Performance Evaluations of Signed and Unsigned Noisy Approximate Quantum Fourier Arithmetic

TL;DR

The paper evaluates QFT-based quantum arithmetic on Noisy Intermediate-Scale Quantum devices, comparing signed and unsigned addition and multiplication using approximate QFT (AQFT) circuits. It systematically varies 1q/2q gate errors, AQFT depth $d$, and operand superposition to identify regimes where AQFT improves performance over full QFT. Key findings show that AQFT often yields higher success under noise, with an optimal depth around $d\approx \log_2 n$, though the best $d$ depends on superposition and noise; 2q errors are the dominant bottleneck, while signed QFA frequently outperforms unsigned. The study highlights algorithmic challenges and proposes directions for error mitigation, modular arithmetic extensions, and larger-scale simulations to move toward a quantum numerical paradigm.

Abstract

The Quantum Fourier Transform (QFT) grants competitive advantages, especially in resource usage and circuit approximation, for performing arithmetic operations on quantum computers, and offers a potential route towards a numerical quantum-computational paradigm. In this paper, we utilize efficient techniques to implement QFT-based integer addition and multiplications. These operations are fundamental to various quantum applications including Shor's algorithm, weighted sum optimization problems in data processing and machine learning, and quantum algorithms requiring inner products. We carry out performance evaluations of these implementations based on IBM's superconducting qubit architecture using different compatible noise models. We isolate the sensitivity of the component quantum circuits on both one-/two-qubit gate error rates, and the number of the arithmetic operands' superposed integer states. We analyze performance, and identify the most effective approximation depths for unsigned quantum addition and quantum multiplication within the given context. We then perform a similar analysis of signed addition and compare to the unsigned results. We observe significant dependency of the optimal approximation depth on the degree of machine noise and the number of superposed states in certain performance regimes. Finally, we elaborate on the algorithmic challenges - relevant to signed, unsigned, modular and non-modular versions - that could also be applied to current implementations of QFT-based subtraction, division, exponentiation, and their potential tensor extensions. We analyze the performance trends in our results and speculate on possible future developments within this computational paradigm.

Figures (7)

• Figure 1: Generalized quantum circuit for the QFT. Gates removed when performing the AQFT at approximation depth $d$ are drawn in red.
• Figure 2: The generalized addition step of the QFA circuit, being applied in the transform domain.
• Figure 3: The QFA (top), sQFA (middle), and cQFA (bottom) composite gates.
• Figure 4: The generalized QFM circuit, drawn such that $m<n-1$. If $n=m$, then $\ket{z_n} = \ket{z_{m+1}}$ and the diagram shifts, and further shifts if $m>n$.
• Figure 5: Success rates (vertical axes) of signed quantum addition of two 8-qubit q-integers performed in the Quantum Fourier Basis with varying gate error rates, number of superposed addend state, and AQFT approximation depth; see Sec. \ref{['sec_results']} for details. The left/right column presents results with varying 1q-gate/2q-gate error rate, ($P_{1q}^{err\%}/P_{2q}^{err\%}$) on the horizontal axes, respectively. The top/middle/bottom row shows results for (1:1)/(1:2)/(2:2) superposed addend states. Color-coded results for each depth of AQFM approximation depth are clustered along the horizontal axes and are only shifted apart horizontally for visibility; the alignment of the center-most point of a cluster reflects the gate error rate used for all calculations in that cluster. The 'full' approximation depth designates that the full QFT circuit is performed in those simulations.