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 progress in classical computer technology has been dramatic Many researchers believe an even greater revolution is coming: quantum computers 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. Number of Atoms in a Useful SystemFrom R. Keyes, IBM J. Res. Develop (1988)# atoms to store a bit # dopant atoms/bipolar transistor

5. 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

6. People First Ideas (1982)” Turing Machine …(1936)” A. Turing R. Feynmann “… Quantum Circuits…(1985)” D. Deutsch P. Shorr “…Factorization …(1997)”

7. 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>

8. Qubit in a Ion Trap

9. Quantum Logic Circuits

10. How a single photon behaves in a beam splitter? Beam splitter 50% 1 Single photon 0 50% Optical sensor

11. … strange behavior 0 1 0 1 Mach-Zehnder apparatus

12. Quantum Gate: square root of NOT 0 1 0 1 1 0 1 0 NOT

13. Simple theory of the 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.

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

15. More experimental data collected to improve the theory

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

17. Probability Amplitude and Measurement If the photon is measured when it is in the state then we get with probability and |1 with probability

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

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

20. Linear Algebra notation for quantum circuits corresponds to corresponds to corresponds to

21. Linear Algebra notation for quantum circuits corresponds to corresponds to

22. Linear Algebra notation for quantum circuits corresponds to

23. What is unitary matrix? is unitary if and only if

24. 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

25. 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

26. - - Re + |0> + Im |1> Register-transfer notation for quantum circuits |0> + |0> - |1> |1>

27. cos - sin cos + sin |0> |1> Register-transfer notation for quantum circuits

28. From physical devices to abstracted quantum circuits An arrangement like is represented with a network like

29. + + + + - - - - cos - sin cos + sin Register-transfer notation for quantum circuits

30. Kronecker Product of Matrices • Superposition property may be mathematically described using the Kronecker product (tensor product) operation  • The Kronecker product of two matrices is defined as follows:

31. Register-transfer diagram for two Hadamard gates in parallel (a) + + + + + + + + - - - - - - - - |0> |0> |00> |0> |00> |01> |1> |01> |1> |1> |0> |10> |10> |11> |1> |11> (b)

32. Quantum Parallelism • 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!!!

33. Kronecker Products for more than one qubit circuits 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

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

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

36. Classical Versus Quantum

37. 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

38. Quantum Circuits are different • 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)

39. Decoherence

40. Classical versus 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

41. More 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

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

43. Classical vs. Quantum Circuits Quantum adder Feynman gate

44. Reversible Circuits

45.       … … 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”

46. 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 inputsm 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.

47. (Toffoli gate) Reversible Circuits • Reversible gate family [Toffoli 1980] • Every Boolean function has a reversible implementation using Toffoli gates. • There is no universal reversible gate with fewer than three inputs

48. Quantum Gates

49. NOT NOT NOT • One-Input gate: NOT • Input state: c0|0 + c1|1 • Output state: c1|0 + c0|1 • Pure states are mapped thus: |0  |1 and |1  |0 • Gate operator (matrix) is • As expected:

50. One-Input gate: “Square root of NOT” • Some matrix elements are imaginary • Gate operator (matrix): • We find: so |0  |0 with probability |i/2|2 = 1/2 and |0  |1 with probability |1/  2|2 = 1/2 Similarly, this gate randomizes input |1 • But concatenation of two gates eliminates the randomness!