Quantum Computer

School of Computational Sciences Korea Institute for Advanced Study KIAS (Korea Institute for Advanced Study)

• Established in 1996 • Located in Seoul, Korea • Pure Theoretical Basic Science • Schools – Mathematics – Physics – Computational Sciences • By the computation, For the computation • 20 Prof’s, 2 Distinguished Prof’s, 20 KIAS Scholars, 70 Research Fellows, 20 Staff Members Quantum Information Physics Science

Quantum Information Science Quantum Parallelism à Quantum computing is exponentially larger and faster than digital computing. Quantum Fourier Transform, Quantum Database Search, Quantum Many-Body Simulation (Nanotechnology)

No Clonability of Quantum Information, Irrevesability of Quantum Measurement à Quantum Cryptography (Absolutely secure digital communication)

Quantum Correlation by Quantum Entanglement à Quantum Teleportation, Quantum Superdense Coding, Quantum Cryptography, Quantum Imaging Twenty Questions

• Animal … • Four legs? Yes(1) or No(0) • Herbivorous? Yes(1) or No(0) . Yes(1) or No(0) . . . . • Tiger? It from Bit

John A. Wheeler Yoot =Ancient Korean Binary Dice

x x x x x x x x x

1 0 1 1

Taegeukgi (Korean National Flag)

0 Qubit j =+ab01

Qubit = quantum bit (binary digit) Businessweek 21 Ideas for the 21st Century Transition to Quantum

• Mathematics: Real à Complex • Physics: Classical à Quantum • Digital Information Processing – Hardwares by Quantum Mechanics – Softwares and Operating Systems by Classical Ideas • Quantum Information Processing – All by Quantum Ideas Quantum Information Processing

• Quantum Computer NT – Quantum Algorithms: Softwares • Simulation of quantum many-body systems • Factoring large integers • Database search IT – Experiments: Hardwares • Ion Traps BT • NMR • Cavity QED, etc. Quantum Information Processing • Quantum Communication – Quantum Cryptography IT • Absolutely secure digital communication • Generation and Distribution of Quantum Key – ~100 km through optical fiber (Toshiba) – 23.4 km wirelessà secure satellite communication – Quantum Teleportation • Photons • Atoms, Molecules • Quantum Imaging and Quantum Metrology Cryptography and QIP Giving disease, -병 주고 약 주고- Giving medicine. Out with the old, In with the new. • Public Key Cryptosystem (Asymmetric) – Computationally Secure – Based on unproven mathematical conjectures – Cursed by Quantum Computation

• One-Time Pad (Symmetric) – Unconditionally Secure – Impractical – Saved by Quantum Cryptography KIAS-ETRI, December 2005. 25 km Quantum Cryptography T.G. Noh and JK Classical Computation

• Hilbert (1900): 23 most challenging math problems – The Last One: Is there a mechanical procedure by which the truth or falsity of any mathematical conjecture could be decided? • Turing – Conjecture ~ Sequence of 0’s and 1’s – Read/Write Head: Logic Gates – Model of Modern Computers Turing Machine

Finite State Machine: Head q¢ ìq¢ = f(,)qs ï ís¢ = gqs(,) q ï îd= dqs(,)

d Bit {0,1} ss® '

Infinitely long tape: Storage Bit and Logic Gate

0à1 “Universal” 1à0 NOT NAND 00à0 01à0 “Reversible” 10à0 AND 11à1 C-NOT 00à0 01à1 10à1 OR 11à1 CCN ( Toffoli ) “Universal” 00à0 “Reversible” 01à1 10à1 XOR 11à0 C-Exch ( Fredkin ) DNA Computing Adleman • Bit { 0, 1 } è Tetrit (?) { A, G, T, C} • Gate è Enzyme

• Parallel Ensemble Computation – Hamiltonian Path Problem Complexity N= (# bits to describe the problem, size of the problem)

(#steps to solve the problem) = Pol(N) à “P(polynomial)”: Tractable, easy

(#steps to verify the solution) = Pol(N) à “NP (nondeterministic polynomial)”: Intractable

NPÉP NP NP¹P or P NP=P? Quantum Information

• Bit {0,1} 2 2 è Qubit a0+b1 with ab+=1 • N bits è 2N states, One at a time Linearly parallel computing AT BEST • N qubits è Linear superposition of 2N states at the same time Exponentially parallel computing è Quantum Parallelism Deutsch But when you extract result, you cannot get all of them. Quantum Algorithms

1. [Feynman] Simulation of Quantum Physical Systems with HUGE Hilber space ( 2N-D ) e.g. Strongly Correlated Electron Systems

2. [Peter Shor] Factoring large integers, period finding 1/3 tq ∝ Pol (N) tcl ∼ Exp (N )

3. [Grover] Searching

tq ∝ÖN átclñ ~ N/2 Digital Computer N bits 1× N N bits Digital Computers in parallel N bits m{ mN× N bits

Quantum Computer : Quantum Parallelism

N bits 2N N bits Quantum Gates Time-dependent Schrödinger Eq. Unitary Transform ¶ iHh yy= Norm Preserving ¶t Reversible y(t)==e-iHt / h yy(0)Ut()(0)

æö10 æ10öæö01 P(q )== IX = ç÷iq ç÷ç÷ èø0 e è01øèø10

æ10öæ0-1ö1 æö11 Z=P(p )=ç÷ Y= XZH=ç÷ = ç÷ è0--1øè10ø2 èø11

1æ11öæ11ö11æö H 0=ç÷ç÷==+ç÷ ( 01) Hadamard 2è1-1øè01ø22èø 1æ11öæ01ö11æö H 1=ç÷ç÷=ç÷=-( 01) 2è1--1øè11ø22èø Hadamard Gate

H12ÄHHÄ...ÄN 012Ä0ÄÄ...0N 111 =( 0+1) Ä( 0+1) Ä+...01( ) 2112222 NN 1 =( 0012...0N+1102...0NN++...1121...1 ) 2N N-1 1 2 = k()binary expression N å 2 k=0 Universal Quantum Gates General Rotation of a Single Qubit æöæqqö-if æö ç÷cosç÷-ie sinç÷ è22øèø V(,)qf =ç÷V ç÷+if æqqöæö ç÷-ie sinç÷cosç÷ èøè22øèø

Xc: CNOT (controlled - NOT) or XOR æ10öæ10öæ00öæö01 ç÷Äç÷ +Ä ç÷ç÷ è00øè01øè01øèø10 = 00ÄIX+Ä11

Xac b=Åaab Quantum Circuit/Network DiVincenzo, Qu-Ph/0002077 Scalable Qubits Unitary evolution : Deterministic : Reversible Initial State Universal Gates

|0ñ X M … X H CX |Yñ |Yñ |0ñ H 1 2 13 M |0ñ t M

Quantum measurement Cohere, : Probabilistic Not Decohere : Irreversible change √ Quantum Key Distribution Single-Qubit Gates : 3 [BB84,B92]

QKD[E91] √ Single- Quantum Repeater : 3 & Two-Qubit Gates Quantum Teleportation√

Quantum Error Correction Single- ³ 40 Quantum Computer & Two-Qubit Gates 7-Qubit QC√ ³ 100 Physical systems actively considered for quantum computer implementation

• Liquid-state NMR • Electrons on liquid He • NMR spin lattices • Small Josephson junctions • Linear ion-trap – “charge”qubits spectroscopy – “flux”qubits • Neutral-atom optical • Spin spectroscopies, lattices impurities in semiconductors • Cavity QED + atoms • Coupled quantum dots • Linear optics with – Qubits: spin,charge, single photons excitons • Nitrogen vacancies in – Exchange coupled, diamond cavity coupled 15=´35 Chuang Nature 414, 883-887 (20/27 Dec 2001) OR QP/0112176 Concept device: spin-resonance transistor R. Vrijen et al, Phys. Rev. A 62, 012306 (2000) Quantum-dot array proposal Ion Traps • Couple lowest centre-of-mass modes to internal electronic states of N ions. Quantum Error Correcting Code Three Bit Code

Encode Recover Decode f channel f

0 Um1m2 0 noise 0 M 0 M Molding a Quantum State

|0ñ X M … X H CX |Yñ |Yñ |0ñ H 1 2 13 M |0ñ t M

Molding Sculpturing a Quantum State - Cluster state quantum computing – - One-way quantum computing --

|+ñ |+ñ |+ñ |+ñ |+ñ |+ñ

1. Initialize each qubit in |+ñ state. 2. Contolled-Phase between the neighboring qubits. 3. Single qubit manipulations and single qubit measurements only [Sculpturing]. No two qubit operations! 00à00 01à01 10à11 XOR 11à10 X=00ÄIX+Ä11 ABBBAAAA Controlled-NOT é10ùé10ùé00ùéù01 =Ä+Äêúêúêúêú ë00ûABABë01ûë01ûëû10 é1000ùéù0000 ê0100úêú0000 =+êúêú ê0000úêú0001 êúêú 00000010 º ëûABëûAB X éù1000 êú0100 = êú êú0001 êú ëû0010AB Qubit Copying Circuit?

x x y =+ab01 x x y =+ab01 ab00+ 11 0 y =+ab01 0 y xyÅ x Entanglement by Two-Qubit Gates

a |0ñ + b |1ñ a |0ñ|0ñ + b |1ñ|1ñ |0ñ

a |0ñ + b |1ñ a |0ñ|0ñ|0ñ + b |1ñ|1ñ|1ñ |0ñ |0ñ Single Particle Entanglement

Single Electron ++ H=-+tc( 1c2cc21) ① ② 1 y =+( 12) 2 1 or y =+(fe12ef12) 2 Single Photon Beam Splitter ②

1 f =+( 1001) ① 2 Quantum Teleportation using a single particle entanglement

Lee, JK, Qu-ph/0007106; Phys. Rev. A 63, 012305 (2001) No Cloning Theorem An Unknown Quantum State Cannot Be Cloned. Zurek, Wootters U ( a0 ) = aa U ( b0) =¹bb ab 1 Let g=+( ab). 2 1 Then U ( g0) =( aa+¹bb) gg 2 “IfIf an unknow quantum state can be cloned … ”• Quantum States can be measured as accurately as possible ??? |yñ Þ |yñ , |yñ , |yñ , |yñ … measure, measure, …

• Commnunication Faster Than Light? |yñ = |0ñA |1ñB - |1ñA |0ñB for “0” = |+ñA |-ñB - |-ñA |+ñB for “1” Communication Faster Than Light? “If”an unknown quantum state can be copied;

|yñ = |0ñA |1ñB - |1ñA |0ñB for “0” = |+ñA |-ñB - |-ñA |+ñB for “1”

• Alice wants to send Bob “1.” |yñ = |+ñA |-ñB - |-ñA |+ñB for “1” Alice mesures her qubit in {|+>,|->}. • Alice’s state will become |+> or |->. • Bob’s state will become |-> or |+>. Let’s assume it is |+>. |+ñ • Bob makes many copies of this. |+ñ, |+ñ, |+ñ, |+ñ, … He measures them in {|+>, |->}, and gets 100% |+>. He measures them in {|0>, |1>}, and gets 50% |0> and 50% |1>.

Thus Bob can conclude that Alice measured her state in {|+>, |->}. Mysterious Connection Between QM & Relativity • Weinberg: Can QM be nonlinear? • Experiments: Not so positive result. • Polchinski, Gisin: If QM is nonlinear, communication faster than light is possible. Cluster State 1. Qubits on lattice sites are initialized to be |+ñ. 01+ +==H 0 2 2. Operate Control-Z on neighboring qubits. Cont-Z =00ÄIX+Ä11 æ10öæ100öæ0öæö01 =Ä+Äç÷ç÷ç÷ç÷ è00øè01øè011øè 0ø æö1000 ç÷0100 = ç÷ ç÷0010 ç÷ è0001- ø 0i æö1000 æöe 000 ç÷ç÷0i 0100 ç÷0e 00 Cont-Z ==ç÷ 0i ç÷0 010 ç÷000e ç ÷ ç÷-pi èø000 -1 èø000e

HI=+ss12zzz -+ss12z g æ10öæ10öæ10öæö10 =ç÷Ä+II-Ä-Äç÷ç÷ç÷I è0-1øè0-1øè0--1øèø01

æö111+--1000æö0000 ç÷0111-+-100ç÷0000 = ç÷= ç÷ ç 00-++-11110÷ç÷0000 ç ÷ç÷ èø0 001----111 èø0004-

p i H Cont-Z = e 4 |0ñ |0ñ |0ñ |0ñ |0ñ |0ñ

|0ñ |0ñ |0ñ |0ñ |0ñ |0ñ

|0ñ |0ñ |0ñ |0ñ |0ñ |0ñ |+ñ |+ñ |+ñ |+ñ |+ñ |+ñ

|+ñ |+ñ |+ñ |+ñ |+ñ |+ñ

|+ñ |+ñ |+ñ |+ñ |+ñ |+ñ |+ñ |+ñ |+ñ |+ñ |+ñ |+ñ

|+ñ |+ñ |+ñ |+ñ |+ñ |+ñ

|+ñ |+ñ |+ñ |+ñ |+ñ |+ñ One-way quantum computing

|+ñ |+ñ |+ñ |+ñ |+ñ |+ñ

|+ñ |+ñ |+ñ |+ñ |+ñ |+ñ Optical Lattice, Quantum Dots, Superconductors, etc. Single qubit manipulations and single qubit measurements only [Sculpturing]. No two qubit operations! Quantum computing with quantum-dot cellular automata Géza Tóth and Craig S. Lent PRA 63, 052315

1-D array Some comments

• 1-D cluster state quantum computing can be efficiently simulated by digital computing. • 2-D cluster state quantum computing is equivalent to quantum circuit quantum computing, but needs more(just polynomially more) qubits. “Proposal”: 2-D array of vertical QDs • 2-D cluster state does not seem to be a ground state of some Hamiltonian. • Once the cluster state is prepared, only single qubit measurements are needed. QUIPU [kí:pu]

1. Quantum Information Processing Unit [JK] 2. A device consisting of a cord with knotted strings of various colors attached, used by the ancient Peruvians for recording events, keeping accounts, etc.

Computation Communication

* Peruvian Information Processing Unit for Computation and Communication

*Rosary *Buddhist Beads