Probing many-body localization by excited-state variational quantum eigensolver

Nonequilibrium physics including many-body localization (MBL) has attracted increasing attentions, but theoretical approaches of reliably studying nonequilibrium properties remain quite limited. In this Letter, we propose a systematic approach to probe MBL phases via the excited-state variational quantum eigensolver (VQE) and demonstrate convincing results of MBL on a quantum hardware, which we believe paves a promising way for future simulations of nonequilibrium systems beyond the reach of classical computations in the noisy intermediate-scale quantum (NISQ) era. Moreover, the MBL probing protocol based on excited-state VQE is NISQ-friendly, as it can successfully differentiate the MBL phase from thermal phases with relatively shallow quantum circuits, and it is also robust against the effect of quantum noises.

Figure 1

The circuit structure for the excited-state VQE and eigenstate witness measurement. In preparation circuit part, we first prepare an antiferromagnetic (AF) state $|0101...\rangle$ from the initial state $|0000...\rangle$ and then apply one layer of two-qubit entangling gates to generate the input state $|{\psi }_0\rangle$ for the PQC. The layer of two-qubit entangling gates is denoted as $U_0\left({\vec{\theta }}^0\right)$ and $U_0\left({\vec{\theta }}^0\right)$ = ${\prod }_{j=0}^{N-1}\exp \left(\frac{i\pi {\theta }_j^0}{4}({\sigma }_j^x{\sigma }_{j+1}^x+{\sigma }_j^y{\sigma }_{j+1}^y)\right)$. We assume two-qubit entangling gates on different qubits sharing the same weight ${\theta }^0$ for simplicity. In the PQC part, each block of the ansatz is defined as $U\left({\vec{\theta }}^k\right)=\left({\prod }_{j=0}^{N-1}\mathrm{Rz}\left({\theta }_{2,j}^k\right)\right)U_0\left({\vec{\theta }}_1^k\right)$, which respects $U\left(1\right)$ symmetry. Two-qubit entangling gates on different qubits have different weights. The depth of the PQC (number of blocks) ${\mathrm{D}}_{\mathrm{VQE}}$ can be adjusted.

Figure 2

EIPR of the input state $|{\psi }_0\rangle$ obtained from the preparation circuit for $W=1.5$ and 8.0, respectively. Here $N=12$.

Figure 3

EIPR of final converged states from excited-state VQE on 12-qubit system with different potential strength $W$.

Figure 4

(a) $ln(1-\mathrm{EIPR})$ of final converged states. (b) $ln(1-r)$ of final converged states, where $r$ is the eigenstate witness and experimentally accessible. Here $N=12$.

Figure 5

(a) $r$ of final converged states from a noisy simulation without Trotter decomposition. (b) $r$ of final converged states from a noisy simulation with one time slice Trotter decomposition. Here $N=8$.

Figure 6

VQC using hardware-efficient ansatz to approximate controlled time evolution. The two-qubit entangling gate is $\mathrm{CZ}$ gate.

Figure 7

Eigenstate witness measured on the noisy simulator and real hardware (IBM_Santiago) (in the inset ${}^{*}$ means the results are processed with readout error mitigation). The estimation error bars due to finite shots of measurement are also included.