a r X i v : q u a n t  p h / 0 5 1 1 0 1 9 v 4 2 0 A u g 2 0 0 6
Synthesis of multiqudit Hybrid and
d
valuedQuantum Logic Circuits by Decomposition
Faisal Shah Khan
a
,
∗
, Marek Perkowski
b
a
Portland State University, Department of Mathematics and Statistics, Portland,Oregon 972070751, USA
b
Portland State University, Department of Electrical and Computer Engineering,Portland, Oregon 972070751, USA
Abstract
Recent research in generalizing quantum computation from 2valued qudits to
d
valued qudits has shown practical advantages for scaling up a quantum computer.A further generalization leads to quantum computing with
hybrid
qudits wheretwo or more qudits have diﬀerent ﬁnite dimensions. Advantages of hybrid and
d
valued gates (circuits) and their physical realizations have been studied in detail byMuthukrishnan and Stroud (Physical Review A, 052309, 2000), Daboul et al. (J.Phys. A: Math. Gen. 36 25252536, 2003), and Bartlett et al (Physical Review A,Vol.65, 052316, 2002). In both cases, a quantum computation is performed whena unitary evolution operator, acting as a quantum logic gate, transforms the stateof qudits in a quantum system. Unitary operators can be represented by squareunitary matrices. If the system consists of a single qudit, then Tilma et al (J. Phys.A: Math. Gen. 35 (2002) 1046710501) have shown that the unitary evolution matrix(gate) can be synthesized in terms of its Euler angle parametrization. However, if the quantum system consists of multiple qudits, then a gate may be synthesizedby matrix decomposition techniques such as QR factorization and the CosinesineDecomposition (CSD). In this article, we present a CSD based synthesis method for
n
qudit hybrid quantum gates, and as a consequence, derive a CSD based synthesismethod for
n
qudit gates where all the qudits have the same dimension.
Key words:
Hybrid Quantum Logic Synthesis, CosineSine Decomposition, Givensrotations, Quantum Multiplexers
PACS:
903.67.Lx, 03.65.Fd 03.65.Ud
∗
Corresponding author.
Email addresses:
faisal@pdx.edu
(Faisal Shah Khan),
mperkows@ece.pdx.edu
(Marek Perkowski).
Preprint submitted to Elsevier Science 1 February 2008
1 Introduction
A
qudit
replaces a classical dit as an information unit in
d
valued quantumcomputing. A qudit is represented as a unit vector in the state space, which is acomplex projective
d
dimensional Hilbert space,
H
d
. In the computational basis, the basis vectors of
H
d
are written in Dirac notation as

0
,

1
,...

d
−
1
,where

i
= (0
,
0
,...,
1
,...,
0)
T
with a 1 in the (
i
+ 1)st coordinate, for0
≤
i
≤
(
d
−
1). An arbitrary vector

a
in
H
d
can be expressed as a linear combination

a
=
d
−
1
i
=0
x
i

i
,
x
i
∈
C
and

x
i

2
= 1. The real number

x
i

2
is the probability that the state vector

a
will be in
i
th basis state uponmeasurement.When the state spaces of
n
qudits of diﬀerent
d
valued dimensions are combined via their algebraic tensor product, the result is a
n
qudit
hybrid
statespace
H
=
H
d
1
⊗H
d
2
⊗···⊗H
d
n
, where
H
d
i
is the state space of the
d
i
valuedqudit. The computational basis for
H
would consist of all possible tensor products of the computational basis vectors of the component state spaces
H
d
i
. If
d
i
=
d
for each
i
, the resulting state space
H
⊗
nd
is that of
n d
valued qudits.The
evolution
of state space changes the state of the qudits via the action of a unitary operator on the qudits. A unitary operator can be represented by aunitary evolution matrix. For the hybrid state space
H
, an evolution matrixwill have size (
d
1
d
2
...d
N
)
×
(
d
1
d
2
...d
N
), while the evolution matrix for
H
⊗
nd
will be of size
d
n
×
d
n
. In the context of quantum logic synthesis, an evolutionmatrix is a quantum logic circuit that needs to be realized by a universal setof quantum logic gates. It is well established that sets of one and two quditquantum gates are universal [3,5,10,15]. Hence, the synthesis of an evolutionmatrix requires that the matrix be decomposed to the level of unitary matricesacting on one or two qudits.Unitary matrix decomposition methods like the QR factorization and the Cosine Sine decomposition from matrix perturbation theory have been used for2valued and 3valued quantum logic synthesis. In these domains, qudits arereferred to as
qubits
and
qutrits
respectively. The Cosine Sine decomposition(CSD) of a unitary matrix, discussed in section 2, has been used by M¨ott¨onenet. al [9] and Shende et. al [12] to iteratively synthesize multiqubit quantumcircuits. The authors of this article recently extended the CSD to iteratedsynthesis of 3valued quantum logic circuits acting on
n
qutrits [8]. Bullocket.al have recently presented a synthesis method for
n
qudit quantum logicgates using a variation of the QR matrix factorization [4]. This article presentsa CSD based method for synthesis of
n
qudit hybrid and
d
valued quantumlogic gates.2
2 The CosineSine Decomposition (CSD)
Let the unitary matrix
W
∈
C
m
×
m
be partitioned in 2
×
2 block form as
W
=
r m
−
rr W
11
W
12
m
−
r W
21
W
22
(2.0.1)with 2
r
≤
m
. Then there exist
r
×
r
unitary matrices
U
and
X
,
r
×
r
realdiagonal matrices
C
and
S
, and (
m
−
r
)
×
(
m
−
r
) unitary matrices
V
and
Y
such that
W
=
U
00
V
C
−
S
0
S C
00 0
I
m
−
2
r
X
00
Y
(2.0.2)The matrices
C
and
S
are the socalled cosinesine matrices and are of theform
C
= diag(cos
θ
1
,
cos
θ
2
,...,
cos
θ
r
),
S
= diag(sin
θ
1
, sin
θ
2
,...,
sin
θ
r
) suchthat sin
2
θ
i
+cos
2
θ
i
= 1 for some
θ
i
, 1
≤
i
≤
r
[13]. Algorithms for computingthe CSD and the angles
θ
i
are given in [2,14]. The CSD is essentially thewell known singular value decomposition of a unitary matrix implementedat the block matrix level [11]. In sections 3 and 4, we give an overview of the CSD based synthesis methods of 2 and 3valued quantum logic circuits,respectively. From now on, we will not distinguish between gates, circuits andtheir corresponding unitary matrices.
3 Synthesis of 2valued (binary) Quantum Logic Circuits
As shown in [8,9,12,16], the CS decomposition gives a recursive method forsynthesizing 2valued and 3valued
n
qudit quantum logic gates. In the 2valued case the CSD of a 2
n
×
2
n
unitary matrix
W
reduces to the form
W
=
U
00
V
C
−
S S C
X
00
Y
(3.0.1)with each block matrix in the decomposition of size 2
n
−
1
×
2
n
−
1
.In terms of synthesis, the block diagonal matrices in (3.0.1) are
quantum multiplexers
[12]. A quantum multiplexer is a gate acting on
n
qubits of whichone is designated as the control qubit. If the control qubit is the highest orderqubit, the multiplexer matrix is block diagonal. Depending on whether thecontrol qubit carries

0
or

1
, the gate then performs either the top left block3
• ∼
= +1
•
+1
•
U
0
M / / U
0
U
1
U
1
Fig. 1.
2valued Quantum Multiplexer
M
controlling the lower (
n
−
1) qubits by the top qubit. The slashsymbol (/) represents (
n
−
1) qubits on the second wire. The gates labeled +1 are shifters (inverters in2valued logic), increasing the value of the qubit by 1 mod 2 thereby allowing for control by the highestqubit value. Depending on the value of the top qubit, one of
U
t
is applied to the lower qubits for
t
∈ {
0
,
1
}
.
or the bottom right block of the
n
×
n
block diagonal matrix on the remaining(
n
−
1) qubits. A circuit diagram for a
n
qubit quantum multiplexer with thehighest order control qubit is given in ﬁgure 1. Observe that we decomposedarbitrary quantum multiplexer to single qubit gates and
n
qubit standard controlled gates. The controlled gates execute the operator in the box when thecontrolling qubit has values 1 (mod 2). Such a quantum multiplexer can beexpressed as
U
0
00
U
1
(

a
1
⊗
a
2
⊗···⊗
a
n
) (3.0.2)where

a
i
is the
i
th qubit in the circuit, and both block matrices
U
0
and
U
1
are of size 2
n
−
1
×
2
n
−
1
. Depending on whether

a
1
=

0
or

a
1
=

1
, theexpression (3.0.2) reduces to

0
⊗
U
0
(

a
2
⊗
a
3
⊗···⊗
a
n
) (3.0.3)or

1
⊗
U
1
(

a
2
⊗
a
3
⊗···⊗
a
n
) (3.0.4)respectively.The cosinesine matrix in (3.0.1) is realized as a
uniformly
(
n
−
1)
controlled
R
y
rotation gate
, a variation of the quantum multiplexer. As shown in ﬁgure 2,a uniformly (
n
−
1)controlled
R
y
rotation gate
R
y
is composed of a sequence of (
n
−
1)fold controlled gates
R
θ
i
y
, all acting on the highest order qubit, where
R
θ
i
y
=
cos
θ
i
−
sin
θ
i
sin
θ
i
cos
θ
i
.
(3.0.5)The control selecting the angle
θ
i
in the gate
R
θ
i
y
depends on which of the(
n
−
1) basis state conﬁgurations the control qubits are in at that particularstage in the circuit. In ﬁgure 2, the open controls represent the basis state

0
and a ﬁlled in control represents basis state

1
. The
i
th (
n
−
1)controlled4
R
y
∼
=
R
θ
0
y
R
θ
1
y
R
θ
2
y
...R
θ
(2
n
−
1)
−
1
y
/
...
•
...
•
... ... ... ... ...
•
...
•
•
...
•
Fig. 2.
A uniformly (
n
−
1)controlled
R
y
rotation for 2valued quantum logic. The lower (
n
−
1) qubitsare the control qubits represented on the left hand side by the symbol / on the second wire. The
◦
controlturns on for control value

0
and the
•
control turns on for control value

1
. It requires 2
n
−
1
one qubitcontrolled gates
R
θ
i
y
to implement a uniformly (
n
−
1)controlled
R
y
rotation.
∼
= +1
• −
1
Fig. 3.
A control by input value 0 (mod 2) realized in terms of control by the highest value 1 (mod 2).
R
y
∼
=
R
θ
0
y
R
θ
1
y
R
θ
2
y
R
θ
3
y
• •
•
•
Fig. 4.
A uniformly 2controlled
R
y
rotation in 2valued logic: the lower two qubits are the control qubits,and the top bit is the target bit.
gate
R
θ
i
y
may be expressed as
cos
θ
i
−
sin
θ
i
sin
θ
i
cos
θ
i

a
1
⊗
(

a
2
⊗···⊗
a
n
) (3.0.6)with
θ
i
taking on values from the set
{
θ
0
,θ
1
,...,θ
2
n
−
1
−
1
}
depending on theconﬁguration of (

a
2
⊗···⊗
a
n
), resulting in a speciﬁc
R
θ
i
y
for each
i
.As an example, consider the 3 qubit uniformly 2controlled
R
y
gate controllingthe top qubit from ﬁgure 4. Then the action of
R
θ
i
y
on the circuit is
cos
θ
i
−
sin
θ
i
sin
θ
i
cos
θ
i

a
1
⊗
(

a
2
⊗
a
3
) (3.0.7)with
θ
i
∈ {
θ
0
,θ
1
,θ
2
,θ
3
}
. As

a
2
⊗
a
3
takes on the values from the set
{
0
⊗
0
,

0
⊗
1
,

1
⊗
0
,

1
⊗
1
}
in order, the expression in (3.0.7) reduces to the following 4 expressions respectively.
cos
θ
0
−
sin
θ
0
sin
θ
0
cos
θ
0

a
1
⊗
(

0
⊗
0
) (3.0.8)5