Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2503.23939

Simulation of Shor algorithm for discrete logarithm problems with comprehensive pairs of modulo p and order q

Kaito Kishi·Junpei Yamaguchi·Tetsuya Izu·Noboru Kunihiro
View original source ↗Read source document ↗Export metadata ↓

Abstract

The discrete logarithm problem (DLP) over finite fields, commonly used in classical cryptography, has no known polynomial-time algorithm on classical computers. However, Shor has provided its polynomial-time algorithm on quantum computers. Nevertheless, there are only few examples simulating quantum circuits that operate on general pairs of modulo $p$ and order $q$. In this paper, we constructed such quantum circuits and solved DLPs for all 1,860 possible pairs of $p$ and $q$ up to 32 qubits using a quantum simulator with PRIMEHPC FX700. From this, we obtained and verified values of the success probabilities, which had previously been heuristically analyzed by Eker\r{a}. As a result, the detailed waveform shape of the success probability of Shor's algorithm for solving the DLP, known as a periodic function of order $q$, was clarified. Additionally, we generated 1,015 quantum circuits for larger pairs of $p$ and $q$, extrapolated the circuit sizes obtained, and compared them for $p=2048$ bits between safe-prime groups and Schnorr groups. While in classical cryptography, the cipher strength of safe-prime groups and Schnorr groups is the same if $p$ is equal, we quantitatively demonstrated how much the strength of the latter decreases to the bit length of $p$ in the former when using Shor's quantum algorithm. In particular, it was experimentally and theoretically shown that when a ripple carry adder is used in the addition circuit, the cryptographic strength of a Schnorr group with $p=2048$ bits under Shor's algorithm is almost equivalent to that of a safe-prime group with $p=1024$ bits.

Explore related subjects

quant-ph

Keep this discovery

BibTeXRIS

Kaito Kishi, Junpei Yamaguchi, Tetsuya Izu, Noboru Kunihiro. 2025-03-31. Simulation of Shor algorithm for discrete logarithm problems with comprehensive pairs of modulo p and order q. https://doi.org/10.1109/tqe.2025.3591213

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Probing the Error-Mitigation Threshold with Matrix Product States

Quantum error mitigation relies on accurate noise characterization, but mismatches between the actual and characterized noise can be amplified and drive a sharp threshold between successful and failed mitigation. In random circuits, this threshold maps onto a random-field Ising transition, but previous exact numerics were limited to small one-dimensional and all-to-all systems, leaving explicit two-dimensional architectures unresolved. We develop a fixed-bond-dimension matrix-product-state method for the replicated transfer dynamics that extends threshold calculations beyond exact propagation while retaining the finite-size signatures of the transition. At system sizes beyond previous exact studies, we recover the predicted absence of a threshold for quenched disorder in 1D, obtain a sharper annealed all-to-all critical point, and resolve architecture-dependent finite-depth thresholds in 2D square and heavy-hex circuits. These results establish replicated tensor-network dynamics as a practical tool for probing error-mitigation thresholds in large and higher-dimensional noisy circuits.

quant-ph↗
arXiv2026

Low-cost algorithm-to-execution framework for surface-code quantum computing

The execution of useful quantum algorithms on fault-tolerant processors requires more than a mapping from logical gates to encoded operations: the spatial organization, non-Clifford resource supply, and execution schedule must also be determined while keeping physical overhead within practical limits. Although the theoretical hierarchy from logical circuits to fault-tolerant operations is well established, these implementation choices are often specified and optimized separately. Here we develop a low-cost algorithm-to-execution framework for surface-code quantum computing. From hierarchical algorithm descriptions, it constructs dependency-preserving logical schedules and an executable workload capturing logical interactions, operation parallelism, and time-resolved non-Clifford demand, thereby linking logical computation to surface-code organization, resource-state preparation, and fault-tolerant execution in a traceable workflow. We apply the framework to twenty benchmark circuits across seven algorithm families and a hierarchically composed application-scale elliptic-curve discrete-logarithm workload. Physical costs vary substantially even for circuits with similar logical resource counts. Under our direct-rotation calibration, non-Clifford implementation selection reduces space-time volume by up to 241.5 times versus an all-synthesis baseline for the QAOA amplitude-amplification workload. Circuit-specific surface-code layouts reduce routed-latency estimates for all twenty benchmarks; thirteen also reduce space-time volume because communication savings outweigh added spatial overhead. These results show that low-cost fault-tolerant execution depends on computation scheduling and organization, not aggregate logical resource counts alone.

quant-ph↗
arXiv2026

Sample-optimal learning of stabilizer states

It is well-known that learning a pure $n$-qubit stabilizer state $|\psi\rangle$ both requires, and can be accomplished with, access to a number of copies of $|\psi\rangle$ linear in $n$. However, the precise constant coefficient of this scaling does not appear to have been determined. Here we prove that $L_\delta(n)$, the smallest number of copies from which a quantum procedure can identify any stabilizer state with failure probability at most $0<\delta<1/8$, satisfies $n+\lceil\log_2(1/\delta)\rceil-3\leq L_\delta(n)\leq n+\left\lceil\log_2(1/\delta)\right\rceil+4$. We present a polynomial-time quantum learning algorithm that saturates this bound, achieving a constant factor improvement in sample-complexity over previously known approaches. As an immediate corollary, we obtain via the Choi-Jamiolkowski isomorphism an algorithm for learning an unknown $n$-qubit Clifford unitary from $2n+\left\lceil\log_2(1/\delta)\right\rceil+4$ queries, the $n$-dependence of which we show to be optimal. Our proof technique, which involves Fourier analysis on the abelian group $\mathbb{Z}_4^n \times \mathbb{F}_2^{n(n-1)/2}$, seems to be qualitatively different to previous approaches to stabilizer state learning, and may be of some independent interest; in particular, it admits natural generalisations to further problems in quantum learning theory.

quant-ph↗