the logarithm of the relative energy difference between the exact ground state and the variational ansatz, $\epsilon$. The two variational ansätze are a product state (red squares), in which a separable solution is forced, and the interconnected solution (blue circles). For presentation purposes, the error of the separable solution is decreased by an order of magnitude. For either vanishing or strong magnetic fields, the exact solution is a product state. However, for intermediate $h_x$ values, the separable solution is far inferior.

Figure 3

VQE results for the one-dimensional $XYZ$ model with 12 spins [see Eq. (3)]. We present a color map of the logarithm of the relative energy differences between the exact ground state and the variational ansatz, $\epsilon$. The panels depict different magnetic field strengths: $h_x=0$, 0.5, and 1.0, and $J_x=1$. Fidelities are usually above 99.9%; the highest reported variation is $1.4\%$.

Figure 4

VQE results for the $S=1$ Heisenberg model, Eq. (4), with up to a six-spin chain represented by $\times 2$ qubits, $N$. We display the log infidelity of the variational state compared to the exact ground state for the separable product states of two $\frac{N}{2}$ qubits (red squares) and the interconnected ansätze (blue circles). The inset shows the relative energy error $\epsilon$, in percent. While the error of the interconnected solution increases with $N$, it still delivers fidelity orders of magnitude better than that of the separable solution.