On Pins and Needles: The Geometry of Quantum Measurements

James Thomas Wheeler

Akira Imada

If we want to break someone else's PIN (personal identification number) of, say, an ATM (automated teller machine), how many trials would be necessary when we want to be efficient? This is a sort of what we calla-needle-in-a-hay-stackproblem. In 1987, in their wonderful paper, Hinton & Nowlan proposed a genetic algorithm with a needle being a unique configuration of 20-bit binary string while all other configurations being a haystack. What they proposed was to exploit alifetimelearningof individuals in their genetic algorithm, calling it the Baldwin effectinacomputer. Since then there has been a fair amount of exploration of this effect, claiming, "this is a-needle-in-a-hay-stack problem, and we've found a more efficient algorithm than a random search." Some of them, however, were found to be the results of an effect oflike-to-hear-what-we-would-like-to-hear. In this talk, we will try a bird's eye view on a few examples we have had so far, and how they were explored, including the approach by means of quantum computation which claims, "The steps to find a needle areO(radicN) while those of exhaustive search by a traditional computer are O(N) whereNis the number of search points."

William K. Wootters

Takayuki Takagi

Quantum Buffon's Needle: Deriving π from the GHZ Structure Constant Overview We demonstrate that the fundamental mathematical constant π can be derived from the quantum structure constant λ₃* = 3/(2π²), which characterizes the phase-space capacity of three-qubit entangled systems. By measuring the structural coherence of GHZ states on IBM Quantum hardware (ibm_torino), we obtained: $$\lambda_3 \approx 0.1482 \quad \Rightarrow \quad \hat{\pi} = \sqrt{\frac{3}{2\lambda_3}} \approx 3.181$$ with only 1.25% deviation from the true value. Key Equation $$\pi = \sqrt{\frac{3}{2\lambda_3^*}}$$ This "Quantum Buffon's Needle" experiment establishes π as a measurable property of quantum geometry, paralleling Buffon's classical probability-based derivation while revealing deeper connections between quantum information theory, spherical geometry, and physical reality. Main Results Measurement Value Mermin inequality violation >700σ above classical bound Measured structure constant λ₃ = 0.1482 ± 0.0006 Reconstructed π 3.181 Relative error 1.25% Platform IBM Quantum ibm_torino (Heron r2) Files Included quantum_buffons_needle_pi_derivation.pdf — Full manuscript quantum_pi_scan.py — Python script for multi-backend π derivation experiments README.md — Documentation Significance This result establishes a novel paradigm: π is not merely an abstract mathematical entity but a measurable physical constant embedded in the geometric fabric of quantum state space. The method enables evaluation of quantum devices by their ability to reproduce fundamental geometric constants—a new direction in quantum benchmarking.

G. Compagno

The Bohr-Rosenfeld measurement theory of the electromagnetic field amplitude is applied to a needle-like detector. The expression of the time average of the self-force due to the measurement process is explicitly obtained. It is shown that the appearance of divergences does not allow the approximations required by the Bohr-Rosenfeld theory.

R. W. Rendell, Michael Steiner