Measurement-based time evolution for quantum simulation of fermionic systems

Quantum simulation using time evolution in phase-estimation-based quantum algorithms can yield unbiased solutions of classically intractable models. However, long runtimes open such algorithms to decoherence. We show how measurement-based quantum simulation uses effective time evolution via measurement to allow runtime advantages over conventional circuit-based algorithms that use real-time evolution with quantum gates. We construct a hybrid algorithm to find energy eigenvalues in fermionic models using only measurements on graph states. We apply the algorithm to the Kitaev and Hubbard chains. Resource estimates show a runtime advantage if measurements can be performed faster than gates, and graph states compactification is fully used. In this letter, we set the stage to allow advances in measurement precision to improve quantum simulation.

Figure 1

(a) Schematic for the hybrid quantum eigenvalue estimation algorithm. (b) Measurement-based quantum computing (MBQC)/circuit-based quantum computing (CBQC)-favorable regimes determined by the hardware-dependent parameter $\delta t_g/\delta t_m$. The point $N_m/N_g$ is obtained by setting the MBQC and CBQC runtimes to be the same $T_{\mathrm{M}}=T_{\mathrm{C}}$. (c) Measurement-based effective time evolution used for a two-site Jordan-Wigner string where information flows from left to right. Each box/circle hosts a single qubit entangled along red-dotted lines. Open (filled) circles are input (output) qubits. Open squares are Pauli-$x$ measurements that can be performed in parallel, and the dotted box around the central star indicates an adaptive measurement with an angle dictating the time step. The left panel uses the square lattice cluster state (SLCS); the right panel is one of many equivalent graph states that can be used instead, see Supplemental Material [30].

Figure 2

(a) Subroutine for implementing $exp(-i{H}_{\text{K}}t)|{\psi }_{\text{I}}\rangle$ for four sites using only measurements on a square lattice cluster state (SLCS), as in Fig. 1. Adaptive measurements are carried out with the angles defined in Eq. (4). The indices $j,k\in \{1,2,3,4\}$ are assigned along the direction of information flow (red arrows). Measurement angles denoted by stars execute effective time evolution, while other shapes denote measurements to perform rotations at the ends of the Jordan-Wigner (JW) strings. All qubits but inputs with Pauli-$x$ measurements (open boxes) can be removed in compactified cluster states (CCSs). (b) The same but for a two-site Hubbard chain with angles defined by Eq. (7) and $j,k,l\in \{1,2,3,4\}$.

Figure 3

Main: Simulation using Eq. (8), where peaks reveal the exact eigenenergies of the four-site Kitaev chain with $g_{\mu }=0.4$, $\eta /w=0.02$, and $\delta \omega /w=0.01$. Trotter error is ${\delta }_{\mathrm{T}}={10}^{-2}$ for $M<8500$, and $L=1272$ is chosen for clarity. $|{\psi }_{\text{I}}\rangle$ is chosen to be the ground state at $g_{\mu }=0$. The blue line indicates the error-free case, and the green and red lines plot the impact of random perturbations [45–56$\%$ (green); 70–82$\%$ (red)] in the measurement angles ${\overline{\varphi }}_j$ and ${\overline{\psi }}_j^3$. Inset: Eigenenergies of Eq. (1), where the energies touching the dashed line match the peak positions in the main panel.