Quantum Circuits for Matrix-Product Unitaries

TL;DR

A large class of MPUs can be implemented with a polynomial-depth quantum circuit and includes nontrivial unitaries that generate long-range entanglement and, in particular, contains a large class of unitaries constructed from representations of $C^*-weak Hopf algebras.

Abstract

Matrix-product unitaries (MPUs) are many-body unitary operators that, as a consequence of their tensor-network structure, preserve the entanglement area law in 1D systems. However, it is unknown how to implement an MPU as a quantum circuit since the individual tensors describing the MPU are not unitary. In this Letter, we show that a large class of MPUs can be implemented with a polynomial-depth quantum circuit. For an $N$-site MPU built from a repeated bulk tensor with open boundary, we explicitly construct a quantum circuit of polynomial depth $T = O(N^α)$ realizing the MPU, where the constant $α$ depends only on the bulk and boundary tensor and not the system size $N$. We show that this class includes nontrivial unitaries that generate long-range entanglement and, in particular, contains a large class of unitaries constructed from representations of $C^*$-weak Hopf algebras. Furthermore, we also adapt our construction to nonuniform translationally-varying MPUs and show that they can be implemented by a circuit of depth $O(N^β \, \mathrm{poly}\, D)$ where $β\le 1 + \log_2 \sqrt{D}/ s_{\min}$, with $D$ being the bond dimension and $s_{\min}$ the smallest nonzero Schmidt value of the normalized Choi state corresponding to the MPU.

Figures (2)

• Figure 1: MPS over $N$ sites admit a sequential circuit decomposition with depth $O(N \mathop{\mathrm{poly}}\nolimits D)$, where $D$ is the bond dimension schon2005sequential. How can MPUs be implemented as quantum circuits, and what is the corresponding depth? The answer is only known for the case of MPUs with uniform bulk and periodic boundary, which correspond to QCA cirac2017matrix1. Here, we introduce a circuit decomposition of $\mathop{\mathrm{poly}}\nolimits(N)$ depth for arbitrary MPUs, when they are well conditioned. We show this holds (i) for MPUs with uniform bulk and open boundary, and (ii) for arbitrary MPUs when $\sqrt{D}/s_{\min} = O(1)$ ($s_{\min}:$ smallest nonzero singular value of the MPU Choi state).
• Figure 2: The MPU implementation algorithm proceeds by first realizing small local isometries and then recursively merging them in a tree-like structure, here indicated by arrows. After $\lceil \log_2 N \rceil$ layers of merging, the global MPU is obtained (left). Each individual merging step is performed deterministically using an amplitude amplification technique (right).

Theorems & Definitions (28)

• Definition 1: Uniform-bulk MPU
• Theorem 1
• Theorem 2
• Definition 2: MPU
• Lemma 1: Local isometries
• proof
• Definition 3: MPU canonical form
• Lemma 2
• proof
• Lemma 3: Subspace amplitude amplification