Depth-Optimal Addressing of 2D Qubit Array with 1D Controls Based on Exact Binary Matrix Factorization

TL;DR

This work forms the depth-optimal rectangular addressing problem as exact binary matrix factorization, an NP-hard problem also appearing in communication complexity and combinatorial optimization, and introduces a satisfiability modulo theories-based solver and a heuristic, row packing, performing close to the optimal solver on various benchmarks.

Abstract

Reducing control complexity is essential for achieving large-scale quantum computing. However, reducing control knobs may compromise the ability to independently address each qubit. Recent progress in neutral atom-based platforms suggests that rectangular (row-column) addressing may strike a balance between control granularity and flexibility for 2D qubit arrays. This scheme allows addressing qubits on the intersections of a set of rows and columns each time. While quadratically reducing controls, it may necessitate more depth. We formulate the depth-optimal rectangular addressing problem as exact binary matrix factorization, an NP-hard problem also appearing in communication complexity and combinatorial optimization. We introduce a satisfiability modulo theories-based solver for this problem, and a heuristic, row packing, performing close to the optimal solver on various benchmarks. Furthermore, we discuss rectangular addressing in the context of fault-tolerant quantum computing, leveraging a natural two-level structure.

Figures (5)

• Figure 1: Rectangular addressing in neutral atom arrays. a) The experimental setup in Bluvstein et al. bluvstein_logical_2023: a 2D acousto-optic deflector (AOD, blue dashes) modulates another laser to realize $R_z$ gates on qubits at the AOD crossing points (colored dots). Different qubits (uncolored dots) are addressed by changing the AOD signal. Qubits not in the pattern (dash circles) should not be addressed. b) Rectangular partition of a). Different markers distinguish 5 rectangles to partition the matrix. The 5 filled markers indicate a fooling set.
• Figure 2: a) Interpreting the matrix as the adjacency matrix of a bipartite graph, the rectangular partition problem becomes biclique partition where the edges are partitioned to form complete bipartite subgraphs (different line types). b) Binary matrix factorization finds low-rank approximations $HW$ of the original matrix where $H$ and $W$ are also required to be binary.
• Figure 3: Two trials of running the row packing heuristic. Rectangles found are represented by different markers. a) needs 5 rectangles but b) needs 4.
• Figure 4: The most time-consuming cases. 'r' means it is a random benchmark, 'g2' means it comes from benchmarks with gap using 2 row pairs, etc.
• Figure 5: Rectangular addressing in fault-tolerant quantum computing. a) An operation $U$ on 2D patterns of logical qubits can be realized by the tensor product of partitions on the logical and physical levels. b) For logical blocks in 1D layout and with different operations, addressing by row is usually enough.