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1 Chaire de Physique Mésoscopique Michel Devoret Année 00, mai - juin INTRODUCTION AU CALCUL QUANTIQUE INTRODUCTION TO QUANTUM COMPUTATION Quatrième Leçon / Fourth Lecture This College de France document is for consultation only. Reproduction rights are reserved. 0-IV- VISIT TE WEBSITE OF TE CAIR OF MESOSCOPIC PSICS then follow Enseignement > Sciences Physiques > Physique Mésoscopique > Site web or PDF FILES OF ALL PAST LECTURES ARE POSTED Questions, comments and corrections are welcome! write to "phymeso@gmail.com" 0-IV-

2 CONTENT OF TIS EAR'S LECTURES QUANTUM COMPUTATION FROM TE PERSPECTIVE OF MESOSCOPIC CIRCUITS. Introduction, c-bits versus q-bits. The Pauli matrices and quantum computation primitives 3. Stabilizer formalism for state representation 4. Clifford calculus 5. Algorithms 6. Error correction 0-IV-3 CALENDAR OF SEMINARS May : Cristian Urbina, (Quantronics group, SPEC-CEA Saclay) Josephson effect in atomic contacts and carbon nanotubes May 8: Benoît Douçot (LPTE / Université Pierre et Marie Curie) Towards the physical realization of topologically protected qubits June : Takis Kontos (LPA / Ecole Normale Supérieure) Points quantiques et ferromagnétisme June 8: Cristiano Ciuti (MPQ, Université Paris - Diderot) Ultrastrong coupling circuit QED : vacuum degeneracy and quantum phase transitions June 5: Leo DiCarlo (ale) Preparation and measurement of tri-partite entanglement in a superconducting quantum circuit June : Vladimir Manucharian (ale) The fluxonium circuit: an electrical dual of the Cooper-pair box? 0-IV-4

3 LECTURE IV : CLIFFORD CALCULUS. Review of stabilizer properties. Number of stabilizers 3. One-qubit Clifford operations 4. Two-qubit interactions 5. Two-qubit Clifford operations 0-IV-5 OUTLINE. Review of stabilizer properties. Number of stabilizers 3. One-qubit Clifford operations 4. Two-qubit interactions 5. Two-qubit Clifford operations 0-IV-5a 3

4 BASIC INGREDIENTS OF STABILIER FORMALISM Primary Pauli operators:... N N =6 example: Stabilizer element: PP P P { I,,, } II Stabilizer: {,,..., N } ± PP... PN = M I M M M { } Condensed form: { M, M,..., M } i j, k, M jmkm jmk = I represents a state of the register α β ν { αβγ} N elements,,,m M ± M α β γ A D-dimensional state manifold of the register is represented by: D = N P { Mα, Mβ,..., Mμ} P elements N : number of qubits 0-IV-6d STABILIER REGISTER STATE A single-state stabilizer can be written as an array of symbols: ± ± P P P P P P ± P P... P + I + I N N N N NN 00 + I + I ( 0 + )( 0 + ) N Pauli words with N+ symbols arranged in "alphabetical" order (I---) The associated state simultaneously diagonalizes the operator-words with + eigenvalue: = I I 0 I + I ( 0 + )( 0 ) = IV-7c 4

5 STABILIER STATE MAPPING (continued) + + remove signs ( 0 + )( 0 + ) + ( 0 )( 0 ) = = ,,, less is more! +? Observe that: + + R y π + since π/ rotation around IV-8e STABILIER STATE MAPPING (continued) + I + I + 00 pivot = = II + II + II + I + I + I pivot pivot + I + I + + I I 000 = = IV-9 5

6 OUTLINE. Review of stabilizer properties. Number of stabilizers 3. One-qubit Clifford operations 4. Two-qubit interactions 5. Two-qubit Clifford operations 0-IV-5b CONSTRUCTION OF A STABILIER CLASS All the Pauli multi-qubit primary operators I I II II II 0-IV-0 6

7 CONSTRUCTION OF A STABILIER CLASS I pick one operator: II I II II II 0-IV-0a CONSTRUCTION OF A STABILIER CLASS I exclude non-com. operators I II II II 0-IV-0b 7

8 CONSTRUCTION OF A STABILIER CLASS II I II I II II II pick next operator II,II 0-IV-0c CONSTRUCTION OF A STABILIER CLASS II I II I II II II exclude non-com. operators II,II 0-IV-0d 8

9 CONSTRUCTION OF A STABILIER CLASS II I II I II II II pick next operator exclude product II,II,I II,II 0-IV-0e CONSTRUCTION OF A STABILIER CLASS II I II I II II II A N D II,II,I S O O N II,II 0-IV-0f 9

10 First choice: NUMBER OF STABILIER CLASSES N Second choice: N Third choice: the - comes from excluding I exclude non.com., last gen. and I N 4 exclude non.com., last gen. comb. and I N k+ k k-th choice: N Total nb. choices: ( N k+ k N k+ ) ( N k = + )( ) k= k= N Last factor corresponds to the number of arrangements of numbers thru N on hyper-cube of stabilizer gen. combinations. It must be divided out to get the classes. The number of stabilizer classes is thus: N N k+ N N ( ) ( )( ) k= + = IV-a NUMBER OF CLIFFORD STATES N CS ( )( ) N N N = N N CS ln N CS 0 Information in selecting one of these states: ln N CS ln N CS grows like N / information is super-extensive! N Gottesman-Knill theorem entanglement, Bell violations, teleportation, error correction, but... 0-IV-c 0

11 OUTLINE 0-IV-5c. Review of stabilizer properties. Number of stabilizers 3. One-qubit Clifford operations 4. Two-qubit interactions 5. Two-qubit Clifford operations TE SINGLE QUBIT CLIFFORD GROUP Consider isomorphisms of the Pauli group into itself Each isomorphism is characterized by the set of the images of the generators: z x g g { } { },,,,,, z x g g z x x z gg gg = The two images must satisfy: There are therefore 6x4=4 isomorphisms, called elements of the -qubit Clifford group: / / = = = Id = = / / = = = / / = = = Why and not? / P [ ] / P The latter is an operator acting on kets. The former is a super-operator acting on operators. 0-IV-3c

12 TE SINGLE QUBIT CLIFFORD GROUP IS ISOMORPIC TO TE OCTAEDRAL GROUP / π rotation around + π/ rotation around / π/ rotation around π/ rotation around / The notation A a means the transformation is a "rotation" around axis A with angle aπ. Attention: A = Id 0-IV-4b GENERATORS OF TE -QUBIT CLIFFORD GROUP (ISOMORPIC TO TE OCTAEDRAL GROUP) Consider S 4 the permutation group on 4 objects, isomorphic to the octahedral group, symmetry group of the cube and the octahedron R R + ( π) ( π /) objects 34 are diagonals R ( π /) R ( π /) = = R+ ( π) = = st choice of generators / duality vertex-face 3 nd choice of generators 3 4 / R ( π /) = = R ( π /) = = 3 4 objects 34 are pairs of faces / 0-IV-5a

13 USEFUL RELATIONS FOR QUANTUM COMPILERS / π/ rot. around / π/ rot. around = / / / / / π/ rot. around π/ rot. around / / / π/ rot. around π/ rot. around 0-IV-6 OUTLINE. Review of stabilizer properties. Number of stabilizers 3. One-qubit Clifford operations 4. Two-qubit interactions 5. Two-qubit Clifford operations 0-IV-5d 3

14 TWO-QUBIT QUANTUM PROCESSOR slide courtesy of Leo DiCarlo & Rob Schoelkopf ale University f I f I Q Q 0-IV-7 TWO-QUBIT QUANTUM PROCESSOR slide courtesy of Leo DiCarlo & Rob Schoelkopf ale University f I [I] β f [I] α I Q Q 0-IV-7a 4

15 TWO-QUBIT QUANTUM PROCESSOR slide courtesy of Leo DiCarlo & Rob Schoelkopf ale University [I] δ f I [I] γ f I Q Q 0-IV-7b TWO-QUBIT QUANTUM PROCESSOR slide courtesy of Leo DiCarlo & Rob Schoelkopf ale University f I f I Q Q 0-IV-7c 5

16 UPGRADING TE PROCESSOR TO 4 QUBITS slide courtesy of Leo DiCarlo & Rob Schoelkopf Q Q Q3 Q4 0-IV-8 CIRCUIT AMILTONIAN qubit qubit α = ω aa+ ( aa) α = ω aa + ( aa) 0-IV-9 6

17 CIRCUIT AMILTONIAN qubit qubit cavity α = ω aa+ ( aa) α = ω aa + ( aa) c = ω aa c c c total = c coupling 0-IV-9a CIRCUIT AMILTONIAN qubit qubit cavity α = ω aa+ ( aa) coupling g RWA α = ω aa + ( aa) ( a ac + aac) + g ( aac + aac) total c = ω aa c c c c = coupling 0-IV-9b 7

18 energy EFFECTIVE QUBIT-QUBIT INTERACTIONS external field 00 OFF ON 00 0-IV-0 EFFECTIVE QUBIT-QUBIT INTERACTIONS OFF ON ON OFF 0-IV-0a 8

19 EFFECTIVE QUBIT-QUBIT INTERACTIONS "FLIP-FLOP" 0 "SECULAR" OFF ON ON OFF 0-IV-0a σ NATURAL ENTANGLING OPERATIONS σ Uˆ ( τ) = exp ( iˆ intτ/ ) * * Secular interaction: ˆ int = g σσ z z with adjustment of gate duration time: [ ] / π τ s = 4 Flip-flop interaction: ˆ int = g ( σσ + + σσ + ) = g ( σσ x x + σσ y y) / τ f = [ ] [ ] π 4g g /4 /4 0-IV- 9

20 REFOCUS SEQUENCE NECESSAR WIT FLIP-FLOP INTERACTION ˆ Consider = g ( t) σ σ + g ( t) σ σ int x x x y y y π 0, 4 0 τ g () t dt/ y π π, 4 4 Green points: [] / equivalent ( 0,0) π,0 4 0 τ g () t dt/ x Need to cut sequence in sections, with a 80 flip of one qubit around x or y. 0-IV- OUTLINE. Review of stabilizer properties. Number of stabilizers 3. One-qubit Clifford operations 4. Two-qubit interactions 5. Two-qubit Clifford operations 0-IV-5e 0

21 GOTTESMAN TABLES FOR QUANTUM OPERATIONS Bit Flip (NOT) π rotation along ˆx [] - Phase Flip π rotation along ẑ [] - adamard π rotation along xˆ + zˆ NOT / π/ rotation along ŷ [] / - Example of a -qubit gate: ct, ct, c control CNOT I phase kick-back I I I I I as in Boolean algebra target 0-IV-3a RULES OF CLIFFORD CALCULUS FOR N QUBITS They are found starting from: B A= [ B] A[ B] α α α B [ A] = [ A] = [ A] / B [ A] = [ B][ A] = [ A] if A and B anticommute if A and B commute if A and B anticommute if A and B commute α NOTE TAT: B = B [ ] [ ] α A = B A= B 0-IV-4a

22 EAMPLE OF CLIFFORD CALCULATIONS / I I = I / I I = I / I I = I / I I = I / I I = I / I I = I / I I = I / I I = I / I I = I / I I = I / I I = I / I I = I / I = / I = / = / = Only need to know: ẑ xˆ = yˆ ŷ ẑ ˆx 0-IV-5 EAMPLE OF POWER OF STABILIER FORMALISM () ow do we go from the Computational basis to the Sign basis? { I, I, } { I, I, } { 00, 0, 0, } { ++, +, +, } For both qubit, must be changed into This is performed by a 90 rotation around (easier than ). I / I / { I, I, } { I, I, } / I I = I / I I = I / I I = I / I I = I 0-IV-6a

23 EAMPLE OF POWER OF STABILIER FORMALISM () ow do we go from the i-sign basis to the Phase basis? / / / { I, I, } ( 0 i )( 0 i ) ( 0 i )( 0 i ) + + +,, ( 0 i )( 0 i ) ( 0 i )( 0 i ) +, We need an entanglement operator: I = I = = {,, } ,, , / / { I, I, } {,, } signs unimportant in basis representation 0-IV-7a VISUALIING TE CLIFFORD MOVES ON TE STABILIER MAP () I I Bell I I I i-bell I I / I / / I / I I I I Comp I I I I 0-IV-8 3

24 VISUALIING TE CLIFFORD MOVES ON TE STABILIER MAP () Phase I I Bell I I I i-bell I I / I / / I I I I Comp I I I I 0-IV-9 END OF LECTURE 4