Lower Bounds on Matrix Rigidity via a Quantum Argument Ronald de - - PowerPoint PPT Presentation

Lower Bounds on Matrix Rigidity via a Quantum Argument

Ronald de Wolf CWI Amsterdam

Lower Bounds on Matrix Rigidity via a Quantum Argument – p.1/6

Rigidity: What and why?

Consider full-rank n × n matrix M

Lower Bounds on Matrix Rigidity via a Quantum Argument – p.2/6

Rigidity: What and why?

Consider full-rank n × n matrix M How many of its entries do we need to change if we want to lower its rank to r?

Lower Bounds on Matrix Rigidity via a Quantum Argument – p.2/6

Rigidity: What and why?

Consider full-rank n × n matrix M How many of its entries do we need to change if we want to lower its rank to r?

RM(r) = min{∆(M, M) | rank( M) ≤ r}

Lower Bounds on Matrix Rigidity via a Quantum Argument – p.2/6

Rigidity: What and why?

Consider full-rank n × n matrix M How many of its entries do we need to change if we want to lower its rank to r?

RM(r) = min{∆(M, M) | rank( M) ≤ r}

Example:

RI(r) = n − r

Lower Bounds on Matrix Rigidity via a Quantum Argument – p.2/6

Rigidity: What and why?

Consider full-rank n × n matrix M How many of its entries do we need to change if we want to lower its rank to r?

RM(r) = min{∆(M, M) | rank( M) ≤ r}

Example:

RI(r) = n − r RM(r) ≈ (n − r)2 for random M

Lower Bounds on Matrix Rigidity via a Quantum Argument – p.2/6

Rigidity: What and why?

Consider full-rank n × n matrix M How many of its entries do we need to change if we want to lower its rank to r?

RM(r) = min{∆(M, M) | rank( M) ≤ r}

Example:

RI(r) = n − r RM(r) ≈ (n − r)2 for random M

Motivation (Valiant 77):

Lower Bounds on Matrix Rigidity via a Quantum Argument – p.2/6

Rigidity: What and why?

Consider full-rank n × n matrix M How many of its entries do we need to change if we want to lower its rank to r?

RM(r) = min{∆(M, M) | rank( M) ≤ r}

Example:

RI(r) = n − r RM(r) ≈ (n − r)2 for random M

Motivation (Valiant 77): Explicit matrix with high rigidity implies size-depth tradeoffs for arithmetic circuits

Lower Bounds on Matrix Rigidity via a Quantum Argument – p.2/6

Rigidity: What and why?

Consider full-rank n × n matrix M How many of its entries do we need to change if we want to lower its rank to r?

RM(r) = min{∆(M, M) | rank( M) ≤ r}

Example:

RI(r) = n − r RM(r) ≈ (n − r)2 for random M

Motivation (Valiant 77): Explicit matrix with high rigidity implies size-depth tradeoffs for arithmetic circuits Good candidate: n × n Hadamard matrix H

Lower Bounds on Matrix Rigidity via a Quantum Argument – p.2/6

Connection with quantum

Renormalized rows |

Hi of matrix H ≈ H

form a quantum communication system!

Lower Bounds on Matrix Rigidity via a Quantum Argument – p.3/6

Connection with quantum

Renormalized rows |

Hi of matrix H ≈ H

form a quantum communication system! To communicate i:

Lower Bounds on Matrix Rigidity via a Quantum Argument – p.3/6

Connection with quantum

Renormalized rows |

Hi of matrix H ≈ H

form a quantum communication system! To communicate i: (1) Alice sends |

Hi in r dimensions

Lower Bounds on Matrix Rigidity via a Quantum Argument – p.3/6

Connection with quantum

Renormalized rows |

Hi of matrix H ≈ H

form a quantum communication system! To communicate i: (1) Alice sends |

Hi in r dimensions

(2) Bob measures in Hadamard basis

Lower Bounds on Matrix Rigidity via a Quantum Argument – p.3/6

Connection with quantum

Renormalized rows |

Hi of matrix H ≈ H

form a quantum communication system! To communicate i: (1) Alice sends |

Hi in r dimensions

(2) Bob measures in Hadamard basis Success probability pi = |

Hi|Hi|2

Lower Bounds on Matrix Rigidity via a Quantum Argument – p.3/6

Connection with quantum

Renormalized rows |

Hi of matrix H ≈ H

form a quantum communication system! To communicate i: (1) Alice sends |

Hi in r dimensions

(2) Bob measures in Hadamard basis Success probability pi = |

Hi|Hi|2

is higher if

Hi is a better approximation of Hi.

Lower Bounds on Matrix Rigidity via a Quantum Argument – p.3/6

Connection with quantum

Renormalized rows |

Hi of matrix H ≈ H

form a quantum communication system! To communicate i: (1) Alice sends |

Hi in r dimensions

(2) Bob measures in Hadamard basis Success probability pi = |

Hi|Hi|2

is higher if

Hi is a better approximation of Hi.

Nayak 99:

n

• i=1

pi ≤ r

Lower Bounds on Matrix Rigidity via a Quantum Argument – p.3/6

Connection with quantum

Renormalized rows |

Hi of matrix H ≈ H

form a quantum communication system! To communicate i: (1) Alice sends |

Hi in r dimensions

(2) Bob measures in Hadamard basis Success probability pi = |

Hi|Hi|2

is higher if

Hi is a better approximation of Hi.

Nayak 99:

n

• i=1

pi ≤ r

Tradeoff between r and the quality of the approximation

Lower Bounds on Matrix Rigidity via a Quantum Argument – p.3/6

Two applications

RH(r) ≥ n2 4r

Lower Bounds on Matrix Rigidity via a Quantum Argument – p.4/6

Two applications

RH(r) ≥ n2 4r

This improves Kashin & Razborov by factor 64

Lower Bounds on Matrix Rigidity via a Quantum Argument – p.4/6

Two applications

RH(r) ≥ n2 4r

This improves Kashin & Razborov by factor 64 If we limit the change-per-entry to θ:

Lower Bounds on Matrix Rigidity via a Quantum Argument – p.4/6

Two applications

RH(r) ≥ n2 4r

This improves Kashin & Razborov by factor 64 If we limit the change-per-entry to θ:

RH(r, θ) ≥ n2(n − r) 2θn + r(θ2 + 2θ)

Lower Bounds on Matrix Rigidity via a Quantum Argument – p.4/6

Two applications

RH(r) ≥ n2 4r

This improves Kashin & Razborov by factor 64 If we limit the change-per-entry to θ:

RH(r, θ) ≥ n2(n − r) 2θn + r(θ2 + 2θ)

Matches earlier results of Lokam and Kashin-Razborov

Lower Bounds on Matrix Rigidity via a Quantum Argument – p.4/6

To be or not to be quantum

Of course, this is all linear algebra

Lower Bounds on Matrix Rigidity via a Quantum Argument – p.5/6

To be or not to be quantum

Of course, this is all linear algebra An anonymous referee suggested an alternative linear algebra proof for the same bounds

Lower Bounds on Matrix Rigidity via a Quantum Argument – p.5/6

To be or not to be quantum

Of course, this is all linear algebra An anonymous referee suggested an alternative linear algebra proof for the same bounds Quantum method is potentially stronger

Lower Bounds on Matrix Rigidity via a Quantum Argument – p.5/6

To be or not to be quantum

Of course, this is all linear algebra An anonymous referee suggested an alternative linear algebra proof for the same bounds Quantum method is potentially stronger Simple proof of RM(r) ≥ n2/4r for H⊗ log n

2

(Midrijanis)

Lower Bounds on Matrix Rigidity via a Quantum Argument – p.5/6

Summary

Reproved best known bounds on rigidity of Hadamard matrix

Lower Bounds on Matrix Rigidity via a Quantum Argument – p.6/6

Summary

Reproved best known bounds on rigidity of Hadamard matrix using quantum information theory

Lower Bounds on Matrix Rigidity via a Quantum Argument – p.6/6

Summary

Reproved best known bounds on rigidity of Hadamard matrix using quantum information theory Fits in a sequence of quantum proofs for classical theorems

Lower Bounds on Matrix Rigidity via a Quantum Argument – p.6/6

Summary

Reproved best known bounds on rigidity of Hadamard matrix using quantum information theory Fits in a sequence of quantum proofs for classical theorems These rigidity bounds are not very good

Lower Bounds on Matrix Rigidity via a Quantum Argument – p.6/6

Summary

Reproved best known bounds on rigidity of Hadamard matrix using quantum information theory Fits in a sequence of quantum proofs for classical theorems These rigidity bounds are not very good But: the connection with quantum gives a fresh look at this 28-year old problem, and may yield more

Lower Bounds on Matrix Rigidity via a Quantum Argument – p.6/6