[Paper Review] Sequential Quantum Circuits as Maps between Gapped Phases

Xie Chen, Arpit Dua+5 more|arXiv (Cornell University)|Jul 3, 2023

Quantum Computing Algorithms and Architecture9 citations

TL;DR

The paper defines and analyzes Sequential Quantum Circuits (SQCs) that map between gapped quantum phases, showing how linear-depth, locality-restricted circuits can generate long-range entanglement while preserving area-law entanglement. It provides constructions for moving between symmetry-breaking, SPT, topological, and fracton phases across dimensions.

ABSTRACT

Finite-depth quantum circuits preserve the long-range entanglement structure in quantum states and map between states within a gapped phase. To map between states of different gapped phases, we can use Sequential Quantum Circuits which apply unitary transformations to local patches, strips, or other sub-regions of a system in a sequential way. The sequential structure of the circuit on the one hand preserves entanglement area law and hence the gapped-ness of the quantum states. On the other hand, the circuit has generically a linear depth, hence it is capable of changing the long-range correlation and entanglement of quantum states and the phase they belong to. In this paper, we discuss systematically the definition, basic properties, and prototypical examples of sequential quantum circuits that map product states to GHZ states, symmetry-protected topological states, intrinsic topological states, and fracton states. We discuss the physical interpretation of the power of the circuits through connection to condensation, Kramers-Wannier duality, and the notion of foliation for fracton phases.

Motivation & Objective

• Motivate and formalize the concept of sequential quantum circuits as locality-restricted, linearly deep maps between gapped phases.
• Show how SQCs preserve area-law entanglement yet enable phase transitions by generating long-range correlations.
• Construct explicit SQC examples that map between: symmetry-protected topological (SPT) states, symmetry-breaking states, 2+1D and 3+1D topological orders, and fracton phases.
• Connect SQC power to physical notions such as condensation, Kramers-Wannier duality, and foliation in fracton models.

Proposed method

• Define Sequential Quantum Circuits as local unitaries acting sequentially on subregions, with overall depth potentially linear in system size.
• Demonstrate that SQCs preserve entanglement area law, hence keeping states gapped, while enabling long-range correlations.
• Provide explicit 1D and higher-dimensional constructions mapping between fixed-point states in various phases (e.g., symmetric vs. symmetry-broken, 1D and 2D SPTs, Toric Code and string-net models).
• Use Majorana swaps to realize phase mappings in 1D Ising models and dress SWAP gates with CCZ to preserve symmetry in 2D SPT circuits.
• Show that all locality-preserving unitaries (Quantum Cellular Automata) can be realized as SQCs.

Experimental results

Research questions

• RQ1 What class of linear-depth or higher-depth circuits can map between gapped quantum phases without leaving the gapped manifold?
• RQ2 How can sequential, locality-restricted circuits realize transitions between symmetric and symmetry-breaking phases, SPT phases, and topological or fracton orders?
• RQ3 Can explicit linear-depth SQCs be constructed for representative fixed-point states in 1D and higher dimensions, preserving global symmetries where required?
• RQ4 What is the relationship between SQC constructions and physical concepts like condensation, dualities, and foliation in complex quantum phases?

Key findings

• SQCs generate mappings between fixed-point states of symmetric and symmetry-broken phases in 1D and higher dimensions.
• A linear-depth SQC can map trivial and non-trivial SPT fixed-point states in 1D and 2D, using symmetric SWAP-like operations (e.g., SWAP and SWAP^{CCZ}) that commute with the protecting symmetry.
• SQCs can prepare 2+1D string-net ground states and 3+1D toric-code-like states, including constructions with gapped boundaries and torus geometries, and their truncations create controlled boundary states.
• The framework links SQC power to dualities and condensation processes, providing a unified view of how local sequential actions build long-range order.
• The results suggest that locality-preserving unitaries (QCA) can all be realized as SQCs, broadening the scope of circuits capable of inter-phase mappings.