Experimental determination of a multiqubit ground state via a cluster mean-field algorithm

Abstract

A quantum eigensolver is designed under a multi-layer cluster mean-field (CMF) algorithm by partitioning a quantum system into spatially-separated clusters. For each cluster, a reduced Hamiltonian is obtained after a partial average over its environment cluster. The products of eigenstates from different clusters construct a compressed Hilbert space, in which an effective Hamiltonian is diagonalized to determine certain eigenstates of the whole Hamiltonian. The CMF method is numerically verified in multi-spin chains and experimentally studied in a fully-connected three-spin network, both yielding an excellent prediction of their ground states. Introduction. — At the dawn of a quantum computing era, applications on quantum simulation and beyond have attracted much attention of the whole quantum community. For example, mixed quantum-classical algorithms have been proposed in the goal of solving unaffordable quantum chemistry problems with quantum computers [1–10]. A variational quantum eigensolver (VQE) was successfully implemented in the determination of electronic states for a hydrogen molecule and multi-atom hydrogen chains [1–5]. The adiabaticity and shortcut-to-adiabaticity (STA) in analog and digitized designs [6–9] can also be used in the quantum eigensolver, where an eigenstate of the target Hamiltonian is obtained by dragging an eigenstate of an initial Hamiltonian through an adiabatic or STA trajectory. Recently, we proposed a ‘leap-frog’ algorithm via the digitized STA and adiabaticity [10]. Through a segmented trajectory of travelling intermediate states, our leap-frog method allows an efficient and relibale quantum eigensolver, as illustrated by our experimental study in H2 and numerical calculation of hydrogen chains. In the architecture of quantum computing, the eigenstructure of a 2 -dimensional (2 D) Hilbert space can be determined in an N -qubit quantum device. However, the number of quantum gates in a digital quantum algorithm quickly increases with the number of qubits [11, 12]. In addition, a multi-qubit quantum gate is realized through a combination of singleand two-qubit gates [13] but the number of the combining gates increases with the gate size. The cost of quantum computing increases in company with the decrease of the fidelity so that a practical quantum eigensolver is still limited by the system size. In the fields of physics and chemistry, cluster-based methods have been applied on various ∗ These authors have contributed equally to this work. † jianlanwu@zju.edu.cn ‡ yiyin@zju.edu.cn § gpguo@ustc.edu.cn 2 problems [14–20]. For example, the concept of block spins was proposed to understand critical phenomena of the Ising model [14]. In the renormalization group (RG) theory, the critical exponents are extracted from the scale invariance around a fixed point [15]. The clustering methods are also utilized in the quantum chemistry computation [16–20]. In the block renormalization group (BRG) method, the total Hamiltonian is reconstructed in a compressed Hilbert space built by a few low-energy block states [16]. In the density matrix renormalization group (DMRG), the compression of the Hilbert space is realized by the diagonalization of reduced density matrices [17, 18]. In this paper, we will apply a multi-layer cluster mean-field (CMF) theory [20] to build a new quantum eigensolver, from which the eigenstructure of a large-scale system can be reliably and efficiently determined in a much smaller-scale quantum device. The product states combined from the eigenstates of reduced cluster Hamiltonians define a compressed Hilbert space, in which the effective Hamiltonian is diagonalized for the eigensolver. This CMF method is numerically verified in N -spin chains and experimentally implemented in a fully-connected three-spin system, both yielding high fidelities for the extracted ground states. Theory. — In a general multi-electron system, the second quantized Hamiltonian can be transformed into a multi-spin form,