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Methods of Quantum Information Processing

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Methods of Quantum Information Processing 1 Methods of Quantum Information Processing (with an emphasis on optical implementations) Adam Miranowicz e-mail: [email protected] http://zon8.physd.amu.edu.pl/»miran summer semester 2008 2 Keywords ² quantum logic gates ² quantum entanglement ² quantum cryptography ² quantum teleportation ² quantum algorithms ² quantum error correction ² quantum tomography ² solid-state implementations of quantum computing 3 Basic textbook: 1. Quantum Computation and Quantum Information by Michael A. Nielsen and Isaac L. Chuang (Cambridge, 2000) Review articles on quantum-optical computing: 1. Linear optical quantum computing, P. Kok, W.J. Munro, K. Nemoto, T.C. Ralph, J. P. Dowling, G.J. Milburn, free downloads at http://arxiv.org/quant-ph/0512071. 2. Linear optics quantum computation: an overview, C.R. Myers, R. Laflamme, free downloads at http://arxiv.org/quant-ph/0512104. 3. Quantum optical systems for the implementation of quantum information processing, T.C. Ralph, free downloads at http://arxiv.org/quant-ph/0609038. 4. Quantum mechanical description of linear optics J. Skaar, J.C.G. Escartin, H. Landro, Am. J. Phys. 72, 1385-1391 (2005). 4 Other Textbooks on Quantum Computing 1. Quantum Computing by Joachim Stolze, Dieter Suter 2. Approaching Quantum Computing by Dan C. Marinescu, Gabriela M. Marinescu 3. Introduction to Quantum Computation and Information edited by Hoi-Kwong Lo, Tim Spiller, Sandu Popescu 4. Quantum Computing by Mika Hirvensalo 5. Explorations in Quantum Computing by Colin P. Williams, Scott H. Clearwater 6. Quantum Information Processing edited by Gerd Leuchs, Thomas Beth 7. Quantum Computing by Josef Gruska 8. Quantum Computing and Communications by Sandor Imre, Ferenc Balazs 5 9. An Introduction to Quantum Computing Algorithms by Arthur O. Pittenger 10. Quantum Computing by M. Nakahara, Tetsuo Ohmi 11. Principles of Quantum Computation and Information - Vol.1 by Giuliano Benenti, Giulio Casati, Giuliano Strini 12. Principles of Quantum Computation And Information: Basic Tools And Special Topics by Giuliano Benenti 13. Quantum Optics for Quantum Information Processing edited by Paolo Mataloni 14. Lectures on Quantum Information edited by Dagmar Bruß, Gerd Leuchs 15. A Short Introduction to Quantum Information and Quantum Computation by Michel Le Bellac 16. Quantum Computation and Quantum Communication by Mladen Pavicic 17. Scalable Quantum Computers: Paving the Way to Realization edited by Samuel L. Braunstein, Hoi-Kwong Lo 6 18. Fundamentals of Quantum Information edited by Dieter Heiss 19. Experimental Aspects of Quantum Computing edited by Henry O. Everitt 20. Quantum Computing: Where Do We Want to Go Tomorrow edited by Samuel L. Braunstein 21. Quantum Information by Gernot Alber et al. 22. The Physics of Quantum Information edited by Dirk Bouwmeester, Artur K. Ekert, Anton Zeilinger 23. Temple of Quantum Computing by Riley T. Perry (free downloads at www.toqc.com) 24. Quantum Computation edited by Samuel J. Lomonaco, Jr. (free downloads at www.cs.umbc.edu/»lomonaco) 25. Lecture Notes for Physics: Quantum Information and Computation by John Preskill (free downloads at www.theory.caltech.edu/people/preskill/ph229). 7 a two-level atom/system jgi - ground state jei - excited state – physical notation · ¸ · ¸ 0 1 jgi = ; jei = 1 0 – information notation · ¸ · ¸ 1 0 jgi = ; jei = 0 1 ² Pauli matrices· ¸ · ¸ · ¸ 0 1 0 ¡i 1 0 σ^ ´ ; σ^ ´ ; σ^ ´ : x 1 0 y i 0 z 0 ¡1 ² rasing (σ^y) and lowering (σ^) energy operators, = atomic-transition operators σ^ + iσ^ σ^ ¡ iσ^ σ^y = x y = jeihgj; σ^ = x y = jgihej; σ^ =σ ^yσ^¡σ^σ^y = jeihej¡jgihgj 2 2 z 8 qubit = quantum bit ² the smallest unit of quantum information ² physically realized by a 2-level quantum system whose two basic states are conventionally labelled j0i and j1i ² By contrast to classical bits, a qubit can be in an arbitrary superposition of ’0’ and ’1’: jÃi = ®j0i + ¯j1i with normalization condition j®j2 + j¯j2 = 1: ² matrix representation of qubit states: · ¸ · ¸ 1 0 j0i = ; j1i = 0 1 · ¸ · ¸ · ¸ 1 0 ® jÃi = ®j0i + ¯j1i = ® + ¯ = 0 1 ¯ 9 qudits = qunits d-dimensional quantum states, generalized qubits dX¡1 jÃid = cnjni n=0 with normalization condition Xd¡1 2 jcnj = 1 n=0 special cases 2D qudit = qubit 3D qudit = qutrit 4D qudit = (qu)quartit = ququart 5D qudit = (qu)quintit ... optical qudits are spanned in d-dimensional Fock space 10 quantum (logic) gates basic quantum circuits operating on a small number of qubits they are for quantum computers what classical logic gates are in conventional computers Pauli X^ gate = quantum NOT gate = bit flip · ¸ 0 1 X^ ´ σ^ = x 1 0 X^ jki = j1 © ki; k = 0; 1 examples · ¸ · ¸ 0 1 X^ j1i = X^ = = j0i 1 0 · ¸· ¸ · ¸ 0 1 ® ¯ X^ (®j0i + ¯j1i) = = = ¯j0i + ®j1i 1 0 ¯ ® 11 Pauli Z^ gate = phase flip · ¸ 1 0 Z^ ´ σ^ = z 0 ¡1 Z^jki = (¡1)kjki; k = 0; 1 example · ¸· ¸ · ¸ 1 0 ® ® Z^(®j0i + ¯j1i) = = = ®j0i ¡ ¯j1i 0 ¡1 ¯ ¡¯ Pauli Y^ gate = phase flip + bit flip · ¸ 0 ¡i Y^ ´ σ^ = X i 0 Y^ jki = i(¡1)kj1 © ki; k = 0; 1 example · ¸· ¸ · ¸ 0 ¡i ® ¡¯ Y^ (®j0i + ¯j1i) = = i = ¡i¯j0i + i®j1i i 0 ¯ ® 12 phase gate · ¸ 1 0 S^ = 0 i Hadamard gate · ¸ 1 1 H^ = p1 2 1 ¡1 examples · ¸· ¸ · ¸ 1 1 1 1 H^ j0i = p1 = p1 = p1 (j0i + j1i) ´ j+i 2 1 ¡1 0 2 1 2 · ¸· ¸ · ¸ 1 1 0 1 H^ j1i = p1 = p1 = p1 (j0i ¡ j1i) ´ j¡i 2 1 ¡1 1 2 ¡1 2 · ¸ · ¸ · ¸ · ¸ 1 1 1 2 1 H^ j+i = p1 p1 = 1 = = j0i 2 1 ¡1 2 1 2 0 0 H^ j¡i = j1i 13 Bloch representation/sphere z |0 θ |ψ |0 + i |1 y 2 φ |0 + |1 x 2 |1 µ ¶ µ ¶ θ θ jÃi = cos j0i + eiÁ sin j1i 2 2 14 qubit rotations ² rotations on Bloch sphere R^n(2θ) ´ exp (¡iθn ¢ σ^ ) =σ ^I cos θ ¡ in ¢ σ^ sin θ =σ ^I cos θ ¡ i(nxσ^x + nyσ^y + nzσ^z) sin θ; ² Pauli matrices (and identity matrix) · ¸ 0 1 σ^ ´ = j0ih1j + j1ih0j; x 1 0 · ¸ 0 ¡i σ^ ´ = ¡ij0ih1j + ij1ih0j; y i 0 · ¸ 1 0 σ^ ´ = j0ih0j ¡ j1ih1j; z 0 ¡1 · ¸ 1 0 σ^ ´ I^ = = j0ih0j + j1ih1j I 0 1 σ^ = (^σx; σ^y; σ^z) 15 ² rotations about k = x; y; z axes – special cases of R^n(θ) ¡ ¢ ^ θ θ θ Rk(θ) = exp ¡i2σ^k =σ ^I cos 2 ¡ iσ^k sin 2 or explicitly · ¸ cos θ ¡i sin θ X^ (θ) ´ R^ (θ) = 2 2 x ¡i sin θ cos θ · 2 ¸2 cos θ ¡ sin θ Y^ (θ) ´ R^ (θ) = 2 2 y sin θ cos θ · 2 2 ¸ e¡iθ=2 0 Z^(θ) ´ R^ (θ) = z 0 eiθ=2 ² Do we need all XYZ-rotations? No! e.g. we can omit Z-rotation as ^ ^ ¼ ^ ^ ¼ ^ ¼ ^ ^ ¼ Z(θ) = X( 2 )Y (θ)X(¡ 2 ) = Y ( 2 )X(¡θ)Y (¡ 2 ): ² any single-qubit gate can be written as a decomposition ZY (or, equivalently, XY or XZ) i® i® U^ = e R^n(θ) = e Z^(θ1)Y^ (θ2)Z^(θ3) 16 Bloch vector (Pauli vector) for a qubit ² in any pure state µ ¶ µ ¶ θ θ jÃi = cos j0i + eiÁ sin j1i 2 2 we have rBloch ´ (rx; ry; rz) = (sin θ cos Á; sin θ sin Á; cos θ) ² in any mixed state · ¸ ½ ½ ½^ = 11 12 ½21 ½22 1 = 2(^σI + rBloch ¢ σ^ ) (so-called Pauli basis) 1 = 2(^σI + rxσ^x + ryσ^y + rzσ^z) we get rBloch = hσ^ i = Tr[^½σ^ ] = (Tr[^½σ^x]; Tr[^½σ^y]; Tr[^½σ^z]) = (2Re½21; 2Im½21; ½11 ¡ ½22) 17 ² Exemplary proof: 1 Tr[^½σ^x] = Tr[2(^σI + rxσ^x + ryσ^y + rzσ^z)^σx] 1 = 2Tr[^σIσ^x + rxσ^xσ^x + ryσ^yσ^x + rzσ^zσ^x] 1 = 2(Tr[^σIσ^x] + rxTr[^σxσ^x] + ryTr[^σyσ^x] + rzTr[^σzσ^x]) 1 = 2(Tr[^σx] + rxTr[^σI] + ryTr[¡iσ^z] + rzTr[iσ^y]) 1 = 2(0 + 2rx + 0 + 0) = rx moreover µ· ¸ · ¸¶ ½11 ½12 0 1 Tr[^½σ^x] = Tr ¢ ½21 ½22 1 0 · ¸ ½ ½ = Tr 12 11 ½22 ½21 ¤ = ½12 + ½21 = ½21 + ½21 = 2Re ½21 = 2Re ½12 ² Properties: jrBlochj = 1 – for pure state, jrBlochj < 1 – for mixed state 18 Kronecker tensor product let · ¸ · ¸ a a b b A = 11 12 ;B = 11 12 a21 a22 b21 b22 then · ¸ · ¸ a a b b A ­ B = 11 12 ­ 11 12 a21 a22 b21 b22 2 · ¸ · ¸ 3 b b b b a 11 12 ; a 11 12 6 11 b b 12 b b 7 = 6 · 21 22 ¸ · 21 22 ¸ 7 4 b11 b12 b11 b12 5 a21 ; a22 b21 b22 b21 b22 2 3 a11b11 a11b12 a12b11 a12b12 6 a b a b a b a b 7 = 6 11 21 11 22 12 21 12 22 7 4 a21b11 a11b12 a22b11 a12b12 5 a21b21 a11b22 a22b21 a12b22 19 two-qubit pure states jÃi = ®j00i + ¯j01i + γj10i + ±j11i notation of two-mode states 0 00 0 00 0 00 0 00 jÃi = jà iA ­ jà iB ´ jà iAjà iB ´ jà à iAB ´ jà à i matrix representation 2 3 · ¸ · ¸ 1 1 1 6 0 7 j00i = ­ = 6 7, 0 0 4 0 5 0 2 3 2 3 2 3 0 0 0 6 1 7 6 0 7 6 0 7 j01i = 6 7, j10i = 6 7, j11i = 6 7, 4 0 5 4 1 5 4 0 5 0 0 1 20 matrix representation jÃi = ®j00i + ¯j01i + γj10i + ±j11i · ¸ · ¸ · ¸ · ¸ · ¸ · ¸ · ¸ · ¸ 1 1 1 0 0 1 0 0 = ® ­ + ¯ ­ + γ ­ + ± ­ 0 0 0 1 1 0 1 1 2 3 2 3 2 3 2 3 1 0 0 0 6 0 7 6 1 7 6 0 7 6 0 7 = ®6 7 + ¯6 7 + γ6 7 + ±6 7 4 0 5 4 0 5 4 1 5 4 0 5 0 0 0 1 2 3 ® 6 ¯ 7 = 6 7 4 γ 5 ± 21 Universal set of gates for quantum computing ² rotations of single qubits ² any nontrivial two-qubit gate (e.g., CNOT, NS, CZ) CZ and CNOT gates CZ = controlled phase gate (CPhase) = controlled sign gate (CSign) control bit target bit CZ CNOT j0i j0i j0; 0i j0; 0i j0i j1i j0; 1i j0; 1i j1i j0i j1; 0i j1; 1i j1i j1i ¡j1; 1i j1; 0i or equivalently q1q2 CZ :jq1; q2i ! (¡1) jq1; q2i CNOT : jq1; q2i ! jq1; q1 © q2i where qk = 0; 1 and q1 © q2 = mod (q1 + q2; 2).
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