2. Advanced Computer Architecture Laboratory EECS 598-1 Fall 2001 Quantum Logic Circuits John P. Hayes EECS Department University of Michigan, Ann Arbor, MI 48109, USA October 2001

3. Advanced Computer Architecture Laboratory Outline Classical vs. Quantum Circuits Reversible Circuits Quantum Gates Quantum Circuits Physical Implementation

4. Advanced Computer Architecture Laboratory 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

5. Advanced Computer Architecture Laboratory 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 vs. Quantum Circuits

6. Advanced Computer Architecture Laboratory 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 Quantum Circuit Characteristics

7. Advanced Computer Architecture Laboratory 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 ≤ 1, 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 Circuit Characteristics

8. Advanced Computer Architecture Laboratory Classical vs. Quantum Circuits Classical adder

9. Advanced Computer Architecture Laboratory Classical vs. Quantum Circuits Quantum adder

10. Advanced Computer Architecture Laboratory Outline Classical vs. Quantum Circuits Reversible Circuits Quantum Gates Quantum Circuits Physical Implementation

11. Advanced Computer Architecture Laboratory 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 … …    n inputs Generic Boolean Circuit m outputs    f(i) i …    Reversible Boolean Circuit …    f(i) … … “Junk” i      

12. Advanced Computer Architecture Laboratory 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

13. Advanced Computer Architecture Laboratory 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

14. Advanced Computer Architecture Laboratory Outline Classical vs. Quantum Circuits Reversible Circuits Quantum Gates Quantum Circuits Physical Implementation

15. Advanced Computer Architecture Laboratory Quantum Gates 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

16. Advanced Computer Architecture Laboratory Quantum Gates 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!

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

18. Advanced Computer Architecture Laboratory Universal One-Input Gate Sets 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 Quantum Gates U  Any state    22 HH

19. Advanced Computer Architecture Laboratory Two-Input Gate: Controlled NOT (CNOT) Quantum Gates 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  >Serves as a “non-demolition” measurement gate xx yy xx  x  y 

20. Advanced Computer Architecture Laboratory 3-Input gate: Controlled CNOT (C 2 NOT or Toffoli gate) Quantum Gates bb cc bb  ab  c  aaaa

21. Advanced Computer Architecture Laboratory General controlled gates that control some 1-qubit unitary operation U are useful Quantum Gates U C(U) U C2(U)C2(U) U U etc.

22. Advanced Computer Architecture Laboratory 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

23. Advanced Computer Architecture Laboratory 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

24. Advanced Computer Architecture Laboratory 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}

25. Advanced Computer Architecture Laboratory Outline Classical vs. Quantum Circuits Reversible Circuits Quantum Gates Quantum Circuits Physical Implementation

26. Advanced Computer Architecture Laboratory 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

27. Advanced Computer Architecture Laboratory 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

28. Advanced Computer Architecture Laboratory Example 1: Simulation 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  ?

29. Advanced Computer Architecture Laboratory Quantum Circuits |1  |x  |1  |x  |1  U|x  VV†V† V = U |1  V|x  |1  |0  |1  |0  V|x  |1  U|x  Example 1: Simulation (contd.) ? Exercise: Simulate the two remaining cases

30. Advanced Computer Architecture Laboratory Quantum Circuits Example 1: Algebraic analysis 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 ) ) ?

31. Advanced Computer Architecture Laboratory Quantum Circuits Example 1 (contd);

32. Advanced Computer Architecture Laboratory Quantum Circuits Example 1 (contd);

33. Advanced Computer Architecture Laboratory 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

34. Advanced Computer Architecture Laboratory 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

35. Advanced Computer Architecture Laboratory Quantum Circuits Half Adder : Generic design (contd.)

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

37. Advanced Computer Architecture Laboratory Quantum Circuits Implementing Deutsch’s Algorithm Problem: Determine whether a given Boolean function f(x) is constant, i.e. f(0) = f(1), or balanced, i.e. f(0) ≠ f(1). Classical algorithms require two evaluations of f. This algorithm uses just one quantum evaluation by, in effect, computing f(0) and f(1) simultaneously The Circuit: M H H H y  f(x) y x x UfUf

38. Advanced Computer Architecture Laboratory Quantum Circuits Deutsch’s Algorithm M H H H y  f(x) y x x UfUf Initialize with |  0  = |01  |0|0 |1|1 |0|0 Create superposition of x states using the first Hadamard (H) gate. Set y control input using the second H gate |1|1 Compute f(x) using the special unitary circuit U f |2|2 Interfere the |  2  states using the third H gate |3|3 Measure the x qubit |0  = constant; |1  = balanced

39. Advanced Computer Architecture Laboratory Quantum Circuits M y  f(x) y x x UfUf |0|0 |1|1 |0|0 |1|1 |2|2 |3|3 HHH if f(0) = f(1) if f(0) ≠ f(1) if f(0) = f(1) if f(0) ≠ f(1)

40. Advanced Computer Architecture Laboratory Quantum Circuits Deutsch-Josza Algorithm Generalization of Deutsch’s algorithm H  n = H  H  …  H H HnHn y  f(x) y x x UfUf HnHn n |0|0 |1|1 |0|0 |3|3

41. Advanced Computer Architecture Laboratory Quantum Circuits Deutsch-Josza Algorithm (contd) This algorithm distinguishes constant from balanced functions in one evaluation of f, versus 2 n–1 + 1 evaluations for classical deterministic algorithms Balanced functions have many interesting and some useful properties >K. Chakrabarty and J.P. Hayes, “Balanced Boolean functions,” IEE Proc: Digital Techniques, vol. 145, pp 52 - 62, Jan. 1998.

42. Advanced Computer Architecture Laboratory Quantum Circuits Quantum Fourier Transform Circuit

43. Advanced Computer Architecture Laboratory Quantum Circuits Quantum Error-Correction Circuit Problem: State |  = a|0  + b |1  is degraded by noise Solution Encode in a suitable EC code such as the 5-bit code: |0  = |00000  + |11000  + |01100  + |00110  + |00011  + |10001  – |10100  – |01010  – |00000  – |10010  – |01001  – |11110  – |01111  – |10111  – |11011  – |11101  |1  = |11111  + |00111  + |10011  + …

44. Advanced Computer Architecture Laboratory Outline Classical vs. Quantum Circuits Reversible Circuits Quantum Gates Quantum Circuits Physical Implementation

45. Advanced Computer Architecture Laboratory Physical Implementation Quantum Technology Requirements [Di Vicenzo ‘01] Scalable with well-characterized qubits Initializable to a pure state such as  00…0  Relatively long decoherence time “Universal” set of quantum gates Qubit-specific measurement capability and Ability to faithfully communicate qubits

46. Advanced Computer Architecture Laboratory Physical Implementation Main Contenders NMR (nuclear magnetic resonance) Ion traps Optical lattices Quantum dots Electrons on liquid helium etc. Main Deficiency Poor scalability

47. Advanced Computer Architecture Laboratory Physical Implementation: NMR Many atoms have a nucleus with quantum “spin” like a tiny bar magnet. Spin up/down =  0  /  1 . Several atoms’ spins can be coupled chemically in a molecule but remain selectively addressable due to different resonant frequencies An RF pulse can rotate an atom’s spin in a manner proportional to the amplitude and duration of the applied pulse A computation such as a gate/circuit operation consists of a sequence of carefully sized and separated RF pulses Many molecules (e..g, 10 18 )can be combined in liquid solution to form a same-state ensemble of macroscopic and manageable size All of Di Vicenzo’s criteria can be met, except that scalability seems to be limited to 20–30 qubits? 00 11

48. Advanced Computer Architecture Laboratory Physical Implementation: NMR Five-qubit NMR computer [Steffen et al. 2001]

49. Advanced Computer Architecture Laboratory Physical Implementation: NMR Five-qubit computer (contd.) >Molecule with 5 flourine atoms whose spins implement the qubits >Experimental 5-qubit circuit to find the order of a permutation

50. Advanced Computer Architecture Laboratory Summary Quantum circuits can implement several important algorithms with fewer operations than any classical algorithm A few generic quantum gate and circuit types have been identified and their properties studied Small quantum circuits have been successfully demonstrated in the lab using several physical technologies, notably NMR Current technologies are slow, fragile, and appear to be limited to tens of qubits and hundreds of gates Huge gaps remain in our understanding of quantum circuit design and implementation techniques