Design of a comparator tree based on reversible logic

The existing design of reversible n-bit binary comparator that compares two n-bit numbers is a serial design [1] having the latency of O(n). In this work, we present a new reversible n-bit binary comparator based on binary tree structure that has the latency of O(log₂(n)). The reversible designs are based on a new reversible gate called the TR gate, the improved quantum cost of which is also derived in this work. In the proposed reversible binary tree comparator each node consists of a 2-bit reversible binary comparator that can compare two 2-bit numbers x(x_i, x_i-1) and y(y_i, y_i-1), to generate two 1-bit outputs Y and Z. Y will be 1 if x(x_i, x_i-1)> y(y_i, y_i-1), and Z will be 1 if x(x_i, x_i-1)<y(y_i, y _i-1). After careful analysis, we modified the logic equations of Y = x₁ȳ₁ + kx₀ȳ₀ and Z = x̄₁y₁ + kx̄₀y₀ to Y = x₁ȳ₁ ⊕ kx₀ȳ₀ and Z = x̄₁y₁ ⊕ kx̄₀y₀, respectively. The replacement of + operator with ⊕ operator without affecting the functionality of the design helped us in reversible mapping of the equations of Y and Z on the third output of the TR gate which is R=AB̄ ⊕ C. Further, TR gate can also efficiently generate functions such as x₀ȳ₀ and x̄₀y₀. In the proposed reversible binary comparator, the leaf nodes will consist of 2-bit reversible binary comparators. Each internal node (2-bit reversible binary comparator) of the binary tree receives the partial comparison results from the left and the right children and propagates the 2-bit output of the comparison to its parent. Finally, the root node which is also a 2-bit reversible binary comparator generates the 2-bit result of the comparison of the n-bit numbers x and y to evaluate whether x>y or x<y. The 2-bit result of the root node are passed to the reversible output circuit designed from a Toffoli gate and 4 NOT gates to generate three signals O₀(x<y), O₁(x>y) and O₂(x=y).