Dichromatic and Trichromatic Calculus for Qutrit Systems Quanlong - PowerPoint PPT Presentation

Dichromatic and Trichromatic Calculus for Qutrit Systems Quanlong Wang Xiaoning Bian School of Mathematics and Systems Science, Beihang University, Beijing, China June 4, 2014

Contents Some Backgrounds Qutrit RG Calculus Qutrit RG Generators Qutrit RG Rules Differences Between Qutrit and Qubit Rules Euler Decomposition Qutrit RGB Calculus Qutrit RGB Generators Qutrit RGB Rules RG to RGB Translation Single Qutrit Quantum Algorithm Qudit ZX Calculus is Universal

Some Backgrounds ◮ [1] B. Coecke, R. Duncan. Interacting quantum observables: Categorical algebra and diagrammatics. ◮ Provided a general framework of dagger symmetric monoidal categories for axiomatising complementarity of quantum observables. ◮ Introduced the intuitive and universal ZX Calculus for qubits. ◮ [2] A. Lang and B. Coecke. Trichromatic open digraphs for understanding qubits. ◮ Introduced a trichromatic graphical calculus. ◮ ‘Dichromatic ZX Calculus + Euler angle decomposition of the Hadamard gate = Trichromatic calculus’.

Qutrit RG Generators We define a category RG where the objects are n -fold monoidal products of an object ∗ , denoted ∗ n ( n ≥ 0). In RG , a morphism from ∗ m to ∗ n is a finite undirected open graph from m wires to n wires, built from δ † ǫ † δ Z = Z = ǫ Z = Z = P Z ( α, β ) = α β δ † ǫ † δ X = X = ǫ X = X = P X ( α, β ) = α β H † = H † H = H where α, β ∈ [0 , 2 π ). For convenience, we denote the frequently used angles 2 π 3 and 4 π 3 by 1 and 2 respectively.

Qutrit RG Rules RG morphisms are also subject to the following equations: 1. Equations in the following figure. 2. All equations hold under flip of graphs, negation of angles, and exchange of H and H † . 3. All equations hold under flip of colours (except for rules K 2 and H 2).

Qutrit RG Rules ... ... ... α β α + η = . (S1) := 0 (S2) . = . η β + θ 0 ... θ ... ... = = (B1) (B2) 1 2 2 = 1 = 1 1 2 2 (K1) 2 2 1 1 β - α - β - β β - α 1 2 1 2 - α - α 2 1 α - β 2 α - β 1 = = = = (K2) α 1 α 2 α 1 α 2 β 2 β 1 β 2 β 1 ... ... H H H = H † = (H1) = (H2) α α β β H † H H † H † ... ... D β D D α := = (P1) (P2) D = = = α D D β D D

Some Derived Rules These equations are very useful when wanting to demonstrates some more complex equalities in describing quantum protocols[4] and algorithms[6]. = (1) β α α = β = (2) = = = (3) D D = = (4)

Dagger Functor � † � † � † � † � � � � = = = = � † � † � † � † � � � � = = = = � † � † � † � † � � � � α = - α α = - α = H † = H H † H β - β β - β RG is a dagger symmetric monoidal category.

RG Interpretation We give an interpretation [ · ] RG : RG → FdHilb Q � � � � � � = | 0 � � 0 | + e i α | 1 � � 1 | + e i β | 2 � � 2 | = | + � = � + | α β � � � � = | 0 � � 00 | + | 1 � � 11 | + | 2 � � 22 | = | 00 � � 0 | + | 11 � � 1 | + | 22 � � 2 | � � � � � � = | + � � ++ | + e i α | ω � � ω | + e i β | ¯ = | 0 � = � 0 | ω � � ¯ ω | α β � � � � = | + � � ++ | + | ω � � ωω | + | ¯ ω � � ¯ ω ¯ ω | = | ++ � � + | + | ωω � � ω | + | ¯ ω ¯ ω � � ¯ ω | � � � � = | + � � 0 | + | ω � � 1 | + | ¯ ω � � 2 | = | 0 � � + | + | 1 � � ω | + | 2 � � ¯ ω | H † H

Differences Between Qutrit and Qubit Rules I ◮ In qubit case, = . For qutrit duliser, D := = ◮ The qubit dualizer (identical permutation) is an even permutation, while the qutrit dualizer is an odd permutation. ◮ There is only one odd permutation π in qubit case satisfying π π - α π π = = = = π α π π π The qutrit dualizer satisfies β D D α D D = α D = = = β D D D

Differences Between Qutrit and Qubit Rules II ◮ In qubit case the K 2 rule still holds when flipping the colours, π - α π - α = = α π α π Whereas it doesn’t hold in qutrit case β - α - β - β β - α 1 2 1 2 - α - α 2 1 2 1 α - β α - β = = = = α 1 α 2 α 1 α 2 β 2 β 1 β 2 β 1

Decomposition of the Hadamard Gate 2 2 ◮ Euler decomposition of the Hadamard gate: H = 2 2 2 2 ◮ The Euler decomposition is not unique: 2 2 2 2 H = ⇒ = 2 2 H 2 2 2 2 2 2 ◮ Proof: 2 2 2 2 2 2 2 2 2 2 = = = = 1 2 2 H H 1 2 2 1 2 2 H 1 2 2 1 1

Euler Decomposition Not Derivable ◮ In the qubit case, Duncan and Perdrix [5] proved that the Euler decomposition is not derivable from ZX calculus. ◮ The Euler decomposition is not derivable from RG . ◮ Proof: We define an alternative interpretation functor [ · ] 0 : RG → FdHilb Q exactly as [ · ] RG with the following change: [ P X ( α, β )] 0 = [ P X (0 , 0)] RG [ P Z ( α, β )] 0 = [ P Z (0 , 0)] RG This functor preserves all the rules, so its image is indeed a valid model of the theory. However we have the following inequality [ H ] 0 � = [ P X (4 π 3 , 4 π 3 )] 0 ◦ [ P Z (4 π 3 , 4 π 3 )] 0 ◦ [ P X (4 π 3 , 4 π 3 )] 0 hence the Euler decomposition is not derivable from RG .

Qutrit RGB Generators Similarly, we define a category RGB where morphism from ∗ m to ∗ n is a finite undirected open graph from m wires to n wires, built from α β α β α β

Qutrit RGB Rules RGB morphisms are subject to the following equations 1. Each colour respects the equations (S1) and (S2). 2. All equations hold under flip of graphs and negation of angles. 3. The following quadruples form bialgebras: � � � � � � , 1 1 , , , 1 1 , , , 1 1 , , � � � � � � , , 2 2 , , , 2 2 , , , 2 2 , 1 2 2 2 1 2 2 2 1 2 2 1 4. = 1 = = = 2 1 2 2 2 1 1 2 2 2 2 2 2 1 1 2 2 1 1 = = = 5. 2 2 2 2 2 2 β - α - β - β β - α 1 2 1 2 - α - α 2 1 α - β 2 α - β 1 = = = = 6. 1 2 1 2 α α α α β 2 β 1 β 2 β 1

Qutrit RGB Rules = 7. D β 1 D D α 1 8. D = = = := α D D β D D 2 2 2 1 1 1 1 = ⊤ := 2 = 2 = 2 := 1 = 1 9. ⊥ 2 2 2 1 1 1 2 2 2 1 1 1 2 ⊥ ⊤ ⊥ ⊤ ⊥ 2 10. = = = = α = α = α β β β 1 ⊤ ⊤ ⊥ ⊥ ⊤ 1

Dagger Functor ◮ Dagger functor (only showing the blue one): � † � † � † � � � = = = � † � † � � = α = - α β - β ◮ RGB is a dagger symmetric monoidal category.

RGB Interpretation We give an interpretation [ · ] RGB : RGB → FdHilb Q � � � � � � = | + � = � + | = | 00 � � 0 | + | 11 � � 1 | + | 22 � � 2 | � � � � = | 0 � � 0 | + e i α | 1 � � 1 | + e i β | 2 � � 2 | = | 0 � � 00 | + | 1 � � 11 | + | 2 � � 22 | α β � � � � � � = | u � = � u | = | ++ � � + | + ω | ωω � � ω | + ω | ¯ ω ¯ ω � � ¯ ω | � � � � = | + � � + | + e i α | ω � � ω | + e i β | ¯ = | + � � ++ | +¯ ω | ω � � ωω | +¯ ω | ¯ ω � � ¯ ω ¯ ω | α ω � � ¯ ω | β � � � � � � = | 0 � = � 0 | = | uu � � u | + | tt � � t | + | vv � � v | � � � � = | u � � u | + e i α | t � � t | + e i β | v � � v | = | u � � uu | + | t � � tt | + | v � � vv | α β

RGB Interpretation 2 4 3 π i , ¯ 3 π i , and where ω = e ω = e  | + � = | 0 � + | 1 � + | 2 �  | ω � = | 0 � + ω | 1 � + ¯ ω | 2 � | ¯ ω � = | 0 � + ¯ ω | 1 � + ω | 2 �  and  | u � = | 0 � + ¯ ω | 1 � + ¯ ω | 2 �  | t � = | 0 � + | 1 � + ω | 2 � | v � = | 0 � + ω | 1 � + | 2 � 

RG to RGB Translation We have a functor T : RG → RGB � � � � � � � � T = T = T = T = � � � � � � � � 1 2 2 1 T = T = T = T = 1 2 2 1 2 1 � � � � � � � � 2 1 2 1 T α = α T α = α T = T = H † H β β β β 2 1 2 1 2 1

Single Qutrit Quantum Algorithm ◮ Recently, Gedik[6] introduces a simple algorithm using only a single qutrit to determine the parity of permutations. ◮ Like Deutsch’s algorithm, a speed-up relative to corresponding classical algorithms is obtained. ◮ The algorithm can be depicted by the dichromatic calculus: (0) (1 2)(0 1)(1 2)(0 2) (1 2) (0 1) (0 2) f 0 1 2 1 2 U f 0 2 1 2 1 D D D 1 1 1 1 1 2 2 2 2 2 1 1 2 = = = 2 2 = = = 1 U f | w � 0 1 2 1 2 0 2 1 2 1 D D D Parity Even Odd

Qudit ZX Calculus is Universal The proof [3] that qudit ZX calculus is universal for quantum mechanics is based on the facts that the d-dimensional phase gates Z d , X d are sufficient to simulate all single qudit unitary transforms, where Z d ( b 0 , b 1 ..., b d − 1 ) : b 0 | 0 � + b 1 | 1 � + ... + b d − 1 | d − 1 � �→ | d − 1 � (the d complex coefficients, b 0 , b 1 ..., b d − 1 are normalized to unity), � | d − 1 � e i φ | d − 1 � �→ X d ( φ ) : | p � �→ | p � for p � = d − 1