Quantum Circuits for Matrix-Product Unitaries

Area of Science:

• Quantum Information Science
• Condensed Matter Theory
• Tensor Network States

Background:

• Matrix-product unitaries (MPUs) are essential quantum operators with tensor-network structures.
• MPUs preserve the entanglement area law in 1D systems.
• Implementing MPUs as quantum circuits is challenging due to non-unitary individual tensors.

Purpose of the Study:

• To demonstrate the feasibility of implementing a broad class of MPUs using polynomial-depth quantum circuits.
• To provide explicit circuit constructions for realizing MPUs.
• To explore the potential of these circuits in generating long-range entanglement.

Main Methods:

• Development of a polynomial-depth quantum circuit construction for N-site MPUs with repeated bulk tensors.
• Explicit circuit design for uniform and non-uniform translationally varying MPUs.
• Analysis of circuit depth scaling with system size (N) and bond dimension (D).

Main Results:

• A polynomial-depth quantum circuit (T=O(N^α)) is constructed for a large class of MPUs.
• The circuit depth depends on tensor properties, not system size N.
• The construction includes non-trivial unitaries generating long-range entanglement, including those from C*-weak Hopf algebras.
• Adaptation for non-uniform MPUs yields circuit depth O(N^β polyD).

Conclusions:

• A significant class of matrix-product unitaries can be efficiently implemented as quantum circuits.
• This work opens avenues for simulating complex quantum phenomena and exploring novel entangled states.
• The findings have implications for quantum computation and the study of quantum many-body systems.