Sparse Selfreducible Sets and Polynomial Size Circuit Lower Bounds

2006 ◽  
pp. 455-468
Author(s):  
Harry Buhrman ◽  
Leen Torenvliet ◽  
Falk Unger
Keyword(s):  
Lower Bounds ◽  
Circuit Lower Bounds ◽  
Polynomial Size

scholarly journals On fixed-polynomial size circuit lower bounds for uniform polynomials in the sense of Valiant

2015 ◽  
Vol 240 ◽  
pp. 31-41
Author(s):  
Hervé Fournier ◽  
Sylvain Perifel ◽  
Rémi de Verclos
Keyword(s):  
Lower Bounds ◽  
Circuit Lower Bounds ◽  
Polynomial Size

scholarly journals Feasibly constructive proofs of succinct weak circuit lower bounds

2020 ◽  
Vol 171 (2) ◽  
pp. 102735
Author(s):  
Moritz Müller ◽  
Ján Pich
Keyword(s):  
Lower Bounds ◽  
Circuit Lower Bounds ◽  
Constructive Proofs

A geometric approach to quantum circuit lower bounds

2006 ◽  
Vol 6 (3) ◽  
pp. 213-262 ◽  
Author(s):  
M.A. Nielsen
Keyword(s):  
Lower Bounds ◽  
Geometric Approach ◽  
Quantum Circuit ◽  
Geodesic Equation ◽  
Initial Position ◽  
Minimal Length ◽  
Finsler Metric ◽  
Finsler Metrics ◽  
Minimal Size ◽  
Circuit Lower Bounds

What is the minimal size quantum circuit required to exactly implement a specified n-qubit unitary operation, U, without the use of ancilla qubits? We show that a lower bound on the minimal size is provided by the length of the minimal geodesic between U and the identity, I, where length is defined by a suitable Finsler metric on the manifold SU(2^n). The geodesic curves on these manifolds have the striking property that once an initial position and velocity are set, the remainder of the geodesic is completely determined by a second order differential equation known as the geodesic equation. This is in contrast with the usual case in circuit design, either classical or quantum, where being given part of an optimal circuit does not obviously assist in the design of the rest of the circuit. Geodesic analysis thus offers a potentially powerful approach to the problem of proving quantum circuit lower bounds. In this paper we construct several Finsler metrics whose minimal length geodesics provide lower bounds on quantum circuit size. For each Finsler metric we give a procedure to compute the corresponding geodesic equation. We also construct a large class of solutions to the geodesic equation, which we call \emph{Pauli geodesics}, since they arise from isometries generated by the Pauli group. For any unitary U diagonal in the computational basis, we show that: (a) provided the minimal length geodesic is unique, it must be a Pauli geodesic; (b) finding the length of the minimal Pauli geodesic passing from I to U is equivalent to solving an exponential size instance of the closest vector in a lattice problem (CVP); and (c) all but a doubly exponentially small fraction of such unitaries have minimal Pauli geodesics of exponential length.

scholarly journals Nonuniform ACC Circuit Lower Bounds

2014 ◽  
Vol 61 (1) ◽  
pp. 1-32 ◽  
Author(s):  
Ryan Williams
Keyword(s):  
Lower Bounds ◽  
Circuit Lower Bounds

Nondeterministic circuit lower bounds from mildly derandomizing Arthur-Merlin games

2014 ◽  
Vol 26 (1) ◽  
pp. 79-118
Author(s):  
Barış Aydınlıoğlu ◽  
Dieter van Melkebeek
Keyword(s):  
Lower Bounds ◽  
Circuit Lower Bounds

Exact Learning Algorithms, Betting Games, and Circuit Lower Bounds

2013 ◽  
Vol 5 (4) ◽  
pp. 1-11 ◽  
Author(s):  
Ryan C. Harkins ◽  
John M. Hitchcock
Keyword(s):  
Lower Bounds ◽  
Learning Algorithms ◽  
Circuit Lower Bounds ◽  
Exact Learning

Lower Bounds on OBDD Proofs with Several Orders

2021 ◽  
Vol 22 (4) ◽  
pp. 1-30
Author(s):  
Sam Buss ◽  
Dmitry Itsykson ◽  
Alexander Knop ◽  
Artur Riazanov ◽  
Dmitry Sokolov
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Communication Complexity ◽  
Multiparty Communication ◽  
Constant Degree ◽  
Polynomial Size ◽  
Explicit Constant ◽  
Open Question ◽  
System 1 ◽  
Exponential Size

This article is motivated by seeking lower bounds on OBDD(∧, w, r) refutations, namely, OBDD refutations that allow weakening and arbitrary reorderings. We first work with 1 - NBP ∧ refutations based on read-once nondeterministic branching programs. These generalize OBDD(∧, r) refutations. There are polynomial size 1 - NBP(∧) refutations of the pigeonhole principle, hence 1-NBP(∧) is strictly stronger than OBDD}(∧, r). There are also formulas that have polynomial size tree-like resolution refutations but require exponential size 1-NBP(∧) refutations. As a corollary, OBDD}(∧, r) does not simulate tree-like resolution, answering a previously open question. The system 1-NBP(∧, ∃) uses projection inferences instead of weakening. 1-NBP(∧, ∃ k is the system restricted to projection on at most k distinct variables. We construct explicit constant degree graphs G n on n vertices and an ε > 0, such that 1-NBP(∧, ∃ ε n ) refutations of the Tseitin formula for G n require exponential size. Second, we study the proof system OBDD}(∧, w, r ℓ ), which allows ℓ different variable orders in a refutation. We prove an exponential lower bound on the complexity of tree-like OBDD(∧, w, r ℓ ) refutations for ℓ = ε log n , where n is the number of variables and ε > 0 is a constant. The lower bound is based on multiparty communication complexity.

scholarly journals Circuit Lower Bounds à la Kolmogorov

1995 ◽  
Vol 123 (1) ◽  
pp. 121-126 ◽  
Author(s):  
L. Fortnow ◽  
S. Laplante
Keyword(s):  
Lower Bounds ◽  
Circuit Lower Bounds
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