17. Oktober QSIT-Course ETH Zürich. Universality of Quantum Gates. Markus Schmassmann. Basics and Definitions.

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1 Quantum Quantum QSIT-Course ETH Zürich 17. Oktober 2007 Two Level

2 Outline Quantum Two Level Two Level

3 Outline Quantum Two Level Two Level

4 Outline Quantum Two Level Two Level

5 (I) Quantum Definition ( ) 0 1 X = 1 0 ( ) = 0 1 ( ) 1 0 S = 0 i ( ) 0 i Y = i 0 ( ) H = ( ) 1 0 T = 0 e iπ/4 H = (X + Z )/ 2 S = T 2 Z = ( ) Two Level

6 (II) Quantum R X (θ) = e iθ/2 X = cos (θ/2) I i sin (θ/2) X R Y (θ) = e iθ/2 Y = cos (θ/2) I i sin (θ/2) Y R Z (θ) = e iθ/2 Z = cos (θ/2) I i sin (θ/2) Z Rˆn (θ) = e iθ/2 ˆn σ = cos (θ/2) I i sin (θ/2) (n X X + n Y Y + n Z Z ) Two Level XYX = Y XZX = Z XR Y (θ)x = R Y ( θ) XR Z (θ)x = R Z ( θ)

7 X-Y decomposition of a single qbit gate Quantum Theorem X-Y decomposition of a single qbit gate U C 2 2 unitary α, β γ, δ R: U = e iα R Z (β)r Y (γ)r Z (δ) Proof. U can be written as U ( = e i(α β/2 δ/2) cos(γ/2) e i(α β/2+δ/2) sin(γ/2) e i(α+β/2 δ/2) sin(γ/2) e i(α+β/2+δ/2) cos(γ/2) also true for any two non-parallel rotation axis Rˆn (θ), R ˆm (θ) ˆn ˆm ) Two Level

8 X-Y decomposition of a single qbit gate Quantum Theorem X-Y decomposition of a single qbit gate U C 2 2 unitary α, β γ, δ R: U = e iα R Z (β)r Y (γ)r Z (δ) Proof. U can be written as U ( = e i(α β/2 δ/2) cos(γ/2) e i(α β/2+δ/2) sin(γ/2) e i(α+β/2 δ/2) sin(γ/2) e i(α+β/2+δ/2) cos(γ/2) also true for any two non-parallel rotation axis Rˆn (θ), R ˆm (θ) ˆn ˆm ) Two Level

9 Corrollary of decomposition Quantum Corollary U C 2 2 unitary α R A, B, C C 2 2 unitary: ABC = I, U = e iα AXBXC Proof. ( A = R Z (β)r Y (γ/2), B = R Y ( γ/2)r Z ), ( C = R δ β Z 2 ( XBX = XR Y ( γ/2)xxr Z R Y (γ/2)r Z ( δ+β 2 ) δ+β 2 ) X = δ+β 2 ), Two Level

10 controled by one Qbit CNOT = Cphase = = e iα e iα = = Quantum Two Level controled U = ( U ) = =

11 controled by several Qbits Quantum =, where V 2 = U = ( ), 1 0 where S = T 2, T = 0 e iπ/4. Expansion to more control Qbits is tedious, but not difficult. Two Level

12 Two Level Theorem Two level gates are universal. U C 3 3 unitary U i C 3 3 : U i = U i 1, U i C 2 2 unitary U = U 1 U 2 U 3 Proof. U = a b c d e f g h j b 0: U 1 = U 1 U =, a b c 0 e f g h j a b a 2 + b 2 a 2 + b a b + b a a + b Quantum Two Level

13 Proof contd. Proof. contd. c 0 U 2 = U 2 U 1 U = a c 0 a 2 + c 2 a 2 + c c a 0 a 2 + c 2 a 2 + c 2 1 b c 0 e f 0 h j d = g = 0 U 3 =, but U 2 U 1 U are unitary e f 0 h j Quantum Two Level U 3 U 2 U 1 U = I U = U 1 U 2 U 3 for higher dimensions similar processes

14 Unitaries of Higher Dimensions Quantum U C d d U = N j=1 (U j 1 d 2), U j C 2 2, N d(d 1) 2 U C d d : N (d 1) ex: U jk = δ jk e 2πi p i, where p j is the j th prime number. With one single qbit gate and CNOTs an arbitrary two-level unitary operation on a state of n qbits can be implemented, where the CNOTs are used to shuffle. Two Level

15 Quantum Therefore CNOTs and unitary single Qbit operations form an universal set of quantum computing. Unfortunately, for most single Qbit operations exists no straightforward method of error correction. Two Level

16 Approximation of Unitaries Quantum Definition error E(U, V ) := max (U V ) ψ ψ E(U m U m 1... U 1, V m V m 1... V 1 ) m j=1 E(U j, V j ) Proof. E(U 2 U 1, V 2 V 1 ) = (U 2 U 1 V 2 V 1 ) ψ = (U 2 U 1 V 2 U 1 ) ψ + (V 2 U 1 V 2 V 1 ) ψ (U 2 U 1 V 2 U 1 ) ψ + (V 2 U 1 V 2 V 1 ) ψ E(U 2, V 2 ) + E(U 1, V 1 ) further by induction Two Level

17 Standard Set of universal Hadamard H, phase S, CNOT, π/8 = T, where π/8 could be replaced by Toffoli. T = R Z (π/4), HTH = R X (π/4) up to a global phase. exp ( iπ/8 Z ) exp ( iπ/8 X) = (cos π 8 I i sin π ) (cos 8 Z π 8 I i sin π ) 8 X = cos 2 π ( 8 I i cos π 8 (X + Z ) + sin π ) 8 Y sin π 8 =Rˆn (θ), Quantum Two Level where ˆn = ( cos π 8, sin π 8, cos π 8 ) and cos θ 2 = cos2 π 8.

18 Multiples of irrational Angles Quantum cos θ 2 = cos2 π 8 = θ 2π / Q, therefore any Rˆn (α) can be arbitrary close approximated. HRˆn (α)h = R ˆm (α), where ˆm = ( cos π 8, sin π 8, cos π ) 8. U C 2 2 unitary α, β γ, δ R: U = e iα Rˆn (β)r ˆm (γ)rˆn (δ) Finally, U C 2 2 unitary, ε > 0 n 1, n 2, n 3 N : E (U, Rˆn (θ) n 1HRˆn (θ) n 2HRˆn (θ) n 3) < ε. Two Level

19 Generic qbit Definition A generic qbit gate is a U C 2n 2 n with eigenvalues e iθ 1, e iθ 2, e iθ 2 n : j, k θ j π / Q θ j θ k / Q. n NU n has eigenvalues e inθ 1, e inθ 2, e inθ 2 n, each n defines therefore a point on a 2 k -torus. If U = e ia λ R ε n : E ( U n, e iλa) < ε. By switching leads we can get another generic qbit gate U = PUP, where might be P = SWAP. It can easily been shown, that { e iλa} have a closed Lie Algebra. U = e ib, B = PAP 1 ; by explicit computation can be shown, that the complete Lie-Algebra of U(4) can be computed by successives commutation, starting by A and B. Quantum Two Level

20 Efficiency of Approximation Quantum Theorem Solovay-Kitaev theorem: Any quantum circuit containing m CNOT s and single qbit gates can be approximatet to an accuracy ε using only O ( m log c (m/ε) ) gates from a discrete set, where c = lim δ 0 δ>0 2 + δ. On one hand U C 2n 2 n : O ( n 2 4 n log c (n 2 4 n /ε) ) operations are sufficient, on the other hand U C 2n 2 n : Ω (2 n log(1/ε)/ log(n)) operations are required for implementing a V : E(U, V ) ε. Two Level

21 Quantum CNOTs and unitary single Qbit operations form an universal set for quantum computing. Unitary single Qbit operations can be approximated to an arbitrary precision by a finite set of gates. This approximation cannot always be done efficiently. Two Level

22 Quantum Michael A. Nielsen, Isaac L. Chuang: Quantum Computation and Quantum Information, Chapter 4: Quantum circuits John Preskill: Lecture Notes for Quantum Information and Computation, Chapter 6.2.3: quantum gates Two Level