Which gates are universal for quantum computation? Although it is well known
that certain gates on two-level quantum systems (qubits), such as the
controlled-not (CNOT), are universal when assisted by arbitrary one-qubit
gates, it has only recently become clear precisely what class of two-qubit
gates is universal in this sense. Here we present an elementary proof that any
entangling two-qubit gate is universal for quantum computation, when assisted
by one-qubit gates. A proof of this important result for systems of arbitrary
finite dimension has been provided by J. L. and R. Brylinski
[arXiv:quant-ph/0108062, 2001]; however, their proof relies upon a long
argument using advanced mathematics. In contrast, our proof provides a simple
constructive procedure which is close to optimal and experimentally practical
[C. M. Dawson and A. Gilchrist, online implementation of the procedure
described herein (2002), http://www.physics.uq.edu.au/gqc/].Comment: 3 pages, online implementation of procedure described can be found at
  http://www.physics.uq.edu.au/gqc

Alexei Gilchrist

Aram W. Harrow

Christopher M. Dawson

D. Deutsch

Duncan Mortimer

J. Preskill

J. L. Brylinski

Jennifer L. Dodd

M. A. Nielsen

Michael A. Nielsen

Michael J. Bremner

P. Wocjan

Tobias J. Osborne

English

arXiv

Crossref

Practical Scheme for Quantum Computation with Any Two-Qubit Entangling Gate

Which gates are universal for quantum computation? Although it is well known that certain gates on two-level quantum systems (qubits), such as the controlled-NOT, are universal when assisted by arbitrary one-qubit gates, it has only recently become clear precisely what class of two-qubit gates is universal in this sense. We present an elementary proof that any entangling two-qubit gate is universal for quantum computation, when assisted by one-qubit gates. A proof of this result for systems of arbitrary finite dimension has been provided by Brylinski and Brylinski; however, their proof relies on a long argument using advanced mathematics. In contrast, our proof provides a simple constructive procedure which is close to optimal and experimentally practical

Bremner, M. J.

Dawson, C. M.

Dodd, J. L.

Gilchrist, A.

Harrow, A. W.

Mortimer, D.

Nielsen, M. A.

Osborne, T. J.

University of Queensland eSpace

VOLUME 89, NUMBER 24 P H Y S I C A L R E V I E W L E T T E R S 9 DECEMBER 2002Practical Scheme for Quantum Computation with Any Two-Qubit Entangling Gate
Michael J. Bremner,1 Christopher M. Dawson,1 Jennifer L. Dodd,1 Alexei Gilchrist,1 Aram W. Harrow,1,2
Duncan Mortimer,1 Michael A. Nielsen,1 and Tobias J. Osborne1
1Centre for Quantum Computer Technology and Department of Physics, The University of Queensland, QLD 4072, Australia
2MIT Physics, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139
(Received 12 July 2002; published 25 November 2002)247902-1Which gates are universal for quantum computation? Although it is well known that certain gates on
two-level quantum systems (qubits), such as the controlled-NOT, are universal when assisted by
arbitrary one-qubit gates, it has only recently become clear precisely what class of two-qubit gates
is universal in this sense. We present an elementary proof that any entangling two-qubit gate is universal
for quantum computation, when assisted by one-qubit gates. A proof of this result for systems of
arbitrary finite dimension has been provided by Brylinski and Brylinski; however, their proof relies on
a long argument using advanced mathematics. In contrast, our proof provides a simple constructive
procedure which is close to optimal and experimentally practical.
DOI: 10.1103/PhysRevLett.89.247902 PACS numbers: 03.67.–aperformed with rapidity and accuracy which are vastly
more demanding than the standard requirements
(iii) A gate U is entangling if it can create entangle-
ment between two systems initially in a product state.A great deal of work has been done to determine what
physical resources are capable of universal quantum com-
putation. It is well known that certain gates on two-level
quantum systems, such as the controlled-NOT (CNOT), are
universal when assisted by arbitrary one-qubit gates [1]. It
has also been shown that almost any gate on two d-level
quantum systems (qudits), together with its swapped
version, is universal for quantum computation without
the aid of one-qudit gates [2,3]. However, this result
does not explicitly specify which two-qudit gates are
universal, requires the ability to apply the given gate in
two different ways, and the resulting procedure is not
practical, requiring large numbers of gates.
Recently, several authors have considered conditions
for universality when only a single fixed multiqudit
Hamiltonian interaction, together with one-qudit gates,
is allowed [4–9]. They have shown that any interaction
that can create entanglement between any pair of qudits is
universal for quantum computation. Theoretical schemes
for quantum computation based on this have been found,
but they are not of practical utility. In order to make the
simulations exact, the given interaction is modified by
one-qudit gates which must be applied so that the period
of evolution between them is infinitesimal. To simulate
evolution for some noninfinitesimal time t, the error is
controlled by concatenating a large number n of periods
of evolution for a small time t=n. Although, in the qubit
case, some such schemes minimize the amount of time
required to do the simulation [7,8], the required number of
one-qubit gates is enormous; an optimistic example of
simulating a CNOT to accuracy only 103 [10] (less strin-
gent than a commonly quoted estimate for the fault-
tolerance threshold, 105–106 —see references at the
end of chapter 10 of [11]) requires approximately 104
one-qubit gates [4]. Thus, the one-qubit gates must be0031-9007=02=89(24)=247902(3)$20.00for quantum computation. We note, however, that
Hammerer,Vidal, and Cirac [12] have obtained a practical
scheme for a restricted class of symmetric Hamiltonians
with no self-energy.
In contrast, the model we consider does not allow an
interaction to be interrupted by one-qubit gates at arbi-
trary times. Following Brylinski and Brylinski [13], but
restricting to the qubit case, we allow only a fixed en-
tangling two-qubit gate U and arbitrary one-qubit gates
between applications of U. Our proof that any such gate is
universal provides an explicit method [14] for implement-
ing a CNOT exactly, using a small number of one-qubit
gates which is fixed for any given gate U. For the opti-
mistic example mentioned above, this method requires
only approximately ten one-qubit gates. This represents a
savings of a factor of 103; typical savings will be much
greater. The only limit to the accuracy achieved in prac-
tice is due to the accuracy with which the required one-
qubit gates are calculated. This limit is inherent in any
procedure for computation, but because of the constant
number of one-qubit gates required by our scheme, the
induced errors will depend only in a constant way on
these inaccuracies. Combined with the fact that the num-
ber of uses of U is near optimal, this suggests that our
scheme will be of practical utility.
We begin the description of our construction with some
convenient definitions:
(i) We say that a two-qubit gate is universal if it can be
used to perform universal quantum computation on two
qubits when assisted by arbitrary one-qubit gates.
(ii) Suppose U  A1  B1VA2  B2. Since we have
the ability to do arbitrary one-qubit gates, being able to
perform U allows us to perform V, and vice versa.
Whenever this is the case, we say that U and V are
equivalent and write U  V. 2002 The American Physical Society 247902-1
VOLUME 89, NUMBER 24 P H Y S I C A L R E V I E W L E T T E R S 9 DECEMBER 2002(iv) Following [13], we define U to be primitive if U is
a product of one-qubit gates or if U is equivalent to the
gate interchanging the two qubits (SWAP); otherwise U is
imprimitive. We will see that, for two qubits, the class of
imprimitive gates is exactly the class of entangling gates.
We now prove the qubit case of the result in [13].
Theorem: A two-qubit gate U is universal if and only
if it is imprimitive, or, equivalently, if and only if it is
entangling.
Proof: A brief summary of our proof is as follows: We
use two nontrivial facts. The first is that CNOT is universal
[1]. The second is the canonical decomposition [15,16] for
any two-qubit gate U:
U  A1  B1eixXXyYYzZZA2  B2; (1)
where X; Y; Z are the Pauli sigma matrices, Aj; Bj are one-
qubit gates, and  4 <   4 (see [16] for a simple
proof). Both of these facts have proofs which are some-
what detailed but elementary and constructive. Our strat-
egy is to show that any imprimitive gate U, together with
one-qubit gates, can be used to implement W  eiZZ
where 0< jj< 2 . We then show that W can be used,
together with one-qubit gates, to exactly implement CNOT,
which proves that W, and therefore U, is universal.
Finally, since any universal gate is entangling, and any
entangling gate is imprimitive, it follows that the class of
entangling gates is exactly the class of imprimitive gates.
We define V  eixXXyYYzZZ  U. First, note
that primitive gates have either x  y  z  0 (corre-
sponding to U being a product of one-qubit gates) or x 
y  z  4 (corresponding to U  SWAP), so we need
not consider these cases.
Suppose U is imprimitive, in which case at least one of
the  is nonzero. We will show that in all cases V, and
hence U, may be used with one-qubit gates to implement a
CNOT and is therefore universal. In each case, we use V to
obtain a gate of the form W  eiZZ, 0< jj< 2 . Note
that we may assume jzj 
 jxj 
 jyj since the  may
be relabeled by conjugating V by the primitive gates H 
H and S  S where H  1
2
p
h
1
1
1
1
i
and S 
h
1
0
0
i
i
.
First, consider the two special cases where either one or
two of x; y; z are 4 and the remainder are 0. Suppose
that z  4 and y  x  0. Then V  ei=4ZZ and is
hence already of the required form. For the second special
case, z  x  4 and y  0. Noting that V8  I, and
thus V7  Vy, we use the one-qubit gate ei=4XI
to obtain Vei=4XIV7  ei=4VXIVy  ei=4YZ 
ei=4ZZ, which is of the required form [17].
Second, consider the more general case, z  4 . Now
I  ZVI  ZV  e2izZZ  eiZZ  W; (2)
where 0< jj< 2 , as required.
Simple algebra shows that W is equivalent to a con-
trolled rotation about the z axis:
eiZZ  j0ih0j  eiZ  j1ih1j  eiZ
 j0ih0j  I  j1ih1j  e2ijjZ: (3)
247902-2Note that, if necessary, we can obtain a positive exponent
in the last line by conjugating by I  X. We introduce the
following notation for a controlled rotation about an
arbitrary axis defined by a vector n, with the direction
specified by n^  n=jnj and the angle of rotation deter-
mined by jnj:
Un  j0ih0j  I  j1ih1j  einX;Y;Z: (4)
In particular, the controlled rotation (3) above is denoted
U0;0;2jj. Conjugation by one-qubit gates on the second
qubit changes the axis of rotation but not the angle of
rotation: Given n0 such that jnj  jn0j, we can find a
one-qubit gate A such that I  AUnI  Ay  Un0 . A
product of two rotations Un and Un0 is clearly another
controlled rotation Um. Both the direction of m and its
magnitude vary depending on n and n0.
In order to implement a CNOT, we need to use U0;0;2jj
to obtain a total rotation U0;0;=2  CNOT. The first step
is to use U0;0;2jj a number of times q  b=22jjc. If =2 is
an exact multiple of 2jj, then we are done. Otherwise,
we must generate a gate to make up the difference; i.e.,
we need to obtain Um with 0< jmj  2  2qjj< 2jj.
To do this, we note that we can easily obtain the follow-
ing controlled rotations: the zero rotation, U0;0;0 
U0;0;2jjU0;0;2jj, and U0;0;4jj  U0;0;2jjU0;0;2jj.
Choose n such that jnj  2jj in which case Un is
equivalent to U0;0;2jj. The product U0;0;2jjUn gives
another controlled rotation Um. jmj varies continuously
as a function of n and so, by the intermediate value
theorem, it must pass through all the angles between 0
and 4jj. As a consequence, it is possible to choose n
such that jmj  2  2qjj. For any given angle , n can
be calculated numerically as the solution to a small set of
equations (these can be found in exercise 4.15 in [11], see
also [18]) [19]. Therefore, since U0;0;jmj  Um, the final
sequence is
U0;0;=2  Uq0;0;2jjI  AUmI  Ay; (5)
where A is an appropriate one-qubit gate.
This completes our proof, since it demonstrates that the
imprimitive gate U together with one-qubit gates can be
used to implement a CNOT, which, in turn, can be used to
perform universal quantum computation [20]. 
It is easy to explore some examples of our procedure
using [14]. As an example, suppose we had a gate whose
canonical decomposition yielded U  ei=6ZZ. Then
A1UA2UA3  CNOT where the gates Aj are primitive:
A1  10 0i
h i
 eiBeiY;
A2  I  ei=6ZeiY; A3  I  ei=6ZeiB;
B 
 
3
5
q
Z

2
5
q
Y

; (6)
where   12 cos113 and   12 cos1 16p .
We conclude with a discussion of the optimality of the
scheme for universal quantum computation described in
our proof. We need to answer two questions: What is the247902-2
VOLUME 89, NUMBER 24 P H Y S I C A L R E V I E W L E T T E R S 9 DECEMBER 2002‘‘optimal’’ use of a given gate U? How optimal is our
scheme? We define a scheme to be optimal if it uses U the
minimal number of times required to implement a CNOT,
with arbitrary one-qubit gates. We will see that, although
our scheme is slightly nonoptimal in usage of the two-
qubit interaction, the number of one-qubit gates used by
our scheme is many orders of magnitude smaller than the
number required to achieve accuracies sufficient for fault-
tolerant quantum computation in the Hamiltonian simu-
lation schemes described at the beginning of this Letter.
It follows from [12] (section D1) that the number of
uses of U required to implement the CNOT in any protocol
using only U and one-qubit gates is bounded below by

4max
where max  maxfjxj; jyj; jzjg.
To compare with our scheme, we obtain an estimate of
the number of uses of U required to implement a CNOT
using our scheme. Recall that we use a controlled rotation
U0;0;2jj q  b 8maxc times to implement a CNOT. (Recall
that we take z  max.) Each controlled rotation uses U
twice, in general (the special cases follow along similar
lines with small changes in the number of uses of U), and
the corrections at the end can require up to four uses of U.
Thus, the CNOT uses U 2q 4 times. The ratio of the
number of uses of U required by our scheme to the
minimum possible number is therefore less than 1
16max=, which is between 1 (for small max) and 5
(for large max). Numerical results suggest that, in prac-
tice, the number of uses of U in our scheme is either
exactly optimal or one greater than the optimal number.
Returning to the comparison of our result with those on
optimal simulation of Hamiltonians [7,8], note that our
fixed given gate U can be thought of as a fixed given
Hamiltonian which always evolves for the same amount
of time between applications of one-qubit gates. Although
our procedure is slightly nonoptimal in the number of
uses of U for large max, the payoff in terms of error
control is enormous. In general, we require only approxi-
mately 6q one-qubit gates, and q depends only on the gate
U, not on the desired accuracy. In the example given
above, only four one-qubit gates are required, compared
to the unbounded number required to achieve arbitrary
accuracy in the Hamiltonian simulation procedures.
We have given a simple algorithm [14] which provides
a near-optimal way of using an arbitrary two-qubit
entangling interaction to do universal quantum computa-
tion. Our scheme makes relatively undemanding require-
ments on local control and, thus, is likely to be
experimentally practical. Our scheme inverts the usual
challenge facing the designer of a quantum computer:
Instead of having to do delicate, system-specific theoreti-
cal calculations to engineer systems to perform gates such
as the CNOT, it will now be possible for physicists to
experimentally determine the character of the available
interaction and then apply our algorithm to use that
interaction to do universal quantum computation.247902-3We thank Tamyka Bell, Carl Caves, Tim Ralph, and
Ru¨diger Schack for helpful comments and suggestions.
A.W. H. thanks the Centre for Quantum Computer
Technology at the University of Queensland for its hos-
pitality and acknowledges support from Army Research
Office. A. G. was supported by The New Zealand
Foundation for Research, Science and Technology.[1] A. Barenco, C. Bennett, R. Cleve, D. DiVincenzo,
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[10] The accuracy is measured by the metric induced by the
operator norm—see [4] for details.
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0205100.
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G. Chen, Computational Mathematics (Chapman & Hall/
CRC Press, Boca Raton, 2002), Chap. II.
[14] C. M. Dawson and A. Gilchrist, online implementation of
the procedure described herein, http://www.physics.uq.
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[15] N. Khaneja, R. Brockett, and S. J. Glaser, Phys. Rev. A
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[16] B. Kraus and J. I. Cirac, Phys. Rev. A 63, 062309 (2001).
[17] Noting that V6  Y  Y allows the number of
uses of V to be reduced to 2: Vei=4XIV7 
Vei=4XIY  YV.
[18] J. Preskill, Physics 229: Advanced Mathematical
Methods of Physics—Quantum Computation and
Information (California Institute of Technology,
Pasadena, CA, 1998), http://www.theory.caltech.edu/
people/preskill/ph229/
[19] The equations are cosjmj  cos22  sin22jjz^  n^
and sin  jm j m^  sin 2 j j  cos 2   z^  n  
sin22jjn^  z^. These are easily solved using standard
numerical techniques.
[20] Note that our construction could be easily modified in
order to simulate any desired two-qubit gate.247902-3


A practical scheme for quantum computation with any two-qubit entangling gate

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