Synthesis of Hybrid and d valued Quantum Logic Circuits

1
Synthesis of Hybrid and d valued Quantum Logic
Circuits
Faisal Shah Khan Department of Mathematics and
Statistics Portland State University
Marek Perkowski Department of Electrical and
Computer Engineering Portland State University
2
Quantum computing

• A quantum computational unit, a qudit, is a state
  vector in the state space, which is a complex,
  projective d- dimensional Hilbert space.
• The Hilbert space is a model for a 2
  dimensional complex Hilbert space with
  computational basis vectors

3
Quantum computing

• Any vector in can be written as a complex
  linear combination of and
• where and are complex numbers.
• Axioms of quantum mechanics require that
• This can be achieved by projectifying

4
Quantum computing

• Projectifying requires that a vector in
  this space and all its non-zero scalar multiples
  be identified

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Quantum computing

• Generalizing the case of the qubit to d
  dimensional complex projective Hilbert space
  , we can write a vector in as
• where

Each of is a complex number
Computational basis. 1 is in the (i1)-st position
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Quantum computing

• A quantum computation is performed when a qudit
  evolves under the action of a special unitary
  operator. Such an operator is interpreted as a
  quantum logic gate.
• For a finite dimensional state space, a special
  unitary operator may be represented as a special
  unitary matrix in the computational basis.

7
Quantum computing

• A special unitary matrix has determinant 1.
• In the case of a single qubit, a quantum logic
  gate would be a special unitary matrix

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Quantum computing

• When n qudits interact, then their combined state
  space is the tensor product space
• The computational basis of this state space is
  generated by the set of all possible tensor
  products of the basis vectors of each component
  space .

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Quantum computing

• Logic gates are then special unitary
  matrices.
• For a quantum logic gate will be a
• special unitary matrix.
• The question of synthesis of such logic gates in
  terms of some set of universal quantum gates
  arises naturally.

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Universal Quantum Logic Gates

• Consider the one qubit NOT gate

The NOT (Shift) gate. Flips the state of the
qubit from to and vice versa.
The minus sign can be ignored due to
projectification.
The special unitary matrix corresponding to the
NOT gate.
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Universal Quantum Logic Gates

• For two qubits, consider the controlled-NOT
  (CNOT) gate.

Control is assumed to be via the highest signal
value, 1 mod 2.
1
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Universal Quantum Logic Gates
Two cases of matrix multiplication shown below
convince us that this is the correct matrix for
CNOT. Other two cases, with projectification in
mind, can be confirmed as well.
1
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Universal Quantum Logic Gates

• It is well established that a set consisting of
  one qubit gates and CNOT gate is universal in the
  sense that any quantum logic gate can be
  approximated to arbitrary accuracy by gate in
  such a set.
• This is in fact true for not just qubits, but
  also for qudits (Brylinski).
• Hence, what is needed is a systematic method to
  synthesize an n qudit quantum gate in terms of
  universal gates.

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Quantum Logic synthesis

• A systematic procedure for such a synthesis of a
  n qudit gate comes form the Cosine-sine
  decomposition (CSD) of the corresponding
  unitary matrix.
• The CSD should be performed until one reaches the
  level of some set of one and two qudit gates.

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2-valued Quantum Logic synthesis
CS decomposition of n qubit quantum gate W
(n-1) - uniformly controlled rotation
Control is assumed to be via the highest signal
value, 1 mod 2.
CS
W
1
1

M2
M1
U0
V0
U1
V1
This wire represents lower (n-1) wires
Quantum Multiplexers
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2-valued Quantum Logic synthesis
An n qubit quantum multiplexer
1
M
Gate U1 is selected if the signal is 1.
U0
U1
Gate U0 is selected if the signal is 0.
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2-valued Quantum Logic synthesis
(n-1) - uniformly controlled rotation for n qubits
CS
0
0
0
0
1
0
0
1
1
1
0
1
0
1
1
0
1
0
1
0
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2-valued Quantum Logic synthesis
Cosine Sine Decomposition

• Given a unitary matrix W, partition W
  into 2 x 2 block form as follows

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Quantum Logic synthesis

• Then W can be decomposed as
• have size and are
  unitary.
• have size
  and are unitary.

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Quantum Logic synthesis

• The cosine-sine matrix
• consists of real diagonal
  matrices
• of the form

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2-valued Quantum Logic synthesis

• For d 2, the size of all the blocks
  and
• in the CSD is the same, namely

Apply the gate U1.
Apply the gate U0.
1
M
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2-valued Quantum Logic synthesis

• The cosine sine matrix in the binary case takes
  on the form
• It is realized as a (n-1) uniformly controlled
  rotation.

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2-valued Quantum Logic synthesis
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2-qubit Example
A 2-qubit quantum multiplexer
1
U
M
U
V
V
A one - uniformly controlled rotation
R0
R1
0
1
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2-qubit Example
(1)
Let Then RHS of (1) becomes
Let Then RHS of (1) becomes
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Hybrid Quantum Logic synthesis

• For d values other than 2, CSD needs to be
  iterated laterally first, and then top to bottom.
• Lateral iteration arises due to the occurrence of
  blocks different sizes when CSD is applied to go
  from n qudits to (n-1) qudits.
• Using the idea of lateral iteration, CSD can in
  fact be generalized to hybrid quantum systems.

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Hybrid Quantum Logic synthesis

• If qudits are allowed to have different
  dimensions, then we get the state space
• , each space of
  a different dimension.
• Quantum logic gates are then special unitary
  matrices of size

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Hybrid Quantum Logic synthesis
CSD for hybrid n qudit quantum logic gates

• Choose a qudit in the hybrid system as the
  control qudit say, .
• Reorder the basis of to
  get the isomorphic state space
  
• so that the
  control qudit is the highest order qudit.
• Apply CSD.

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Hybrid Quantum Logic synthesis
CSD algorithm for Hybrid quantum logic synthesis
Iterated lateral decomposition
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Hybrid Quantum Logic synthesis
CS decomposition of a Hybrid n qudit quantum gate
W
(n-1) - uniformly controlled Givens rotation
Control is assumed to be via highest signal value
W
G

M
M
Quantum Multiplexers
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Hybrid Quantum Logic synthesis
A hybrid n qudit quantum multiplexer
0
1
M
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Hybrid Quantum Logic synthesis
Hybrid (n-1) uniformly controlled Givens
rotation
Target qudit of dimension .
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Hybrid Example
An arbitrary 6 valued hybrid Quantum logic Gate
2-valued (binary) control qudit wire
3-valued (ternary) wire
1-Uniformly controlled rotation
2-valued Quantum Multiplexer
1
1
1
M
M
M
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Hybrid Example
An arbitrary 6 valued hybrid Quantum logic Gate
3-valued (ternary) wire
2-valued (binary) wire
0
1
2
0
1
2
0
1
2
2
2
2
2
2
2
2
2
2
M
M
M
1
0
1
0
1
0
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Future Research

• Code Hybrid CSD and establish benchmarks for
  various d-valued and hybrid systems.
• Explore connections between quantum (Parrondo)
  games and CSD Logic synthesis.