Elements of quantum information processing and quantum computers (with trapped ion examples) D. J. Wineland, Dept. of Physics, U Oregon, Eugene, OR; & Research Associate, NIST, Boulder, CO Summary: * Why “quantum” for computers? * Logic gates and algorithms * Manipulating quantum states for computers * Quantum computers and Schrödinger’s cat
h e q u a n t u m c o m p u t e r Data storage : • classical: computer bit: (0) or (1) • quantum: “qubit” Ψ = α 0 |0 〉 + α 1 |1 〉 superposition: |0 〉 AND |1 〉 Scaling : Consider 3-bit register (N = 3): Classical register: (example): (101) Quantum register: (3 qubits): Ψ = α 0 |0,0,0 〉 + α 1 |0,0,1 〉 + α 2 |0,1,0 〉 + α 3 |1,0,0 〉 + α 4 |0,1,1 〉 + α 5 |1,0,1 〉 + α 6 | 1,1,0 〉 + α 7 |1,1,1 〉 (represents 2 3 numbers simultaneously) For N = 300 qubits, store 2 300 ≈ 10 90 numbers simultaneously (more than all the classical information in universe!) Parallel processing : single gate operates on all 2 N inputs simultaneously
h e q u a n t u m c o m p u t e r Data storage : • classical: computer bit: (0) or (1) • quantum: “qubit” Ψ = α 0 |0 〉 + α 1 |1 〉 superposition: |0 〉 AND |1 〉 measurement: α 0 |0 〉 + α 1 |1 〉 “collapses” to Scaling : Consider 3-bit register (N = 3): |0 〉 OR |1 〉 with probabilities | α 0 | 2 and | α 1 | 2 Classical register: (example): (101) Quantum register: (3 qubits): Ψ = α 0 |0,0,0 〉 + α 1 |0,0,1 〉 + α 2 |0,1,0 〉 + α 3 |1,0,0 〉 + α 4 |0,1,1 〉 + α 5 |1,0,1 〉 + α 6 | 1,1,0 〉 + α 7 |1,1,1 〉 (represents 2 3 numbers simultaneously) For N = 300 qubits, store 2 300 ≈ 10 90 numbers simultaneously (more than all the classical information in universe!) Parallel processing : single gate operates on all 2 N inputs simultaneously But! : quantum measurement rule: measured register gives only one number
h e q u a n t u m c o m p u t e r Data storage : • classical: computer bit: (0) or (1) • quantum: “qubit” Ψ = α 0 |0 〉 + α 1 |1 〉 superposition: |0 〉 AND |1 measurement: α 0 |0 〉 + α 1 |1 〉 “collapses” to Scaling : Consider 3-bit register (N = 3): |0 〉 OR |1 〉 with probabilities | α 0 | 2 and | α 1 | 2 Classical register: (example): (101) Quantum register: (3 qubits): Ψ = α 0 |0,0,0 〉 + α 1 |0,0,1 〉 + α 2 |0,1,0 〉 + α 3 |1,0,0 〉 + α 4 |0,1,1 〉 + α 5 |1,0,1 〉 + α 6 | 1,1,0 〉 + α 7 |1,1,1 〉 (represents 2 3 numbers simultaneously) For N = 300 qubits, store 2 300 ≈ 10 90 numbers simultaneously (more than all the classical information in universe!) Parallel processing : single gate operates on all 2 N inputs simultaneously But! : quantum measurement rule: measured register gives only one number Factoring : Peter Shor’s Algorithm (1994)
Logic gates for Universal computation Classical: 1-bit NOT 2-bit AND 00 → 0 01 → 0 0 → 1 10 → 0 1 → 0 11 → 1 Quantum: 2-qubit logic gate “rotation” 1 R(θ,φ) U 1,2 ( π ) 2 e.g., |0 〉 |0 〉 → |0 〉 |0 〉 |0 〉 → cos( θ /2)|0 〉 + e i φ sin( θ /2)|1 〉 |0 〉 |1 〉 → |0 〉 |1 〉 |1 〉 → -e -i φ sin( θ /2)|0 〉 + cos( θ /2)|1 〉 〉 |1 〉 |0 〉 → |1 〉 |0 〉 like rotating a spin-1/2 particle |1 〉 |1 〉 → - |1 〉 |1 〉 states: | ↓〉 (= |0 〉 ) and | ↑〉 (= |1 〉 ) phase gate
Universal logic gates for processing Classical: 1-bit NOT 2-bit AND 00 → 0 01 → 0 entanglement! 0 → 1 10 → 0 1 → 0 11 → 1 Quantum: 2-qubit logic gate “rotation” 1 R(θ,φ) U 1,2 ( π ) 2 e.g., |0 〉 |0 〉 → |0 〉 |0 〉 |0 〉 → cos( θ /2)|0 〉 + e i φ sin( θ /2)|1 〉 |0 〉 |1 〉 → |0 〉 |1 〉 |1 〉 → -e -i φ sin( θ /2)|0 〉 + cos( θ /2)|1 〉 〉 |1 〉 |0 〉 → |1 〉 |0 〉 like rotating a spin-1/2 particle |1 〉 |1 〉 → - |1 〉 |1 〉 states: | ↓〉 (= |0 〉 ) and | ↑〉 (= |1 〉 ) phase gate
Quantum computer algorithm Peter Shor to efficiently factorize large numbers (1994) N-qubits: N 2 1 e.g., for N = 3, Ψ in = C N [ |0,0,0 〉 + |0,0,1 〉 + |0,1,0 〉 + |1,0,0 〉 ψ i in + |0,1,1 〉 + |1,0,1 〉 + |1,1,0 〉 + |1,1,1 〉 ] i 0 Process all inputs simultaneously bit no. 0 1 two-qubit gates 2 3 U = U r,s ( π ) U p,q ( π ) …… R k (θ,φ) U i,j ( π ) single qubit bit gate (manipulate superpositions) N
Quantum computer algorithm Peter Shor to efficiently factorize large numbers (1994) N-qubits: N 2 1 e.g., for N = 3, Ψ in = C N [ |0,0,0 〉 + |0,0,1 〉 + |0,1,0 〉 + |1,0,0 〉 ψ i in + |0,1,1 〉 + |1,0,1 〉 + |1,1,0 〉 + |1,1,1 〉 ] i 0 analogous to Process all inputs simultaneously waterwave interference (photo,Hannover Univ.) bit no. 0 1 two-qubit gates 2 3 U = U r,s ( π ) U p,q ( π ) …… R k (θ,φ) U i,j ( π ) incident waves single qubit bit gate (rotation) N
Quantum computer algorithm Peter Shor to efficiently factorize large numbers (1994) N 2 1 ψ i measure in ψ i i 0 qubits out Process all possible small selection inputs simultaneously bit no. 0 1 two-qubit gates use measured “i” 2 3 in classical algorithm to determine factors U = U r,s ( π ) U p,q ( π ) …… R k (θ,φ) U i,j ( π ) single qubit bit gate (rotation) e.g., factorize 150 digit decimal number N ⇒ ~ 10 9 operations
“spin” rotations (superpositions of internal energy states of ion) | 2 D 5/2 〉 ≡ | ↑〉 e.g., | ↓〉 → α 0 | ↓〉 + α 1 | ↑〉 | 2 S 1/2 〉 ≡ | ↓〉 λ = 282 nm 199 Hg + 1 mm describe as rotations: | ↓〉 → cos( θ /2)| ↓〉 + e i φ sin( θ /2)| ↑〉 | ↑〉 → -e -i φ sin( θ /2)| ↓〉 + cos( θ /2)| ↑〉 〉
detection: α 0 | ↓〉 + α 1 | ↑〉 → | ↓〉 2 P 1/2 | ↑〉 Hg + | ↓〉 194 nm 1 mm photomultiplier
α 0 | ↓〉 + α 1 | ↑〉 → | ↑〉 2 P 1/2 | ↑〉 Hg + | ↓〉 194 nm 1 mm photomultiplier
Hyperfine qubits: e.g. 9 Be + 2 P 3/2 2 S 1/2 electronic ground level hyperfine states 2 P 1/2 |↓〉 ≡ | F = 2, m F = 0 〉 |↑〉 ≡ | F = 1,m F = -1 〉 B ≅ 119 G (coherence time > 10 s) two-photon coherent stimulated-Raman transitions two copropagating beams • focused beams (ion motion insensitive) ⇒ individual ion addressability e.g., rotations | ↓〉 → cos( θ /2)| ↓〉 + e i φ sin( θ /2)| ↑〉 ~1.21 GHz | ↑〉 → -e -i φ sin( θ /2)| ↓〉 + cos( θ /2)| ↑〉 〉
Hyperfine qubits: e.g. 9 Be + 2 P 3/2 2 S 1/2 electronic ground level hyperfine states 2 P 1/2 |↓〉 ≡ | F = 2, m F = 0 〉 |↑〉 ≡ | F = 1,m F = -1 〉 B ≅ 119 G (coherence time ~ 10 s) n' • spin/motion coupling with • • non-copropagating beams. (e.g., ∆ m = -1) 1 n 0 • • ~1.21 GHz 2 1 ~5 MHz 0
A bit more history : Peter Shor: algorithm for efficient number factoring on a quantum computer (~ 1994) Artur Ekert: presentation at the 1994 International Conference on Atomic Physics Boulder, Colorado
Atomic ion quantum computer: J. I. Cirac, P. Zoller, Phys. Rev. Lett. 74 , 4091 (1995) 1. START MOTION IN GROUND STATE 2. SPIN → MOTION MAP Ignacio Cirac Peter Zoller MOTION “DATA BUS” (e.g., center-of-mass mode) INTERNAL STATE “SPIN” QUBIT • • • • • • • • | ↑〉 = |1 〉 |m = 3 〉 “m” for |m = 2 〉 motion |m = 1 〉 |m = 0 〉 | ↓〉 = |0 〉 Motion qubit states
Atomic ion quantum computation: J. I. Cirac, P. Zoller, Phys. Rev. Lett. 74 , 4091 (1995) 1. START MOTION IN GROUND STATE 2. SPIN → MOTION MAP 3. SPIN ↔ MOTION GATE Ignacio Cirac Peter Zoller MOTION “DATA BUS” (e.g., center-of-mass mode) INTERNAL STATE “SPIN” QUBIT • • • • • • • • | ↑〉 |m = 3 〉 “m” for |m = 2 〉 motion |m = 1 〉 |m = 0 〉 | ↓〉 Motion qubit states
Step 2. Spin → motion map 1 0 λ ≅ 282 nm 2 1 1 – 10 MHz (motional 0 mode frequency)
Step 2: Spin-qubit/motion-qubit logic gate (Chris Monroe et al., Phys. Rev. Lett. 75 , 4714 (1995)). Hyperfine states of 9 Be + coupled with stimulated-Raman transitions Complete Cirac-Zoller gate (two ions): (F. Schmidt-Kaler et al., Nature, 422 , 408 (2003), Innsbruck) |1 〉 π phase shift, |↑〉| 1 〉 → - |↑〉| 1 〉 | ↑〉 |0 〉 Cirac – Zoller gates: need precise control of motion quantum states. Alternative gates suppress this problem. |1 〉 | aux 〉 |0 〉 |1 〉 | ↓〉 |0 〉
“Motion-insensitive” gates (in Lamb-Dicke limit: wave function << λ ) K. Mølmer and A. Sørensen, Phys. Rev. Lett. 82 , 1835 (1999). A. Sørensen and K. Mølmer, Phys. Rev. Lett. 82 , 1971 (1999). E. Solano, R. L. de Matos Filho, and N. Zagury, Phys. Rev. A 59 , 2539 (1999). A. Sørensen and K. Mølmer, Phys. Rev. A 62 , 02231 (2000). G. J. Milburn, S. Schneider, and D. F. V. James, Fortschr. Physik 48 , 801 (2000). X.Wang, A. Sørensen, and K. Mølmer, Phys. Rev. Lett. 86 , 3907 (2001). n (2-photon stimulated-Raman For 2 ions transitions for hyperfine qubits) H ∝ σ x 1 σ x 2 n 1 n 1 n n n 1 n 1 n - dependence disappears when summing the paths (in Lamb-Dicke limit) n
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