“Both Toffoli and CNOT need
little help to do universal QC”
(following a paper by the same
title by Yaoyun Shi)
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
The Agenda
• Background
– Completeness vs. Universality
– Kitaev-Solovay Theorem
– Another result by Kitaev
• Completeness (existence) proofs
• Completeness: an explicit construction
• Conclusion
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
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
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
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 )
The Agenda
• Background
• Completeness (existence) proofs
– CNOTs and Rotations
– Eigenvectors & Eigenvalues
– Who’s Dense
• Completeness: an explicit construction
• Conclusion
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
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.
A sketch of the proof:
• Let U be the operator be computed by
S
S
S
S
• Apply the Kitaev lemma several times
– Q.E.D.
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 
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)
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
The Agenda
• Background
• Completeness (existence) proofs
• Completeness: an explicit construction
– Barenko’s Reduction
– the Z gate
– Grover’s Algorithm
• Conclusion
An Explicit Construction
• Recall {CNOT, S} is complete
– when S2 doesn’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
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
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
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
The Z Gate
• How to use S to flip a sign
– Suppose θ = pi/4
– One can use a well known trick:
Z
=
1
S
S†
1
– This works because: XUpi/41=-Upi/41
The Z Gate
• For arbitrary θ, it’s more difficult
– XUθ1 could be anywhere relative to Uθ1
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
The W /2 Gate
• Want: W /20k = U/2 0  0k-1
• Idea ?
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
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)
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
The Agenda
• Background
• Completeness (existence) proofs
• Completeness: an explicit construction
• Conclusion
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
Questions?