Matrix Product State on a Quantum Computer

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.

💡 Research Summary

The paper introduces a quantum‑adapted version of the matrix product state (MPS) called quantum MPS (qMPS) and demonstrates how it can be used to run the variational MPS (vMPS) algorithm—equivalent to the density‑matrix renormalization group (DMRG)—on near‑term noisy intermediate‑scale quantum (NISQ) devices. A qMPS site is composed of three entangled groups of qubits: a left auxiliary block (l), a physical block (p), and a right auxiliary block (r). The auxiliary blocks encode the bond dimension χ of a classical MPS using ⌈log₂χ⌉ qubits each, while the physical block holds the actual degrees of freedom of the simulated many‑body system. By tiling many such sites, a global N‑qubit state is assembled with a total qubit count that scales only with the sum of the physical dimensions, dramatically reducing the hardware requirements compared with a full‑state VQE approach.

To make qMPS suitable for variational optimization, the authors develop two quantum sub‑routines that bring each site into left‑ or right‑canonical form. The first sub‑routine is a Quantum Singular Value Decomposition (QSVD). The global state is split into two subsystems A and B (which may have unequal sizes). A variational circuit consisting of single‑qubit rotations and interleaved CNOT layers is trained to find unitaries U_A and U_B such that U_A⊗U_B|ψ⟩ becomes a Schmidt‑decomposed state Σ_i s_i|i⟩|i⟩. The loss function L_SVD combines σ_z⊗σ_z correlation measurements on paired qubits with single‑qubit σ_z expectations on the remaining qubits, ensuring that the measurement outcomes of the two subsystems match. Numerical experiments show exponential decay of the loss with circuit depth.

The second sub‑routine is a quantum “reshape” operation that converts the columns of U_A (or U_B) into a quantum state |Ψ_rs⟩ whose amplitudes are the matrix elements of the unitary. A direct circuit implementation would be deep, so the authors propose a variational circuit that approximates |Ψ_rs⟩. The associated loss L_reshape = Σ_q (1‑⟨σ_z^q⟩) drives the output toward the all‑zero computational basis; when minimized, the circuit prepares the desired reshaped state. Simulations up to seven qubits achieve >99 % fidelity.

With sites in canonical form, the variational quantum MPS (vqMPS) algorithm proceeds exactly like classical vMPS/DMRG: a sweeping schedule updates one (or a few) sites at a time while treating the rest of the chain and the Hamiltonian MPO as an effective environment. For a given site i, the local effective Hamiltonian is constructed from the surrounding qMPS sites and the MPO representation of the target observable (e.g., the system Hamiltonian). The local expectation value is measured as a rank‑6 tensor V_i by projecting auxiliary qubits onto computational basis states and measuring Pauli strings on the physical qubits. Global expectation values are then obtained by classical tensor contraction of the V_i tensors. The overall computational cost scales as O(K·m·χ⁴), where K is the number of Pauli terms in the observable, m the number of qMPS sites, and χ the bond dimension.

The key advantages highlighted are: (1) a logarithmic reduction in required qubits, enabling larger problem instances on NISQ hardware; (2) locality of the optimization, which mitigates barren‑plateau gradients because each sub‑problem lives in a much smaller parameter space; (3) natural compatibility with distributed quantum computing, allowing different physical platforms (superconducting, photonic, etc.) to host different qMPS sites and cooperate via classical tensor contraction. The authors also discuss limitations: the depth of QSVD and reshape circuits can still be substantial, leading to error accumulation; MPO decomposition and mapping to sites incurs preprocessing overhead; and extending the approach to higher‑dimensional tensor networks (e.g., PEPS) remains an open challenge.

In conclusion, the work provides a concrete pathway to translate the powerful classical DMRG methodology into the quantum domain, offering a scalable, hardware‑efficient alternative to global variational algorithms such as VQE. By leveraging tensor‑network insights and tailored quantum sub‑routines, qMPS and the vqMPS algorithm open new avenues for accurate quantum many‑body simulations and for the development of distributed quantum computing architectures.