Quantum search by measurements assisted by pre-trained tensor network states for Hamiltonian simulations

TL;DR

A quantum algorithm for simulating complex many-body systems and finding their ground states, combining the use of tensor networks and density matrix renormalization group (DMRG) techniques, based on von Neumann's measurement prescription.

Abstract

We present a quantum algorithm for simulating complex many-body systems and finding their ground states, combining the use of tensor networks and density matrix renormalization group (DMRG) techniques. The algorithm is based on von Neumann's measurement prescription, which serves as a conceptual building block for quantum phase estimation. We describe the implementation and simulation of the algorithm, including the estimation of resources required and the use of matrix product operators (MPOs) to represent the Hamiltonian. We highlight the potential applications of the algorithm in simulating quantum spin systems and electronic structure problems.

Figures (9)

• Figure 1: The circuit of the quantum algorithm consists of $r$ ancillary qubits, referred to as pointers, along with $N$ qubits representing the system. The system is initialized in a preoptimized DMRG state that approximates its ground state ($\hat{\mathcal{U}}_{mps}$). The quantum Fourier transform (QFT) is applied to the $r$ pointer qubits, coupling them to the system via the interaction $\hat{H} \otimes \hat{p}$. Both the pointers and the system then undergo unitary time evolution. After a certain time $t$, an inverse QFT is performed, allowing for the measurement of the pointers. These measurements are subsequently post-processed to estimate the ground state of the system.
• Figure 2: Here in (a) and (b), we illustrate a standard sequential circuit construction for loading an MPS into a quantum register. Panel (a) shows a representative example for bond dimension $\chi=2$, where each site is generated by applying a single local two-qubit gate $U_i$ in a staircase (sequential) pattern along the chain. For larger bond dimensions, the same sequential structure applies, with $U_i$ acting on a correspondingly larger local register; compiling these higher-dimensional unitaries into a fixed native gate set yields an approximate decomposition whose depth depends on the chosen compilation strategy and increases with bond dimension.
• Figure 3: The schematic of the algorithm as the MPS representation. (a). MPO representation of system and pointer. (b). Attaching the MPOs of the system and pointer together to create the composite system. (c). After preparing the system in its MPS state output from the DMRG algorithm, we prepare the MPS representation of ancillary qubits as a product state. By attaching the MPS of the system and ancillary qubits together, we have the initial state for the quantum algorithm. (d). Here, we use the TDVP algorithm to evolve both systems using the operator $e^{-it\hat{H} \otimes \hat{p}}$.
• Figure 4: Ground state of Heisenberg model on $4\times3$ triangular lattice with periodic boundary condition. All the units are in coupling $J$ in the Eq. \ref{['eq:heisenberg']}.
• Figure 5: Ground-state energy estimation for octahydrogen ($\mathrm{H}_8$) in the sto-3g basis using the geometry specified in Appendix \ref{['appendix: molecular geometry']}. (Top) Pointer-readout distributions at different total evolution times $t$, obtained from the same pre-trained DMRG/MPS initial state after the controlled evolution and inverse QFT. (Bottom left) Fitted energy $\hat{E}$ as a function of the total evolution time $t=n\delta t$ (bottom axis); the top axis shows the corresponding Trotter depth $n=t/\delta t$ for the fixed step size $\delta t$. The shaded band (or dashed lines) indicates the chemical-accuracy window $E_{\mathrm{ref}}\pm \epsilon_{\rm chem}$, where $\epsilon_{\rm chem}=1$ kcal/mol $\approx 1.5936\times 10^{-3}$ Hartree. (Bottom right) Absolute energy error $\Delta E=|\hat{E}-E_{\mathrm{ref}}|$ on a logarithmic scale as a function of $t$ (bottom axis) and the corresponding Trotter depth $n$ (top axis). The horizontal dashed line indicates chemical accuracy. All energies are in Hartree.