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. Classical Versus Quantum

10. 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

11. 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)

12. Decoherence

13. 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

14. 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

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

16. Classical vs. Quantum Circuits Quantum adder Feynman gate

17. Reversible Circuits

18.       … … 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”

19. 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.

20. (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

21. Quantum Gates

22. 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:

23. 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!

24. H Quantum Gates • One-Input gate:Hadamard • Maps |0  1/  2 |0 + 1/  2 |1 and |1  1/  2 |0 – 1/  2 |1. • Ignoring the normalization factor 1/  2, we can write |x  (-1)x|x – |1 –x • One-Input gate:Phase shift 

25. U Any state|y |0 H H 2 Universal One-Input Quantum Gate Sets • Requirement: • Hadamard and phase-shift gates form a universal gate set • Example: The following circuit generates|y = cos |0 + ei sin |1 up to a global factor

26. |00> |00> |01> |01> |10> |10> |11> |11> 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>.

27. |x |x |x |x CNOT |x  y |y |y |x  y Quantum XOR gate • CNOT maps |x|0  |x||xand |x|1  |x||NOT x • |x|0  |x||x looks like cloning, but it’s not. • These mappings are valid only for the pure states|0 and |1

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

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

30. Quantum Gates 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 (n34n) gates [Reck et al. 1994] • CNOT and the (infinite) set of all 1-qubit gates is universal

31. Quantum Gates 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

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

33. Quantum Circuits Simulation

34. Quantum Circuits • 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!

35. = V V† V U Quantum Circuits • Ad hoc designs known for many specific functions and gates • Example 1 illustrating a theorem by [Barenco et al. 1995]: Any C2(U) gate can be built from CNOTs, C(V), and C(V†) gates, where V2 = U

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

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

38. Analysis of Quantum Circuits based on Unitary Matrices

39. x1 x2 x3 ? = U4 U2 U3 U1 U5 U0 V V† V U Algebraic Analysis of Quantum Circuits • Is U0(x1, x2, x3) = U5U4U3U2U1(x1, x2, x3) • = (x1, x2, x1x2 U(x3) ) ?

40. Quantum Circuits • Example 1 (contd);

41. Quantum Circuits • Example 1 (contd);

42. Quantum Circuits • Example 1 (contd); • U5 is the same as U1 but has x1and x2 permuted (tricky!) • It remains to evaluate the product of five 8 x 8 matrices U5U4U3U2U1 using the fact that VV† = I and VV = U

43. Quantum Circuits • Implementing a Half Adder • Problem: Implement the classical functions sum = x1 x0 and carry = x1x0 • Generic design: |x1 |x1 |x0 |x0 Uadd |y1 |y1 carry |y0 |y0  sum

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

45. Quantum Circuits • Half Adder: Specific (reduced) design |x1 |x1 CNOT C2NOT (Toffoli) |x0 sum |y |y carry

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

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

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.