Ultra-Precise Quantum Projective Designs in Constant Depth
Abstract
Random quantum objects are powerful resources for quantum information processing, yet exact Haar randomness is costly and typically unnecessary. We introduce an explicit sparse commuting circuit ensemble on n qubits that reproduces low-order Haar moments in the stringent relative-error sense. The circuit consists of a sparse Clifford phase layer followed by independent single-qubit Clifford gates. Acting on a simple product state, the resulting ensemble forms ε-approximate projective $2$- and $3$-designs in relative error, with the required logarithmic interaction degree being asymptotically optimal within this circuit family. It admits an ancilla-free implementation of quantum depth O(log(n/ε)) on an all-to-all architecture, as well as an adaptive constant-depth implementation---in fact, depth seven---using O(nlog(n/ε)) ancilla qubits. Departing from existing shallow-design paradigms, our analysis exploits the intrinsic moment structure of commuting phase circuits; at third order, this requires a new block decomposition and combinatorial analysis that also suggests a route toward higher-order shallow designs. Our results show that precise Haar-like statistics can emerge from sparse commuting dynamics with remarkably low quantum resources, with applications to randomized characterization, quantum metrology, quantum algorithms, and many-body physics.
Related
- TDCOSMO XXVIII. The Hubble constant from the quadruply lensed quasar J1537−3010 with precise time delays
- Lattice moments and quantum designs
- Moment-Selective Quantum Designs in Aperiodic Temporal Ensembles
- Can Entanglement-Induced Feature-Map Designs Improve Projected Quantum Kernels?
- Quantum state learning beyond approximate unitary designs
BibTeX
@misc{zhang2026ultraprecise,
title = {{Ultra-Precise Quantum Projective Designs in Constant Depth}},
author = {Qingyue Zhang and Junjie Chen and Zhou You and You Zhou},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.03925},
eprint = {2609.03925},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.03925},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Ultra-Precise Quantum Projective Designs in Constant Depth
AU - Qingyue Zhang
AU - Junjie Chen
AU - Zhou You
AU - You Zhou
PY - 2026
DA - 2026/09/03
PB - arXiv
DO - 10.48550/arxiv.2609.03925
UR - https://arxiv.org/abs/2609.03925
AB - Random quantum objects are powerful resources for quantum information processing, yet exact Haar randomness is costly and typically unnecessary. We introduce an explicit sparse commuting circuit ensemble on n qubits that reproduces low-order Haar moments in the stringent relative-error sense. The circuit consists of a sparse Clifford phase layer followed by independent single-qubit Clifford gates. Acting on a simple product state, the resulting ensemble forms ε-approximate projective $2$- and $3$-designs in relative error, with the required logarithmic interaction degree being asymptotically optimal within this circuit family. It admits an ancilla-free implementation of quantum depth O(log(n/ε)) on an all-to-all architecture, as well as an adaptive constant-depth implementation---in fact, depth seven---using O(nlog(n/ε)) ancilla qubits. Departing from existing shallow-design paradigms, our analysis exploits the intrinsic moment structure of commuting phase circuits; at third order, this requires a new block decomposition and combinatorial analysis that also suggests a route toward higher-order shallow designs. Our results show that precise Haar-like statistics can emerge from sparse commuting dynamics with remarkably low quantum resources, with applications to randomized characterization, quantum metrology, quantum algorithms, and many-body physics.
ER -