Outline

• Main result

• Quantum computation and quantum circuits

• Feynman’s sum over paths Polynomials

• QuPol program “Quantum Polynomials”

• Quantum polynomials

• Conclusion

1. Quantum computing and polynomial equations over the finite field Z2

by C. M. Dawson et al

arXiv:quant-ph/0408129 20 Aug 2004

Using ideas published in [1] we have written a C# program tool enabling us to assemble an arbitrary quantum circuit in a particular gate basis and to construct the corresponding set of polynomial equations over Z2. The number of solutions of the set defines the matrix elements of the circuit and therefore the output value of the circuit for any input value.

Quantizing classical bit

2

2 2

| , || || 1

| | 0 |1

,

| | | | 1

1 0| , | 0 , |1

0 1

In computational basis

{0,1}b

qubit

any superposition of basis states

Evolving quantum bit

U

2 2 † 1 0,

0 1U UU I

unitary transformation

Observing quantum bit

0| | 0 |1

1

{0,1}b

2| |with probability

with probability

2| |

• quantum bit (qubit) = superposition of basis states

• computation = unitary (reversable) evolution

• probabilistic results

• quantum parallelism, interference, and entanglement

• quantum circuit calculates a classical Boolean function using the peculiarities of quantum computation

Distinctions of Quantum Computation

Quantum computation

fUa fUb a

2 2: n nf Z ZUnitary transformation computing function

2 2 2, n a b

Initial state Final state

tensor products of qubits

Quantum circuit

1 2 1f m mU U U U U

a bmU1U 1mU 2U

Quantum circuit is a sequence of elementary unitary transformations called quantum gates

Quantum gate acts on a few qubitsnot changing the others

Quantum gate bases

Quantum gate basis is a set of universal quantum gates:any unitary transformation can be presented as a composition

of gates of the basis

1 2 1- unitary: , Gate Basism m iU U U U U U U

Hadamard gate, Toffoli gate

We work with the following universal gate basis

Hadamard gate, acts on one qubit

1 11

1 12H

1: 0 ( 0 1 )

21

: 1 ( 0 1 )2

H

H

Toffoli gate, acts on three qubits

000 000

001 001

010 010

011 011:

100 100

101 101

110 111

111 110

Toffoli

y

z2 2x y z

x x

y

Matrix elements of quantum circuit

fU U U U i

m m-1 m-1 m-1 m-2 1 1a

b a b a a a a a

1 2 1f m mU U U U U

1,2, 1i m

fUb a

initial statefinal state

From quantum to classical circuit

We wish to use famous Feynman’s sum-over-paths method to calculate the matrix element for a quantum circuit built of the Toffoli and Hadamard gates (these two constitute a universal basis).

To do that, we replace the quantum circuit under consideration by its classical version where the quantum Toffoli and Hadamard gates are replaced by their classical counterparts.

1 2 3 1 2 3 1 2, , , ,a a a a a a a aClassical Toffoli

Classical Hadamard

1a x 2,ia x

Output may be 0 or 1 for any input

- path variablex

Example circuit

H

H

H

H

1a

2a

3a

1b

2b

3b

2a 2x

4x

H

H

H

H

1a

3a

1x

2x

3 1 2a x x

3x

4x

3 2 4x x x1( )b x

2( )b x

3( )b x

41 2 3 4 2( , , , )Tx x x x x

quantum circuit

classical circuit

path variables

Admissible classical paths

A classical path is a sequence of classical bit strings

obtained after each classical gate has been applied.

2a 2x

4x

H

H

H

H

1a

3a

1x

2x

3a 1 2x x

3x

4x

3 2 4x x x1( )b x

2( )b x

3( )b x

1 2, , , , m a a a a b

a 1a 2a 3a 4 a b

A choice of the path variables

determines an admissible classical path.

31 2 4, , xx ,x x 2ix

Phase of admissible classical path

2a 2x

4x

H

H

H

H

1a

3a

1x

2x

3 1 2a x x

3x

4x

3 2 4x x x1( )b x

2( )b x

3( )b x

2Hadamard gates

( ) input output x

1 1 2 2 1 3 4 3 1 2( ) ( )a x a x x x x a x x x

0 1

1 1 01

1 1 12

Toffoli gates do not contribute to phase

Quantum circuit’s matrix element

0 1

1

2f h

U N N b a

0 | ( ) & ( ) 0N x x xb b

1 | ( ) & ( ) 1N x x xb b

this equations count solutions to a system of

n+1 polynomials in h variables over the field Z2

11

2f h

U

x

x: xb b

b a

number of Hadamard gates

admissible paths form a to b

number of positive terms

number of negative terms

Elementary decomposition of a circuit

11 12 1

21 22 2

1 2

m

m

n n nm

u u u

u u u

u u u

1

2

n

a

a

a

1 1 1( 1) 12 11 1

2 2 2( 1) 22 21 2

( 1) 2 1

m m

m m

n nm n m n n n

b u u u u a

b u u u u a

b u u u u a

| , 1, ,T

j ij ijU u u E i n

1 2 mU U Uelementary gates

Elementary gates enable us to assemble a classical form of a quantum circuit

E - elementary gates

I

I

I

I

M

M

Identities

A

A

HH

Multiplication modulo 2

Addition modulo 2

Hadamard

Elementary gates - Identities

( )I

( )I

( )I

( )I

Elementary gates - Operations

( )M

2( , )A

2( , )A

( ) ,x xH H

{ , , , , , , , , }E I I I I M M A A H

2( , ) *M

( )M

2( , ) *M

Assembling circuit, step 0

1a

2a

3a

1 ?b

2 ?b

3 ?b

?

Let we want to assemble a circuits with 3 rows and 4 columns

Assembling circuit, step 1

H

H

H

1a

2a

3a

1a

2a

3a

1b

2b

3b

1 1 1b Ha x

2 2 2b Ha x

3 3 3b Ia a H

I

1 1 2 2a x a x

We place elementary gates in cells

Assembling circuit, step 2

H

H

H

1a

2a

3a

1a

2a

3a

1b

2b

3b

1 1 1b I Ha x

2 2 2b M Ha x

3 3 3 1 2b A Ia a x x

H

I

I

M

A

1 1 2 2a x a x

Assembling circuit, step 3

H

H

H

H

H

1a

2a

3a

1a

2a

3a

1b

2b

3b

1 1 3b H I Ha x

2 2 2b I M Ha x

3 3 4b H AIa x

H

H

HI

I

M

I

A

1 1 2 2 1 3 4 3 1 2( )a x a x x x x a x x

Assembling circuit, step 4

H

H

H

H

H

1a

2a

3a

1a

2a

3a

1b

2b

3b

1 1 3 2 4b AH I Ha x x x

2 2 2b M I M Ha x

3 3 4b I H AIa x

H

H

HI

I

I

M

M

I

A

A

1 1 2 2 1 3 4 3 1 2( )a x a x x x x a x x

QuPol - General View

elementary gates toolbar

menu toolbar

window for assembling circuit

circuit polynomials

QuPol - Assembling circuit

• New circuit

• Selecting gates

• Placing gates

• Constructing polynomials

• Saving circuit

• Opening circuit

• New circuit

• Selecting gates

• Placing gates

• Constructing polynomials

• Saving circuit

• Opening circuit

New circuit dialogue

click

New circuit

The simplest possible circuit: only identities are used

Placing gates, step 1

click – placing gate

click – selecting gate

Placing gates, step 2

click

click

Placing gates, step 3

click

click

Placing gates, step 4

many clicks to select and place gates

Constructing polynomials

click

Saving circuit

click

Opening circuit

click

Opening circuit, finished

Symbolic form of polynomials

x_1 + a_2*x_2-b_1,

x_4 + a_4*x_7 + x_1*x_4*x_7 + a_2*x_2*x_4*x_7-b_2,

x_5 + a_4*x_3 + a_4^2*x_2 + x_1*x_4 + a_2*x_2*x_4-b_3,

a_4 + x_1*x_4 + a_2*x_2*x_4-b_4,

x_7-b_5,

x_6 + a_4*x_3 + a_4^2*x_2 + x_1*x_3*x_4 + a_4*x_1*x_2*x_4 + a_2*x_2*x_3*x_4 + a_2*a_4*x_2^2*x_4-b_6,

a_1*x_1 + a_3*x_2 + a_5*x_3 + a_2*x_4 + x_2*x_5 + a_6*x_6 + x_3*x_7 + a_4*x_2*x_7;

C# name space “Polynomial_Modulo_2”

• class Polynomial - list of monomials

• class Monomial - list of letters

• class Letter - letter with index and power

• class Polynomial - list of monomials

• class Monomial - list of letters

• class Letter - letter with index and power

Circuit Matrix

A system generated by the program is a finite set of polynomials in the ring

2 1

2

: [ , ][ ,..., ]

, , , 1,...i j h

i j

R Z a b x x

a b Z i j n

F R

0 1{ ,..., , }, { ,..., , 1}k kF f f F f f

One has to count the number of roots N0 and N1 in Z2 of the sets

Then the matrix is 0 1

1( )

2j i hb U a N N

Computing Matrix Elements

To count the number of roots we convert F0 and F1 into the triangular form by computing the lexicographical Gröbner basis by means of the Buchberger algorithm or by involutive algorithm (Gerdt’04).

1 2 4 3 1

2 2 2

3 4 3

1 2 1 3 1 1 2 2 3 4

f x x x b

f x b

f x b

x x x x a x a x a x

We illustrate this by example from Dawson et al

Lexicographical Gröbner basis with and for F0 and F1

Computing Matrix Elements (cont)

1 1 1 1 2 2 3 3

2 2 2

3 3 1 2 3

3 4 3

( ) ( 1)g a b x a b a b

g x b

g x b b b

g x b

1 1000 000 , 000 001 , 000 111 0

2 2U U U

1 2 3 4x x x x

1 1 2 2 3 3

1 1 2 2 3 3

0 & 0

0 & 1

a b a b a b

a b a b a b

0 12 (0) roots of ( )F F

0 10(2) roots of ( )F F

0 11root of and F Fother cases

some matrix elements

References

• Christopher M. Dawson et al.,Quantum computing and polynomial equations over the finite field Z2,arXiv:quant-ph/0408129, 2004.

• Gerdt V.P. Involutive Algorithms for Computing Gröbner Bases, Proceedings of the NATO Advanced Research Workshop "Computational commutative and non-commutative algebraic geometry" (Chishinau, June 6-11, 2004), IOS Press, to appear.

• Microsoft Visual C# .net Standard, Version 2003

• The first version of a program tool for assembling arbitrary quantum circuits and for constructing quantum polynomial systems has been designed.

• There is the algorithmic Gröbner basis approach to converting the system of quantum polynomials into a triangular form which is useful for computing the number of solutions.

• The number of solutions uniquely determines the circuit matrix.

• Thus the above software and algorithmic methods provide a tool for simulating quantum circuits.