Complex polynomials and circuit lower bounds for modular counting

2005 ◽  
pp. 24-31 ◽  
Author(s):  
David A. Mix Barrington ◽  
Howard Straubing
Keyword(s):  
Lower Bounds ◽  
Complex Polynomials ◽  
Circuit Lower Bounds ◽  
Modular Counting

Complex polynomials and circuit lower bounds for modular counting

1994 ◽  
Vol 4 (4) ◽  
pp. 325-338 ◽  
Author(s):  
David A. Mix Barrington ◽  
Howard Straubing
Keyword(s):  
Lower Bounds ◽  
Complex Polynomials ◽  
Circuit Lower Bounds ◽  
Modular Counting

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

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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