Motivation Classical Technique: For AND-OR-NOT circuit for function - PowerPoint PPT Presentation

Matrix Decompositions and Quantum Circuit Design Stephen S. Bullock (joint with Vivek V.Shende,Igor L.Markov, U.M. EECS) M athematical and C omputational S ciences D ivision Division Seminar National Institute of Standards and Technology September 15, 2004 1

� ✁ � � � Motivation Classical Technique: For AND-OR-NOT circuit for function ϕ on bit strings Build AND-NOT circuit firing on each bit-string with ϕ 1 Connect each such with an OR Restatement: Produce a decomposition of the function ϕ Produce circuit blocks accordingly 2

� � � Motivation, Cont. Quotation, Feynman on Computation , 2.4: However, the approach described here is so simple and general that it does not need an expert in logic to design it! Moreover, it is also a standard type of layout that can easily be laid out in silicon. (ibid.) Remarks: Analog for quantum computers? Simple & general? 3

� � � Motivation, Cont. 2 n unitary matrix Quantum computation, n quantum bits: 2 n Matrix decomposition: Algorithm for factoring matrices – Similar strategy: decomposition splits computation into parts – Divide & conquer: produce circuit design for each factor 4

Outline I. Introduction to Quantum Circuits II. Two Qubit Circuits (CD) III. Circuits for Diagonal Unitaries IV. Half CNOT per Entry (CSD) V. Differntial Topology & Lower Bounds 5

✄ ✍ ✠ � ✟ ✁ ✁ ✞ ✁ ✂ ✂ ☛ ✞ � ✁ ✁ � � � ✂ ✁ ☛ ✁ ✁ � � ✁ ✁ � ✌ ✁ ✟ ✁ ✂ � ✆ ✁ ✂ � ✠ ✁ Quantum Computing replace bit with qubit: two state quantum system, states , 0 1 2 – Single qubit state space H 1 0 1 ✁✝✆ ✁☎✄ 1 2 ψ ψ – e.g. or 1 2 0 i 1 i 2 ✁☎☞ ✟✡✠ 2 n n ¯ – n -qubit state space H n 1 H 1 b b an n bit string ¯ – Kronecker (tensor) product entanglement 6

✁ ✞ ✁ ☛ ✠ ✞ ☛ ✁ ☛ ✞ � ✟ ☛ ✁ ✂ � ☞ ✁ ✂ � ✟ ✞ ☛ ☛ ✂ ✂ ✂ ✂ ✂ ✁ � ✞ ☛ � ✠ ✟ � ✞ ✁ � ✁ � ✁ ✞ ✂ ✂ � � ✁ ☛ ✂ ✞ ✁ ✁ ☛ � � � � ✆ ✁ ✁ � � ✁ ✂ ✂ ✠ � ✟ � ✞ ✁ ✁ ✂ � ✂ ✂ ✟ � ✂ � ✂ � ✂ ☞ ✁ � � Nonlocality: Entangled States ∑ 2 n ψ 0 α j α j α j 1 ∑ N 2 2 von Neumann measurement: , Prob j meas j j j 0 ψ Standard entangled state: 1 2 00 11 – Prob 00 meas Prob 11 meas 1 2 Also , GHZ 1 2 00 0 11 1 ✁☎☞ W 1 n 100 0 010 0 0 01 ✁☎☞ ✁☎☞ ψ ψ quantum computations: apply unitary matrix u , i.e. u ✁ ☎✄ 7

� ✁ ✁ � � � ✁ ✁ � � ✌ � � � ✁ ✁ ✁ � ✁ ✁ � � � ☛ � � � ✁ ✁ � � ✁ � ✁ ✌ ✁ ✞ � ✌ ✁ ✁ � � ✌ Tensor (Kronecker) Products of Data, Computations φ ψ H 1 , 0 i 1 0 1 ✁ ✁� ✁ ✄✂ ✁☎☞ – interpret etc. 10 1 0 φ ψ – composite state in H 2 : 00 01 i 10 i 11 ✁ ✁� ✁☎☞ Most two-qubit states are not tensors of one-qubit states. α β If A is one-qubit, B one-qubit, then the two-qubit tensor β ¯ α ¯ α B β B B is . Most 4 4 unitary u are not local. A A B β B α B ¯ ¯ 8

� � � ✌ � � � ✌ � Complexity of Unitary Evolutions n Easy to do: 1 u j for 2 2 factors, j Slightly tricky: two-qubit operation v 4 , some 4 4 unitary v I 2 n Optimization problem: Use as few such factors as possible Visual representation: Quantum circuit diagram Thm: (’93, Bernstein-Vazirani) The Deutsch-Jozsa algorithm proves quan- tum computers would violate the strong Church-Turing hypothesis. 9

� � � � � � � � � � � � � � � � � � � � � � � � ✁ � ✆ � ✆ ✁ ✁ � ✌ � ✌ � � � � � � � � � Complexity of Unitary Evolutions Cont. � � � � u 1 u 4 u 7 v 2 u 2 u 5 u 8 U v 3 v 1 u 3 u 6 u 9 � � � � Outlined box is Kronecker (tensor) product u 1 u 2 u 3 Common practice: not arbitrary v 1 , v 2 , v 3 but CNOT, 10 11 10

✆ � � � ✁ ✁ ✂ ✄ ✁ � ☎ ☎ ✁ ✆ Quantum Circuit Design 0 1 For , sample quantum circuit: 1 0 0 1 0 0 1 0 0 0 is implemented by u ���� ���� ���� ���� 0 0 1 0 0 0 0 1 ���� ���� good quantum circuit design: find tensor factors of computation u 11

� ✁ � � � ✞ ✠ � ☛ ☛ ✠ � ✟ ✁ ✞ ✁ ☎ ✁ ☎ � � � ☛ ✠ � ✟ ✞ � ✁ � � ✆ ✆ ✆ ✆ ✄ � ✁ � � � ✁ ✄ ✄ � � ✁ ✁ ✄ ✁ ✂ � � � ✁ � � Example: F the Two-Qubit Fourier Transform in 4 , the discrete Fourier transform F : Relabelling as 00 11 0 3 ✂☎✄ ✂☎✄ 1 1 1 1 3 1 1 1 i 1 i ∑ F jk or F j 1 k 1 1 1 1 2 2 k 0 1 i 1 i 1 1 1 0 one-qubit unitaries: H , S 1 2 1 2 1 1 0 i H S ���� ���� F H ���� ���� ���� ���� 12

Outline I. Introduction to Quantum Circuits II. Two Qubit Circuits (CD) III. Circuits for Diagonal Unitaries IV. Half CNOT per Entry (CSD) V. Differntial Topology & Lower Bounds 13

� � ✁ ☛ ✟ ✠ � ✁ ✟ ✁ ✁ ✞ ☛ � ✞ ✁ � � ✠ � � ✁ ✞ ✁ ✁ ☛ � ✟ ✠ ✟ ☛ ✁ ✠ � � ☞ ✌ ✁ ✁ ✁ ✂ ✁ ✁ ✁ ✄ ☛ ✝ ✁ ✁ ✞ � ✁ ✁ ✍ ✞ � ✞ ✁ ✁ ✞ ✞ ☞ ✠ ✟ ☛ ✁ ✁ � ✁ The Magic Basis of Two-Qubit State Space 00 11 2 � ✆☎ 01 10 2 � ✆☎ i 00 i 11 2 � ✆☎ 01 10 2 i i � ✆☎ Remark: Bell states up to global phase; global phases needed for theorem Theorem (Lewenstein, Kraus, Horodecki, Cirac 2001) Consider a 4 4 unitary u , global-phase chosen for det u 1 Compute matrix elements in the magic basis All matrix elements are real u a b ☛ ☛✡ 14

✁ ✞ � ✌ � ☛ ✁ ✌ ✁ ☎ ✌ ☛ � ✌ � ✞ ✁ ✂ � ✂ Two-Qubit Canonical Decomposition Two-Qubit Canonical Decomposition: Any u a four by four unitary admits a matrix decomposition of the following form: u d f a b c 0 e i θ j ∑ 3 for b f are tensors of one-qubit computations, a c d j � ✆☎ Note that a applies relative phases to the magic or Bell basis. Circuit diagram: For any u a two-qubit computation, we have: b d u a f c 15

� ✂ � � � � � Application: Three CNOT Universal Two-Qubit Circuit Many groups: 3 CNOT circuit for 4 4 unitary: (F .Vatan, C.P .Williams), (G.Vidal, C.Dawson), (V.Shende, I.Markov, B-) – Implement a somehow, commute SWAP through circuit to cancel – Earlier B-,Markov: 4 CNOT circuit w/o SWAP , CD & na¨ ıve a R z B ���� ���� ���� ���� D u R y R y ���� ���� C F 16

✡ ✞ ☛ � ✞ ✞ ☛ ✁ ☛ ✂ ☞ � ✞ ☛ ✁ ☛ ✄ ☛ ✞ ✞ ☛ ✍ � ✡ � ✍ � � ✞ ☛ � ☎ ✁ ✞ ☛ ☛ ✁ ✞ ✞ ✍ ✁ ✞ ✞ � ✡ ☛ ✞ ✞ � ✌ ☛ ✞ ✂ ☛ Two-Qubit CNOT-Optimal Circuits Theorem:(Shende,B-,Markov) Suppose v is a 4 4 unitary normalized so 1 . Label γ i σ y i σ y 2 v 2 v T . Then any v admits a circuit det v v ☛ ✁� ☛ ✁� 2 and 3 CNOT’s, up to global phase. Moreover, holding elements of SU 2 λ λ I 4 γ the characteristic poly of γ for p det : v v λ ( v admits a circuit with 2 CNOT’s) ( p has real coefficients) λ λ λ 2 2 ) ( v admits a circuit with 1 CNOT) ( p i i ( γ ( v ) I 4 ) SU 2 SU 2 v 17

� ✟ � ✁ ✁ � ✁ ✞ ☛ � � � ✂ � � � ✂ Optimal Structured Two-qubit Circuits R x B D B D R z C ���� ���� ���� ���� F C ���� ���� F Quantum circuit identities: All 1 2 CNOT diagrams reduce to these Computing parameters: useful to use operator E , E j � ✆☎ π R x 2 E S † S ���� ���� 18

Outline I. Introduction to Quantum Circuits II. Two Qubit Circuits (CD) III. Circuits for Diagonal Unitaries IV. Half CNOT per Entry (CSD) V. Differntial Topology & Lower Bounds 19

✞ ✆ � ✁ � ✂ ✍ � ✞ ✁ ✁ ☛ � ✁ � � ✞ ☛ ✞ ✁ ☛ ✁ ☛ ✆ ✞ ✂ ✄ ☛ Relative Phase Group Easiest concievable n -qubit circuit question: How to build circuits for 2 n 1 ∑ e i θ j 2 n ; θ j A j j ? j 0 2 n commutative vector group A 2 n 2 n – log : A carries matrix multiplication to vector sum – Strategy: build decompositions from vector space decompositions – Subspaces encoded by characters, i.e. continuous group maps χ : A 2 n e it 20