A Geometrical Design Tool for Building Cost-Effective Layout-Aware n-Bit Quantum Gates Using the Bloch Sphere Approach

TL;DR

The paper tackles the high quantum cost of conventional $n$-bit gate design by introducing the Bloch sphere approach (BSA), a geometry-based method that constructs gates from XY-plane rotations to align with hardware connectivity. By avoiding unitary-matrix multiplication and leveraging Clifford+T gates with symmetrical circuit structures, the BSA yields layout-aware designs and a technology-dependent WTQC cost metric, $WTQC = W_1 N_1 + W_2 N_2 + W_3 XC + W_4 D$. The authors implement the GALA-$n$ and CALA-$n$ libraries for IBM-like architectures and demonstrate consistently lower WTQC on IBM devices, illustrating a practical route to cost-effective $n$-bit gates such as the Toffoli, FE, and Boolean operators. This work enables more efficient, hardware-aware quantum circuit design and opens avenues for extending the geometric framework to other planes and larger operators, including arithmetic circuits and complex Boolean functions.

Abstract

The conventional design technique of any n-bit quantum gate is mainly achieved using unitary matrices multiplication, where n >= 2 and 1 <= m <= n-1 for m target qubits and n-m control qubits. These matrices represent quantum rotations by an n-bit quantum gate. For a quantum designer, such a conventional technique requires extensive computational time and effort, which may generate an n-bit quantum gate with a too high quantum cost. The Bloch sphere is only utilized as a visualization tool to verify the conventional design correctness for quantum rotations by a quantum gate. In contrast, this paper introduces a new concept of using the Bloch sphere as a "geometrical design tool" to build cost-effective n-bit quantum gates with lower quantum costs. This concept is termed the "Bloch sphere approach (BSA)". In BSA, a cost-effective n-bit quantum gate is built without using any unitary matrices multiplication. Instead, the quantum rotations for such a gate are visually selected using the geometrical planar intersections of the Bloch sphere. The BSA can efficiently map m targets among n-m controls for an n-bit quantum gate, to satisfy the limited layout connectivity for the physical neighboring qubits of a quantum computer. Experimentally, n-bit quantum gates built using the BSA always have lower quantum costs than those for such gates built using the conventional quantum design techniques.

Figures (8)

• Figure 1: The Bloch sphere is a three-dimensional space with three spatial axes (X, Y, and Z) for illustrating six distinct base states (the stars) of a qubit: (i) the $|0\rangle$ and $|1\rangle$ states intersecting the Z-axis with the Bloch sphere, (ii) the $|+\rangle$ and $|-\rangle$ states intersecting the X-axis with the Bloch sphere, and (iii) the $|+i\rangle$ and $|-i\rangle$ states intersecting the Y-axis with the Bloch sphere.
• Figure 2: Arbitrary $n$-bit symmetrical quantum gates: (a) a 2-bit symmetrical gate, (b) a 3-bit symmetrical gate, (c) a 4-bit symmetrical gate, and (d) a 4-bit symmetrical gate, where $2 \leq n \leq 4$ and the colored rectangles denote symmetries al2025layoutal2024bsa.
• Figure 3: Schematics of coordinate and parallel planes ($P$) perpendicular to various axes intersected with the Bloch sphere: (a) the XY-plane perpendicular to the Z-axis intersected with the Bloch sphere, as the $P_3$ in (d), (b) the XZ-plane perpendicular to the Y-axis intersected with the Bloch sphere, as the $P_3$ in (e), (c) the YZ-plane perpendicular to the X-axis intersected with the Bloch sphere, as the $P_3$ in (f), (d) $m$ parallel planes perpendicular to the Z-axis, (e) $m$ parallel planes perpendicular to the Y-axis, (f) $m$ perpendicular to the X-axis, and (g) three collections of $m$ parallel planes, each collection perpendicular to one of the three axes of the Bloch sphere, where $m \geq 1$ and the black dots subdividing their axial intersections with the Bloch sphere into the semicircles, quadrants, and octants al2024pal2025layoutal2024bsa.
• Figure 4: Schematics of a SWAP gate (left side) and its equivalent decompositions using three CNOT gates (middle and right sides), for swapping the states of two arbitrary indexed physical neighboring qubits ($\text{q}_\text{a}$ and $\text{q}_\text{b}$).
• Figure 5: The heavy-hex layout of ibm_torino QPU of 133 qubits baglio2024datahung2025improved illustrating arbitrary sets of $n$ physical neighboring qubits: (i) sets of two physical neighboring qubits ($n = 2$) as denoted in green, (ii) sets of three physical neighboring qubits ($n = 3$) as denoted in blue, (iii) sets of four physical neighboring qubits ($n = 4$) as denoted in orange, and (iv) sets of seven physical neighboring qubits ($n = 5$ with two additional ancilla qubits cala) as denoted in red.