Authors

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• Authors
• Faisal Shah Khan
• Marek A. Perkowski

Slides prepared by Faisal Shah Khan
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Overview
? A qudit replaces a classical dit as an
information unit in d-valued quantum computing. ?
A qudit represented as a unit vector in the
state space, which is a complex projective
d-dimensional Hilbert space, ? In the
computational basis, the basis vectors of
are written in Dirac notation as where
1 is in the i-th position
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Overview
? An arbitrary vector in can be
expressed as a linear combination ? The real
number is the probability that the state
vector will be in i-th basis state upon
measurement. ? 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 state space where is the
state space of the qudit.
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Overview
? The computational basis for H will consist of
all possible tensor products of the computational
basis vectors of the component state spaces
. ? If all the different are assigned the
same value d, the resulting state space
is that of n d-valued qudits. ? The
evolution of state space changes the state of the
qudits via the action of a unitary (length
preserving) operator on the qudits. ? A unitary
operator can be represented by a square unitary
evolution matrix. ? For the hybrid state space
H, an evolution matrix will have size
while the evolution
matrix for will be of size
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Overview
? In the context of quantum logic synthesis, an
evolution matrix is a quantum logic circuit that
needs to be realized by a universal set of
quantum logic gates. ? It is well established
(Brylinski, Muthukrishnan) that sets of one and
two qudit quantum gates are universal. Hence, the
synthesis of an evolution matrix requires that
the matrix be decomposed to the level of unitary
matrices acting on one or two qudits. ? Unitary
matrix decomposition methods like the QR
factorization and the Cosine Sine decomposition
(CSD) from matrix perturbation theory have been
used for 2-valued (binary) and 3-valued (Ternary)
quantum logic synthesis. ? Mottonen et. al,
Shende et. al Binary CSD sythesis ? Khan and
Perkowski Ternary CSD synthesis
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Cosine-Sine Decomposition (CSD)
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n-qubit (Binary) Quantum Logic Synthesis via CSD

• ? In this case, unitary W matrices are of size
  .
• ? Let so that
• ? Now the CSD gives W decomposed as
• ? Each block in the block matrices of the CSD of
  W are of size

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n-qubit (Binary) Quantum Logic Synthesis via CSD
(continued)

• ? Note that the CSD can be iteratively applied
  to the unitary block diagonals that occur in the
  decomposition at each stage.
• ? The iteration stops when the block are of size
  2 x 2.
• ? However, there may be local optimizations
  involving a CNOT (4x4 matrix) and 2 x 2 gates.

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n-qubit (Binary) Quantum Logic Synthesis via CSD
(continued)
? Shende et al and Mottonen et al give the
following realization of the factors in the CSD
at each iterative level a) Block diagonal
matrices are Quantum Multiplexers. b) The
cosine-sine matrices are uniformly controlled
rotations, a variation of the multiplexer.
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n-qubit Quantum Multiplexer
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Uniformly (n-1)-controlled rotation
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n-qubit Quantum Multiplexer in Dirac / Matrix
notation
(M-1)
where is the i-th qubit in he circuit, and
both block matrices are of
size ? Depending on whether
, (M-1) reduces to
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(n-1)-controlled rotation in Dirac /
Matrix notation
(R-1)
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n-qubit (Binary) Quantum Logic Synthesis via CSD
(continued)
A 2-qubit quantum multiplexer
U
M
U
0
V
1
V
A 2-qubit uniformly 1-controlled rotation
R0
R1
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n-qubit (Binary) Quantum Logic Synthesis via CSD
(continued)
Example 1. A 2-qubit quantum multiplexer matrix.
The first qubit controls the second.
? If the first qubit is 0, then U is applied to
the second qubit. ? If the first qubit is 1, then
V is applied to the second qubit. ? For n-qubits,
a quantum multiplexer will control the lowest
(n-1) qubits via the top qubit.
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n-qubit (Binary) Quantum Logic Synthesis via CSD
(continued)

• Example 2.
• A 2-qubit uniformly controlled rotation matrix.

This 4 x 4 matrix acts on the tensor product of
the two qubits
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n-qubit (Binary) Quantum Logic Synthesis via CSD
(continued)
Calculations give
(1)
Let Then (1) becomes
Let Then (1) becomes
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n-qubit (Binary) Quantum Logic Synthesis via CSD
(continued)
? A uniformly controlled rotation is essentially
a multiplexer. ? In our example, the bottom qubit
controls the top. ? For n-qubits, a uniformly
controlled rotations is a multiplexer in which
the lowest qubit controls the top (n-1) qubits.
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n-qutrit (Ternary) Quantum Logic Synthesis via CSD

• ? In this case, unitary matrices W are of size
• ? Let so that
• ? Now the CSD gives W decomposed as
• ? The top corner blocks in the diagonal matrices
  are of size
• ? The lower corner blocks, are of
  size
• Both C and S matrices are of size

(2)
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n-qutrit (Ternary) Quantum Logic Synthesis via
CSD (continued)
Uniformly controlled rotation around z-axis in
Uniformly controlled rotation around x-axis in
Multiplexer
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n-qutrit Quantum Multiplexer
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(n-1)-controlled R_x rotation
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d-valued Quantum Logic Synthesis vis CSD
? Consider the matrix
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