Sequential Quantum Circuits as Maps between Gapped Phases

TL;DR

This paper introduces sequential quantum circuits (SQC) as a controlled, locality-respecting yet depth-efficient toolkit for connecting ground states across a variety of gapped phases. By building state maps layer-by-layer on subregions, SQCs preserve entanglement area laws while enabling linear-depth transformations that generate long-range correlations, enabling mappings between symmetry-breaking, SPT, topological, and fracton phases. The authors construct explicit SQCs for 1D and 2D symmetry-breaking and SPT states, 2+1D TC and string-net topological orders, 3+1D TC and Walker-Wang models, and two fracton X-cube realizations, with careful treatment of boundaries and Morita dualities. They further show that any quantum cellular automaton can be realized as an SQC, establishing a hierarchical landscape among FDQC, QCA, SQC, and LDQC, and discuss limitations and open questions, including connections to MERA and chiral states.

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.

Figures (14)

• Figure 1: Examples of Sequential Quantum Circuits in 2D where (a) a sequence of local unitary gates are applied to local patches, or a sequence of finite depth quantum circuits are applied to (b) strips or (c) annuli.
• Figure 2: Sequential circuit that maps from the symmetric state to the symmetry breaking GHZ state of the $1+1$D transverse field Ising model. Each blue box represents the unitary $U_{i,i+1}$ in Eq. \ref{['eq:U1D']}. Site $N+1$ is the same as site $1$.
• Figure 3: Sequential circuit that maps from symmetric state to the symmetry breaking GHZ state of the $2+1$D transverse field Ising model on a $N\times M$ lattice. The blue ovals correspond to the gate set in the blue boxes in Fig. \ref{['fig:SBcircuit']}. The numbers in the ovals indicate the layer numbers in the sequential circuit. The horizontal gates act on the top layer only.
• Figure 4: (a) The trivial (bottom) and non-trivial (top) fixed-point states for 1D SPTs on a ring of length 8. Circles indicate qubits, and two circles connected by a black line are in the maximally entangled state $|\Omega\rangle$. The red lines indicate the linear-depth circuit of $\mathrm{SWAP}$s which map between the two states. (b) The 1D states created after applying each of the three gates in (a) to the initial trivial state. The red region indicates the region spanned by the non-trivial SPT with periodic boundaries, which grows with each applied gate. (c) The trivial (upper) and non-trivial (lower) fixed-point 2D SPT states. The shaded regions depict single sites, and solid lines indicate which qubits are entangled. (d) The linear-depth circuit which maps between the trivial and non-trivial 2D SPT states. Each ellipse indicates one application of $\mathrm{SWAP}^{CCZ}$.
• Figure 5: Quantum circuits for creating Toric Code ground states with $L_x=L_y=4$. The lattice has periodic boundary conditions in both directions. Each circle represents the gate defined in Eq. \ref{['mapping_between_gate_loop_notation']}, while the blue arrows represent CNOT gates coupling two adjacent vertical edges. Qubits on solid (dashed) edges are initialized in the state $|0\rangle$ ($|+\rangle$), then the gates are applied in the order shown. (a) Circuit which leaves a smooth boundary when truncated. The gates in the bottom row "zip" the smooth boundaries together to create the periodic boundary conditions. (b) Circuit which leaves periodic boundaries when truncated.