Unitary RQL Equals RQL — Draconitic Research

Abstract

Intermediate measurements let quantum computations discard information and reuse their workspace. Deferring all measurements can require storing their entire history, which need not preserve logarithmic space. Fefferman and Remscrim proved that measurements can nevertheless be eliminated with two-sided bounded error, and asked whether the same holds with one-sided error [FR21]. We prove RQLΓ = RQULΓ for the standard gate set Γ = {H, T, CNOT}, preserving polynomial time, logarithmic space, and exactly zero acceptance on no-instances. Our proof extends the density-matrix doubling method of Girish, Raz and Zhan [ GRZ21 ] to general channels, including resets and classical memory. We also prove measurement elimination for broader finite gate sets with exact inverses, including suitable gates with transcendental entries, and establish gate-set independence for a specified family of algebraic gate sets. A separate history-checking construction proves QMAL1,G = QUMAL1,G under explicit gate assumptions: measurements can also be eliminated from logarithmic-space quantum verification while preserving perfect completeness

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BibTeX RIS

BibTeX

@misc{tupker2026unitary,
  title = {{Unitary RQL Equals RQL}},
  author = {Quinten Tupker},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.06231},
  eprint = {2610.06231},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.06231},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Unitary RQL Equals RQL
AU  - Quinten Tupker
PY  - 2026
DA  - 2026/10/05
PB  - arXiv
DO  - 10.48550/arxiv.2610.06231
UR  - https://arxiv.org/abs/2610.06231
AB  - Intermediate measurements let quantum computations discard information and reuse their workspace. Deferring all measurements can require storing their entire history, which need not preserve logarithmic space. Fefferman and Remscrim proved that measurements can nevertheless be eliminated with two-sided bounded error, and asked whether the same holds with one-sided error [FR21]. We prove RQLΓ = RQULΓ for the standard gate set Γ = {H, T, CNOT}, preserving polynomial time, logarithmic space, and exactly zero acceptance on no-instances. Our proof extends the density-matrix doubling method of Girish, Raz and Zhan [ GRZ21 ] to general channels, including resets and classical memory. We also prove measurement elimination for broader finite gate sets with exact inverses, including suitable gates with transcendental entries, and establish gate-set independence for a specified family of algebraic gate sets. A separate history-checking construction proves QMAL1,G = QUMAL1,G under explicit gate assumptions: measurements can also be eliminated from logarithmic-space quantum verification while preserving perfect completeness
ER  -

Abstract

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