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Non-Identity Check Remains QMA-Complete for Short Circuits

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arxiv 0906.5416 v1 pith:IOE7LTOH submitted 2009-06-30 quant-ph

classification quant-ph
keywords problemcheckcircuitdepthnon-identityqma-completegateremains
verification ladder T0 review T1 audit T2 compute T3 formal
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abstract

The Non-Identity Check problem asks whether a given a quantum circuit is far away from the identity or not. It is well known that this problem is QMA-Complete \cite{JWB05}. In this note, it is shown that the Non-Identity Check problem remains QMA-Complete for circuits of short depth. Specifically, we prove that for constant depth quantum circuit in which each gate is given to at least $\Omega(\log n)$ bits of precision, the Non-Identity Check problem is QMA-Complete. It also follows that the hardness of the problem remains for polylogarithmic depth circuit consisting of only gates from any universal gate set and for logarithmic depth circuit using some specific universal gate set.

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Forward citations

Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Q-Sylvan: A Parallel Decision Diagram Package for Quantum Computing

    quant-ph 2025-08 conditional novelty 6.0 of 10

    Q-Sylvan parallelizes edge-valued decision diagrams for quantum circuit simulation and equivalence checking, reaching up to 18x speedup on 64 cores on certain circuit classes.

  2. Predicting symmetries of quantum dynamics with optimal samples

    quant-ph 2025-02 conditional novelty 6.0 of 10

    Optimal failure probabilities for detecting identity, diagonal, and real symmetries of unknown qubit unitaries are exactly computed and achieved by parallel strategies.

  3. Position: Quantum Program Generation Must Prioritize Validity Over Probabilistic Scaling

    cs.LG 2026-07 conditional novelty 5.0 of 10

    The paper argues that probabilistic scaling alone cannot fix the validity gap in quantum circuit generation, so quantum code assistants must build verification into generation rather than filter outputs after the fact.

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