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![Quantum Algorithms (I)
14
• Factoring, discrete log [Shor 94]
• Unstructured search [Grover 96]
• Various hidden subgroup problems [Long List]
• Pell’s equation [Hallgren 02]
• Hidden shift problems [van Dam, Hallgren, Ip 03]
• Graph traversal [CCDFGS 03]
• Spatial search [AA 03, CG 03/04, AKR 04]
• Element distinctness [Ambainis 03]
• Various graph problems [DHHM 04, MSS 03,…]
• Testing matrix multiplication [Buhrman,Špalek 04]
• Hidden subgroup problem [Bacon, Childs, van Dam 05]
• Certain hidden shift problems [Childs, van Dam 05]](https://image.slidesharecdn.com/bolhasaniqc20200120-200120084531/85/An-Introduction-to-Quantum-Computers-Architecture-14-320.jpg)
![Quantum Algorithms (I)
14
• Factoring, discrete log [Shor 94]
• Unstructured search [Grover 96]
• Various hidden subgroup problems [Long List]
• Pell’s equation [Hallgren 02]
• Hidden shift problems [van Dam, Hallgren, Ip 03]
• Graph traversal [CCDFGS 03]
• Spatial search [AA 03, CG 03/04, AKR 04]
• Element distinctness [Ambainis 03]
• Various graph problems [DHHM 04, MSS 03,…]
• Testing matrix multiplication [Buhrman,Špalek 04]
• Hidden subgroup problem [Bacon, Childs, van Dam 05]
• Certain hidden shift problems [Childs, van Dam 05]](https://image.slidesharecdn.com/bolhasaniqc20200120-200120084531/75/An-Introduction-to-Quantum-Computers-Architecture-14-2048.jpg)
Uploaded byHamidreza Bolhasani
An Introduction to Quantum Computers Architecture
This document provides a comprehensive overview of quantum computer architecture, detailing the history and development of quantum mechanics and its distinct features compared to classical mechanics. It explains key concepts such as qubits, quantum gates, and algorithms like Shor's and Grover's, highlighting their significance in advancing computational capabilities. Additionally, it addresses the challenges faced in quantum computing and future aspirations for improving qubit technology and operational efficiency.
In this document
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Overview of the presentation by Hamid Reza Bolhasani, covering topics from quantum history to key concepts.
Major contributions to quantum mechanics from Feynman, Deutsch, Shor, and Grover that shaped quantum computing.
Comparison of classical and quantum mechanics, focusing on behavior and properties of systems at different scales.
Discussion of the Bohr model of the atom and the concept of wave functions in quantum mechanics.
Comparison between traditional bits and qubits in quantum computing, including qubit operations.
Basics of quantum computation, operations of qubits, and introduction to quantum gates.
Overview of key quantum algorithms such as Shor's for factoring and Grover's for search problems.
Different architectures of quantum computers including ion traps, superconducting qubits, and photon-based systems.
Identifies significant challenges in quantum computing such as decoherence and operational control.
Explores potential advancements in quantum computing technology aiming for error reduction and operational improvements.
Wrap-up of presentation and contact information for further discussion.
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An Introduction to Quantum Computers Architecture
- 1. Quantum Computers Architecture Hamid Reza Bolhasani PhDStudent / Data Scientist Computer Architecture JAN 2020 1
- 2. Table of Contents -History - Classical vs Quantum Mechanics - Breakdown of Classical Mechanics - Quantum Mechanics Introduction - Wave Function - Quantum Bit (Qubit) - Quantum Gates - Quantum Computer - Challenges - Q & A
- 3. Richard Feynman “I thinkI can safely say that no one understand quantum mechanics!” 1982: Feynman proposed the first idea. 1985: Deutsch developed the quantum Turing Machine. 1994: Shor Algorithm to factor very large numbers in polynomial time. 1997: Grover proposed a quantum search algorithm. …Future! 3
- 4. Classical Mechanics vsQuantum Mechanics Mechanics: the study of the behavior of physical bodies when subjected to forces or displacements Classical Mechanics: describing the motion of macroscopic objects. Macroscopic: measurable or observable by naked eyes Quantum Mechanics: describing behavior of systems at atomic length scales and smaller . 4
- 5. Classical Mechanics 5 position r= (x,y,z) velocity v Property Behaviour mass momentum Particles position collisions velocity Waves wavelength diffraction frequency interference x m F k x = 0 k m(d2x/dt2) = kx +A A x time period = 1/ position x(t) = Asin(t) of particle frequency = /2 = (of oscillation) m π2 1 k
- 6. Breakdown of ClassicalMechanics 6 ePhotelectrons- h Metal surface work function = F e Photoelectrons ejected with kinetic energy: Ek = h - F a) Black Body Radiation b) Photoelectric Effect Max Planck (1900) Albert Einstein (1921)
- 7. The Bohr Modelof the Atom 7 E = E2 E1 = h h E1 E2 h E1 E2 Absorption Emission n2 n1 e p+ Niels Bohr, 1913 i s e p=h/s p=mev i s cos1λλΔλ is cm h e The Compton Effect (1923) Electron Diffraction(1925)
- 8. Partial Wave Duality 8 TwoSplit Experiment t tx jtxxV x tx m ),( ),()( ),( 2 2 22 Wave Function (x,y,z) = (r) = (r,,)
- 9. Bit vs QunatumBit (Qubit) 9 1 0 1 0 ≡ ≡ 1 1 0 0 0 1 2221 1211 S=(Sφ Sθ SR=const) • Two states classical bit • Equalities • Two levels quantum system (qubit) • Single qubit operations Polarization vector: Density matrix: // )0()( iHtiHt eet
- 10. Computation with Qubit(I) 10 0 1 1 0 Classical Computation Operations: logical Valid operations: AND = 0 i -i 0 1 0 0 -1 1 1 1 -1 0 1 0 1 0 0 0 1 NOT = 0 1 1 0 in out out in in 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 1-bit 2-bit Quantum Computation Operations: unitary Valid operations: σX = σy = σz = Hd = CNOT = √2 1 1-qubit 2-qubit
- 11. Computation with Qubit(II) 11 1 0 0 0 u11 u12 u21u22 Single qubit c1 c2 c1 c2 Two qubits H2 = 1 0 0 1, |0,|1 H2 2 = H2H2 = , |00,|01,|10,|11 0 1 0 0 , 0 0 1 0 , 0 0 0 1 c1 c2 c3 c4 c1 c2 c3 c4 u11 u12 u13 u14 u21 u22 u23 u24 u31 u32 u33 u34 u41 u42 u43 u44 Hilbert space U| = U| =Operator | = c1|0 + c2|1 = | c1|00 + c2|01 + c3|10 + c4|11 == Arbitrary state
- 12. Quantum Gates(I) 12 pure state→ mixed state Only 1 classical single-bit gate, but ∞single-qubit gates H² = 1 Hadamard Gate (H)
- 13.
- 14. Quantum Algorithms (I) 14 •Factoring, discrete log [Shor 94] • Unstructured search [Grover 96] • Various hidden subgroup problems [Long List] • Pell’s equation [Hallgren 02] • Hidden shift problems [van Dam, Hallgren, Ip 03] • Graph traversal [CCDFGS 03] • Spatial search [AA 03, CG 03/04, AKR 04] • Element distinctness [Ambainis 03] • Various graph problems [DHHM 04, MSS 03,…] • Testing matrix multiplication [Buhrman,Špalek 04] • Hidden subgroup problem [Bacon, Childs, van Dam 05] • Certain hidden shift problems [Childs, van Dam 05]
- 15. Quantum Algorithms (II) 15 Shor’sAlgorithm 18819881292060796383869723946165043 98071635633794173827007633564229888 59715234665485319060606504743045317 38801130339671619969232120573403187 9550656996221305168759307650257059 4727721461074353025362 2307197304822463291469 5302097116459852171130 520711256363590397527 3980750864240649373971 2550055038649119906436 2342526708406385189575 946388957261768583317 Best classical algorithm takes time Shor’s quantum algorithm takes time Peter Shor 1994
- 16. Quantum Algorithms (II) 16 Shor’sAlgorithm
- 17. Quantum Algorithms (III) 17 Grover’sAlgorithm (1996) n qubit 1qubit Suppose we have a black box with the property Problem: find with as few queries as possible.
- 18. Quantum Algorithms (V) 18 Grover’sAlgorithm Repeated application of the Grover iterate Grover’s algorithm: 1. start with 2. repeatedly apply Grover’s iterate to rotate to near
- 19. Quantum Algorithms (IV) 19 Grover’sAlgorithm We have identified marked item using only queries!𝑂 𝑛
- 20.
- 21. Quantum Computers Architecture(II) 21 • Ion traps and neutral atoms E0 E1 E2 • Superconducting qubit • Semiconductor charge qubit • Spin qubit Nuclear spin (liquid state NMR, solid state NMR) I Electron spin S SQUIDCooper pair box Double QD e 0 1 N pairs - 0 1N+1 pairs - Single QD F i E0 E1 e • Photon based QC P 0 1
- 22. Quantum Computers Architecture(III) 22 Trapped Ion motion head target pushing laser
- 23. Quantum Computers Architecture(IV) 23 Individual Photon A B |1 = |0A|1B + |1A|0B Quantum Entanglement! send single photons weak laser Ultraviolet () X c(2) nonlinear crystal
- 24. Quantum Computer Challenges 24 01 106 eV CLASSICAL |0 |1 10-6 eV QUANTUM - Dephasing - Decoherence - Control of Operations - Isolation from environment - Superpositions are very fragile!
- 25. QC Future Works… 25 -More Qubits: 64, 128, 192, 256, … - Much Greater Connectivity - Much Lower Error Rate - Much Longer Coherence - True Fault Tolerance - Much Lower Cost - Non-cryogenic Operating Temperature
- 26.