outline main result quantum computation and quantum circuits feynman’s sum over paths polynomials...
TRANSCRIPT
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.