From Quantum Gates to Quantum Learning: recent research and open problems in quantum circuits

1. From Quantum Gates to Quantum Learning:recent research and open problems in quantum circuits Marek A. Perkowski, Portland Quantum Logic Group, Department of Electrical Engineering and Computer Science, Korea Advanced Institute of Science and Technology, and Department of Electrical and Computer Engineering, Portland State University, USA.

2. The computer as we know it? 1999 Pentium IIIB www.icknowledge.com 1947 First point contact transistor by Bardeen and Brattain http://www.pbs.org/transistor/science/events/pointctrans.html

3. 1 meter 10 mm 1 mm 10 nm 1nanometer 0.1 nm 1 picometer 1 femtometer Size of red blood cell = a millionth of a meter Size of polio virus = a billionth of a meter Size of the hydrogen atom = a trillionth of a meter = 10 -15 m, size of a proton Nano-systemHow small is a nanometer?

4. History • 1970s and 1980s, introduction of quantum computers (Richard Feynmann, David Deutsch, and Paul Benioff) • 1994, Peter Shor’s factoring algorithm • 1996, Lov Grover, searching algorithm • 1998, 1999, 2001 Isaac L. Chuang, developed the world's first 2-qubit, 3-qubit, 5-qubit and 7-qubit quantum computer

5. People First Ideas…(1982)” Turing Machine …(1936)” A. Turing R. Feyemann “… Quantum Circuits…(1985)” “…Factorization …(1997)” D. Deutsch P. Shor

6. Number of Atoms in a Useful SystemFrom R. Keyes, IBM J. Res. Develop (1988)# atoms to store a bit # dopant atoms/bipolar transistor

7. EX: Quantum Parallelism • Quantum: • Put all 7-bits into a superposition state • superposition allows quantum computer to make calculations on all 128 possible numbers (27) in ONE iteration i.e. finishes in 1 second. • Tremendous possibilities… imagine doing computations on even larger sample spaces all at the same time!!!

8. Jiffy Quantum Theory Info unit: 1 bit. Physical system: 2 states • Quantum nature: a combination of both. • In preparing the initial state: only one of the 2 states • On measurement: only one state found. • Probability: the state’s component in the mix • Both preparation and measurement in contact with a macro system |1> |0> |0> and |1>

9. Qubits as binary Qudits • In multi-valued (MV) Quantum Computing (QC), the unit of memory (information) is qudit. • For instance, ternary logic values of 0, 1, and 2 are represented by a set of distinguishable different basis states of a qutrit. • These states can be a photon’s polarizations or an elementary particle’s spins. • After encoding these distinguishable quantities into multiple-valued values, qutrit states are represented by basis states |0>, |1> and |2> , respectively. • A qubit, used in binary QC uses only two basis states, |0> and |1> • Qubit and qutrit are then special cases of qudits

10. Qudits • Qudits exist in a linear superposition of states, and are characterized by a wave function . • As an example (), it is possible to have light polarizations other than purely horizontal or vertical, such as slant 45 corresponding to the linear superposition of . • In ternary logic, the notation for the superposition is , where , , and  are complex numbers. • These intermediate states cannot be distinguished, rather a measurement will yield that the qutrit is in one of the basis states, , , or . • The probability that a measurement of a qutrit yields state is , state is , and state is . • The sum of these probabilities is one. • The absolute values are required since, in general, ,  and γ are complex quantities. • Pairs of qutrits are capable of representing nine distinct states,, , , , , , , , and , as well as all possible superpositions of these states.

11. Quantum Logic Circuits

12. Quantum Logic Single photon Specchio 50% 1 0 50% Optical sensor

13. … strange behavior 0 1 0 1

14. Quantum Gate 0 1 0 1 1 0 1 0 NOT

15. Qubit

16. Qubit in a Ion Trap

17. Deterministic Turing Machine Initial State Final State Deterministic Turing Machine transits deterministically from initial to final state.

18. Probabilistic Turing Machine Probabilistic output states P4 Probabilities of final output states P5 P1 P6 P2 P = P2P7 + P3P8 P7 P3 P8 P9

19. Quantum Computation A = A1A2 + A3A4 P = |A1A2 + A3A4|2 = |A1A2 + A3A4|2 +2Re(A1A2A3A4) A1 A2 A3 A4

20. Decoherence

21. A beam-splitter The simplest explanation is that the beam-splitter acts as a classical coin-flip, randomly sending each photon one way or the other.

22. Quantum Interference The simplest explanation must be wrong, since it would predict a 50-50 distribution.

23. More experimental data

24. A new theory The particle can exist in a linear combination or superposition of the two paths

25. Probability Amplitude and Measurement If the photon is measured when it is in the state then we get with probability

26. Quantum Operations The operations are induced by the apparatus linearly, that is, if and then

27. Quantum Operations Any linear operation that takes states satisfying and maps them to states satisfying must be UNITARY

28. Linear Algebra corresponds to corresponds to corresponds to

29. Linear Algebra corresponds to corresponds to

30. Linear Algebra corresponds to

31. Linear Algebra is unitary if and only if

32. Abstraction The two position states of a photon in a Mach-Zehnder apparatus is just one example of a quantum bit or qubit Except when addressing a particular physical implementation, we will simply talk about “basis” states and and unitary operations like and

33. (b) (a) - - Re + |0> + (c) Im |1> |0> + |0> (d) - |1> |1>

34. cos - sin cos + sin |0> |1> (b) (a) (c) (d)

35. An arrangement like is represented with a network like

36. + + + + - - - - (a) cos - sin cos + sin (b)

37. (a) + + + + + + + + - - - - - - - - |0> |0> |00> |0> |00> |01> |1> |01> |1> |1> |0> |10> |10> |11> |1> |11> (b)

38. More than one qubit If we concatenate two qubits we have a 2-qubit system with 4 basis states and we can also describe the state as or by the vector

39. More than one qubit In general we can have arbitrary superpositions where there is no factorizationinto the tensor product of two independent qubits. These states are called entangled.

40. Measuring multi-qubit systems If we measure both bits of we get with probability

41. Classical Versus Quantum

42. Classical vs. Quantum Circuits • Goal: Fast, low-cost implementation of useful algorithms using standard components (gates) and design techniques • Classical Logic Circuits • Circuit behavior is governed implicitly by classical physics • Signal states are simple bit vectors, e.g. X = 01010111 • Operations are defined by Boolean Algebra • No restrictions exist on copying or measuring signals • Small well-defined sets of universal gate types, e.g. {NAND},{AND,OR,NOT}, {AND,NOT}, etc. • Well developed CAD methodologies exist • Circuits are easily implemented in fast, scalable and macroscopic technologies such as CMOS

43. Classical vs. Quantum Circuits • Quantum Logic Circuits • Circuit behavior is governed explicitly by quantum mechanics • Signal states are vectors interpreted as a superposition of binary “qubit” vectors with complex-number coefficients • Operations are defined by linear algebra over Hilbert Space and can be represented by unitary matrices with complex elements • Severe restrictions exist on copying and measuring signals • Many universal gate sets exist but the best types are not obvious • Circuits must use microscopic technologies that are slow, fragile, and not yet scalable, e.g., NMR

44. Quantum Circuit Characteristics • Unitary Operations • Gates and circuits must be reversible (information-lossless) • Number of output signal lines = Number of input signal lines • The circuit function must be a bijection, implying that output vectors are a permutation of the input vectors • Classical logic behavior can be represented by permutation matrices • Non-classical logic behavior can be represented including state sign (phase) and entanglement

45. Quantum Circuit Characteristics • Quantum Measurement • Measurement yields only one stateX of the superposed states • Measurement also makes X the new state and so interferes with computational processes • X is determined with some probability, implying uncertainty in the result • States cannot be copied (“cloned”), implying that signal fanout is not permitted • Environmental interference can cause a measurement-like state collapse (decoherence)

46. cn–1 a0 s0 b0 s1 a1 b1 s2 a2 b2 s3 a3 Sum b3 cn Carry Classical vs. Quantum Circuits Classical adder

47. Classical vs. Quantum Circuits Quantum adder

48. Reversible Circuits

49.       … … Generic Boolean Circuit … … …       f(i)       … n inputs m outputs Reversible Circuits • Reversibility was studied around 1980 motivated by power minimization considerations • Bennett, Toffoli et al. showed that any classical logic circuit C can be made reversible with modest overhead i i “Junk” Reversible Boolean Circuit f(i) “Junk”

50. a b c a b f 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 1 0 0 1 1 0 1 1 1 0 0 1 0 0 1 0 1 1 0 1 1 1 0 1 1 1 1 1 1 1 1 0 a a Reversible AND gate b b c f = ab c Reversible Circuits • How to make a given f reversible • Suppose f :i  f(i) has n inputs m outputs • Introduce n extra outputs and m extra inputs • Replace f by frev: i, j  i, f(i) j where  is XOR • Example 1: f(a,b) = AND(a,b) • This is the well-known Toffoli gate, which realizes AND when c = 0, and NAND when c = 1.