A Logarithmic Depth Quantum Carry-Lookahead Modulo $(2^n-1)$ Adder

TL;DR

This work tackles the challenge of implementing modulo addition in quantum circuits under NISQ constraints by introducing the quantum carry-lookahead modulo $(2^n-1)$ adder (QCLMA), which achieves $O(\log n)$ depth through a tree-based carry path. The design combines an out-of-place carry-generation stage with an in-place carry-lookahead truncated adder to compute the sum, trading some additional ancilla qubits for substantially reduced depth. Experimental evaluation on the 27-qubit IBM Cairo platform demonstrates a 47.21% improvement in the Quantum State Fidelity Ratio (QSFR) for a 4-qubit modulo adders over a QRCA-based baseline, indicating improved noise resilience. The results suggest that shorter depth and distributed carry propagation enhance fidelity, making QCLMA suitable for practical quantum modulo arithmetic in cryptography and quantum image processing, and scalable to fault-tolerant regimes.

Abstract

Quantum Computing is making significant advancements toward creating machines capable of implementing quantum algorithms in various fields, such as quantum cryptography, quantum image processing, and optimization. The development of quantum arithmetic circuits for modulo addition is vital for implementing these quantum algorithms. While it is ideal to use quantum circuits based on fault-tolerant gates to overcome noise and decoherence errors, the current Noisy Intermediate Scale Quantum (NISQ) era quantum computers cannot handle the additional computational cost associated with fault-tolerant designs. Our research aims to minimize circuit depth, which can reduce noise and facilitate the implementation of quantum modulo addition circuits on NISQ machines. This work presents quantum carry-lookahead modulo $(2^n - 1)$ adder (QCLMA), which is designed to receive two n-bit numbers and perform their addition with an O(log n) depth. Compared to existing work of O(n) depth, our proposed QCLMA reduces the depth and helps increase the noise fidelity. In order to increase error resilience, we also focus on creating a tree structure based Carry path, unlike the chain based Carry path of the current work. We run experiments on Quantum Computer IBM Cairo to evaluate the performance of the proposed QCLMA against the existing work and define Quantum State Fidelity Ratio (QSFR) to quantify the closeness of the correct output to the top output. When compared against existing work, the proposed QCLMA achieves a 47.21% increase in QSFR for 4-qubit modulo addition showcasing its superior noise fidelity.

Figures (5)

• Figure 1: The CNOT and Toffoli gates.
• Figure 2: Quantum ripple-carry adder (QRCA) based quantum modulo (2n - 1) adder by Kim et al. in 4-qubit configuration kim2021quantum. The inputs are (a0:a3) and (b0:b3) while (S0:S3) represent Sum qubits and (c0:c3) represent the Carry qubits. The QRCA mechanism causes higher depth and longer paths in both Carry and the Sum generation, increasing the noise susceptibility. The LSB qubits are more noise prone as they spend significantly longer time idling than the MSB qubits jayashankar2022achieving.
• Figure 3: Proposed quantum carry-lookahead modulo adder (QCLMA) in 4-qubit configuration. First, the Carry generation logic takes the inputs (a0:a3) and (b0:b3) to generate Carry qubits (c0:c4). The out-of-place carry-lookahead generation logic makes sure that the inputs are passed intact for next stage. The in-place carry-lookahead truncated full adder generates the Sum (S0:S3) in place of input b. Both the Carry and Sum are generated using a tree-based structure which has a shorter O(log n) depth. Lesser variation in idling time among the qubits makes the proposed QCLMA noise resilient.
• Figure 4: Comparison of Average QSFR of Proposed QCLMA and Kim et al.'s QRCA based modulo (2n - 1) adder, obtained keeping constant A and varying B. Proposed adder has 47.21% higher average QSFR establishing superior noise fidelity.
• Figure 5: The above graphs illustrate the output of the modulo adders generated from IBM Cairo, arranged in descending order of frequency. Identical input A = 10 and B = 2 is supplied, with both adders yielding S = 12 or 1100 as the output at the same 11th position among 16 possible outputs. The output profile of our proposed QLCMA adder exhibits a slower descent rate than the version proposed by Kim et al., reducing the relative gap with the top output and improving its chances of improvement.