[Paper Review] Preparation of matrix product states with log-depth quantum circuits

Daniel Malz, Georgios Styliaris+2 more|arXiv (Cornell University)|Jul 4, 2023

Quantum Computing Algorithms and ArchitectureComputer Science3 citations

TL;DR

This paper presents an optimal quantum circuit protocol to prepare translation-invariant matrix product states (MPS) using log-depth quantum circuits. By leveraging renormalization-group (RG) transformations and isometric circuits, it achieves depth T = O(log(N/ϵ)) for normal MPS and T = O(log log(N/ϵ)) with measurements—proving a lower bound of T = Ω(log N) and demonstrating asymptotic optimality for both unitary and measurement-assisted protocols.

ABSTRACT

We consider the preparation of matrix product states (MPS) on quantum devices via quantum circuits of local gates. We first prove that faithfully preparing translation-invariant normal MPS of $N$ sites requires a circuit depth $T=Ω(\log N)$. We then introduce an algorithm based on the renormalization-group transformation to prepare normal MPS with an error $ε$ in depth $T=O(\log (N/ε))$, which is optimal. We also show that measurement and feedback leads to an exponential speedup of the algorithm, to $T=O(\log\log (N/ε))$. Measurements also allow one to prepare arbitrary translation-invariant MPS, including long-range non-normal ones, in the same depth. Finally, the algorithm naturally extends to inhomogeneous MPS.

Motivation & Objective

• To establish a tight lower bound on the circuit depth required to prepare translation-invariant normal matrix product states (MPS).
• To develop a quantum circuit protocol that saturates this lower bound using strictly local gates.
• To extend the protocol to inhomogeneous and non-normal MPS using measurement and feedback.
• To demonstrate that measurement-assisted circuits can achieve exponentially faster preparation of arbitrary translation-invariant MPS.

Proposed method

• The algorithm uses iterative renormalization-group (RG) transformations to block sites and converge to a fixed-point state with only nearest-neighbor entanglement.
• It constructs isometries that encode the local structure of the MPS and proves they can be implemented with a strictly local quantum circuit of depth O(log(N/ϵ)).
• The isometries are optimized via a local variational algorithm using singular value decomposition of environment operators to maximize fidelity with the target state.
• Measurement and feedback are integrated to implement the fixed-point state and isometries in constant depth, enabling exponential speedup.
• The protocol is extended to inhomogeneous MPS by applying the same RG and isometry construction to locally canonical forms.
• Gate teleportation is used to implement non-local isometries efficiently in measurement-based schemes.

Experimental results

Research questions

• RQ1 What is the minimal circuit depth required to faithfully prepare translation-invariant normal matrix product states using local quantum gates?
• RQ2 Can a quantum circuit prepare normal MPS in depth O(log(N/ϵ)) using only local gates, and is this depth optimal?
• RQ3 How does access to measurements and feedback affect the circuit depth for preparing both normal and non-normal MPS?
• RQ4 Can the proposed protocol be generalized to inhomogeneous MPS with short-range correlations?

Key findings

• The paper establishes a lower bound of T = Ω(log N) for preparing translation-invariant normal MPS using local quantum circuits, proving that logarithmic depth is necessary.
• An explicit quantum circuit construction achieves preparation of normal MPS in depth T = O(log(N/ϵ)) using only local gates, saturating the lower bound and proving asymptotic optimality.
• With measurement and feedback, the circuit depth is reduced to T = O(log log(N/ϵ)), enabling exponentially faster preparation of arbitrary translation-invariant MPS, including long-range correlated ones.
• The protocol naturally extends to inhomogeneous MPS that are short-range correlated, maintaining efficient depth scaling via local variational optimization of isometries.
• The finite-range MERA is shown to approximate normal translation-invariant MPS in O(log log(N/ϵ)) layers, establishing a new connection between MERA and MPS preparation.
• Numerical results confirm exponential decay of error per block with blocking range, supporting the efficiency of the RG-based approximation and isometry construction.