We give new quantum algorithms for evaluating composed functions whose inputs may be shared between bottom-level gates. Let f be an m -bit Boolean function and consider an n -bit function F obtained by applying f to conjunctions of possibly overlapping subsets of n variables. If f has quantum query complexity Q ( f ) , we give an algorithm for evaluating F using O ~ ( Q ( f ) ⋅ n ) quantum queries. This improves on the bound of O ( Q ( f ) ⋅ n ) that follows by treating each conjunction independently, and our bound is tight for worst-case choices of f . Using completely different techniques, we prove a similar tight composition theorem for the approximate degree of f .By recursively applying our composition theorems, we obtain a nearly optimal O ~ ( n 1 − 2 − d ) upper bound on the quantum query complexity and approximate degree of linear-size depth- d AC 0 circuits. As a consequence, such circuits can be PAC learned in subexponential time, even in the challenging agnostic setting. Prior to our work, a subexponential-time algorithm was not known even for linear-size depth-3 AC 0 circuits.As an additional consequence, we show that AC 0 ∘ ⊕ circuits of depth d + 1 require size Ω ~ ( n 1 / ( 1 − 2 − d ) ) ≥ ω ( n 1 + 2 − d ) to compute the Inner Product function even on average. The previous best size lower bound was Ω ( n 1 + 4 − ( d + 1 ) ) and only held in the worst case (Cheraghchi et al., JCSS 2018).
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KRW Composition Theorems via Lifting
One of the major open problems in complexity theory is proving super-logarithmic lower
bounds on the depth of circuits (i.e., P 6⊆ NC1). Karchmer, Raz, and Wigderson [KRW95]
suggested to approach this problem by proving that depth complexity behaves “as expected” with respect to the composition of functions f ⋄ g. They showed that the validity of this conjecture would imply that P 6⊆ NC^1
.
Several works have made progress toward resolving this conjecture by proving special cases. In particular, these works proved the KRW conjecture for every outer function f, but only for few inner functions g. Thus, it is an important challenge to prove the KRW conjecture for a wider range of inner functions.
In this work, we extend significantly the range of inner functions that can be handled. First,
we consider the monotone version of the KRW conjecture. We prove it for every monotone
inner function g whose depth complexity can be lower bounded via a query-to-communication lifting theorem. This allows us to handle several new and well-studied functions such as the s-t-connectivity, clique, and generation functions.
In order to carry this progress back to the non-monotone setting, we introduce a new notion of semi-monotone composition, which combines the non-monotone complexity of the outer function f with the monotone complexity of the inner function g. In this setting, we prove the KRW conjecture for a similar selection of inner functions g, but only for a specific choice of the outer function f.
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- Award ID(s):
- 1900460
- PAR ID:
- 10169734
- Date Published:
- Journal Name:
- Electronic colloquium on computational complexity
- ISSN:
- 1433-8092
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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