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

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 ✫ ✪ ❅ ❆ ❅ ❆ ❅ ❆ ❅ ❆ ❅ ❆ ❅ ❆ ❅ ❆ ❅ ❆ ❆ ❅ ❆ ❅ ❅ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆

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

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

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

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

Single-qubit circuits � 1 0 � I = 0 1

Single-qubit circuits � 1 0 � I = 0 1 Two square roots: � 0 1 � X = 1 0 and � 1 � 0 Z = 0 − 1

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)

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 � 1 0 � N (0) = = I 0 1 � 0 1 � √ N ( π ) = = I = X 1 0 � 1 − i 1 + i � N ( π/ 2) = 1 √ = X = V 1 + i 1 − i 2 √ N ( π/ 4) = V = W

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 1 2 − 1   2 exp( iθ ) 2 exp( iθ ) =   1 2 − 1 1 2 + 1 2 exp( iθ ) 2 exp( iθ )

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 )

Single-qubit circuits � 1 0 � I = 0 1 Two square roots: � 0 1 � X = 1 0 and � 1 � 0 Z = 0 − 1

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

Single-qubit circuits PHASOR = Φ( θ ) = � 1 � 0 0 exp( iθ ) with special values � 1 0 � Φ(0) = = I 0 1 � 1 � √ 0 Φ( π ) = = I = Z 0 − 1 � 1 0 � √ Φ( π/ 2) = = Z = S 0 i √ Φ( π/ 4) = S = T

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

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

Two-qubit circuits • N ( θ )

Two-qubit circuits • N ( θ ) Φ( θ ) ���� ����

Multiple-qubit circuits • ���� ���� N ( θ )   1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0     0 0 1 0 0 0 0 0     0 0 0 1 0 0 0 0     i sin( θ/ 2) e − iθ/ 2 0 0 0 0 0 0 cos( θ/ 2) e − iθ/ 2     0 0 0 0 i sin( θ/ 2) e − iθ/ 2 cos( θ/ 2) e − iθ/ 2  0 0     0 0 0 0 0 0 1 0    0 0 0 0 0 0 0 1

Multiple-qubit circuits • ���� ���� N ( θ )   1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0     0 0 1 0 0 0 0 0     0 0 0 1 0 0 0 0     i sin( θ/ 2) e − iθ/ 2 0 0 0 0 0 0 cos( θ/ 2) e − iθ/ 2     0 0 0 0 i sin( θ/ 2) e − iθ/ 2 cos( θ/ 2) e − iθ/ 2  0 0     0 0 0 0 0 0 1 0    0 0 0 0 0 0 0 1 XU( n ) ⊂ U( n )

Multiple-qubit circuits • ���� ���� N ( θ )   1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0     0 0 1 0 0 0 0 0     0 0 0 1 0 0 0 0     i sin( θ/ 2) e − iθ/ 2 0 0 0 0 0 0 cos( θ/ 2) e − iθ/ 2     0 0 0 0 i sin( θ/ 2) e − iθ/ 2 cos( θ/ 2) e − iθ/ 2  0 0     0 0 0 0 0 0 1 0    0 0 0 0 0 0 0 1 XU( n ) ⊂ U( n ) ���� ���� ���� ���� • • • • •

Multiple-qubit circuits • ���� ���� Φ( θ )   1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0     0 0 1 0 0 0 0 0     0 0 0 1 0 0 0 0     0 0 0 0 1 0 0 0     0 0 0 0 0 exp( iθ ) 0 0     0 0 0 0 0 0 1 0   0 0 0 0 0 0 0 1

Multiple-qubit circuits • ���� ���� Φ( θ )   1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0     0 0 1 0 0 0 0 0     0 0 0 1 0 0 0 0     0 0 0 0 1 0 0 0     0 0 0 0 0 exp( iθ ) 0 0     0 0 0 0 0 0 1 0   0 0 0 0 0 0 0 1 ZU( n ) ⊂ U( n )

Multiple-qubit circuits • ���� ���� Φ( θ )   1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0     0 0 1 0 0 0 0 0     0 0 0 1 0 0 0 0     0 0 0 0 1 0 0 0     0 0 0 0 0 exp( iθ ) 0 0     0 0 0 0 0 0 1 0   0 0 0 0 0 0 0 1 ZU( n ) ⊂ U( n ) ���� ���� ���� ���� • • • • •

Within the unitary group U( n ), two subgroups : • the group XU( n ) of all n × n unitary matrices with all line sums equal to 1 • the group ZU( n ) of all n × n unitary diagonal matrices with first entry equal to 1.

✬ ✩ ✬ ✩ U( n ) ZU( n ) ✬ ✩ • ✫ ✪ ⋆ XU( n ) ✫ ✪ ✫ ✪ Whereas U( n ) has n 2 dimensions, XU( n ) has ( n − 1 ) 2 dimensions and ZU( n ) has ( n − 1 ) dimensions.

An arbitrary member of U( n ) can be decomposed U = exp( iβ ) Z 1 X Z 2

An arbitrary member of U( n ) can be decomposed U = exp( iβ ) Z 1 X Z 2 where Z 1 ∈ ZU( n ) X ∈ XU( n ) Z 2 ∈ ZU( n )

An arbitrary member of U( n ) can be decomposed U = exp( iβ ) Z 1 X Z 2

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

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

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

We conjectured ( arXiv:math - ph 1401.7883 ) on 30 Jan 2014. Idel and Wolf proved ( arXiv:math - ph 1408.5728 ) on 25 Aug 2014

Thus U = exp( iβ ) Z 1 X Z 2 e iβ Z 1 Z 2 X

The two ZU( n ) parts : = Z

The two ZU( n ) parts : = Z ���� ���� ���� ���� • • ���� ���� • ���� ���� ... • • ���� ���� • • Φ( α 7 ) Φ( α 5 ) Φ( α 3 ) ���� ���� ���� ���� ���� ���� • • ���� ���� ���� ���� ���� ���� ... • • • ���� ���� Φ( α 8 ) Φ( α 6 ) Φ( α 4 ) Φ( α 2 ) • •

The XU( n ) part : = X

The XU( n ) part : = X ... C 2 n − 2 C 1 C 2 C 3

The XU( n ) part : = X ... C 2 n − 2 C 1 C 2 C 3 where C j is a block-circulant XU( n ) matrix

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

E.g.   1 0 0 0 0 − 1 / 3 2 / 3 2 / 3   C =   0 2 / 3 − 1 / 3 2 / 3   0 2 / 3 2 / 3 − 1 / 3 ���� ���� N 2 N 3 N 1 • • • ���� ���� ... N 1 N 1 N 4 • • N 1 N 3 N 2 • • • ���� ���� ... N 5 N 1 N 1 •

E.g.   1 0 0 0 0 − 1 / 3 2 / 3 2 / 3   C =   2 / 3 − 1 / 3 2 / 3 0   0 2 / 3 2 / 3 − 1 / 3 ���� ���� N 2 N 3 N 1 • • • ���� ���� ... N 1 N 1 N 4 • • N 1 N 3 N 2 • • • ���� ���� ... N 5 N 1 N 1 • where N 1 = N ( π/ 2) , N 2 = N ( π/ 4) N 3 = N (7 π/ 4) N 4 = N ( π + Arccos (1 / 3)) N 5 = N ( − π − Arccos (1 / 3)) .

Conclusion : U = exp( iβ ) Z 1 X Z 2 e iβ Z 1 Z 2 X

Conclusion : U = exp( iβ ) Z 1 X Z 2 e iβ Z 1 Z 2 X Z 1 with PHASOR s X with NEGATOR s Z 2 with PHASOR s

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