From classical circuits to quantum circuits Alexis De Vos and Stijn - - PDF document

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From classical circuits to quantum circuits Alexis De Vos and Stijn - - PDF document

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From classical circuits to quantum circuits Alexis De Vos and Stijn De Baerdemacker Waterloo, 9 June 2015 Un bonjour de Waterloo, Belgique pour Waterloo, Ontario

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From classical circuits to quantum circuits Alexis De Vos and Stijn De Baerdemacker

Waterloo, 9 June 2015

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✬ ✩ ✪ ✫

Un bonjour de Waterloo, Belgique pour Waterloo, Ontario

❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆

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From classical circuits to quantum circuits Alexis De Vos and Stijn De Baerdemacker

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U(n) = the unitary group = the group of n × n unitary matrices

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U(n) = the unitary group = the group of n × n unitary matrices U(2w) = the quantum circuits acting on w qubits

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U(n) = the unitary group = the group of n × n unitary matrices U(2w) = the quantum circuits acting on w qubits Thus: U(2) = the quantum circuits acting on 1 qubit = the group of 2 × 2 unitary matrices

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Single-qubit circuits I = 1 0 0 1

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Single-qubit circuits I = 1 0 0 1

  • Two square roots:

X = 0 1 1 0

  • and

Z = 1 0 −1

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Single-qubit circuits NEGATOR = N(θ) = cos(θ/2) exp(−iθ/2) i sin(θ/2) exp(−iθ/2) i sin(θ/2) exp(−iθ/2) cos(θ/2) exp(−iθ/2)

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Single-qubit circuits NEGATOR = N(θ) = cos(θ/2) exp(−iθ/2) i sin(θ/2) exp(−iθ/2) i sin(θ/2) exp(−iθ/2) cos(θ/2) exp(−iθ/2)

  • with special values

N(0) = 1 0 0 1

  • = I

N(π) = 0 1 1 0

  • =

√ I = X N(π/2) = 1 2 1 − i 1 + i 1 + i 1 − i

  • =

√ X = V N(π/4) = √ V = W

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Single-qubit circuits NEGATOR = N(θ) = cos(θ/2) exp(−iθ/2) i sin(θ/2) exp(−iθ/2) i sin(θ/2) exp(−iθ/2) cos(θ/2) exp(−iθ/2)

  • =

 

1 2 + 1 2 exp(iθ) 1 2 − 1 2 exp(iθ) 1 2 − 1 2 exp(iθ) 1 2 + 1 2 exp(iθ)

 

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Single-qubit circuits NEGATOR = N(θ) = cos(θ/2) exp(−iθ/2) i sin(θ/2) exp(−iθ/2) i sin(θ/2) exp(−iθ/2) cos(θ/2) exp(−iθ/2)

  • N(θ)
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Single-qubit circuits I = 1 0 0 1

  • Two square roots:

X = 0 1 1 0

  • and

Z = 1 0 −1

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Single-qubit circuits PHASOR = Φ(θ) = 1 0 exp(iθ)

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Single-qubit circuits PHASOR = Φ(θ) = 1 0 exp(iθ)

  • with special values

Φ(0) = 1 0 0 1

  • = I

Φ(π) = 1 0 −1

  • =

√ I = Z Φ(π/2) = 1 0 0 i

  • =

√ Z = S Φ(π/4) = √ S = T

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Single-qubit circuits PHASOR = Φ(θ) = 1 0 exp(iθ)

  • Φ(θ)
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Single-qubit circuits PHASOR = Φ(θ) = 1 0 exp(iθ)

  • Φ(θ)

NEGATOR = N(θ) = cos(θ/2) exp(−iθ/2) i sin(θ/2) exp(−iθ/2) i sin(θ/2) exp(−iθ/2) cos(θ/2) exp(−iθ/2)

  • N(θ)
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Two-qubit circuits

  • N(θ)
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Two-qubit circuits

  • N(θ)

Φ(θ)

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Multiple-qubit circuits

  • N(θ)

             1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 cos(θ/2)e−iθ/2 i sin(θ/2)e−iθ/2 0 0 0 0 0 0 i sin(θ/2)e−iθ/2 cos(θ/2)e−iθ/2 0 0 0 0 0 0 1 0 0 0 0 0 0 1             

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Multiple-qubit circuits

  • N(θ)

             1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 cos(θ/2)e−iθ/2 i sin(θ/2)e−iθ/2 0 0 0 0 0 0 i sin(θ/2)e−iθ/2 cos(θ/2)e−iθ/2 0 0 0 0 0 0 1 0 0 0 0 0 0 1              XU(n) ⊂ U(n)

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Multiple-qubit circuits

  • N(θ)

             1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 cos(θ/2)e−iθ/2 i sin(θ/2)e−iθ/2 0 0 0 0 0 0 i sin(θ/2)e−iθ/2 cos(θ/2)e−iθ/2 0 0 0 0 0 0 1 0 0 0 0 0 0 1              XU(n) ⊂ U(n)

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Multiple-qubit circuits

  • Φ(θ)

            1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 exp(iθ) 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1            

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Multiple-qubit circuits

  • Φ(θ)

            1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 exp(iθ) 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1             ZU(n) ⊂ U(n)

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Multiple-qubit circuits

  • Φ(θ)

            1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 exp(iθ) 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1             ZU(n) ⊂ U(n)

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Within the unitary group U(n), two subgroups :

  • the group XU(n)
  • f all n×n unitary matrices

with all line sums equal to 1

  • the group ZU(n)
  • f all n×n unitary diagonal

matrices with first entry equal to 1.

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✬ ✫ ✩ ✪ ✬ ✫ ✩ ✪ ✬ ✫ ✩ ✪

U(n) XU(n) ZU(n)

  • ⋆

Whereas U(n) has n2 dimensions, XU(n) has (n−1)2 dimensions and ZU(n) has (n−1) dimensions.

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An arbitrary member of U(n) can be decomposed U = exp(iβ) Z1 X Z2

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An arbitrary member of U(n) can be decomposed U = exp(iβ) Z1 X Z2 where Z1 ∈ ZU(n) X ∈ XU(n) Z2 ∈ ZU(n)

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An arbitrary member of U(n) can be decomposed U = exp(iβ) Z1 X Z2

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An arbitrary member of U(n) can be decomposed U = exp(iβ) Z1 X Z2 1 + (n−1) + (n−1)2 + (n−1)

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An arbitrary member of U(n) can be decomposed U = exp(iβ) Z1 X Z2 n2 = 1 + (n−1) + (n−1)2 + (n−1)

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We conjectured (arXiv:math-ph 1401.7883)

  • n 30 Jan 2014.
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We conjectured (arXiv:math-ph 1401.7883)

  • n 30 Jan 2014.

Idel and Wolf proved (arXiv:math-ph 1408.5728)

  • n 25 Aug 2014
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Thus U = exp(iβ) Z1 X Z2 eiβ Z1 X Z2

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The two ZU(n) parts : Z =

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The two ZU(n) parts : Z =

  • ...
  • • Φ(α7)

Φ(α5) Φ(α3)

  • ...
  • •
  • Φ(α8)

Φ(α6) Φ(α4) Φ(α2)

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The XU(n) part : X =

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The XU(n) part : X = C1 C2 C3 C2n−2 ...

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The XU(n) part : X = C1 C2 C3 C2n−2 ... where Cj is a block-circulant XU(n) matrix

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E.g. C =     1 0 −1/3 2/3 2/3 2/3 −1/3 2/3 2/3 2/3 −1/3    

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E.g. C =     1 0 −1/3 2/3 2/3 2/3 −1/3 2/3 2/3 2/3 −1/3    

  • N2
  • N3
  • N1
  • N1
  • N1
  • N4
  • ...

N1

  • N3
  • N2
  • ...

N5

  • N1
  • N1
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E.g. C =     1 0 −1/3 2/3 2/3 2/3 −1/3 2/3 2/3 2/3 −1/3    

  • N2
  • N3
  • N1
  • N1
  • N1
  • N4
  • ...

N1

  • N3
  • N2
  • ...

N5

  • N1
  • N1

where N1 = N(π/2), N2 = N(π/4) N3 = N(7π/4) N4 = N(π + Arccos(1/3)) N5 = N(−π − Arccos(1/3)).

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Conclusion : U = exp(iβ) Z1 X Z2 eiβ Z1 X Z2

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Conclusion : U = exp(iβ) Z1 X Z2 eiβ Z1 X Z2 Z1 with PHASORs X with NEGATORs Z2 with PHASORs

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The end / La fin Waterloo (Belgique), 18 June 1815