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Quantum Circuits for Matrix-Product Unitaries.

Georgios Styliaris1, Rahul Trivedi1, J Ignacio Cirac1

  • 1Munich Center for Quantum Science and Technology (MCQST), Max Planck Institute of Quantum Optics, Hans-Kopfermann-Straße 1, Garching 85748, Germany and , Schellingstraße 4, 80799 München, Germany.

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This summary is machine-generated.

We present a method to implement matrix-product unitaries (MPUs) as quantum circuits. This approach allows for polynomial-depth circuits, enabling the study of complex quantum systems and long-range entanglement.

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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.