Shallow Unitary Circuits for Kramers-Wannier Dualities

Technical Summary: Shallow Unitary Circuits for Kramers-Wannier Dualities

Problem Statement
The Kramers-Wannier (KW) duality is a fundamental transformation in quantum many-body physics that maps short-range entangled (SRE) states, such as paramagnetic ground states, to long-range entangled (LRE) states, such as Greenberger-Horne-Zeilinger (GHZ) states or topologically ordered states. Physically realizing this duality as a quantum circuit is desirable for preparing LRE states and simulating exotic excitations. However, the circuit complexity of this transformation is constrained by available quantum resources:

• Local Unitaries: Under the Lieb-Robinson bound, spatially local unitary circuits require a depth scaling linearly with system size ($O(N)$) to generate LRE states from SRE states.
• Measurement-Based Approaches: Recent work has shown that mid-circuit measurements and classical feed-forward can achieve $O(1)$ depth, but these require non-unitary operations and real-time classical control.
• The Open Question: Modern quantum platforms (e.g., Rydberg atom arrays, trapped ions) support spatially nonlocal interactions. It remains an open question whether purely unitary circuits with nonlocal connectivity can implement KW duality efficiently. Existing no-go results indicate that $O(1)$ depth is impossible even with all-to-all connectivity (as GHZ and toric code states cannot be prepared in $O(1)$ depth unitary circuits). The optimal unitary compilation depth for KW duality was previously unknown.

Methodology
The authors explicitly construct shallow, spatially nonlocal unitary circuits that realize exact KW duality maps. The methodology relies on an inductive recursive construction:

1. 1D Construction ($Z_2$):

   • The authors define a "rotated" KW duality circuit, $\tilde{U}_{1d}$, related to the standard duality $U_{1d}$ by a layer of Hadamard gates.
   • They employ a divide-and-conquer strategy. For a chain of $2M$ sites, the circuit is constructed by applying CNOT gates to couple odd and even sites, recursively applying the duality circuit to the even sublattice, and then applying CNOT gates to decouple them.
   • The base case is a single site (Hadamard gate).
   • For general system sizes $N$ (not powers of 2), the lattice is split into a power-of-2 sublattice and a remainder, with specific CNOT patterns connecting them to preserve the duality map.
   • The resulting circuit depth is $O(\log_2 N)$.

2. 2D Construction ($Z_2$):

   • The 2D duality maps qubits on vertices to qubits on edges of a square lattice.
   • The construction uses a similar inductive approach, doubling the lattice size in one dimension (e.g., from $N_1 \times M$ to $N_1 \times 2M$).
   • The step involves three isometries/unitaries:
     • $V_1$: CNOT gates coupling rows.
     • $U$: A map acting on the even rows (effectively a 1D duality on those rows).
     • $V_2$ and $V_3$: CNOT gates and ancilla qubits (initialized to $|0\rangle$) to handle vertical edges and enforce the "flux-free" condition ($\prod Z_e = 1$) required for the toric code ground state.
   • The base case is a $N_1 \times 1$ lattice, which reduces to the 1D rotated duality with added ancilla qubits for vertical edges.
   • The total depth scales as $O(\log_2 N)$, where $N$ is the total number of vertices.

3. Generalization to $Z_n$:

   • The construction is extended to arbitrary $Z_n$ KW dualities ($n > 2$).
   • This involves replacing standard Pauli gates with generalized $Z_n$ operators:
     • CNOT gates are replaced by controlled-$X^\dagger$ (or $CX^{-1}$) gates.
     • Hadamard gates are replaced by the Quantum Fourier Transform ($F$) and its inverse.
   • The recursive structure remains identical to the $Z_2$ case.

Key Results

• Logarithmic Depth: The authors demonstrate that exact KW duality maps in both 1D and 2D can be implemented by purely unitary circuits with depth $O(\log_2 N)$ using at-most-two-qubit gates with nonlocal spatial connectivity.
• Exact Duality Maps: Unlike algorithms designed to prepare specific target states (e.g., ideal GHZ or toric code states), these circuits implement the complete duality map.
• Sector Restriction: The circuits realize the duality exactly when acting on the symmetric (charge-neutral) sector (states with eigenvalue $+1$ under the global symmetry operator $\prod X_j$). Within this sector, the circuits map arbitrary non-fixed-point SRE states to their corresponding LRE duals, preserving finite correlation lengths and transforming local excitations into anyons.
• Operator Transformation: The circuits explicitly satisfy the operator mapping rules (e.g., $(Z_j Z_{j+1}, X_j) \to (X_{j+1}, Z_j Z_{j+1})$ in 1D) up to global symmetry dressing, which vanishes on the charge-neutral sector.

Significance and Claims
The paper claims to establish an efficient, purely coherent pathway for exploring phase transitions and topological dualities on modern quantum hardware equipped with nonlocal connectivity.

• Bridging the Gap: The work fills the gap between the linear-depth limit of local circuits and the constant-depth limit of measurement-based circuits, providing the optimal speed limit for purely unitary transformations.
• Hardware Relevance: The results are motivated by the fact that executing nonlocal unitary gates on platforms like Rydberg atom arrays is often less resource-intensive than performing fast, non-destructive mid-circuit measurements.
• Generality: The protocol is not restricted to specific analytical states but applies to arbitrary states within the symmetric sector, making it a versatile tool for simulating topological matter and duality transformations.
• Future Outlook: The authors suggest the results generalize to higher dimensions and note that extending these nonlocal circuits to arbitrary non-abelian discrete symmetries remains a future challenge, as even adaptive circuits currently lack sub-linear depth protocols for such cases.