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Analytical Lower Bound on Query Complexity for Transformations of Unknown Unitary Operations.

Tatsuki Odake1, Satoshi Yoshida1, Mio Murao1,2

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This study establishes analytical lower bounds for the query complexity of unitary operations like inversion and conjugation. The findings demonstrate the optimality of existing protocols and reveal limitations for certain methods.

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Area of Science:

  • Quantum computing
  • Linear algebra
  • Information theory

Background:

  • Unitary operations are fundamental in quantum computation.
  • Efficient protocols for manipulating unknown unitary operations are crucial.
  • Previous work established deterministic protocols for complex conjugation, inversion, and transposition.

Purpose of the Study:

  • To establish analytical lower bounds for the query complexity of unitary inversion, transposition, and complex conjugation.
  • To assess the optimality of existing deterministic exact protocols.
  • To explore the possibility of catalytic protocols for unitary complex conjugation.

Main Methods:

  • Derivation of analytical lower bounds using a novel differentiation framework.
  • Analysis of query complexity for general differentiable functions f: SU(d) → SU(d).
  • Extension of the framework to partially known and probabilistic settings.

Main Results:

  • Established a lower bound of d^2 for unitary inversion, proving asymptotic optimality of O(d^2) inversion protocols.
  • Demonstrated the impossibility of catalytic protocols for unitary complex conjugation.
  • Extended the analysis to scenarios with partial knowledge and probabilistic outcomes.

Conclusions:

  • The established lower bounds provide fundamental limits on the efficiency of manipulating unknown unitary operations.
  • Deterministic exact inversion protocols are asymptotically optimal.
  • Catalytic protocols are not viable for unitary complex conjugation, highlighting the unique challenges of this operation.