A Reproducible Walk-Through of RSA, Classical Factoring, and Shor's Algorithm on IBM Heron r2 Hardware

Bilisuma Merga

Farhan K A, Raghavan G

Abstract Recent research on quantum walk-based optimization has shown theoretical potential as an alternative to variational algorithms, avoiding local minima entrapment or barren plateaus by utilizing physically motivated structure rather than unconstrained parameter search. Despite this, the Guided Quantum Walk (GQW) has never been used for portfolio optimization or tested on existing noisy quantum hardware. In this paper we propose a hybrid quantum-classical solver for portfolio optimization that integrates Guided Quantum Walks (GQW) with domain-specific knowledge elicited from real Indian equity data. The core technical contributions are threefold. First, we implement a parameterized GQW circuit in Qiskit (v2.2) with a B\'{e}zier-curve-parameterized hopping rate schedule, mapping the portfolio selection problem directly to the GQW Hilbert space via an Ising Hamiltonian encoding. Second, we introduce a problem-aware warm-start strategy that initializes the energy scale parameters from classical sampling of the feasible portfolio subspace, bypassing the high-energy exploration phase that characterizes naive initialization. Third, we incorporate Conditional Value-at-Risk (CVaR) aggregation into the quantum walk measurement to improve solution space sampling, as well as a hardware-adaptive pipeline that adjusts concentration to per-backend circuit fidelity for IBM Heron devices.\\Our simulation results demonstrate near-optimal approximation ratios exceeding 0.99 for problem sizes up to N=20 on a noiseless setting with CVaR concentration \(((\alpha \in \{0.1,0.25\}))\), significantly outperforming standard energy expectation values \(((\alpha=1.0))\). Hardware validation on IBM Heron R2 processors, confirms that parameters optimized on noiseless simulation transfer effectively to real devices without re-optimization, with median approximation ratios above 0.85 across all backends for N=12 drawn from live NIFTY 50 market data. This constitutes the first end-to-end GQW implementation for financial portfolio optimization using IBM quantum hardware, and its application as a scalable quantum solver for real-world financial decision support systems.

Bhupendra Singh, Mohankumar Mylsamy, Thamaraimanalan Thangarajan

“Shor's Algorithm: Factoring and Cryptanalysis” focuses on Shor's Algorithm, particularly its application to factoring and cryptanalysis. It begins with an overview of Shor's algorithm, detailing the classical extraction of periods from measured values and the quantum gates required for the Quantum Fourier Transform (QFT). Further, this chapter explores the applications and limitations of Shor's algorithm, including its impact on RSA encryption. A literature review follows, discussing Shor's algorithm in the context of quantum computation, its effectiveness on RSA, and the effects of imperfections on its factorization process. The section also includes an experimental study of Shor's algorithm using IBM Q, followed by a practical implementation of factoring numbers N=21 and N=35. Further, it covers resource estimation, quantum simulations, and different types of simulators in IBM Qiskit, including state vector, QASM, Matrix Product State (MPS), and stabilizer simulators, providing insights into their use cases and the limits of classical simulation.

Anand Ghugare, Atharva Rajoba, Dinesh Kumar Saini

James P. Galasyn, Claude Théodore

Experimental companion (Paper 21b) of the NWT program. The substrate-canonical Steane [[7,1,3]] preparation circuits are executed on real quantum hardware (IBM Heron r2, IQM Garnet, AQT Ibex-Q1), reproducing the predicted stabilizer-syndrome structure cross-vendor and cross-architecture, including the cross-sector syndrome equivalences. Honest scope: the experiments establish that the predicted syndrome structure is preparable, mutually distinguishable, and architecture-independent on real hardware (realizability), not framework validation. Companion to Paper 21a (theory) and Paper 22 (cosmogenesis).

憲 栗山, S. Sano, S. Furuichi