Experimental Realization of Shor’s Quantum Factoring Algorithm‡ Outline The quantum limit The promise of Quantum Computation Classical vs. Quantum

Experimental Realization of Shor’s Quantum Factoring Algorithm‡ Outline The quantum limit The promise of Quantum Computation Classical vs. Quantum www.phwiki.com

Experimental Realization of Shor’s Quantum Factoring Algorithm‡ Outline The quantum limit The promise of Quantum Computation Classical vs. Quantum

Myers, Steve, Managing Editor has reference to this Academic Journal, PHwiki organized this Journal Experimental Realization of Shor’s Quantum Factoring Algorithm‡ ‡V in addition to ersypen L.M.K, et al, Nature, v.414, pp. 883 – 887 (2001) M. Steffen1,2,3, L.M.K. V in addition to ersypen1,2, G. Breyta1, C.S. Yannoni1, M. Sherwood1, I.L.Chuang1,3 1 IBM Almaden Research Center, San Jose, CA 95120 2 Stan as long as d University, Stan as long as d, CA 94305 3 MIT Media Laboratory, Cambridge, MA 02139 © http://www.spectrum.ieee.org/WEBONLY/resource/mar02/nquant.html © Outline Background Introduction to Quantum Computation Classical bits vs. Quantum bits Quantum Algorithms NMR Quantum Computation Shor’s algorithm in addition to Factoring 15 Shor’s Quantum circuit Molecule Results (spectra, innovations) Conclusions The quantum limit 1bit = 1 atom Can we exploit quantum mechanics as long as ultra-fast computation

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The promise of Quantum Computation Searching databases1 unsorted list of N entries how many queries 1 month 27 minutes Factoring Integers2 N = pq N has L digits given N, what are p in addition to q 10 billion years 3 years 400 digits [1] L.K. Grover, PRL, 79, 4709 (1997) [2] P. Shor, Proc. 35th Ann. Symp. On Found. Of Comp. Sci., p.124 (1994) Classical vs. Quantum Classical bits transistors 0 or 1 Quantum bits quantum systems 0 or 1or in-between NAND, NOT, CNOT NAND, NOT, CNOT Sqrt(NOT) These quantum gates allow operations that are impossible on classical computers! Quantum bits One qubit: Multiple qubits: Evolution of quantum states: Conservation of probabilities allows only reversible, unitary operations U is a 2n x 2n unitary matrix, i.e. UU†=I

Quantum bits cont’d Let a function f(x) (implemented by unitary trans as long as ms) act on an equal superposition: Parallel operation, BUT a measurement collapses the wave function to only one of the states with probability ai2 Need to design clever algorithms Quantum Algorithms Example: 2 qubit Grover search 1. Create equal superposition 2. Mark special element 3. Inversion about average One query = marking & inversion In general, need N queries Physical Realization of QCs Requirements as long as Quantum Computers1: A quantum system with qubits Individually addressable qubits Two qubit interactions (universal set of quantum gates) Long coherence times Initialize quantum system to known state Extract result from quantum system Meeting all of these requirements simultaneously presents a significant experimental challenge. Nuclear Magnetic Resonance (NMR) techniques largely satisfies these requirements in addition to have enabled experimental exploration of small-scale quantum computers [1] DiVincenzo D.P., Fortschr. Physik, 48 (9-11), 771 – 783 (2000)

NMR Quantum Computing 1,2 B0 spin ½ particle in magnetic field: 0 1 Characterize all Hamiltonians [1] Gershenfeld, N. et al., Science, 275, 350 – 356 (1997) [2] Cory D. et al., Proc. Natl. Acad. Sci., 94, 1634 – 1639 (1997) Multiple spin ½ nuclei Heteronuclear spins: Homonuclear spins: Chemical shift Spin-spin coupling Dipolar couplings (averaged away in liquids) J-coupling (through shared electrons) Lamour frequency of spin i shifts by –Jij/2 if spin j is in 0 in addition to by +Jij/2 if spin j is in 1

Single qubit rotations Radio-frequency (RF) pulses tuned to 0 0 1 Two qubit gates Lamour frequency of spin i shifts by –Jij/2 if spin j is in 0 in addition to by +Jij/2 if spin j is in 1 2-bit CNOT Put video here State initialization Thermal Equilibrium: highly mixed state Effective pure state: still mixed but: Spatial Labeling Temporal Labeling Logical Labeling Schulman-Vazirani a b c d a c d b a d b c 3a b+c+d b+c+d b+c+d

NMR Setup Shor’s Factoring Algorithm Quantum circuit to factor an integer N gcd(ar/2±1,N) m = log2(N) – n = 2m – ‘a’ is r in addition to omly chosen in addition to can’t have factors in common with N. The algorithm fails as long as N even or equal to a prime power (N=15 is smallest meaningful instance). Factoring 15 Challenging experiment: synthesis of suitable 7 qubit molecule requires interaction between almost all pairs of qubits coherent control over qubits sounds easy, but

Shor’s Factoring Algorithm where xk are the binary digits of x. a = 2, 7, 8, 13 a = 4, 11, 14 a4 mod 15 = 1 a2 mod 15 = 1 “hard case” “easy case” Three qubits in the first register are sufficient to factor 15. Factoring N = 15 a = 11 ‘easy case’ a = 7 ‘hard case’ mod exp QFT The molecule

Pulse Sequence Init. mod. exp. QFT ~ 300 RF pulses ~ 750 ms duration Experimental detail in addition to innovations Modified state initialization procedure Gaussian shaped /2 pulses (220 – 900 s) Hermite 180 shaped pulses (~ 2 ms) 4 channels, 7 spins: 6 spins always off resonance transient Bloch-Siegert shifts used technique as long as simultaneous soft pulses1 refocus T2 effects correct J-coupling during pulses [1] Steffen M., JMR, 146, 369 – 374 (2000) Results: Spectra qubit 3 qubit 2 qubit 1 Mixture of 0,2,4,6 23/2 = r = 4 gcd(74/2 ± 1, 15) = 3, 5 Mixture of 0,4 23/4 = r = 2 gcd(112/2 ± 1, 15) = 3, 5 15 = 3 · 5 a = 11 a = 7

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Results: Predictive Decoherence Model Generalized Amplitude Damping Operator sum representation: kEk Ek† Phase Damping Decoherence Model cont’d GAD ( in addition to PD) acting on different spins commute Ek as long as GAD commute with Ek as long as PD on arb. Pauli matrices PD commutes with J-coupling, in addition to z-rotations GAD ( in addition to PD) do NOT commute with RF pulses Pulse: time delay / GAD / PD / ideal pulse Results: Circuit Simplifications control of C is 0 control of F is 1 E in addition to H inconsequential to outcome targets of D in addition to G in computational basis ‘Peephole’ optimization

Conclusions First experimental demonstration of Shor’s factoring algorithm Developed predictive decoherence model Methods as long as circuit simplifications

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