Renormalization-group-based preparation of matrix product states on up to 80 qubits

Abstract

A key challenge for quantum computers is the efficient preparation of many-body entangled states across many qubits. In this work, we demonstrate the preparation of matrix product states (MPS) using a renormalization-group(RG)-based quantum algorithm on superconducting quantum hardware. Compared to sequential generation, it has been shown that the RG-based protocol asymptotically prepares short-range correlated MPS with an exponentially shallower circuit depth (when scaling system size), but it is not yet clear for which system sizes it starts to convey an advantage. We thus apply this algorithm to prepare a class of MPS exhibiting a phase transition between a symmetry-protected topological (SPT) and a trivial phase for systems of up to 80 qubits. We find that the reduced depth of the RG-based circuits makes them more resilient to noise, and that they generally outperform the sequential circuits for large systems, as we showcase by measuring string-order-like local expectation values and energy densities. We thus demonstrate that the RG-based protocol enables large-scale preparation of MPS and, in particular, SPT-ordered states beyond the fixed point.

Figures (6)

• Figure 1: Experimental results comparing the local string-order parameters $\mathcal{S}_a^\mathbb{I}$ and $\mathcal{S}_a^{ZY}$$\mathbf{(a,b)}$ and the energy density $\eta\coloneqq\langle H\rangle/n$$\mathbf{(c,d)}$ measured on states prepared sequentially [\ref{['eq:seq-circ']}, $\mathbf{(a,c)}$] or with the RG-based algorithm [\ref{['eq:approximated_mps']}, $\mathbf{(b,d)}$], as a function of the tuning parameter $g$ and evaluated for varying number of qubits $n$. The (statistical) error bars in this plot are smaller than the plot markers. For reference, we also include an additional scale showing the correlation length $\xi$ corresponding to the parameter $g$.
• Figure 2: $\mathbf{(a)}$ and $\mathbf{(b)}$ show the local string-order parameters $\mathcal{S}^\mathbb{I}_a$ and $\mathcal{S}^{ZY}_a$ and the energy density $\eta$, respectively, as a function of the number of qubits $n$, evaluated for tuning parameters $g=0.5$ and $g=-0.5$. The blue x-markers denote the crossover point of the sequential and the RG-based preparation. Each plot also shows the ideal value.
• Figure 3: Quantum circuit implementing the $16 \times 4$ isometries of \ref{['eq:approximated_mps']}. With the implementation of $8 \times 2$ isometries proposed in Ref. Iten_2016, the overall CNOT count of 13 gates is a significant improvement over the 54 CNOT gates for an arbitrary $16 \times 4$ isometry Iten_2016.
• Figure 4: Layout of the ibm_fez device. 156 superconducting qubits are arranged on a heavy-hex lattice, which is a hexagonal lattice with an additional node on each edge. Both the RG-based and sequential protocols are mapped to rings of qubits, as depicted in purple and blue, respectively. The ring used for the RG-based preparation consists of eight-qubit blocks, as illustrated in \ref{['fig:qubit_mapping']}.
• Figure 5: Mapping of identical eight-qubit blocks of the RG-based preparation circuit to ibm_fez. This layout can be repeated to form a closed ring on the heavy-hex lattice for all $n \in \{16, 32, 48, 64, 80\}$.