We consider the implementation of two-qubit unitary transformations by means
of CNOT gates and single-qubit unitary gates. We show, by means of an explicit
quantum circuit, that together with local gates three CNOT gates are necessary
and sufficient in order to implement an arbitrary unitary transformation of two
qubits. We also identify the subset of two-qubit gates that can be performed
with only two CNOT gates.Comment: 3 pages, 7 figures. One theorem, one author and references added.
  Change of notational conventions. Minor correction in Theorem 

Dawson, C. M.

Vidal, G.

English

arXiv

We consider quantum circuits made of controlled-NOT (CNOT) gates and single-qubit unitary gates and look for constructions that minimize the use of CNOT gates. We show, by means of an explicit quantum circuit, that three CNOT gates are necessary and sufficient in order to implement an arbitrary unitary transformation of two qubits. We also identify the subset of two-qubit gates that can be performed with only two CNOT gates and provide a simple characterization for them. 5555 2004 The American Physical Society

University of Queensland eSpace

Universal quantum circuit for two-qubit transformations with three controlled-NOT gates

We consider quantum circuits made of controlled-NOT (CNOT) gates and single-qubit unitary gates and look for constructions that minimize the Use Of CNOT gates. We show, by means of an explicit quantum circuit, that three CNOT gates are necessary and sufficient in order to implement an arbitrary unitary transformation of two qubits. We also identify the subset of two-qubit gates that can be performed with only two CNOT gates and provide a simple characterization for them

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PHYSICAL REVIEW A 69, 010301~R! ~2004!Universal quantum circuit for two-qubit transformations with three controlled-NOT gates
G. Vidal1 and C. M. Dawson2
1Institute for Quantum Information, California Institute of Technology, Pasadena, California 91125, USA
2Center for Quantum Computer Technology and Department of Physics, The University of Queensland, Brisbane 4072, Australia
~Received 12 August 2003; published 8 January 2004!
We consider quantum circuits made of controlled-NOT ~CNOT! gates and single-qubit unitary gates and look
for constructions that minimize the use of CNOT gates. We show, by means of an explicit quantum circuit, that
three CNOT gates are necessary and sufficient in order to implement an arbitrary unitary transformation of two
qubits. We also identify the subset of two-qubit gates that can be performed with only two CNOT gates and
provide a simple characterization for them.
DOI: 10.1103/PhysRevA.69.010301 PACS number~s!: 03.67.Lx, 03.67.MnIn the context of establishing the existence of universal
sets of two-qubit gates for quantum computation @1#,
Barenco et al. @2# showed that any unitary transformation on
n qubits can be decomposed into a sequence of controlled-
NOT ~CNOT! and single-qubit gates. Since then it is customary
in quantum information to use CNOT and single-qubit gates in
order to express any unitary transformation in the quantum
circuit model @3#. As a result, the CNOT gate has acquired a
special status as the standard hallmark of multiqubit control.
Among quantum information experimentalists, achieving a
CNOT gate is one of the most coveted goals @4#. In turn,
exhaustive theoretical studies on the optimal use of two-
qubit interactions and of entangling gates to perform a CNOT
gate have been conducted @5–8#.
Here we shall consider the construction of quantum cir-
cuits that minimize the use of CNOT gates. Such optimal con-
structions are relevant in two separated scenarios. First, they
play a role in determining the algorithmic complexity of a
given quantum computation, that is, the number of elemen-
tary gates required to implement the corresponding n-qubit
unitary evolution. A most remarkable result of Ref. @2# is the
explicit decomposition of an arbitrary UPU(2n) as a se-
quence of CNOT and single-qubit gates. This general con-
struction, however, unavoidably requires exp(n) CNOT gates,
which renders the resulting quantum circuit inefficient, while
the transformations relevant for quantum computation are
precisely those that can be decomposed into only poly(n)
elementary gates. Thus, given a unitary transformation U
PU(2n), in quantum computation it is important to deter-
mine how many CNOT gates are required for its implementa-
tion.
Algorithmic complexity is typically concerned with the
asymptotic scaling of computational resources, and thus with
gates involving a large number of qubits. Here, instead, we
shall analyze unitary gates of just two qubits, n52. Quan-
tum circuits that minimize the use of CNOT gates are also of
interest in this case, but for another, more practical reason. In
present day experiments, two-qubit gates as the CNOT gate
are implemented in an imperfect way due to technological
limitations. Therefore, in order to minimize the probability
that an error occurs in performing a certain unitary evolution
U on n52,3, . . . qubits, it is instrumental that the number of
times the qubits interact is as small as possible. Unfortu-1050-2947/2004/69~1!/010301~4!/$22.50 69 0103nately, there is a general lack of results concerning how to
optimally decompose UPU(2n) into CNOT and single-qubit
gates.
In this paper we describe a universal quantum circuit for
two-qubit unitary transformations UPSU(4) consisting of
only three CNOT gates and four rounds of local gates. The
shortest circuit previously known requires four CNOT gates
@9#. In addition, we show that three CNOT gates are necessary
in order to perform a generic two-qubit gate, thereby estab-
lishing the optimality of the proposed universal quantum cir-
cuit. We also characterize the subset of two-qubit gates
whose implementation requires only two CNOT gates and
construct an alternative, smaller quantum circuit for them. In
this way we give a complete classification of two-qubit gates
in terms of their CNOT complexity.
We consider two qubits, labeled A and B, and an arbitrary
unitary transformation UPSU(4). Let ul ,v lPSU(2) denote
single-qubit unitary gates acting, respectively, on qubits A
and B, and let UCNOT denote a CNOT gate that has qubit A as
control and qubit B as target,
UCNOTuzm&A ^ uzn&B5uzm&A ^ uzn % m&B , m ,n50,1,
~1!
where i % j denotes sum modulo 2 and uz0& (uz1&) is the
eigenvector with eigenvalue 1 (21) of the third of the Pauli
matrices
sx5S 0 11 0 D , sy5S 0 2ii 0 D , sz5S 1 00 21 D . ~2!
Theorem 1. An arbitrary unitary gate UPSU(4) can be de-
composed in terms of three CNOT gates and single-qubit uni-
tary gates ul ,v l ~to be specified below! as
~3!
An important element in order to prove Theorem 1 is the
decomposition of UPSU(4) derived by Khaneja et al. @5#
and Kraus et al. @10#, namely,©2004 The American Physical Society01-1
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G. VIDAL AND C. M. DAWSON PHYSICAL REVIEW A 69, 010301~R! ~2004!~4!
H[hxsx ^ sx1hysy ^ sy1hzsz ^ sz , ~5!
where p/4>hx>hy>uhzu. An explicit protocol to extract the
single-qubit gates u1 ,v1 ,u48 ,v48PSU(2) and the coefficients
hx ,hy ,hzPR from U was presented in Ref. @10#. In what
follows we show that e2iH can be further decomposed as
~6!
where
u2[
i
A2
~sx1sz!e
2i(hx2p/4)sx, v2[e
2ihzsz, ~7!
u3[
2i
A2
~sx1sz!, v3[e
ihysz, ~8!
w[
I2isx
A2
, ~9!
so that u4 and v4 in Eq. ~3! are
u45u48w , v45v48w
21
. ~10!
Let us introduce the Bell basis
ug00&[
1
A2
~ u00&1u11&), ug01&[
1
A2
~ u01&1u10&),
ug10&[
1
A2
~ u00&2u11&), ug11&[
1
A2
~ u01&2u10&),
where umn& denotes uzm&A ^ uzn&B . Operator H in Eq. ~5! can
be rewritten as
H5 (
m ,n50
1
lmnugmn&^gmnu, ~11!
with lmn defined as
l00[hx2hy1hz , l01[hx1hy2hz , ~12!
l10[2hx1hy1hz , l11[2hx2hy2hz . ~13!
Then e2iH becomes
e2iH5 (
m ,n50
1
e2ilmnugmn&^gmnu. ~14!
Direct inspection shows that circuit ~6! indeed acts on the
Bell basis ugmn& as @11#01030ugmn&→e2ilmnugmn&. ~15!
Next we describe a way to check this fact. The first ~left-
most! CNOT in Eq. ~6! maps the Bell basis into a product
basis, namely,
ugmn&→uxm&A ^ uzn&B , ~16!
where ux0&[(uz0&1uz1&)/A2, ux1&[(uz0&2uz1&)/A2. The
local transformations u2 ^ v2 introduce convenient phases
e2ifmn,
fmn[~21 !mS hx1 p2 D1~21 !nhz , ~17!
into this product basis, and map the latter into a new product
basis,
uxm&A ^ uzn&B→e2ifmnuzm&A ^ uzn&B . ~18!
The second CNOT gate exchanges only two elements of the
new product basis @recall Eq. ~1!#,
uz1&A ^ uz0&B↔uz1&A ^ uz1&B , ~19!
after which local gates u3 ^ v3 switch back to the uxm&A
^ uzn&B basis and introduce more phases e2ifn8,
fn8[~21 !n11hy . ~20!
The rightmost CNOT in Eq. ~6! maps the basis uxm&A ^ uzn&B
back into the original Bell basis,
uxm&A ^ uzn&B→ugmn& , ~21!
and the final local gates w ^ w† exchange vectors ug10& and
ug11& in order to undo the permutation ~19! @and also add a
p/2 phase to each of them#, so that circuit ~6! implements
transformation ~15!. This finishes the proof of Theorem 1.
As shown in Ref. @2#, a nontrivial subset of two-qubit
unitary transformations, namely, control-V transformations
for VPU(2), can be performed by using only two CNOT
gates and single-qubit gates. For these gates, one finds hy
5hz50 in its decomposition ~4! and ~5!, so that they are
locally equivalent to a control-phase gate Uw ,
Uwuzm&A ^ uzn&B5e2imnwuzm&A ^ uzn&B , ~22!
for an arbitrary phase w . Theorem 2 characterizes the set of
all two-qubit transformations that can be performed with
only two CNOT gates. They correspond to hz50 in Eqs. ~4!
and ~5!, and are therefore a subset of zero measure in the
space of two-qubit gates.
Theorem 2. A two-qubit gate U¯ PSU(4) can be decom-
posed in terms of two CNOT gates and single-qubit gates
u¯ l ,v¯ l as1-2
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UNIVERSAL QUANTUM CIRCUIT FOR TWO-QUBIT . . . PHYSICAL REVIEW A 69, 010301~R! ~2004!~23!
if and only if hz50 in its decomposition ~4! and ~5!.
First we prove that if hz50, then U¯ can be decomposed
as in Eq. ~23!. Decomposition ~4! and ~5! becomes
~24!
H¯ [hx sx ^ sx1hy sy ^ sy , hx>hy>0. ~25!
One can now express e2iH¯ as
~26!
where w has been defined in Eq. ~9! and where u¯ l ,u¯ l8 ,v¯ l ,v¯ l8
in Eqs. ~23! and ~24! are related through
u¯ 15w
†u¯ 18 , v¯ 15wv¯ 18 , u¯ 35u¯ 38w , v¯ 35v¯ 38w
†
.
~27!
The validity of circuit ~26! can be checked by reasoning
similarly as we did in the proof of Theorem 1. In particular,
the first local gates wA
†
^ wB permute vectors ug10& and ug11&.
Then the leftmost CNOT gate maps the Bell basis ugmn& into
the product basis uxm&A ^ uzn&B . Gates u¯ 2 ^ v¯ 2 introduce con-
venient phases e2if¯ mn to uxm&A ^ uzn&B ,
f¯ mn[~21 !mhx2~21 !nhy . ~28!
Finally, the rightmost CNOT and wA ^ wB
† gates map the local
basis back into the original Bell basis. This shows that any U¯
with hz50 in its decomposition ~4! and ~5! can be imple-
mented with local gates and two CNOT gates.
To prove the converse we will argue that circuit ~26! is,
up to local unitary gates, the most general form of a gate
implementable with two CNOT gates, as that in Eq. ~23!. The
initial and final gates u¯ 1 ,v¯ 1 ,u¯ 3 ,v¯ 3 in circuit ~23! do not
change the parameters hx ,hy ,hz for U¯ and may be ignored.
As a rotation on the Bloch sphere, single-qubit gates can be
written as a series of three rotations around two perpendicu-
lar axes. In particular we may write
u¯ 25e
2ia1sze2iasxe2ia2sz, ~29!
v¯ 25e
2ib1sxeibsze2ib1sx ~30!
~see, for example, Theorem 4.1 in Ref. @3#!. Recalling that
the CNOT gate ~1! can be expressed as
UCNOT5
I1sz
2 ^ I1
I2sz
2 ^ sx , ~31!01030it is clear that the leftmost and rightmost exponentials in Eqs.
~29! and ~30! commute with the contiguous CNOT gates in
circuit ~26!. This implies that the hx ,hy ,hz parameters of U¯
are the same as those of the gate,
UCNOT~e2iasx ^ eibsz!UCNOT . ~32!
But up to local gates this is circuit ~26! with a5hy and b
5hx . Therefore U¯ has hz50 in the decomposition ~4! and
~5!, completing the proof of Theorem 2.
Motivated by current experimental difficulties in the real-
ization of CNOT gates, in this work we have analyzed the
construction of quantum circuits that achieve two-qubit uni-
tary transformations by using the smallest number of CNOT
gates possible. For an arbitrary n-qubit unitary transforma-
tion UPU(2n), both characterizing its exact CNOT complex-
ity and constructing an optimal quantum circuit seem very
ambitious tasks, even by numerical analysis. Surprisingly, in
the case of two-qubit transformations it has been possible to
characterize the CNOT complexity of an arbitrary gate ana-
lytically, and to express this complexity just in terms of the
coefficients (hx ,hy ,hz) of the decomposition presented in
@5,10#, as summarized in Table I.
A possible extension of the present analysis is to also
optimize the use of single-qubit unitary transformations in
the above constructions. However, in order to perform this
optimization, a sensible cost function for single-qubit gates is
first required. In a given experimental setup it could well
happen that rotations in the Bloch sphere around, say, axis x
are simpler to perform than around axis y; or it could be
easier to perform rotations by a fixed angle and axis than by
arbitrary ones. All these details need to be properly reflected
in a cost function, to be optimized in order to obtain the most
convenient quantum circuit for the given experimental setup.
However, the following two facts in such optimization are
independent of the specific experimental details: ~i! 15 inde-
pendent angles must be specified by means of local opera-
tions in the case of an arbitrary transformation UPSU(4),
and ~ii! at least three of these angles correspond to local
gates that are performed between the three CNOT gates.
Finally, one may also seek to generalize the present re-
sults by considering the CNOT complexity of specific trans-
formations of n.2 qubits. In the case of three qubits, for
instance, the Toffoli gate is known to require at most six
CNOT gates @3#, but only five CNOT gates have so far been
proved to be necessary @12#. In order to study this problem,
techniques different from the ones employed in this work
TABLE I. Number of CNOT gates necessary and sufficient for
the implementation of a two-qubit gate UPSU(2), in terms of the
vector of coefficients (hx ,hy ,hz) of Eq. ~5!, where p/4>hx>hy
>uhzu. We use h1 and h6 to denote h.0 and uhuÞ0, respectively.
(hx ,hy ,hz) CNOT complexity
(p/4,0,0) 1
(hx1 ,hy ,0)a 2
(hx1 ,hy1 ,hz6) 3
aExcept for (p/4,0,0).1-3
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G. VIDAL AND C. M. DAWSON PHYSICAL REVIEW A 69, 010301~R! ~2004!are required, since no decomposition analogous to decompo-
sition ~4! and ~5! is known for unitary transformations in-
volving more than two qubits.
G.V. thanks J. I. Cirac and J. Pachos for comments and for
hospitality at the Max Planck Institute for Quantum Optics,01030Garching, Germany, December 2001, and Debbie Leung for
important corrections in an earlier version of this paper. C.D.
acknowledges the Institute for Quantum Information, Cali-
fornia Institute of Technology for hospitality. This work was
supported by the National Science Foundation of USA under
Grant No. EIA–0086038.@1# D.P. Divincenzo, Phys. Rev. A 51, 1015 ~1995!; T. Sleator and
H. Weinfurter, Phys. Rev. Lett. 74, 4087 ~1995!; A. Barenco,
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@4# Fortschr. Phys. 48~9–11! ~2000!.
@5# N. Khaneja, R. Brockett, and S.J. Glaser, Phys. Rev. A 63,
032308 ~2001!.@6# G. Vidal, K. Hammerer, and J.I. Cirac, Phys. Rev. Lett. 88,
237902 ~2002!.
@7# M.J. Bremner et al., Phys. Rev. Lett. 89, 247902 ~2002!.
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@11# In Eq. ~15! we have neglected a physically irrelevant, global
phase eip/4 originating in that det(UCNOT)521 in Eq. ~1!, i.e.,
UCNOT„SU(4).
@12# J.L. Dodd and G. Vidal ~unpublished!.1-4


A universal quantum circuit for two-qubit transformations with three CNOT gates

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