A new quantum ripple-carry addition circuit

TL;DR

The paper presents a reversible, linear-depth quantum ripple-carry adder that uses only a single ancilla, improving on the Vedral-Barenco-Ekert design by reducing depth and gate counts. It achieves this via in-place MAJ and UMA gates to propagate and erase carries, recording sums in memory while maintaining reversibility. Extensions show variants for modulo arithmetic, incoming carries, and high-bit-only computation, with detailed resource counts and depths. The work highlights open questions about optimality and lower bounds for ancilla usage and depth, and contrasts with transform-based adders in practicality and implementation. Overall, it provides a practical, ancilla-efficient framework for quantum addition with broad applicability in quantum algorithms requiring reversible arithmetic.

Abstract

We present a new linear-depth ripple-carry quantum addition circuit. Previous addition circuits required linearly many ancillary qubits; our new adder uses only a single ancillary qubit. Also, our circuit has lower depth and fewer gates than previous ripple-carry adders.

Figures (6)

• Figure 1: The in-place majority gate ${\mathop{\rm MAJ}\nolimits}$
• Figure 2: Two implementations of the ${\mathop{\rm UMA}\nolimits}$ gate
• Figure 3: Combining the ${\mathop{\rm MAJ}\nolimits}$ and ${\mathop{\rm UMA}\nolimits}$ gates
• Figure 4: A simple ripple-carry adder for $n = 6$.
• Figure 5: The ripple-carry adder for $n \ge 4$. Each line of pseudocode corresponds to a single time-slice.