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 1999 Pentium IIIB www.icknowledge.com http://www.pbs.org/transistor/scie nce/events/pointctrans.html 1947 First point contact transistor by Bardeen and Brattain Many researchers believe an even greater revolution is coming: quantum computers

3. Nano-system How small is a nanometer? 1 meter 10  m 1  m 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

4. Number of Atoms in a Useful System From 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 Turing Machine …(1936)” “… Quantum Circuits…(1985)” First Ideas (1982) ” “…Factorization …(1997)” A. Turing R. Feynmann D. Deutsch P. Shorr

7. Jiffy Quantum Theory 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 |0> |1> |0> and |1> Info unit: 1 bit. Physical system: 2 states

8. Qubit in a Ion Trap

9. Classical Versus Quantum

10. 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 Classical vs. Quantum Circuits

11. Quantum Measurement –Measurement yields only one state X 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) Quantum Circuits are different

12. Decoherence

13. 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 Classical versus Quantum Circuits

14. 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 More Quantum Circuit Characteristics

15. Classical vs. Quantum Circuits Classical adder

16. Classical vs. Quantum Circuits Quantum adder Feynman gate

17. Reversible Circuits

18. 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 … …    n inputs Generic Boolean Circuit m outputs    f(i) i …    Reversible Boolean Circuit …    f(i) … … “Junk” i      

19. 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 f rev : 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. Reversible Circuits Reversible AND gate a b f = ab  c a b c a b c a b f 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1 1 1 1 1 1 0

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

21. Quantum Gates

22. One-Input gate: NOT One-Input gate: NOT –Input state: c 0  + c 1  –Output state: c 1  + c 0  –Pure states are mapped thus:    and    –Gate operator (matrix) is –As expected: NOT

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

24. Quantum Gates One-Input gate: Hadamard –Maps   1/  2  + 1/  2  and   1/  2  – 1/  2 . –Ignoring the normalization factor 1/  2, we can write x x x–x  x   (-1) x  x  –  –  x  One-Input gate: Phase shift H 

25. Requirement: Hadamard and phase-shift gates form a universal gate set Example: The following circuit generates  = cos   0  + e i  sin   1  up to a global factor Universal One-Input Quantum Gate Sets U  Any state    22 HH

26. Quantum XOR gate Called also Feynman gate or Controlled Not gate. This gate allows inputs of |00> and |01> to appear unchanged at the outputs, but interchanges the pairs |10> and |11>. For example, consider the quantum XOR gate’s operation for an input |10>. |00> |01> |10> |11>

27. xx yy xx  x  y  CNOT –CNOT maps  x    x  x  and  x    x  NOT x  –  x    x  x  looks like cloning, but it’s not. –These mappings are valid only for the pure states  and  xx yy xx  x  y  Quantum XOR gate

28. bb cc bb  ab  c  aaaa |000> |001> |110> |010> |111> |011> |010> |011> |100> |101> |100> |101> |110> |111> 3-Input gate: Controlled CNOT (C2NOT or Toffoli gate)

29. General controlled gates that control some 1-qubit unitary operation U are useful Controlled Quantum Gates U C(U) U C2(U)C2(U) U U etc.

30. Universal Gate Sets To implement any unitary operation on n qubits exactly requires an infinite number of gate types The (infinite) set of all 2-input gates is universal –Any n-qubit unitary operation can be implemented using  (n 3 4 n ) gates [Reck et al. 1994] CNOT and the (infinite) set of all 1-qubit gates is universal Quantum Gates

31. Discrete Universal Gate Sets The error on implementing U by V is defined as If U can be implemented by K gates, we can simulate U with a total error less than  with a gate overhead that is polynomial in log(K/  ) A discrete set of gate types G is universal, if we can approximate any U to within any  > 0 using a sequence of gates from G Quantum Gates

32. Discrete Universal Gate Set Example 1: Four-member “standard” gate set Quantum Gates HS  /8 CNOT Hadamard Phase  /8 (T) gate Example 2: {CNOT, Hadamard, Phase, Toffoli}

33. Quantum Circuits Simulation

34. A quantum (combinational) circuit is a sequence of quantum gates, linked by “wires” The circuit has fixed “width” corresponding to the number of qubits being processed Logic design (classical and quantum) attempts to find circuit structures for needed operations that are –Functionally correct –Independent of physical technology –Low-cost, e.g., use the minimum number of qubits or gates Quantum logic design is not well developed! Quantum Circuits

35. Ad hoc designs known for many specific functions and gates Example 1 illustrating a theorem by [Barenco et al. 1995]: Any C 2 (U) gate can be built from CNOTs, C(V), and C(V † ) gates, where V 2 = U Quantum Circuits VV†V† V = U

36. Simulation of Quantum Circuits |0  |1  |x  |0  |1  |x  |0  |1  |x  VV†V† V = U |0  |1  V|x  |0  |1  |0  |1  |x  |0  |1  |0  |1  |x  ?

37. |1  |x  |1  |x  |1  U|x  VV†V† V = U |1  V|x  |1  |0  |1  |0  V|x  |1  U|x  Simulation continued ? Simulation of Quantum Circuits

38. Analysis of Quantum Circuits based on Unitary Matrices

39. Algebraic Analysis of Quantum Circuits U4U4 U2U2 U3U3 U1U1 U5U5 U0U0 VV†V† V = U ? x1x2x3x1x2x3 Is U 0 (x 1, x 2, x 3 ) = U 5 U 4 U 3 U 2 U 1 (x 1, x 2, x 3 ) = (x 1, x 2, x 1 x 2  U (x 3 ) ) ?

40. Quantum Circuits Example 1 (contd);

41. Quantum Circuits Example 1 (contd);

42. Quantum Circuits Example 1 (contd); –U 5 is the same as U 1 but has x 1 and x 2 permuted (tricky!) –It remains to evaluate the product of five 8 x 8 matrices U 5 U 4 U 3 U 2 U 1 using the fact that VV † = I and VV = U

43. Quantum Circuits Implementing a Half Adder –Problem: Implement the classical functions sum = x 1  x 0 and carry = x 1 x 0 Generic design: |x 1  U add |x 0  |y 1  |y 0  |x 1  |x 0  |y 1   carry |y 0   sum

44. Quantum Circuits Half Adder : Generic design (contd.)

45. Quantum Circuits Half Adder : Specific (reduced) design |x 1  |x 0  |y  |x 1  |y   carry sum C 2 NOT (Toffoli) CNOT

46. 1.Walsh Transform using classical logic circuits is expensive, 2.we need many adders and subtractors. NOT 1.Walsh Transform using quantum logic circuits is NOT expensive, 2.we need only quantum Hadamard gates, 3.each gate costs only two pulses 1.We can go from standard space to spectral space, back to standard space and so on many times. 2.This would be very expensive in classical spectral- based logic circuits. 3.These methods can be used for filtering.

47. Walsh Transform for two binary-input many- valued variables + - + - + - + - You need a butterfly H Classical logic Quantum logic H Butterfly is created automatically by tensor product corresponding to superposition minterms Variable 1

48. Tremendous potential for truly innovative research

49. Potential new research areas in quantum circuits

50. Research Potential Our community should develop a systematic methodology and CAD tools for synthesizing, verifying, testing and simulating of quantum computers. These methods and tools will be counterparts of what exists now in binary CMOS. –Re-use spectral approaches, DDs, XOR logic, etc. Development of these tools will require understanding of real quantum circuit technology.

51. New Frontiers Quantum Computer

52. Open Problems in Quantum Circuits Synthesis of binary quantum cascades with no garbage or small garbage –(Maslov, Dueck, Miller, Kerntopf, Perkowski, Khlopotine, Mishchenko, Curtis, Khan, Jha and Agrawal, Hayes, Markov) Synthesis of multiple-valued quantum cascades –(Muthukrishnan and Stroud, Miller et al, Khan, Perkowski, Curtis, Lee, Denler) –Universal gates, what are the counterparts of Toffoli and Fredkin gates? Fredkin Toffoli

53. Open Problems in Quantum Circuits What is the Fault Model for quantum circuits? –Technology dependent? Formal Verification of quantum circuits Test Generation for quantum circuits Fault Localization of quantum circuits Synthesis of testable quantum circuits Synthesis of fault-tolerant, error correcting quantum circuits.

54. Open Problems in Quantum Circuits What are universal gates? How to calculate costs of elementary gates for each quantum technology such as NMR or ion trap? What are the gates that can be truly realized in a quantum technology? What are the synthesis, analysis and test methods for sequential quantum circuits?

55. Open Problems in Quantum Circuits Invent new quantum algorithms. What are the principles to create quantum algorithms The nature of entanglement. Quantum computer architectures. Quantum formalisms. (Clifford algebras). Quantum Logic.

56. Example 1: First method to realize MV quantum circuits Analogous to binary quantum circuits. As an example, consider two qutrits. When the two qutrits are considered to represent a state, that state is the superposition of all possible combinations of the original qutrits, where: MV Tensor Products This approach to multi-valued quantum circuits requires measurements with more than two basis states. Also, new gates should be defined as well as the synthesis methods for these gates.

57. Quantum MV Superposition The superposition property allows the qubit states to grow much faster in dimension than classical bits, and the qudits faster than qubits. In a classical system, n bits represent distinct states, whereas n qubits correspond to a superposition of 2 n states and n qutrits correspond to a superposition of 3 n states. Because in contemporary quantum technologies every qubit is costly, higher radices than 2 give an advantage of improved processing and storage power at the same realization cost. This is a strong argument for realization of multi-valued logic in quantum circuits. In addition to standard advantages of mv logic, quantum mv logic may be superior to binary one because of different nature of entanglement.

58. Bloch Sphere The normalization |  | 2 + |  | 2 = 1 admits the parametrization  = cos(  /2) e j ,  = sin(  /2) e j . |  = e j  (cos (  / 2) |0  + e j  sin (  / 2) |1  ). Since the global phase of |  has no observable effect, we may write |  = cos(  /2) |0  + e j  sin(  /2) |1 . The angles  and  define a point on the surface of a unit sphere – the Bloch sphere, see Fig. 1. The Bloch sphere provides an excellent tool to visualize the state vector of a qubit. This is a binary Bloch sphere, but a multi-valued counterpart of it can be also created. Example 2: Second Method to realize MV quantum circuits.

59. Second method to realize multi-valued logic using binary quantum computing (cont). Figure shows the location of 6 points, that may correspond to values of some multi- valued algebras. For binary logic we use |0  and |1 . For quaternary logic we use |0 , |1 , |0  +|1 , and |0  -|1 . For 6-valued logic we may use additionally |0  + j |1  and |0  - j|1 . A rotation or a combination of rotations leads from one value to any other value.

60. Second Method to realize MV quantum circuits (cont). Above we showed how multiple-valued logic can be encoded in binary quantum computing. Quaternary logic requires two binary measurements (readings). The first reading distinguishes states |0  and |1 , and the second reading uses additional rotation gates to distinguish between states |0  +|1 , and |0  -|1 . It can be shown that the logic with 2 n values requires n readings.

61. Example 3: Quantum Circuit Simulation Simulation of quantum circuits plays absolutely fundamental role in many areas of quantum physics and engineering. Simulation is used to: –verify correctness of the design, –analyze its properties and –find some interesting aspects that cannot be found by “hand and pencil” methods. –Fault simulation –Evolutionary algorithms Researchers routinely use quantum simulators to help them with inventions and verify their design guesses.

62. Fast simulation is extremely important Matrix methods are slow. Acceleration is attempted to be achieved by two fundamental methods: –(1) acceleration of standard operations by using special hardware emulators, parallel computers or processor networks, – (2) creating new advanced data structures to represent quantum data more efficiently using standard computers.

63. Example 4: Quantum Decision Diagrams New data structures, such as QUIDDs [Viamontes, Markov, Hayes] allow for implicit parallelism when executing Kronecker multiplications on them. QUIDDs are based on ADDs and MTBDDs, so hopefully in future other decision diagrams may be used to represent quantum circuits. It is also expected that basic software engines used successfully in classical CAD (such as for instance SAT or ATPG methods) may be used to deal with quantum circuits. Also, the fast simulators based on new types of decision diagrams should be in future parallelized and possibly accelerated in FPGA-based boards. Even before quantum computers will be available, their emulations on standard computers and ASIC/FPGA may prove useful to solve some practical problems.

64. Example 5 : Testing and diagnosis of quantum circuits Patel, Markov, and Hayes showed that reversible circuits are much better testable than irreversible circuits. –This is because every test covers half faults and every fault is covered by half tests. –The reversible circuits are then “transparent” to faults, making them well observable and controllable. We showed that fault localization in reversible circuits is easier. We presented preliminary results on testing binary quantum circuits and on fault localization of quantum circuits.

65. Testing Quantum Circuits (1) The good circuit is simulated. Next every possible quantum fault is inserted (our fault model is inserting arbitrary matrix in place of fault, this allows to simulate many different types of faults) and the circuit with fault is simulated in Hilbert space (no measurement). All possible measurement values are calculated with their probabilities. The comparison of a measurement from the unitary matrix of a correct circuit and a circuit with fault determines which input combinations (tests) give different measurements. In some cases the circuit is modified for multi- valued realization in order to distinguish the values.

66. Testing Quantum Circuits (2) Observe that in contrast to standard testing and reversible circuits testing, there are three types of faults in quantum domain: –(1) faults that can be detected deterministically, –(2) faults that cannot be detected (like global phase faults), and –(3) faults that can be detected by repeated application of tests, possibly with special measuring gates. These faults can be detected only with certain probability. Thus, quantum testing is probabilistic testing.

67. Research Challenges in Quantum Test Open problems include basically everything: –fault models, –fault simulation, –test generation, –test minimization, –fault coverage, –fault localization using probabilistic adaptive trees.

68. Quantum Computational Intelligence (QCI) The two most famous quantum algorithms to date were created by Peter Shor and Lov Grover. Shor’s algorithm is for factoring integers: –It produces an exponential computational speedup over classical algorithms –It can break the RSA encryption techniques. Grover’s algorithm searches an unordered list of data, to find a particular item. –It has a provable quadratic speedup over the best classical algorithm. –It is like looking for name of a person in yellow pages knowing only his telephone number.

69. Research Challenges in Quantum Algorithms for Computational Intelligence “How these algorithms can be used in the field of Computational Intelligence?”. Quantum computing is in every particular instance at least as powerful as standard computing. It is therefore very reasonable to look for quantum counterparts to all concepts created in past in: – algorithm design, –Artificial Intelligence, –Machine Learning, –Computational Intelligence, –Soft Computing.

70. Future Applications in Structured Search Grover algorithm for searching an unstructured database started many practical applications because of the generality of its main idea – phase amplification. Grover himself extended his algorithm for the structured search problem, one of the main tough research issues in AI, with a multitude of important and practical applications, including in EDA. Many interesting papers about quantum search using problem structure were written by Hogg and collaborators. Boyer developed bound for quantum searching algorithms. The class of NP-complete problems includes: –graph coloring, –satisfiability, –planning, –set covering, –combinatorial optimization, –tautology verification –and many other problems that are useful for instance to solve the synthesis and optimization problems.

71. Generalizations of gates, circuits and automata Because gates, the basic concept of quantum computing, are a powerful generalization of gates in standard computing, researchers are systematically generalizing all the fundamental concepts of computing to involve quantum concepts in one way or another. And thus; –a quantum circuit is a generalization of a combinational Boolean circuit, –Quantum Automata (various formalizations) generalize Finite State Machines, –Quantum Turing machine generalizes Turing Machines and Probabilistic Turing Machines, –and so on.

72. From CI to QCI The same tendency is seen in Computational Intelligence. Its concepts and algorithms are being generalized to those of the Quantum Computational Intelligence (QCI). And thus; –Quantum Neural Networks, –Quantum Associative Memories, –Quantum Bayesian Nets, –Quantum Games, –Quantum Markets, –Quantum Agents, –Quantum Formulas, –Quantum Fuzzy Networks, –Quantum Spectral Transforms and Networks, –Quantum Evolutionary Algorithms, –Quantum Braitenberg Vehicles, –and many others have been created and are actively investigated both theoretically, using software simulators, hardware emulators and in real quantum circuits.

73. 理论计算机科学中的几个问题 应明生 清华大学计算机科学与技术系 智能技术与系统国家重点实验室 Importance of intelligent learning

74. Research Challenges in NP problems Because laws of quantum mechanics proved useful to improve algorithmic performance of some NP problems, there is a high probability that more problems will find efficient solutions in quantum domain.

75. Quantum-Neural Algorithms: Quantum Associative Memories of Ventura and Martinez, Competitive Learning in Quantum System by Ventura and Perus. While neural net processes real values, quantum NN processes complex values. It includes therefore standard NN and binary computers as special cases Thanks to superposition and entanglement can do much more. Weights that are complex values will allow to express much more and higher order information. Totally new algorithms can be invented for learning and using such nets. QuAM is analogous to a linear associative memory but all neurons are quantum mechanical gates.

76. Research Challenges in QCI There are dual influences of CI and quantum computing. –1. The quantum ideas can be used to create powerful quantum-inspired algorithms to solve many types of problems in EDA, QDA and robotics. –2. The ideas and algorithms from many classical computer science areas can be now used in quantum domain or transformed and extended to quantum domain. Very little operational software packages that use these ideas.

77. Quantum Computational Intelligence Quantum Neural Nets Quantum Associative Memories Quantum Inspired Genetic Algorithms Learning by synthesis of quantum circuits Other models of learning based on quantum concepts. Quantum Braitenberg Vehicles.

78. In 2020 quantum computing will be a reality As a community, we have a unique chance to work on the forefront of the future dominating technology. Logic design community did not have this opportunity in the past. Quantum Circuit Design And Technology Mathematics and logic Quantum Design Automation Quantum Information and Quantum Computational Intelligence

79. Conclusions (1) Emerging new area of Quantum Design Automation (QDA). Similarly as in design automation, there will appear sub-areas of: –high level quantum synthesis, –logic level quantum synthesis, –quantum test, –quantum verification, –quantum simulation, –quantum software-hardware co-design, –quantum physical design, –automatic learning from examples, –data mining, –and so on.

80. Conclusions (2) At the moment, even a single paper has been not published in many of these areas But surely they will appear in the forthcoming 10 years. We outlined some subjective choice of recent papers as a potential base of future research in QDA. Conventional logic synthesis, test and machine learning methods, for both binary and multiple- valued logic, form a powerful base of new approaches in quantum engineering.

81. Conclusions (3) Similarly the data structures like decision diagrams or fundamental algorithms such as satisfiability or reachability analysis continue to have their role. Because of high numerical demands of quantum logic there exist even higher expectations on these methods. Growing mutual influence of QDA and QCI, leading in long term to their unification.

82. My husband has not taken his decision yet, he is not sure if he should work on quantum computing

83. What to remember? 1.Types of quantum gates 2.Analysis of circuits based on various types of gates. 3.Six important states on the Bloch Sphere. How to use them? 4.Open Research areas in quantum circuits. 5.Simulation of quantum circuits (arrays) 6.Formal analysis of quantum circuits. 7.Decoherence.