How Few Bits Per Amplitude? A Rate–Distortion Floor for Quantum State-Vector Formats, and Where Per-Block Scaling Wins
Year: 2026
Type: article
Abstract: Full state-vector simulation of quantum circuits is limited by the memory and bandwidth needed to hold 2^n complex amplitudes. Two published simulators store amplitudes in two bytes rather than sixteen: JUQCS-A (De Raedt et al., 2019), which uses a global linear magnitude grid with an eight-bit phase, and the polar-logarithmic format of Betelú (2020). Betelú closes by asking whether any format uses less storage for a given error. We answer that question. We first establish an information-theoretic floor. The amplitudes of a maximally entangled random state are independent complex Gaussians, so the Shannon rate-distortion bound applies and no format can achieve squared error below 2^-B at B bits per amplitude. Measured against it, JUQCS-A sits 2.17 bits above the floor and Betelú's format 4.21 bits. Block floating point with a shared power-of-two exponent per 32 amplitudes — the MXINT8 layout that current machine-learning hardware executes natively — sits 2.24 bits above it, and 1.20 bits with an exact scale. Block floating point beats JUQCS-A by 1.95x at an exact scale, sits within 17% of it at a power-of-two scale with the ordering set by qubit parity, and beats it on every concentrated state we tested on both metrics. Against Betelú it splits along a boundary we locate and explain. It wins on random states by 3.90x in squared error, a margin that grows as ln N because a global scale is set by an extreme value of the sample. It loses above three or four spikes spread across distinct blocks — the output shape of Shor's algorithm — and it loses the quantum Fourier transform by 2.16x, where Betelú's phase field renders most controlled rotations exact. But the choice of metric decides the winner: across every concentrated state we tested, total variation distance between measurement distributions reverses each squared-error loss, because a block scale flattens amplitudes carrying O(1/N) of the norm that no measurement resolves. All results are at 16-24 qubits on a single core. Every number is produced by a released script. v2 — 2026-09-04 No results or numbers changed from v1. Figure 1: legend moved clear of the curves it was overlapping, and the "Betelú" label in both figure legends corrected — a LaTeX escape had been rendering literally as Betel\'u. Section 4: replaced "per-block scaling recovers most of the remaining gap" with the measured fractions (71% with an exact scale, 47% with a power-of-two scale), which the following paragraph contradicted. Source archive now includes LICENSE and CORRECTIONS.md, so the deposit carries the license it names and the corrections record its README points at.
Author: Merrill, Christopher K.
Language: English
Subfield: Artificial Intelligence
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7208821314