Circuit lower bounds and linear codes

R. Paturi; P. Pudlák

doi:10.1007/s10958-006-0119-5

Circuit lower bounds and linear codes

Journal of Mathematical Sciences ◽

10.1007/s10958-006-0119-5 ◽

2006 ◽

Vol 134 (5) ◽

pp. 2425-2434 ◽

Cited By ~ 3

Author(s):

R. Paturi ◽

P. Pudlák

Keyword(s):

Lower Bounds ◽

Linear Codes ◽

Circuit Lower Bounds

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On Exponential-Time Hypotheses, Derandomization, and Circuit Lower Bounds: Extended Abstract

2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS) ◽

10.1109/focs46700.2020.00010 ◽

2020 ◽

Author(s):

Lijie Chen ◽

Ron D. Rothblum ◽

Roei Tell ◽

Eylon Yogev

Keyword(s):

Lower Bounds ◽

Exponential Time ◽

Circuit Lower Bounds

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Feasibly constructive proofs of succinct weak circuit lower bounds

Annals of Pure and Applied Logic ◽

10.1016/j.apal.2019.102735 ◽

2020 ◽

Vol 171 (2) ◽

pp. 102735

Author(s):

Moritz Müller ◽

Ján Pich

Keyword(s):

Lower Bounds ◽

Circuit Lower Bounds ◽

Constructive Proofs

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A geometric approach to quantum circuit lower bounds

Quantum Information and Computation ◽

10.26421/qic6.3-2 ◽

2006 ◽

Vol 6 (3) ◽

pp. 213-262 ◽

Cited By ~ 1

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.

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