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Residue Number System (RNS) based Distributed Quantum Addition

Bhaskar Gaur, Travis S. Humble, Himanshu Thapliyal

TL;DR

To address noise and depth constraints in quantum arithmetic on NISQ devices, the paper proposes a distributed quantum computing framework that uses Residue Number System representations to partition quantum addition into independent modulo adders. The Quantum Superior Modulo Addition tool (QSMART) automatically selects RNS moduli and generates QDMA-based modulo adders, enabling distributed execution across multiple QPUs. Simulations on Quantinuum's H1 show 11.36% to 133.15% higher output probabilities for 6–10 bit additions compared to non-distributed full adders, illustrating improved noise resilience and scalability beyond 20-qubit ranges. The approach provides a pathway to scalable quantum arithmetic in the NISQ and FTQ eras and suggests extensions to multiplication and data-processing tasks leveraging RNS parallelism.

Abstract

Quantum Arithmetic faces limitations such as noise and resource constraints in the current Noisy Intermediate Scale Quantum (NISQ) era quantum computers. We propose using Distributed Quantum Computing (DQC) to overcome these limitations by substituting a higher depth quantum addition circuit with Residue Number System (RNS) based quantum modulo adders. The RNS-based distributed quantum addition circuits possess lower depth and are distributed across multiple quantum computers/jobs, resulting in higher noise resilience. We propose the Quantum Superior Modulo Addition based on RNS Tool (QSMART), which can generate RNS sets of quantum adders based on multiple factors such as depth, range, and efficiency. We also propose a novel design of Quantum Diminished-1 Modulo (2n + 1) Adder (QDMA), which forms a crucial part of RNS-based distributed quantum addition and the QSMART tool. We demonstrate the higher noise resilience of the Residue Number System (RNS) based distributed quantum addition by conducting simulations modeling Quantinuum's H1 ion trap-based quantum computer. Our simulations demonstrate that RNS-based distributed quantum addition has 11.36% to 133.15% higher output probability over 6-bit to 10-bit non-distributed quantum full adders, indicating higher noise fidelity. Furthermore, we present a scalable way of achieving distributed quantum addition higher than limited otherwise by the 20-qubit range of Quantinuum H1.

Residue Number System (RNS) based Distributed Quantum Addition

TL;DR

To address noise and depth constraints in quantum arithmetic on NISQ devices, the paper proposes a distributed quantum computing framework that uses Residue Number System representations to partition quantum addition into independent modulo adders. The Quantum Superior Modulo Addition tool (QSMART) automatically selects RNS moduli and generates QDMA-based modulo adders, enabling distributed execution across multiple QPUs. Simulations on Quantinuum's H1 show 11.36% to 133.15% higher output probabilities for 6–10 bit additions compared to non-distributed full adders, illustrating improved noise resilience and scalability beyond 20-qubit ranges. The approach provides a pathway to scalable quantum arithmetic in the NISQ and FTQ eras and suggests extensions to multiplication and data-processing tasks leveraging RNS parallelism.

Abstract

Quantum Arithmetic faces limitations such as noise and resource constraints in the current Noisy Intermediate Scale Quantum (NISQ) era quantum computers. We propose using Distributed Quantum Computing (DQC) to overcome these limitations by substituting a higher depth quantum addition circuit with Residue Number System (RNS) based quantum modulo adders. The RNS-based distributed quantum addition circuits possess lower depth and are distributed across multiple quantum computers/jobs, resulting in higher noise resilience. We propose the Quantum Superior Modulo Addition based on RNS Tool (QSMART), which can generate RNS sets of quantum adders based on multiple factors such as depth, range, and efficiency. We also propose a novel design of Quantum Diminished-1 Modulo (2n + 1) Adder (QDMA), which forms a crucial part of RNS-based distributed quantum addition and the QSMART tool. We demonstrate the higher noise resilience of the Residue Number System (RNS) based distributed quantum addition by conducting simulations modeling Quantinuum's H1 ion trap-based quantum computer. Our simulations demonstrate that RNS-based distributed quantum addition has 11.36% to 133.15% higher output probability over 6-bit to 10-bit non-distributed quantum full adders, indicating higher noise fidelity. Furthermore, we present a scalable way of achieving distributed quantum addition higher than limited otherwise by the 20-qubit range of Quantinuum H1.
Paper Structure (12 sections, 3 equations, 5 figures, 2 tables, 2 algorithms)

This paper contains 12 sections, 3 equations, 5 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Quantum Modulo Addition
  4. Residue Number System
  5. Distributed Quantum Computing
  6. Proposed Quantum Diminished-1 Modulo (2n + 1) Adder
  7. Proposed QSMART Tool
  8. Results
  9. Quantum Modulo Adders
  10. Noise-Resilience of QSMART Tool
  11. Discussion
  12. Conclusion

Figures (5)

  • Figure 1: Quantum Modulo 2n adders used in this work: (a) Mod 4 Adder. (b) Mod 8 Adder. The inputs are A and B. The output M is Modulo Sum, while input B is passed unchanged.
  • Figure 2: Comparison of Quantum Modulo 3 adders used in this work: (a) Quantum Modulo (2n - 1) adder for n = 2 configuration kim2021quantum. (b) Quantum Modulo (2n + 1) adder for n = 1 configuration. The inputs are A (A0:A1) and B (B0:B1), while the output is Modulo Sum M (M0:M1). Input A is passed without modification. Quantum Modulo (2n + 1) adder's Toffoli count and CNOT count are lower by 3 and 6 respectively. Also, its Toffoli depth and CNOT depth are lower by 3 and 5 respectively, showcasing its better design.
  • Figure 3: Proposed Quantum Superior Modulo Addition based on RNS Tool (QSMART) uses a hybrid classical-quantum flow to achieve Distributed Quantum Computing (DQC) for quantum addition. QSMART accepts the range of addition, planned efficiency of representation, and inputs to deliver multiple independent Residue Number System (RNS) based quantum modulo adders that can be distributed across multiple quantum computers or jobs.
  • Figure 4: Proposed Quantum Modulo (2n + 1) Adder for n = 3 configuration yielding Quantum Modulo 9 Adder. The inputs are A (A0:A3) and B (B0:B3), while the output is Modulo Sum M (M0:M3). Input B (B0:B3) and MSB of A (A3) are passed unchanged. The Sum (S0:S2:Carry3) represents the sum of (n-1) qubits of inputs A and B. As shown in Algorithm \ref{['alg:1']}, first the Sum is calculated, with the Carry3 used to compute $\overline{A\textsubscript{3}}.\overline{B\textsubscript{3}}.\overline{Carry\textsubscript{3}}$. Finally, quantum half adder stage calculates the Modulo Sum M after adding (A3.B3, S2 ... S0) + $\overline{A\textsubscript{3}}.\overline{B\textsubscript{3}}.\overline{Carry\textsubscript{3}}$
  • Figure 5: Comparison of Output Probability between TPL13 based quantum full adders and QSMART Tool's RNS based distributed quantum addition for output sizes 6 to 10 qubit.