1. Introduction to Quantum Computation Neil Shenvi Department of Chemistry Yale University

2. Talk Outline Background What is Quantum Computation? Quantum Algorithms Decoherence and Noise Implementations Applications Quantum Random Walks O Noise in Grover’s Algorithm Decoherence in Spin Systems

3. Background: Classical Computation C:\Hello.exe Hello World! InputComputationOutput What is the essence of computation? 2 + 2 4

4. Classical Computation Theory Church-Turing Thesis: Computation is anything that can be done by a Turing machine. This definition coincides with our intuitive ideas of computation: addition, multiplication, binary logic, etc… What is a Turing machine? …0100101101010010110… Infinite tape Read/Write head Finite State Automaton (control module) …0000001011111111100… Computation …1110010110100111101… Output …0100101101010010110… Input

5. Classical Complexity P NP NP-complete Some problems are more difficult than others. Polynomial hierarchy Require polynomial time to solve Require exponential time to solve All Turing machine-equivalent computers have an identical hierarchy. Require exponential(?) time to solve

6. Classical Complexity P NP NP-complete Some important problems do not have known classical polynomial algorithm and or a known place in the hierarchy. Polynomial hierarchy Factoring Graph Isomorphism ? ? Best known algorithm to factor N-digit number: Time ~ Exp(N 1/3 ) Best known algorithm to compare two N-node graphs: Time ~ Exp(N)

7. Classical Computation Theory What kind of systems can perform universal computation? Desktop computers Billiard balls DNA Cellular automata These can all be shown to be equivalent to each other and to a Turing machine! The Big Question: What next?

8. Talk Outline Background What is Quantum Computation? Quantum Algorithms Decoherence and Noise Implementations Applications

9. What Is Quantum Computation? Conventional computers, no matter how exotic, all obey the laws of classical physics. On the other hand, a quantum computer obeys the laws of quantum physics.

10. The Bit The basic component of a classical computer is the bit, a single binary variable of value 0 or 1. 1 0 0 1 The state of a classical computer is described by some long bit string of 0s and 1s. 0001010110110101000100110101110110... At any given time, the value of a bit is either ‘0’ or ‘1’.

11. The Qubit A quantum bit, or qubit, is a two-state system which obeys the laws of quantum mechanics. =|1  =|0  Valid qubit states: |  = |0  |  = |1  |  = (|0  - e i  /4 |1  )/  2 |  = (2|0  - 3e i5  /6 |1  )/  13 Spin-½ particle The state of a qubit |  can be thought of as a vector in a two-dimensional Hilbert Space, H 2, spanned by the Basis vectors |0  and |1 .

12. Computation with Qubits How does the use of qubits affect computation? Classical Computation Data unit: bit x = 0 x = 1 0 1 0 1 Valid states: x = ‘0’ or ‘1’ |  = c 1 |0  + c 2 |1  Quantum Computation Data unit: qubit Valid states: |  = |0  |  = |1  |  = (|0  + |1  )/√2 =|1  =|0  = ‘1’ = ‘0’

13. Computation with Qubits 01 10 How does the use of qubits affect computation? Classical Computation Operations: logical Valid operations: AND = 0i -i0 10 0 11 1 0 1 0101 00 01 NOT = 0 1 10 in out in 1000 0100 0001 0010 1-bit 2-bit Quantum Computation Operations: unitary Valid operations: σ X = σ y = σ z = H d = CNOT = √2 1 1-qubit 2-qubit

14. Computation with Qubits How does the use of qubits affect computation? Classical Computation Measurement: deterministic x = ‘0’ State Result of measurement ‘0’ x = ‘1’ ‘1’ Quantum Computation Measurement: stochastic |  = |0  |  = |0  - |1  State Result of measurement |  = |1  22 ‘0’ ‘1’ ‘0’ 50% ‘1’ 50%

15. More than one qubit 1 0 0 0 u 11 u 12 u 21 u 22 Single qubit c1c1 c2c2 c1c1 c2c2 Two qubits H 2 = 1 0 0 1, |0,|1|0,|1 H 2  2 = H 2  H 2 =, |00 ,|01 ,|10 ,|11  0 1 0 0, 0 0 1 0, 0 0 0 1 c1c1 c2c2 c3c3 c4c4 c1c1 c2c2 c3c3 c4c4 u 11 u 12 u 13 u 14 u 21 u 22 u 23 u 24 u 31 u 32 u 33 u 34 u 41 u 42 u 43 u 44 Hilbert space U|  = U|  = Operator |  = c 1 |0  + c 2 |1  = |  c 1 |00  + c 2 |01  + c 3 |10  + c 4 |11  = = Arbitrary state

16. Quantum Circuit Model 1 0 0 0 0010 0001 1000 0100 σ x  I = 0 0 1 0 1000 0100 0001 0010 CNOT = 0 0 0 1 0 0 0 1 |0|0 |0|0 |1|1 |0|0 |1|1 |1|1 ‘1’ Example Circuit σxσx One-qubit operation CNOT Two-qubit operation Measurement

17. Quantum Circuit Model 1/√2 0 0 1 0 0 0 σxσx CNOT |0  + |1  |0|0 Example Circuit √2 ______ 1/√2 0 0 0 0 0 0 0 1 |0  + |1  |0|0 √2 ______ ‘0’ or ‘1’ or 50% Separable state: can be written as tensor product |  = |   |  Entangled state: cannot be written as tensor product |  ≠ |   |  ? ?

18. Some Interesting Consequences Quantum Superordinacy All classical quantum computations can be performed by a quantum computer. U No cloning theorem It is impossible to exactly copy an unknown quantum state |  |0|0 Reversibility Since quantum mechanics is reversible (dynamics are unitary), quantum computation is reversible. |00000000  |  |00000000 

19. Talk Outline Background What is Quantum Computation? Quantum Algorithms Decoherence and Noise Implementations Applications

20. Quantum Algorithms: What can quantum computers do? Grover’s search algorithm Quantum random walk search algorithm Shor’s Factoring Algorithm

21. Grover’s Search Algorithm Imagine we are looking for the solution to a problem with N possible solutions. We have a black box (or ``oracle”) that can check whether a given answer is correct. 78 Question: I’m thinking of a number between 1 and 100. What is it? Oracle No 3 Oracle Yes

22. Grover’s Search Algorithm The best a classical computer can do on average is N/2 queries. 1 Oracle No... 2 Oracle No 3 Oracle Yes Classical computer Oracle 1+2+3+...No+No+Yes+No+... Quantum computer Using Grover’s algorithm, a quantum computer can find the answer in  N queries! Superposition over all N possible inputs.

23. Grover’s Search Algorithm Pros: Can be used on any unstructured search problem, even NP-complete problems. Cons: Only a quadratic speed-up over classical search. The circuit is not complicated, but it doesn’t provide an immediately intuitive picture of how the algorithm works. Are there any more intuitive models for quantum search? O σzσz O σzσz … … … … |0|0 |0|0 |0|0 O(  N) iterations HdHd HdHd HdHd … HdHd HdHd HdHd … HdHd HdHd HdHd … HdHd HdHd HdHd … HdHd HdHd HdHd

24. Quantum Random Walk Search Algorithm Idea: extend classical random walk formalism to quantum mechanics Classical random walk:

25. Quantum Random Walk Search Algorithm To obtain a search algorithm, we use our “black box” to apply a different type of coin operator, C 1, at the marked node C0C0 C1C1 1 1 1 1 C0=C0= 1 2 C1=C1= 0 0 0 0 0 0 0 0 0 0 0 0

26. Quantum Random Walk Search Algorithm Pros: As general as Grover’s search algorithm. Cons: Same complexity as Grover’s search algorithm. Slightly more complicated in implementation Slightly more memory used Interesting Feature: Search algorithm flows naturally out of random walk formalism. Motivation for new QRW- based algorithms?

27. Shor’s Factoring Algorithm Find the factors of: 57 3 x 19 Find the factors of: 16238476016501762387610762691722612171239872103974621876187 12073623846129873982634897121861102379691863198276319276121 whimper All known algorithms for factoring an n-bit number on a classical computer take time proportional to O(n!). But Shor’s algorithm for factoring on a quantum computer takes time proportional to O(n 2 log n). Makes use of quantum Fourier Transform, which is exponentially faster than classical FFT.

28. # bits102420484096 factoring in 200610 5 years5x10 15 years3x10 29 years factoring in 202438 years10 12 years7x10 25 years factoring in 20423 days3x10 8 years2x10 22 years with a classical computer # bits102420484096 # qubits 51241024420484 # gates3x10 9 2X10 11 X10 12 factoring time4.5 min36 min4.8 hours with potential quantum computer (e.g., clock speed 100 MHz) R. J. Hughes, LA-UR-97-4986 Shor’s Factoring Algorithm The details of Shor’s factoring algorithm are more complicated than Grover’s search algorithm, but the results are clear:

29. Talk Outline Background What is Quantum Computation? Quantum Algorithms Decoherence and Noise Implementations Applications

30. Decoherence and Noise What happens to a qubit when it interacts with an environment? Quantum computer Environment V Quantum information is lost through decoherence. σ1σ1 σ2σ2 σ3σ3 σNσN …

31. Types of Decoherence T 1 processes: longitudinal relaxation, energy is lost to the environment V T 2 processes: transverse relaxation, system becomes entangled with the environment V + + What are the effects of decoherence?

32. Effects of Environment on Quantum Memory Fidelity of stored information decays with time. T 1 – timescale of longitudinal relaxation T 2 – timescale of transverse relaxation

33. Effects of Environment on Quantum Algorithms Errors accumulate, lowering success rate of algorithm Grover’s algorithm success rate n = # of qubits O O Ideal oracle Noisy oracle

34. Suppressing Decoherence 1. Remove or reduce V, i.e. build a better computer System isolated from environment 2. Increase B, i.e. increase level splitting B E |0|0 |1|1 When  E >> V, decoherence is small EE 3.Use decoherence free subspace (DFS) 4. Use pulse sequence to remove decoherence

35. Talk Outline Background What is Quantum Computation? Quantum Algorithms Decoherence and Noise Implementations Applications

36. Some Proposed Implementations for QC NMR B Ion trap Optical Lattice Kane Proposal

37. The Loss-Divincenzo Proposal D. Loss and D.P. DiVincenzo, Phys. Rev. A 57, 120 (1998); G. Burkhard, H.A. Engel, and D. Loss, Fortschr. der Physik 48, 965 (2000).

38. Solid State Electron Spin Qubit Silicon lattice Phosphorus impurity Electron wavefunction Si 28 (no spin) Si 29 (spin ½) External Magnetic Field, B Hyperfine coupling Dipolar coupling

39. System Hamiltonian Electron spin N nuclear spins Hyperfine couplingDipolar coupling ~10 5 Hz~10 2 Hz ~10 7 Hz / T ~10 11 Hz / T

40. Hyperfine-Induced Longitudinal Decay For B > B c, T 1 is infinite Critical field for electron spin relaxation:

41. Hyperfine-Induced Transverse Decay Free evolution Spin echo pulse sequence Spin echo pulse sequence removes nearly all dephasing!

42. Talk Outline Background What is Quantum Computation? Quantum Algorithms Decoherence and Noise Implementations Applications

43. Factoring – RSA encryption Quantum simulation Spin-off technology – spintronics, quantum cryptography Spin-off theory – complexity theory, DMRG theory, N-representability theory

44. Acknowledgements Dr. Julia Kempe, Dr. Ken Brown, Sabrina Leslie, Dr. Rogerio de Sousa Dr. K. Birgitta Whaley Dr. Christina Shenvi Dr. John Tully and the Tully Group