Universal Quantum Computing: CNOT and Toffoli Gate Help

1. “Both Toffoli and CNOT need little help to do universal QC” (following a paper by the same title by Yaoyun Shi)

2. Abstract • Well known fact: • {CNOT,S} is universal when S is an irrational one qubit rotation • Less well known fact: • S really only needs to not square to something classical • Another less well known fact: • {Toffoli, Hadamard} is universal

3. The Agenda • Background • Completeness vs. Universality • Kitaev-Solovay Theorem • Another result by Kitaev • Completeness (existence) proofs • Completeness: an explicit construction • Conclusion

4. Universality • A (real) gate library G is universal if • it can approximate any unitary (orthogonal) operator if constant inputs from the computational basis are allowed • for example, a TOFFOLI gate can approximate a CNOT gate in this sense

5. Completeness • A gate library G is complete if • it can approximate any unitary operator in U(2k) for some k • no extra wires or constant inputs allowed • Completeness => Universality

6. Why completeness? • The Kitaev-Solovay Theorem: • Any complete gate library can efficiently approximate any 1 qubit unitary operator • specifically, one can get within ε in polylog(1/ε) gates

7. Another theorem of Kitaev • Suppose: • M is a (real) Hilbert space of dimension > 2 •  is a unit vector • H  SO(M ) is the stabilizer of span() • v  O(M ),  not an eigenvector of v • Then: • the subgroup generated by H  v-1Hv is dense in SO(M )

8. The Agenda • Background • Completeness (existence) proofs • CNOTs and Rotations • Eigenvectors & Eigenvalues • Who’s Dense • Completeness: an explicit construction • Conclusion

9. A CNOT and a rotation • Fix an arbitrary one qubit rotation S about an angle θ • if θ/π is irrational, we know from general theory that {CNOT, S} is complete • So, suppose θ is a rational multiple of pi

10. A CNOT and a rotation • Finally, suppose S2 does not have both 0 and 1 as eigenvectors • a theorem of Gottesman-Knill implies that: • for an S failing this condition, any {S, CNOT} circuit may be efficiently simulated by a classical computer • thus, such an S is not universal for QC • Then {S, CNOT} is complete.

11. S S S S A sketch of the proof: • Let U be the operator be computed by • Apply the Kitaev lemma several times • Q.E.D.

12. Eigenvectors & Eigenvalues • Calculating U’s eigenvalues gives them as • 1, 1, ei, e-i •  is incommensurable with pi • Let i be the orthonormal eigenvectors • U restricted to span(1, 2) is the identity • U restricted to span(3, 4):=H1 is a rotation through the angle 

13. Who’s Dense • U generates a dense subgroup of H1 • Call SO(span(2, 3, 4)) H2 • H1  H2 is the stabilizer of span(2) • one CNOT, C1 fixes 1, and moves span(2)

14. Who’s Dense • The Kitaev lemma applies: {U, C1} generates a dense subset of H2 • A similar argument shows {U, C1, C2} generates a dense subset of SO(4) • So, {U, C1, C2} is complete

15. The Agenda • Background • Completeness (existence) proofs • Completeness: an explicit construction • Barenko’s Reduction • the Z gate • Grover’s Algorithm • Conclusion

16. An Explicit Construction • Recall {CNOT, S} is complete • when S2doesn’t have both basis states as eigenvectors • It is true that {TOFFOLI, S} is complete • when S doesn’t have both basis states as eigenvectors • a similar proof exists

17. An Explicit Construction • Additionally, Shi explicitly {TOFFOLI, S} approximates an arbitrary one qubit gate • By Barenko’s decomposition, this is sufficient to approximate an arbitrary unitary matrix

18. Some preliminaries • Define Ut to be rotation by the angle t • Let S be the one-qubit gate in our library • define θ by S = Uθ • Let W be the desired one qubit operator • define  by W = U

19. Reduction of the problem • It suffices to approximate • the Z gate • a gate W/2 s.t. W /20k = U/2 0  0k-1 • Using these gates and the TOFFOLI, one may simulate a gate W satisfying • W (  0k-1) = U  0k-1

20. Z S S† = 1 1 The Z Gate • How to use S to flip a sign • Suppose θ = pi/4 • One can use a well known trick: • This works because: XUpi/41=-Upi/41

21. The Z Gate • For arbitrary θ, it’s more difficult • XUθ1 could be anywhere relative to Uθ1

22. The Z Gate • A similar construction exists, however • Uθ0Uθ1 = a(11-00) + b01 + c10 • swap the basis vectors 11, 00 • this is within sqrt(b2+c2) of a sign flip • sqrt(b2+c2) < 1, so do a lot of these

23. The W /2 Gate • Want: W /20k = U/2 0  0k-1 • Idea ?

24. Prelude to Grover’s Algorithm • Let 0 = 02k • Use S, CNOT, to build a T such that • 0T0 is small and positive • define φ = T0 • Let 1 be the vector perpendicular to 0 in the plane spanned by 0 , φ

25. Using Grover’s Algorithm • The system begins in the state 00 • apply IT • the state = 0φ • Iteratively reflect φ about 1 ala Grover • want: φ -> cos(/2)1 + sin(/2)0 • state = 0(cos(/2)1 + sin(/2)0)

26. Using Grover’s Algorithm • Apply an appropriately conjugated 2k-cnot to flip the first bit if the remaining 2k are orthogonal to 0 • state = 11cos(/2) + 00sin(/2) • Apply a controlled-T-1 : 11 -> 10 • state = (cos(/2)1 + sin(/2)0)0

27. The Agenda • Background • Completeness (existence) proofs • Completeness: an explicit construction • Conclusion

28. Conclusion • The CNOT needs only a one qubit rotation whose square is nonclassical to form a complete library • The Toffoli can partner with any nonclassical gate for a complete library • In the second case, we have an explicit approximation algorithm

29. Questions?