A Software Package to Construct Polynomial Sets

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A Software Package to Construct Polynomial
Sets over Z_2 for Determining the Output of
Quantum Computations
Vladimir Gerdt Vasily Severyanov
Laboratory of Information Technologies Joint
Institute for Nuclear Research Dubna, Russia
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Outline

• Main result
• Quantum computation and quantum circuits
• Feynmans sum over paths Polynomials
• QuPol program Quantum Polynomials
• Quantum polynomials
• Conclusion

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Main Result
1. Quantum computing and polynomial equations
over the finite field Z2 by C. M. Dawson et
al arXivquant-ph/0408129 20 Aug 2004
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Quantum Computation and Circuits
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qubit
Quantizing classical bit
any superposition of basis states
In computational basis
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Evolving quantum bit
unitary transformation
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Observing quantum bit
with probability
with probability
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Distinctions of Quantum Computation

• 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

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Quantum computation
Unitary transformation computing function
Initial state
Final state
tensor products of qubits
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Quantum circuit
Quantum circuit is a sequence of elementary
unitary transformations called quantum gates
Quantum gate acts on a few qubits not changing
the others
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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
We work with the following universal gate basis
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Hadamard gate, acts on one qubit
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Toffoli gate, acts on three qubits
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Matrix elements of quantum circuit
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Feynman's sum over paths
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From quantum to classical circuit
We wish to use famous Feynmans 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.
Classical Toffoli
Classical Hadamard
Output may be 0 or 1 for any input
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Example circuit
quantum circuit
classical circuit
path variables
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Admissible classical paths
A classical path is a sequence of classical bit
strings obtained after each classical gate has
been applied.
A choice of the path variables determines an
admissible classical path.
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Phase of admissible classical path
Toffoli gates do not contribute to phase
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Quantum circuits matrix element
admissible paths form a to b
number of Hadamard gates
number of positive terms
number of negative terms
this equations count solutions to a system of n1
polynomials in h variables over the field Z2
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Program Quantum Polynomials
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Elementary decomposition of a circuit
elementary gates
Elementary gates enable us to assemble a
classical form of a quantum circuit
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E - elementary gates
Identities
Multiplication modulo 2
Addition modulo 2
Hadamard
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Elementary gates - Identities
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Elementary gates - Operations
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Assembling circuit, step 0
Let we want to assemble a circuits with 3 rows
and 4 columns
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Assembling circuit, step 1
We place elementary gates in cells
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Assembling circuit, step 2
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Assembling circuit, step 3
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Assembling circuit, step 4
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QuPol - General View
elementary gates toolbar
menu toolbar
window for assembling circuit
circuit polynomials
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QuPol - Assembling circuit

• New circuit
• Selecting gates
• Placing gates
• Constructing polynomials
• Saving circuit
• Opening circuit

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New circuit dialogue
click
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New circuit
The simplest possible circuit only identities
are used
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Placing gates, step 1
click selecting gate
click placing gate
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Placing gates, step 2
click
click
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Placing gates, step 3
click
click
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Placing gates, step 4
many clicks to select and place gates
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Constructing polynomials
click
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Saving circuit
click
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Opening circuit
click
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Opening circuit, finished
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Symbolic form of polynomials
x_1 a_2x_2-b_1, x_4 a_4x_7
x_1x_4x_7 a_2x_2x_4x_7-b_2, x_5
a_4x_3 a_42x_2 x_1x_4
a_2x_2x_4-b_3, a_4 x_1x_4
a_2x_2x_4-b_4, x_7-b_5, x_6 a_4x_3
a_42x_2 x_1x_3x_4 a_4x_1x_2x_4
a_2x_2x_3x_4
a_2a_4x_22x_4-b_6, a_1x_1 a_3x_2
a_5x_3 a_2x_4 x_2x_5 a_6x_6 x_3x_7
a_4x_2x_7
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C name space Polynomial_Modulo_2

• class Polynomial - list of monomials
• class Monomial - list of letters
• class Letter - letter with index and power

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Quantum Polynomials
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Circuit Matrix
A system generated by the program is a finite set
of polynomials in the ring
One has to count the number of roots N0 and N1
in Z2 of the sets
Then the matrix is
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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
(Gerdt04).
We illustrate this by example from Dawson et al
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Computing Matrix Elements (cont)
Lexicographical Gröbner basis with and for F0
and F1
some matrix elements
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References

• Christopher M. Dawson et al.,Quantum computing
  and polynomial equations over the finite field
  Z2,arXivquant-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

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Conclusion

• 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.

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Thanks for Your Attention !