Matrix Product State on a Quantum Computer

TL;DR

The paper introduces a quantum Matrix Product State (qMPS) framework and a variational quantum MPS (vqMPS) algorithm to simulate quantum many-body systems on near-term devices. It develops quantum analogues of canonical MPS steps, including quantum singular value decomposition (QSVD) and quantum reshape, to enable site-by-site optimizations within a distributed quantum–classical tensor-network approach. Numerical results show high fidelity in QSVD for imbalanced partitions and competitive ground-state energies for Heisenberg XXZ models compared with full-scale VQE, illustrating reduced qubit requirements and potential scalability. Overall, the work offers a practical path to transplant classical tensor-network techniques onto quantum hardware, mitigating barren plateaus and enabling hybrid quantum-classical processing across distributed platforms.

Abstract

Solving quantum many-body systems is one of the most significant regimes where quantum computing applies. Currently, as a hardware-friendly computational paradigms, variational algorithms are often used for finding the ground energy of quantum many-body systems. However, running large-scale variational algorithms is challenging, because of the noise as well as the obstacle of barren plateaus. In this work, we propose the quantum version of matrix product state (qMPS), and develop variational quantum algorithms to prepare it in canonical forms, allowing to run the variational MPS method, which is equivalent to the Density Matrix Renormalization Group method, on near term quantum devices. Compared with widely used methods such as variational quantum eigensolver, this method can greatly reduce the number of qubits required, and thus can mitigate the effects of Barren Plateaus while obtain comparable or even better accuracy. Our method holds promise for distributed quantum computing, offering possibilities for fusion of different computing systems.

Figures (5)

• Figure 1: Quantum Matrix Product State model (qMPS). (a) A diagram of implementing a qMPS site on a superconducting processor. A site of qMPS is a quantum state divided into three entangled groups of qubits, labeled by $l$, $p$ and $r$. Qubits in groups $l$ and $r$ are used as auxiliary dimensions of the qMPS, and the qubits in group $p$ are used ad the physical dimensions of the qMPS. (b) The global state represented by a qMPS is determined by contracting all quantum dimensions, as represented by Eq. \ref{['eq:qmps_full_state']}. The number of qubits contained in the global states is the sum of qubits in physical dimension of each site. (c) The MPO sites can be divided into $m$ groups to match the number of physical qubits in each qMPS site. (d) The local expectation is represented by a rank-6 classical tensor, with each element calculated through Eq. \ref{['eq:expectation']}.
• Figure 2: Turning a qMPS site into left-canonical form. (a) Diagram of preparing a classical MPS site into left-canonical form using SVD algorithm. (b) Schematic view of preparing a qMPS site into left-canonical form with a QSVD lagorithm. (c) The variational quantum circuits for finding two unitary matrices to diagonalize the state. The expectation value of observables include $\sigma_z\sigma_z$ on qubits lined with dashed lines, and $\sigma_z$ on the rest qubits. (d) Quantum circuit for quantum reshape operation, transferring the data of a unitary into a quantum state, namely a new qMPS site. It turns the elements of $U$ into the amplitudes of a quantum state $|\Psi_{\rm rs}\rangle = \frac{1}{\sqrt{2^{n_{B}}}}\sum_{i=0}^{2^{n_B} - 1}|i\rangle (U|i\rangle)$. (e) Variational algorithm for quantum reshape. The idea for reshape is to generate the state $|\Psi_{\rm rs}\rangle$ through a variational circuits, and then turn it into maximally entangled state by $(I\otimes U^\dagger)|\Psi_{\rm rs}\rangle=\sum_{i=0}^{2^{n_B} - 1}|i\rangle |i\rangle$.
• Figure 3: The schematic view of vqMPS method. The process is similar to the classical DMRG method, by sweeping back-and-forth. The global qMPS is initially prepared in left-canonical form (the blue sites), and updated site-by-site from right to left. After this sweeping, the qMPS would be converted into right-canonical (the green sites), and can be updated by sweeping from left to right. The ground energy is estimated during these sweeps. The vqMPS facilitates quantum-classical hybrid computing. The sites on both sides can be updated on classical computers, while updating strongly correlated sites can be accelerated using quantum computing.
• Figure 4: Results of quantum singular value decomposition. For $n$ qubits, we test to decomposing it to $n_A + n_B$ qubits through variational algorithms, rebuild it through the obtained singular values and unitaries, and then measure the fidelity. The overall fidelity recovered reaches over 96% and it achieves better accuracy when the system is imbalanced.
• Figure 5: Results of numerical simulations for Heisenberg spin-chain Hamiltonian with $\Delta = 0.5$, $1.0$ and $1.5$, respectively. The base-line is the result of full-scale VQE, and $E_{\rm vqMPS}-E_{VQE}$ is the difference between the ground energy found from the sweeps of vqMPS and that of full-scale VQE. The vqMPS method reaches a comparable accuracy with VQE when $n_\chi = 1$, which requires only $2n_\chi + 1 = 3$ qubits per VQE process. Better accuracy can be achieved when $n_\chi = 2$ or $3$ because all the data are below the baseline. However, for some cases using $n_\chi=3$ can be less accurate compared with that of $n_\chi = 2$.