Ana sayfa Algoritma Tensörler

Tensörler

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PAYLAŞ

Giriş.

Vektörler bir uzaydan başka bir uzaya taşındığında bileşenleri değişmektedir. Koordinat dönüşümleri altında fiziksel bağımlıkları birimsel bileşenleri değişmektedir. Bu nedenle bilim insanları evrendeki temel kanunların uzayın her yerinde geçerli olması gerekir. Fiziksel nesneler bir uzaydan başka bir uzaya geçildiğinde, bileşenleri değişmeyen niceliklere ihtiyaç duyarlar ki bunun adı tensörlerdir. Bir uzaydan başka bir uzaya geçildiğinde bileşenleri değişmeyen niceliklerdir. Matematiksel olarak bir nesnenin koordinatlarını bir sayı ile tanımlarız. x bileşeni, y bileşeni, z bileşeni (3,2,-5). Fizikte bu hep böyle değildir. Koordinat dönüşümleri altında bu değerlerin beleşenleri temsil eden değerler değişmektedir. Reel düzlemde matematiksel bir doğru denklemi complex uzayda bir bir çemberi temsil eder. Bir nesnenin konum değerlerini koordinat eksenleri ile tanımlarız ve bunu vektörel olarak gösterebiliriz. Bu nesnenin hız, ağırlık, sıcaklık, hacim gibi birimlerini de sayı olarak ifade ederiz. Ama bu değerler fiziksel olarak başka düzleme geçtiğinde bileşenlerindeki değişimleri tanımlarsak bunu tensörler ile ifade ederiz. Kuantum bilgisayarlarda bilginin ifadesi tensörel çarpım şeklinde olmaktadır. Klasik bilgisayarlarda bu işlem kartezyen çarpım şeklinde iken kuantum bilgisayarlarda bu işlem tensörel çarpım şeklinde olmaktadır. Tensörler ile ilgili detalı bilgiye http://kitaplar.ankara.edu.tr/dosyalar/pdf/736.pdf adresinden ulaşabilirsiniz.

Sütun Vektörlerinin Tensörel Çarpmı

Sutun vektörlerin Tensörel Çarpım

Yukarıda görüldüğü gibi tensörel çarpım sonucunda oluşan boyut her iki vektörün bulunduğu uzayın boyutlarının çarpımına eşittir..

Örnek olarak tensörel çarpım için

Matrislerin tensörel çarpımı da sütun vektörlerinin çarpımına benzemektedir. Bu bağlamda matrislerin tensörel çarpımı her bir eleman diğer matrisin tüm elemanları ile çarpılarak çarpım sonucu bulunduğu satır ve sütun yazılır.

Matrislerin Tensörel Çarpımı

olsun.

Python için geliştirilen QUTIP modülü tensörler için biçilmiş kaftandır. Çok kolay yapılmaktadır.

>>tensor(sigmax(), sigmaz())

şeklinde hazır fonksiyon bu işlemi yapmaktadır. Tensörlerin uygulamaları ile ilgili çok geniş bir alan bulunmaktadır. Özellikle Kuantum bilgisayarlarda çok geniş bir alan bulunmaktadır.  

>>> tensor(op1, op2, op3)

H = tensor(sigmaz(), identity(2)) + tensor(identity(2),
  …:          sigmaz()) + 0.05 * tensor(sigmax(), sigmax())

 

H = (tensor(sigmaz(), identity(2), identity(2)) +
  …:    tensor(identity(2), sigmaz(), identity(2)) +
  …:    tensor(identity(2), identity(2), sigmaz()) +
  …:    0.5 * tensor(sigmax(), sigmax(), identity(2)) +
  …:    0.25 * tensor(identity(2), sigmax(), sigmax()))

 

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