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Quantum Logic

- Part 2 Quantum Circuits

Reversible Computation

- Developing Reversible Logic
- A short example on developing reversible logic

gates based on primitives and combinational logic

Reversible Computation

- Developing Reversible Logic
- The fundamental Theorem
- For every finite function
- f Bm ? Bn, there exists an invertible function
- f Br x Bm ?Bn x Brm-n r n, such that
- f fi
- (? r ?) (i 1, , n)

Reversible Computation

- Developing Reversible Logic
- Reversible AND gate
- We have m2 and n 1 ? r 1
- To obtain the invertible realization we develop
- Br x Bm ?Bn x Brm-n B1 x B2 ? B1x B2
- This provides the number of Source and Sink
- We need to add 1 source input and 2 sink outputs

Reversible Computation

- Developing Reversible Logic
- Reversible AND gate
- Set the Source line to a constant 0 value (1 for

a NAND) and let the Sink lines return the inputs

A (S1) and B (S2) - The developed gate is now 1-1 and onto

Reversible Computation

- Developing Reversible Logic
- Reversible OR gate
- We have m2 and n 1 ? r 1
- To obtain the invertible realization we develop
- Br x Bm ?Bn x Brm-n B1 x B2 ? B1x B2
- This provides the number of Source and Sink
- We need to add 1 source input and 2 sink outputs

Reversible Computation

- Developing Reversible Logic
- Reversible OR gate
- Set the Source line to a constant 0 value (1 for

a NOR) and let the Sink lines return the inputs A

(S1) and B (S2) - The developed gate is now 1-1 and onto

Reversible Computation

- Developing Reversible Logic
- Reversible Comb. Logic gate
- We have m4 and n 1 ? r 1
- To obtain the invertible realization we develop
- B1 x B4 ? B1x B4 We need 1 source input and 4

sink outputs

Reversible Computation

- Developing Reversible Logic
- Reversible Comb. Logic gate
- Set the Source line to a constant 0 value and let

the Sink lines return the inputs A (S1), B (S2),

C (S3), D (S4), - The developed gate is now 1-1 and onto

Reversible Computation

- Developing Reversible Logic
- Reversible Comb. Logic gate

Quantum Circuits

- Quantum Logic
- Quantum Logic was introduced in the 30s by

Birkhoff and von Neumann - They organized it top-down starting with von

Neumanns Hilbert Space formalism of quantum

mechanics - As a second step, certain entities of Hilbert

spaces are identified with propositions, partial

order relations and lattice operations

Quantum Circuits

- A Quantum Circuit
- Behavior of Quantum Circuits are governed by

quantum mechanics - Signal states are qubit vectors (Dirac Notation)
- Operations are defined by linear algebra over a

Hilbert Space and represented by unitary matrices - Quantum Circuits are required to be reversible
- Quantum Circuits do not allow loops, they are

acyclic - Quantum bits cannot be copied
- No Cloning Theorem (No Fan-Out)
- Quantum bits cannot be erased
- No Fan-In
- Number of output lines number of input lines

(rev) - Experiments have been conducted with a seven

qubit quantum circuit implementing Shors

Algorithm

Quantum Circuits

- Quantum Logic
- Hilbert Space
- Any closed linear subspace of a Hilbert space

corresponds to an elementary proposition - (i.e. The physical system has a property

corresponding to the associated closed linear

subspace) - The logical AND operation is identified with the

set theoretical intersection of two propositions

n (i.e. with the intersection of two

subspaces). It is denoted by the symbol . - For example, two propositions p and q and their

associated closed linear subspaces Mp and Mq, - Mpq xx ? Mp, x ? Mq

Quantum Circuits

- Quantum Logic
- Hilbert Space
- The logical OR operation is identified with the

closure of the linear span ? of the subspaces

corresponding to the two propositions. It is

denoted by the symbol v. - For example, two propositions p and q and their

associated closed linear subspaces Mp and Mq, - Mpvq Mp ? Mq xx ay bz, a,b ? C, y ?

Mp, z ? Mq - The symbol ? is used to indicate the closed

linear subspace spanned by two vectors - For example,
- u ? v ww au bv, a,b ? C, u,v ? H

Quantum Circuits

- Quantum Logic
- Hilbert Space
- More generally, the symbol ? indicates the closed

linear subspace spanned by two linear subspaces - For example, if u,v ? C(H), where C(H) stands

for the set of all subspaces of the Hilbert

space, then - u ? v ww au bv, a,b ? R, u,v ? C(H)
- The logical NOT operation, or the complement is

identified with the operation of taking the

orthogonal subspace -. It is denoted by the

symbol . - For example, a proposition p and its associated

closed subspace Mp, - Mp x (x,y) 0, y ? Mp

Quantum Circuits

- Quantum Logic
- Hilbert Space
- The logical implication relation is identified

with the set theoretical subset relation ?. It

is denoted by the symbol ? - For example, two propositions p and q and their

associated closed linear subspaces Mp and Mq, - p ? q ?? Mp ? Mq
- The trivial statement which is always true

(tautology) is denoted by 1. It is represented

by the entire Hilbert space H - The trivial statement which is always false is

denoted by 0. It is represented by the zero

vector (0).

Quantum Circuits

- Dirac Bra-Ket notation
- The Bra is denoted by
- The Ket is denoted by B which is a column

vector - The Bra-Ket denotes which is the

inner product of A, B

Quantum Circuits

- Dirac Bra-Ket notation
- As discussed in the presentation on Quantum

Computing, Quantum Mechanics uses Probability

Amplitude to describe superposition or qubit

state - A Hydrogen Atom with an electron in the ground

state (0) and an excited state (1) is commonly

used to describe a qubit

Quantum Circuits

- Dirac Bra-Ket notation
- For the Hydrogen atom, the electron state is

described by - ? a0 b1
- The electron exists, thus the probability must be

1, which is described by - a² b² 1
- a, b are Complex
- In addition, a or b can
- be zero, thus resulting in either
- the ground state (0) or excited
- state (1)

Quantum Circuits

- Dirac Bra-Ket notation
- Multiple qubit systems
- For a two qubit system there are 4 classical

states 00, 01, 10, 11 - The two qubit system is then described in the

Dirac notations by - ? a00 b01 c10 d11
- N qubit systems will be developed based on the

superposition of 2n classical state - In General, N qubits are represented by
- ? Si Ci i (i0, 2N-1)
- ( ? is in binary form)

Quantum Circuits

- Unitary Matrices
- A unitary matrix is a square matrix, is

invertible and has the following property - Ut U-1 Ut U I (identity)
- t stands for conjugate transpose

Quantum Circuits

- Single Qubit gates
- Classically the only non-trivial single bit gate

is the NOT gate where 0 ? 1 and 1 ? 0 - The qubit NOT is described by
- Where the quantum operations are,
- 0 ? 1 and 1 ? 0
- This is often referred to as the Pauli-X gate

Quantum Circuits

- Single Qubit gates
- This can be understood more clearly by

visualizing the actual unitary matrix describing

the single qubit system - 0 1
- ? a0 b1
- NOT
- Now the NOT operation (or Pauli X) can be

expressed as a matrix multiplication - NOT ( a0 b1 )
- b0 a1

Quantum Circuits

- Single Qubit gates
- Pauli Z or commonly referred to as Phase-Flip
- Z Z ( a0 b1 ) a0 -

b1 - Pauli Y
- Y Y ( a0 b1 ) i ( a1

- b0 )

Quantum Circuits

- Single Qubit gates
- Hadamard ( sq. root of NOT )
- H
- H ( a0 b1 )
- Other Single Bit Gates
- Rotation Phase-Shift
- R P

Quantum Circuits

- Single Qubit gates
- Construction of Pauli-Z gate
- By a certain ordering the operations of a

Hadamard and NOT the result can produce the

Pauli-Z gate - ?

Quantum Circuits

- Single Qubit gates
- Construction of NOT gate
- By the definition of a Hadamard gate (sq. root of

NOT) it should be obvious how the NOT gate can be

constructed from two Hadamard gates - ?

H

Quantum Circuits

- Single Qubit gates
- Generalization of Pauli-gates
- The product of any two Pauli matrices is either a

(scaled version) Pauli matrix or the Identity - XX ? I YX ? aZ ZX ? aY
- XY ? aZ YY ? I ZY ? aX
- XZ ? aY YZ ? aX ZZ ? I
- Since each gate must be represented by a Unitary

Matrix, and by definition of a Unitary Matrix it

is obvious that XX, YY, ZZ will all result in the

Identity - In this generalization, a, is used to represent

the scalar value associated with the resulting

Pauli matrix

Quantum Circuits

- Multiple Qubit gate
- Basic Principals
- Each qubit is represented by a horizontal line
- The most significant qubit is shown on top and

the least significant qubit is shown at the

bottom - The black dot ? depicts the control bit of the

quantum gate - The crossed circle ? depicts the target bit of

the quantum gate

Quantum Circuits

- Multiple Qubit gate
- Basic Principals (2 Qubit)
- A line represents do nothing
- A box represents an operation
- U
- This maps the states as
- 00 ? 0U0
- 01 ? 0U1
- 10 ? 1U0
- 11 ? 1U1

Quantum Circuits

- Multiple Qubit gate
- Basic Principals (2 Qubit)
- The output is the result of the tensor products

of the operations - The first operation is the do nothing which is

represented by the Identity Matrix (I) - The second operation is U
- I ? U

Quantum Circuits

- Multiple Qubit gate
- Basic Principals (2 Qubit)
- Implementing interaction between qubits
- Given the following system
- This is a controlling circuit
- It maps Resulting Matrix
- 00 ? 00
- 01 ? 01
- 10 ? 1U0
- 11 ? 1U1

Quantum Circuits

- Multiple Qubit gate
- Basic Principals (2 Qubit)
- If the controlling qubit is 0 the operation U

is not implemented - If the controlling qubit is 1 the operation U

is implemented - Control Qubit
- Target Qubit

Quantum Circuits

- Multiple Qubit gate
- Controlled NOT (CNOT)
- CNOT is a generalization of the classical XOR
- CNOT has one control qubit and one target

qubit - CNOT maps
- x0 ? xx
- x1 ? xNOTx

Quantum Circuits

- Multiple Qubit gate
- Controlled NOT (CNOT)
- CNOT A, B A, A ? B
- CNOT Unitary Matrix
- CNOT circuit
- A A
- ?
- B A ? B
- From the CNOT gate we can build another very

interesting gate SWAP

Quantum Circuits

- Multiple Qubit gate
- SWAP
- SWAP A, B B, A
- SWAP Unitary Matrix
- SWAP Circuit
- ?
- A B
- B A

Quantum Circuits

- Multiple Qubit gate
- SWAP
- A ? ?
- A ?
- B ?
- A ? B ? ?
- This circuit design is implemented by having 3

CNOT gates. - The first CNOT gate has A as the controller and

B as the target. - This results is what we expect from a CNOT gate.

Quantum Circuits

- Multiple Qubit gate
- SWAP
- A A ?(A ? B) B

? - A ?
- B ?
- A ? B A ? B

? - The next (second) CNOT gate takes the result of

the target from the first CNOT gate as the

controller and takes the result of the controller

from the first CNOT as the target.

Quantum Circuits

- Multiple Qubit gate
- SWAP
- A A ?(A ? B) B

B - A ?
- B ?
- A ? B A ? B B (A

? B) A - The last (third) CNOT gate takes the result of

the controller from the second CNOT gate as the

target and takes the result of the target from

the second CNOT as the controller.

Quantum Circuits

- Multiple Qubit gate
- SWAP
- A A ?(A ? B) B

B - A B
- B A
- A ? B A ? B B (A

? B) A - So now it can be seen that the qubits A and B

have been swapped.

Quantum Circuits

- Multiple Qubit gate
- The Tofolli Gate
- Classically, the Tofolli gate is a universal and

reversible gate consisting of three input and

three outputs

Quantum Circuits

- Multiple Qubit gate
- The Tofolli Gate
- The Tofolli gate can be implemented in developing

half and full adders - The Tofolli gate is referred to in Quantum Logic

as a Controlled-Controlled-NOT - a a
- b b
- c c ?(ab)

Quantum Circuits

- Multiple Qubit gate
- Half-adder
- This circuit computes the sum and carry for two

bits X0 and X1 - Classical
- This circuit is not reversible (test inputs 0,1

and 1,0)

Quantum Circuits

- Multiple Qubit gate
- Half-adder
- This circuit computes the sum and carry for two

qubits X0 and X1 - Quantum
- x1 x1
- x0 x0 ? x1 (SUM)
- 0 0 ? (x0x1) (CARRY)
- The quantum half adder consists of a Tofolli gate

(CCNOT) followed by a CNOT gate - In comparison of the classical half-adder, the

quantum half adder is reversible

Quantum Circuits

- Multiple Qubit gate
- Full-adder
- This circuit computes the sum and carry for two

bits X0 and X1 - Classical
- This circuit is not reversible

Quantum Circuits

- Multiple Qubit gate
- Full-adder
- This circuit computes the sum and carry for three

qubits X0, X1, C (Carry-in qubit). D is an

auxiliary input (Source) for reversibility - Quantum

x1 x1 x0 x0 C SUM D

CARRY

Quantum Circuits

- Multiple Qubit gate
- Full-adder
- Quantum

x1 x1 x0 x0 C SUM D

CARRY T1 CNOT1 T2

CNOT2 x1 x1 x1 x1

x1 x1 x0 x0 x0 x0

x0 x0 C C B?C B?C

x1?(B?C) SUM D D?(x0C)

D?(x0C) (D?(x0C))?(x0?(C x1))

CARRY

D?(x1x0)?(x0C)?(Cx1) (Maj A,B,C)

Quantum Circuits

- Multiple Qubit gate
- No Cloning Theorem
- The No Cloning Theorem states that a qubit cannot

be copied - First, lets take the classical XOR gate with an

additional garbage (or Sink) line - Set B0
- Z A ? B A
- S1 A

Quantum Circuits

- Multiple Qubit gate
- No Cloning Theorem
- The output result will always produce a copy of

the input A - Therefore, classically a copy circuit does exist

Quantum Circuits

- Multiple Qubit gate
- No Cloning Theorem
- As stated before, the quantum gate equivalent of

the classical XOR is the CNOT - Is it possible to develop a qubit copy of by

using the CNOT gate? - a0 b1 a0 b1
- 0 a00 b10
- So, ( a0 b1 ) ? 0 a00 b10
- This results in entanglement, thus the answer is

NO, qubits cannot be directly copied.

Quantum Circuits

- Analyzing Quantum Circuits
- Suppose we are given a circuit
- Assume the initial state of the three qubits is
- The first operation is the CNOT which has the

qubit in position 2 as the controller and qubit

in position 1 as the target - The second operation is another CNOT which has

the qubit in position 0 as the controller and the

qubit in position 2 as the target - Qubits are operated on in regards to superposition

Quantum Circuits

- Designing Quantum Circuits
- Suppose we are designing a circuit that has the

unitary map - 00 ? 00
- 01 ? 10
- 10 ? 01
- 11 ? 11
- From analyzing the unitary map, we see that the

action of the mapping is a swap of qubits - Implementing a SWAP consists of 3 CNOT gates

Quantum Circuits

- Designing Quantum Circuits
- Suppose we are to design a circuit with a

prepared state - Assume that the initial state of the qubit is

0000 - First, we must apply the H gate to the most

significant qubit - This result is obtained by
- 0000 0 ? 0 ? 0 ? 0
- Applying the H gate to the most significant bit

results in

Quantum Circuits

- Designing Quantum Circuits
- Now, after applying the H gate, the current state

is - Next, apply a CNOT gate with the most significant

bit as the controller and let the target qubit be

position 2, current state is - Next, apply a CNOT gate with the most significant

bit as the controller and let the target qubit be

position 1, current state is - Next, apply a CNOT gate with the most significant

bit as the controller and let the target qubit be

position 0, current state is

Quantum Circuits

- Designing Quantum Circuits
- We have now reached the desired state
- The quantum circuit is given by
- Other circuits will also implement the desired

quantum state

Quantum Circuits

- References
- I. L. Markov, An Introduction to Reversible

Circuits, Univ. of Michigan EECS Dept. - A. Klappenecker, Quantum Algorithms, Texas AM

University, Spring 2003 - P. Gossett, Quantum Carry-Save Arithmetic,

Silicon Graphics, Inc, August 29, 1998 - D. P. DiVincenzo, Quantum Gates and Circuits, IBM

Research Division, Thomas J. Watson Research

Center, Proc. R Soc. London. A, 1998 - A. Barenco, C. H. Bennett, Elementary Gates for

Quantum Computation, Physical Review, Vol 52,

Num. 5, Nov 1995 - B. Travaglione, Designing and Implementing Small

Quantum Circuits and Algorithms, Univ. of

Cambridge Center for Quantum Computation, June 5,

2003 - J. P. Hayes, Tutorial Basic Concepts in Quantum

Circuits, Advanced Computer Architecture

Laboratory, EECS Dept. Univ. of Michigan, 2003 - B. Vesenmayer, Quantum Gates, Universality,

Laboratory of Mathematical Logic Steklov

Institute of Mathematics at St. Petersburg,

August 2003

Quantum Circuits

- References
- K. Svozil, Quantum Logic. A Brief Outline,

University of Technology Vienna, February 1999 - C. Tseng, C. Chen, C. Huang, Quantum Gates

Revisited A Tensor Product Based Interpretation

Model, Dept. of Information Engineering and

Computer Science, Feng Chia Univ. - R. Cleve, Introduction to Quantum Information

Processing, lecture 5, Institute for Quantum

Computing, University of Waterloo, 2004 - N. Papanikolaou, Quantum Computation and Quantum

Information, Lecture 3, Dept. of Computer

Science, Univ. of Warwick

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