Entanglement from Tensor Networks on a Trapped-Ion Quantum Computer

The ability to selectively measure, initialize, and reuse qubits during a quantum circuit enables a mapping of the spatial structure of certain tensor-network states onto the dynamics of quantum circuits, thereby achieving dramatic resource savings when simulating quantum systems with limited entanglement. We experimentally demonstrate a significant benefit of this approach to quantum simulation: the entanglement structure of an infinite system—specifically the half-chain entanglement spectrum—is conveniently encoded within a small register of “bond qubits” and can be extracted with relative ease. Using Honeywell’s model H0 quantum computer equipped with selective midcircuit measurement and reset, we quantitatively determine the near-critical entanglement entropy of a correlated spin chain directly in the thermodynamic limit and show that its phase transition becomes quickly resolved upon expanding the bond-qubit register.

Figure 1

Using Honeywell’s H0 quantum computer described in Ref. [29] and shown in (a), we execute a state-preparation algorithm for arbitrary matrix-product states (MPS) (b). (c) The algorithm consists of repeatedly entangling a single system qubit (gray) with a register of $\log \chi$ “bond” qubits (purple) in order to encode a bond dimension $\chi$ MPS (here we show the case of $\chi =2$, which requires a single bond qubit). The spatial extent of the MPS is encoded in the temporal extent of the algorithm, with properties of the $n$th site corresponding to the measurement record of the system qubit prior to reset at the $n$th iteration.

Figure 2

Quantum MPS. The tensors of a matrix product state (a) can be embedded in unitary matrices (b) in order to generate the MPS as the output of a quantum circuit (c). By executing any desired measurements as soon as possible, the system qubit representing site $n$ can be reset and reused as the system qubit on site $n+1$, leading to an equivalent circuit with just a single system qubit (d). Entanglement entropy of the MPS across a bipartition (dashed vertical line in the figures) becomes entanglement entropy of the bond qubits immediately after crossing that partition (e), as future gates cannot affect the state prior to the partition.

Figure 3

(a),(b) Circuits for extracting bipartite entanglement entropy of a $\chi =2$ (a) or $\chi =4$ (b) MPS. (c) Measured entanglement entropy for the TFIM. Open purple circles are data taken with $n_b=1$, and the purple dashed line shows the entanglement entropy of the lowest-energy MPS at $\chi =2^{n_b}=2$. Results using $n_b=2$ are shown as filled green circles, and the solid green line corresponds to the lowest-energy $\chi =4$ MPS. The black-dotted line is the exact entanglement entropy computed following Ref. [46]. All error bars represent $\pm \sigma$ confidence intervals obtained by bootstrapping. (a) Circuits to extract energy estimates from a bond dimension 2 MPS. (b) Solid line is the exact ground-state energy of the infinite-size TFIM. Experimental data (purple, $1\sigma$ error bars) agrees reasonably well from deep inside the ordered phase into the disordered phase. The gray shaded region is a $1\sigma$ confidence interval from a numerical simulation with the same shot-count as the experiment.