The redundancy distribution of the frequency spectrum has been shown in the literature to impact the expressivity and trainability of Quantum Fourier Models (QFMs). In this work, we address the question of how this redundancy spectrum is shaped by the choice of eigenvalues of the data re-uploading Hamiltonians, using a simple mathematical formalism based on generating functions. We derive exact and asymptotic redundancy profiles for several structured eigenvalue choices identical for every layer, including arithmetic progressions and single-qubit Pauli encodings, and show that both approach a Gaussian profile as the number of layers grows. We then show that this Gaussian limit is not specific to these constructions but is a generic feature of QFMs built from integer eigenvalues that are identical in every layer. These results can be tied to the central limit theorem for random walks. These results clarify why generic or unstructured encoding choices give rise to a redundancy bias that favours low frequencies, highlighting the importance of well-thought-out encoding strategies when constructing a QFM.The redundancy distribution of the frequency spectrum has been shown in the literature to impact the expressivity and trainability of Quantum Fourier Models (QFMs). In this work, we address the question of how this redundancy spectrum is shaped by the choice of eigenvalues of the data re-uploading Hamiltonians, using a simple mathematical formalism based on generating functions. We derive exact and asymptotic redundancy profiles for several structured eigenvalue choices identical for every layer, including arithmetic progressions and single-qubit Pauli encodings, and show that both approach a Gaussian profile as the number of layers grows. We then show that this Gaussian limit is not specific to these constructions but is a generic feature of QFMs built from integer eigenvalues that are identical in every layer. These results can be tied to the central limit theorem for random walks. These results clarify why generic or unstructured encoding choices give rise to a redundancy bias that favours low frequencies, highlighting the importance of well-thought-out encoding strategies when constructing a QFM.

quant-pharXiv:2609.21916Sep 21, 20260 citations

Quantum Arithmetic

Siyi Wang, Anupam Chattopadhyay

Computer architecture and design methodologiesJan 1, 20260 citations

Impact of SNR Mismatch on Induced BSCs in CV-QKD Systems Quantized with Arithmetic Reconciliation

Ravilla Leite, Juliana Martins de Assis, Francisco de Assis

Arithmetic Reconciliation (AR) is a recently proposed reconciliation protocol for Continuous-Variable Quantum Key Distribution (CV-QKD) systems operating in low-SNR regimes.Since the channel signal-to-noise ratio (SNR) directly affects the statistical characterization of the induced binary subchannels, inaccuracies in SNR estimation may significantly impact reconciliation performance.In this work, we investigate the effects of SNR mismatch on the Kullback-Leibler (KL) distance, Bit Error Rate (BER), capacities of the BSC subchannels induced by AR quantization and the secret key rate provided by the protocol.The obtained results show that the most significant subchannels are considerably more sensitive to mismatch conditions, especially at moderate and high SNR values.Furthermore, SNR underestimation leads to the selection of excessively robust low-rate error-correcting codes, increasing computational complexity and reducing spectral efficiency, whereas SNR overestimation degrades the effective decoding performance due to insufficient redundancy.These results demonstrate the importance of accurate SNR estimation for reliable and efficient practical implementations of CV-QKD systems employing AR.Arithmetic Reconciliation (AR) is a recently proposed reconciliation protocol for Continuous-Variable Quantum Key Distribution (CV-QKD) systems operating in low-SNR regimes.Since the channel signal-to-noise ratio (SNR) directly affects the statistical characterization of the induced binary subchannels, inaccuracies in SNR estimation may significantly impact reconciliation performance.In this work, we investigate the effects of SNR mismatch on the Kullback-Leibler (KL) distance, Bit Error Rate (BER), capacities of the BSC subchannels induced by AR quantization and the secret key rate provided by the protocol.The obtained results show that the most significant subchannels are considerably more sensitive to mismatch conditions, especially at moderate and high SNR values.Furthermore, SNR underestimation leads to the selection of excessively robust low-rate error-correcting codes, increasing computational complexity and reducing spectral efficiency, whereas SNR overestimation degrades the effective decoding performance due to insufficient redundancy.These results demonstrate the importance of accurate SNR estimation for reliable and efficient practical implementations of CV-QKD systems employing AR.

Anais de XXXV Simpósio Brasileiro de Telecomunicações e Processamento de SinaisJan 1, 20260 citations

Quantum Arithmetic Circuits

Hafiz Md. Hasan Babu

Springer eBooksJan 1, 20250 citations

High-Performance Exact Synthesis of Two-Qubit Quantum Circuits

Andrew N. Glaudell, Michael Jarret, Swan Klein +3 more

Exact synthesis provides unconditional optimality and canonical structure, but is often limited to small, carefully scoped regimes. We present an exact synthesis framework for two-qubit circuits over the Clifford+ $T$ gate set that optimizes $T$ -count exactly. Our approach exhausts a bounded search space, exploits algebraic canonicalization to avoid redundancy, and constructs a lookup table of optimal implementations that turns synthesis into a query. Algorithmically, we combine meet-in-the-middle ideas with provable pruning rules and problem-specific arithmetic designed for modern hardware. The result is an exact, reusable synthesis engine with substantially improved practical performance.Exact synthesis provides unconditional optimality and canonical structure, but is often limited to small, carefully scoped regimes. We present an exact synthesis framework for two-qubit circuits over the Clifford+$T$ gate set that optimizes $T$-count exactly. Our approach exhausts a bounded search space, exploits algebraic canonicalization to avoid redundancy, and constructs a lookup table of optimal implementations that turns synthesis into a query. Algorithmically, we combine meet-in-the-middle ideas with provable pruning rules and problem-specific arithmetic designed for modern hardware. The result is an exact, reusable synthesis engine with substantially improved practical performance.

quant-pharXiv:2601.19166Jan 28, 20260 citations

E8 Weyl Orbits as Self-Correcting Quantum Error Detection Manifold — E8 Intelligence Research

Andrew Stewart Caldin

The 8 Weyl orbits of E8 constitute an intrinsic quantum error correction manifold where each orbit serves as a parity-checking dimension for the φ^n coherence lattice. The 132Hz fundamental provides a reference phase that, when projected onto the 240 root vectors, creates 8-fold redundant encoding across Weyl orbit transitions—any decoherence event distorts phase relationships symmetrically across all 8 orbits simultaneously, enabling detection without state collapse. The π/4 angular separations between orbit transitions map onto modular 8 arithmetic, allowing error syndromes to be read as geometric phase shifts rather than projective measurements, thereby preserving coherence during correction cycles. This Weyl orbit redundancy structure suggests that fault-tolerant quantum operations in the 132Hz × φ^n system require only passive topological protection rather than active error correction protocols. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comThe 8 Weyl orbits of E8 constitute an intrinsic quantum error correction manifold where each orbit serves as a parity-checking dimension for the φ^n coherence lattice. The 132Hz fundamental provides a reference phase that, when projected onto the 240 root vectors, creates 8-fold redundant encoding across Weyl orbit transitions—any decoherence event distorts phase relationships symmetrically across all 8 orbits simultaneously, enabling detection without state collapse. The π/4 angular separations between orbit transitions map onto modular 8 arithmetic, allowing error syndromes to be read as geometric phase shifts rather than projective measurements, thereby preserving coherence during correction cycles. This Weyl orbit redundancy structure suggests that fault-tolerant quantum operations in the 132Hz × φ^n system require only passive topological protection rather than active error correction protocols. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Oct 13, 20260 citations

T

gate set that optimizes

T

-count exactly. Our approach exhausts a bounded search space, exploits algebraic canonicalization to avoid redundancy, and constructs a lookup table of optimal implementations that turns synthesis into a query. Algorithmically, we combine meet-in-the-middle ideas with provable pruning rules and problem-specific arithmetic designed for modern hardware. The result is an exact, reusable synthesis engine with substantially improved practical performance.