Linear Depth QFT over IBM Heavy-hex Architecture

TL;DR

The paper addresses the challenge of compiling QFT circuits to IBM heavy-hex hardware under limited qubit connectivity. It introduces a coupling-graph design that isolates a major line with dangling qubits and leverages a recursive QFT mapping inspired by linear-nearest-neighbor methods to achieve linear-depth scheduling. Key contributions include a tailored coupling-graph construction with repeating patterns, a recursive QFT mapping that yields a time bound of $5N+O(1)$ in a special dangling-qubit pattern and $6N+O(1)$ in general, plus formal correctness and upper-bound analyses. The approach reduces circuit depth and compilation time, enabling more practical deployment of QFT-based algorithms on current IBM devices.

Abstract

Compiling a given quantum algorithm into a target hardware architecture is a challenging optimization problem. The compiler must take into consideration the coupling graph of physical qubits and the gate operation dependencies. The existing noise in hardware architectures requires the compilation to use as few running cycles as possible. Existing approaches include using SAT solver or heuristics to complete the mapping but these may cause the issue of either long compilation time (e.g., timeout after hours) or suboptimal compilation results in terms of running cycles (e.g., exponentially increasing number of total cycles). In this paper, we propose an efficient mapping approach for Quantum Fourier Transformation (QFT) circuits over the existing IBM heavy-hex architecture. Such proposal first of all turns the architecture into a structure consisting of a straight line with dangling qubits, and then do the mapping over this generated structure recursively. The calculation shows that there is a linear depth upper bound for the time complexity of these structures and for a special case where there is 1 dangling qubit in every 5 qubits, the time complexity is 5N+O(1). All these results are better than state of the art methods.

Figures (11)

• Figure 1: All necessary CPHASE gate operations over a QFT. H represents the Hadamard gate and Ri represents the CPHASE gate.
• Figure 2: Turn heavy-hex (left-hand side) to our transformed coupling graph (right-hand side) with repeated patterns by removing some connection links (highlight by X). The transformed coupling graph consists of a major line (labeled by 0) with dangling points (labeled by 1).
• Figure 3: Initial mapping for Heavy-hex with dangling qubits: for any node $i$ that has both a node connected to it from the below and a node connected to it from the right (a "T" junction node), the below node has index $i+1$, and the right node has index $i+2$. For any node $i$ that is not involved in a "T" junction does not have a node below, its left node will be $i-1$, and the right node will be $i+1$.
• Figure 4: The concrete steps in QFT mapping for LNN over a line with 4 qubits. Each pair consists of one CPHASE step and SWAP step occurring in the same connection links.
• Figure 5: Rules of gate operations for the QFT mapping with one dangling point. The LHS shows the process of turning the QFT mapping into 3 phases: before $q_0$ moves to the place above $q_4$, interaction between $q_0$ and $q_4$, all steps afterwards. The initial state of the third phase is an intermediate step of RHS where we do QFT for $q_1$ to $q_{n-1}$, realizing the reduction from one dangling point to none.