Reversible Logic Circuit Synthesis

1. Reversible Logic Circuit Synthesis Vivek V. Shende, Aditya K. Prasad, Igor L. Markov and John P. Hayes University of Michigan

2. Outline • Motivation • Real-world Applications • Theoretical Advantages • Links to Quantum Computation • Background • Theoretical Results • Synthesis of Optimal Circuits • An Application to Quantum Computing

3. Real-world Applications • Many inherently reversible applications • Info. is re-coded, but none is lost or added • Digital signal processing • Cryptography • Communications • Computer graphics • Network congestion modeling

4. Theoretical Advantages • Information conservation laws in physics • Thermodynamics ties irreversibility to dissipated heat: every lost bit causes an energy loss • C. Bennett, 1973, IBM J. of R & D • Energy-lossless circuits (Time  ∞ ) • must be information-lossless • have been built:S. Younis and T. Knight, 1994, Workshop on Low Power Design

5. Links to Quantum Computation • Quantum operations are all reversible • M. Nielsen and I. Chuang, Quantum Computation and Quantum Information, Cambridge Univ. Press 2000 • Every (classical) reversible circuit may be implemented in quantum technology, with overhead • “Pseudo-classical” subroutines of quantum algos • Can be implemented in classical reversible logic circuits • S. Betteli, L. Serafini, and T. Calarco, 2001, http://xxx.lanl.gov/abs/cs.PL/0103009

6. Outline • Motivation • Background • Reversibility • Permutations • Known Facts • Theoretical Results • Synthesis of Optimal Circuits • An Application to Quantum Computing

7. Reversibility in Logic Gates • Definition: reversible logic gate • #input wires = #output wires • Permutes the set of input values • Examples • Inverter • 2-input, 2-output SWAP (S) gate • k-CNOT gate • (k+1)-inputs and (k+1)-outputs • Values on the first k wires are unchanged • The last value is flipped if the first k were all 1

8. Reversibility in Logic Circuits • Definition:A combinational logic circuit is reversible iff • It contains only reversible gates • It has no fan-out • It is acyclic (as a directed multi-graph) • Theorem:A reversible circuit must • Have as many input wires as output wires • Permute the set of input values

9. A Reversible Circuit and Truth Table Equivalent to a single CNOT gate

10. Circuit Equivalences • Circuit equivalences: useful in synthesis • More will be shown later

11. Reversible Circuits & Permutations • A reversible gate (or circuit) with n inputs and n outputs has • 2n possible input values • 2n possible output values • The function it computes on this set must, by definition, be a permutation • The set of such permutations is called S2n

12. Basic Facts About Permutations • Permutations are multiplied by first applying one, then the other • example: (1,2)◦(2,3) = (1,3,2) • A transposition • permutes exactly two elements • does not change any others • Every permutation can be writtenas a product of transpositions

13. Even Permutations • Consider all possible decompositionsof a permutation into transpositions • Theorem: The parity of the numberof transpositions is constant • Definition: Even permutations are those for which the number of transpositions is even

14. Known Facts • Fact 1: Consider a reversible circuit • n+1 inputs and n+1 outputs • Built from gates which have at most n inputs and n outputs • Must compute an even permutation • Fact 2: A universal gate library • CNOT, NOT, and TOFFOLI (“CNT”) • Temporary storage may be required

15. Temporary Storage

16. Outline • Motivation • Background • Theoretical Results • Zero-storage Circuits • Reversible De Morgan’s Laws • Synthesis of Optimal Circuits • An Application to Quantum Computing

17. Minimizing Temporary Storage • Consider CNT circuits • Theorem: even permutations computableby circuits without temporary storage • Theorem: odd permutations computablewith one line of temporary storage • Same holds for NT- and CNTS-circuits • The proof is constructive and may be used as a synthesis heuristic

18. C- and N-circuits • N-circuits need at most one gate per wire • Can cancel inverters • C-circuits on k wires act as k × k matrices • By reversibility, the matrix must be invertible • A C(x,y) gate is a row-addition matrix which adds the x-th row to the y-th • Can get all invertible matrices

19. T-circuits • T-circuits fix 0, 2i • These have only one 1 in their binary expansion • T-circuits on 4+ wires compute even perms. • In fact, any circuit in which no gate involves all the wires computes an even permutation • T. Toffoli, "Reversible Computing", Tech. Memo MIT/LCS/TM-151, MIT Lab for Comp. Sci, 1980. • Any permutation satisfying the above constraints can be computed in a T-circuit

20. T-circuits • Assume #wires > 3 • Decompose an even permutation into a product of disjoint transpositions • (a,b)(c,d) is okay, (a,b)(b,c) is not • But (a,b)(b,c) = [(a,b)(d,e)] [(d,e)(b,c)] • Explicitly construct a circuit to computean arbitrary pair of disjoint transpositions

21. Computing Disjoint Trans. Pairs • Use an (n-2) generalized Toffoli gate on n wires • Can compute z = (2n-1, 2n-2)(2n-3, 2n-4) • Can be simulated by T gates (no temporary storage) • A. Barenco et al.: ``Elementary Gates For Quantum Computation'', Physi. Review A (52), 3457-3467, 1995 • Can compute a permutation p sending: • a→2n-1, b→2n-2, c→2n-3, d→2n-4 • Use these constructions to compute a given pair of disjoint transpositions • p-1zp = (a,b)(c,d)

22. CT|N-circuits • A CT|N-circuit is a CNT-circuit with all the N gates at the right end • To convert a CNT-circuit into CT|N form • Write down rules for interchanging N gates with C and T gates • Push all the N gates to the end • Like pushing inverters to the front of (irreversible) AND/OR/NOT-circuits

23. Reversible De Morgan’s Laws

24. T|C-circuits • Similar rules exist for interchanging TOFFOLI and CNOT gates

25. Reversible De Morgan’s Laws (2)

26. T|C-circuits • However, it is not always possibleto push all C gates to the inputs • A T|C-circuit’s permutation must fix 0, and map the elements 2i to linearly independent vectors • Recall: T-circuits fix 0, 2i, and C-circuits compute invertible linear transformations • A permutation can be computed by a T|C-circuit if it satisfies the above requirements • The permutation (1,2)(3,4) does not • We will see that it can be computed by a CT-circuit

27. T|C|T-circuits • Given any even permutation fixing zero • Can multiply it by an even permutation fixing 0, 2iso that the product fixes 0, and takes inputs 2i to linearly independent outputs • A T-circuit can simulate the former • A T|C circuit can simulate the latter • So, any zero-fixing even permutation can be computed by a T|C|T-circuit

28. T|C|T|N-circuits • Let p be an even permutation • There is an N-circuit taking 0 to p(0) • Unique up to canceling redundant gates • Say this N-circuit computes the permutation n • The permutation pn-1 fixes 0, and can be computed by a T|C|T-circuit P’ • Then the circuit P’N computes p • Theorem: any even permutation can be computed by a T|C|T|N-circuit

29. Comments on Circuit Length • For “long” CNT-circuits, this algorithm produces circuits which are suboptimal by at worst a logarithmic factor (in the number of wires) • Converting a CNT-circuit to a CT|N-circuit increases circuit size by about a factor of 3 • We do not know any good algorithm to convert CNT-circuits to T|C|T|N-circuits, or even if this conversion may be done with polynomial length increase

30. Outline • Motivation • Background • Theoretical Results • Synthesis of Optimal Circuits • Optimality • DFID Search Algorithm • Circuit Libraries • An Application to Quantum Computing

31. Optimality • The cost of a circuit is its gate count • Other cost functions can be considered • Definition:optimal reversible circuit • no circuit with fewer gates computes the same permutation • Theorem: a sub-circuit of an optimal circuit is optimal • Proof: otherwise, can improve the sub-circuit

32. The Search Procedure • Depth First Iterative Deepening Search • Checks all possible circuits of cost 1, then all possible circuits of cost 2, etc… • Avoids the memory blowup of BFS • Still finds optimal solutions (unlike DFS) • Checking circuits of cost less than n • Is much faster than processing cost-n circuits

33. Dynamic Prog + Circuit Libraries • DFID search requires a subroutine to checkall circuits of cost n, for arbitrary n • Called iteratively for 1…n • Only need to check locally optimal circuits • Build optimal circuit library bottom up by DP • Index optimal circuits by computed permutation • In practice use hash_map datastruct from STL

34. Synthesis Algorithm

35. Empirical Circuit Synthesis • Consider all reversible functions on 3 wires(8! = 40,320 functions) • For each gate library fromN, C, T, NC, CT, NT, CNT, CNTS • Is it universal? • How many functions can it synthesize? • How long does it take to synthesize circuits? • What are largest optimal circuits?

36. Optimal Circuit Sizes

37. Largest Optimal Circuits

38. Why Circuit Libraries? • Large speedup relative to just branching • Can be calculated from previous table • Calculated values are very large • In practice, the table cannot be generated in several hours without circuit libraries • With libraries, the table takes less than 10 min

39. Outline • Motivation • Background • Theoretical Results • Synthesis of Optimal Circuits • An Application to Quantum Computing • Grover’s Search • Pseudo-classical Synthesis

40. Quantum Circuits • Necessarily reversible • Information stored on qubits • Superposition allows linear combinations of 0 and 1 to be stored • All reversible gates still allowed • Many other gates used as well

41. Grover’s Search • A quantum algorithm for associative search(input is not sorted) • Search criterion: a classical one-output function f • L. K. Grover, “A Framework For Fast Quantum Mechanical Algorithms”, STOC 1998 • M. Nielsen and I. Chuang, 2000 • Runs in time O(√N) • any classical algorithm provably requires (N ) time • Requires a subroutine (oracle) that • changes the phase (sign) of all basis states (bit-strings)that match the search criterion f

42. Grover Oracle Circuits • To change the sign of a bit-string • Initialize a qubit to |0> - |1> • Compute the classical one-output function f • XOR the qubit with f • Whenever f=1, the sign (phase) will change

43. Sample Grover Oracle Circuit

44. Grover Oracle Circuit Synthesis • Thus, the design of Grover search circuitsfor a givenf • Is reduced to reversible synthesis • Can be solved optimally by our methods

45. ROM-based Circuits • Desired circuits must alter phase of basis states • All bits except one must be restored to input values • Previous work studied ROM-based circuits • Constraint: ROM qubits can never change • B. Travaglione et al., 2001, http://xxx.lanl.gov/abs/quant-ph/0109016 • Theorems + heuristic synthesis algorithms • Our work: synthesis of pseudo-classical circuits • 3 read-only “ROM” wires that can never change • 1 wire that can be changed during computation, but must be restored by end • 1 wire on which function is computed

46. Synthesis Algorithms Compared • Heuristic synthesis of ROM-based circuits • Proposed by Travaglione et al, 2001 • Based on EXOR-sum decomposition (“XOR”) • Imposed a restriction: at most one control bit per gate can be on a ROM bit • Optimal synthesis (as described earlier) • with restriction from Travaglione (“OPT T”) • without this restriction (“OPT”)

47. Sizes of 3+2 ROM-circuits

48. Discussion of Empirical Results • The EXOR-SUM heuristic is sub-optimal • All methods able to synthesize all 256 fns • “OPT T” can synthesize as many as “OPT”: B. Travaglione et al., 2001 • “OPT” results symmetrical about 5-6 gates • Function x requires one fewer gate than 256-x • Explanation yet to be found • “XOR” results symmetrical about 13 gates

49. Conclusions • Classical reversible circuitsas special-case quantum circuits • Existence theorems • Reversible De Morgan’s laws • Future research on optimization heuristics • Algorithm for synthesis of optimal circuits • Applicable to Grover’s search

50. Thank You For Your Attention