Quantum lower bounds by polynomials

Robert Beals, Harry Buhrman, Richard Cleve, Michele Mosca, Ronald De Wolf

Research output: Contribution to journalArticlepeer-review

315 Scopus citations


We examine the number of queries to input variables that a quantum algorithm requires to compute Boolean functions on {0, } Nin the black-box model. We show that the exponential quantum speed-up obtained for partial functions (i.e., problems involving a promise on the input) by Deutsch and Jozsa, Simon, and Shor cannot be obtained for any total function: if a quantum algorithm computes some total Boolean function fwith small error probability using T black-box queries, then there is a classical deterministic algorithm that computes f exactly with O(T 6) queries. We also give asymptotically tight characterizations of T for all symmetric f in the exact, zero-error, and bounded-error settings. Finally, we give new precise bounds for AND, OR, and PARITY. Our results are a quantum extension of the so-called polynomial method, which has been successfully applied in classical complexity theory, and also a quantum extension of results by Nisan about a polynomial relationship between randomized and deterministic decision tree complexity.

Original languageEnglish (US)
Pages (from-to)778-797
Number of pages20
JournalJournal of the ACM
Issue number4
StatePublished - Jul 2001
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Software
  • Control and Systems Engineering
  • Information Systems
  • Hardware and Architecture
  • Artificial Intelligence


  • Black-box model
  • Lower bounds
  • Polynomial method
  • Quantum computing
  • Query complexity


Dive into the research topics of 'Quantum lower bounds by polynomials'. Together they form a unique fingerprint.

Cite this