Optimal Quantum Circuits for Nearest-Neighbor Architectures

We show that the depth of quantum circuits in the realistic architecture where a classical controller determines which local interactions to apply on the kD grid Z^k where k >= 2 is the same (up to a constant factor) as in the standard model where arbitrary interactions are allowed. This allows minimum-depth circuits (up to a constant factor) for the nearest-neighbor architecture to be obtained from minimum-depth circuits in the standard abstract model. Our work therefore justifies the standard assumption that interactions can be performed between arbitrary pairs of qubits. In particular, our results imply that Shor’s algorithm, controlled operations and fanouts can be implemented in constant depth, polynomial size and polynomial width in this architecture. We also present optimal non-adaptive quantum circuits for controlled operations and fanouts on a kD grid. These circuits have depth Theta(n^(1 / k)), size Theta(n) and width Theta(n). Our lower bound also applies to a more general class of operations.

💡 Research Summary

The paper investigates the relationship between the abstract quantum circuit model, which permits two‑qubit gates between any pair of qubits, and a realistic nearest‑neighbor (NN) architecture where qubits are placed on a k‑dimensional integer lattice Z^k (k ≥ 2) and only adjacent qubits may interact. A classical controller is assumed to decide, at each time step, which local interactions to apply. The central result, the “Depth Equivalence Theorem,” shows that any circuit of depth d in the abstract model can be simulated on the NN architecture with depth O(d). The simulation proceeds by a parallel routing scheme: qubits that must interact are first brought next to each other using a constant‑depth swap network that exploits the geometry of the lattice. Because the routing overhead depends only on the dimension k, the overall depth increases by at most a constant factor.

From this equivalence, the authors derive several important corollaries. First, any algorithm that has a minimum‑depth implementation in the abstract model—most notably Shor’s factoring algorithm—can be realized on a NN grid with constant depth, polynomial size, and polynomial width. Second, controlled operations (Controlled‑U) and fan‑out gates can also be performed in constant depth under the same model, because the necessary routing of control and target qubits can be done in O(1) steps.

The paper further addresses non‑adaptive circuits, i.e., circuits where the sequence of gates is fixed in advance and does not depend on intermediate measurement outcomes. For this class, the authors construct optimal circuits for controlled operations and fan‑out on a k‑dimensional grid. These constructions achieve depth Θ(n^{1/k}), size Θ(n), and width Θ(n), where n is the number of logical qubits involved. A matching lower bound is proved by analyzing the minimum time required to propagate information across the lattice; any circuit that must broadcast a value to n locations must incur at least Ω(n^{1/k}) depth, which coincides with the upper bound. Consequently, the presented non‑adaptive circuits are provably optimal for any dimension k.

The lower‑bound technique extends to a broader class of operations beyond fan‑out, showing that any operation requiring global communication on the grid cannot beat the Θ(n^{1/k}) depth barrier. The authors discuss practical implications for quantum compiler design: because the depth penalty for mapping abstract circuits to NN hardware is only a constant factor, existing compiler optimizations and algorithmic designs can be transferred to realistic hardware with minimal loss of performance. Moreover, the constant‑depth overhead suggests that error‑correction overheads remain manageable, as the additional swaps do not dramatically increase the circuit’s exposure to noise.

In the discussion, the authors note that higher‑dimensional architectures (e.g., 3‑D or 4‑D ion‑trap or superconducting lattices) further reduce the routing cost, making the constant‑depth simulation even more attractive. They also point out that while the model assumes a powerful classical controller capable of instantaneously selecting local gates, this assumption aligns with current experimental control systems that can schedule parallel operations across many qubits.

Overall, the paper provides a rigorous justification for the widespread theoretical assumption that arbitrary two‑qubit interactions can be treated as free in algorithmic analysis. It demonstrates that, in a physically realistic nearest‑neighbor setting, this assumption incurs at most a constant‑factor depth overhead, and for many fundamental primitives (Shor’s algorithm, controlled gates, fan‑out) it yields truly constant‑depth implementations. The optimal non‑adaptive constructions and matching lower bounds further enrich our understanding of the fundamental limits imposed by locality in quantum computation.