Span Programs and Quantum Space Complexity

Author Stacey Jeffery

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Stacey Jeffery
  • CWI, Amsterdam, The Netherlands
  • QuSoft, Amsterdam, The Netherlands


I am grateful to Tsuyoshi Ito for discussions that led to the construction of approximate span programs from two-sided error quantum algorithms presented in Section 3.2, and to Alex B. Grilo and Mario Szegedy for insightful comments. I thank Robin Kothari for pointing out the improved separation between certificate complexity and approximate degree in [M. Bun and J. Thaler, 2017], which led to an improvement in from (log n)^(7/6) (using [S. Aaronson et al., 2016]) to (log n)^(2-o(1)) in Theorem 32.

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Stacey Jeffery. Span Programs and Quantum Space Complexity. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 4:1-4:37, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


While quantum computers hold the promise of significant computational speedups, the limited size of early quantum machines motivates the study of space-bounded quantum computation. We relate the quantum space complexity of computing a function f with one-sided error to the logarithm of its span program size, a classical quantity that is well-studied in attempts to prove formula size lower bounds. In the more natural bounded error model, we show that the amount of space needed for a unitary quantum algorithm to compute f with bounded (two-sided) error is lower bounded by the logarithm of its approximate span program size. Approximate span programs were introduced in the field of quantum algorithms but not studied classically. However, the approximate span program size of a function is a natural generalization of its span program size. While no non-trivial lower bound is known on the span program size (or approximate span program size) of any concrete function, a number of lower bounds are known on the monotone span program size. We show that the approximate monotone span program size of f is a lower bound on the space needed by quantum algorithms of a particular form, called monotone phase estimation algorithms, to compute f. We then give the first non-trivial lower bound on the approximate span program size of an explicit function.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum computation theory
  • Quantum space complexity
  • span programs


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