Subhrendu Sarkar

1
Satisfiability by Quantum Decision Diagrams
(QUIDDs) and Quantum Simulation Circuits(Ising
Model Analysis)

• By
• Subhrendu Sarkar
• ss3295_at_columbia.edu
• COMS 4117 Compilers and Interpreters

2
Outline

• Satisfiability (SAT)
• BDDs and SAT
• Quantum Circuits and QuIDDs

3
Satisfiability

• Satisfiability or SAT
• Q. What is the Problem ?
• A. To determine whether there exists a truth
  assignment to variable appearing in a Boolean
  expression F in Conjunctive Normal Form (CNF)
  such that F is satisfied (true).
• Example
• (X V Y V Z) ? (X V Y) ? (Y V Z) ? (Z V X)
  ?(X V Y V Z)
• No. of clauses 5 (m)
• No. of variables 3 (n)
• No. of literals 12 (p)
• Is the expression satisfiable ?
• NO

• A Boolean expression in CNF is
• A set of clauses logically AND-ed together
• Each clause is a disjunction of literals (logical
  OR)
• A literal is simple a positive or negated
  variable (A or A )

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SAT

• Naive approach to solving SAT brute force
• Truth statements 2n
• Literals p
• O(p. 2n) operations.
• SAT ,in general, is an NP-complete problem.
• However
• Restrictions on the type of Boolean expressions
  can give specific formulations of SAT which can
  be solved in polynomial time.

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Davis-Putnam Algorithm

• The original Algorithm for solving SAT.
• Example (A V C)(A V C)(B V C)(A V B)
• Pick any one literal, say A, set it to true.
• Simplify the formula by removing all clauses in
  which A appears and also by removing all
  occurrences of the literals negation. (which here
  is A).
• This is called chasing the literal.
• Chasing A above gives us
• (C) (B V C)(B).
• Keep chasing literals until
• If empty clause results, we abort since
  expression cannot be satisfied.
• If all clauses are deleted, then the expression
  is satisfied with the truth assignments selected
  while chasing the literals.

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The Algorithm

• Introduced in 1960
• Let F be a CNF formula and Xx1,..,xn be its
  set of variables.

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More on DP Procedure

• Virtually all SAT solvers use the DP procedure in
  their core.
• The DP procedure performs a backtracking depth
  first search in the space of all truth
  assignments to find a satisfying assignment for
  the CNF formula.
• Constraints on Boolean expressions can actually
  help us solve SAT problems in polynomial time.
• Example Horn-SAT , 2-SAT

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BDDs and SAT

• How do we use BDDs on SAT problems
• The number of backtracks using the DP Procedure
  and number of paths in the corresponding BDD is
  proven to have a relation.
• The Relations
• Given a BDD B with number of paths P and a CNF
  formula F for some logic circuit C (or Boolean
  expression) then if the variable ordering
  strategy of the DP procedure follows the same
  ordering for every path of B, then DP proves
  equivalence of C against an equivalent version in
  P-1 backtracks.
• Given a DP procedure, the optimal number of
  backtracks needed to prove equivalence of 2
  equivalent circuits is bounded by the number of
  paths in the corresponding minimal path BDD.

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Quantum Computation

• Some background.
• Visualize all quantum computing operations as
  matrix operations in the Hilbert Space.
• Qubits
• A qubit has some similarities to a classical bit,
  but is overall very different. Like a bit, a
  qubit can have only two possible valuesnormally
  a 0 or a 1. The difference is that whereas a bit
  must be either 0 or 1, a qubit can be 0, 1, or a
  superposition of both.
• A pure qubit state is a linear superposition of
  those two states. This means that the qubit can
  be represented as a linear combination of 0 and
  1 a0 ß1
• When we measure this qubit in the standard basis,
  the probability of outcome is a 2 and the
  probability of outcome is ß 2.
• a 2 ß 2 1
• Qubits and the Bloch Sphere
• http//en.wikipedia.org/wiki/Bloch_sphere

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Quantum Decision Diagrams

• QuIDD can be viewed as Multi Terminal Binary
  Decision Diagrams (MTBDDs) or Algebraic Decision
  Diagrams (ADDs) with the following properties
• The values with terminal nodes are complex nos.
• Rather than contain the values explicitly, QuIDD
  terminal nodes contain integer indices which map
  into a separate array of complex numbers. This
  allows the use
• of a simpler integer function for Apply-based
  operations.
• 3. The variable ordering of QuIDDs interleaves
  row and column variables, which favors
  compression of repeated sub-structure.

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An alternative approach to QuIDDs

• QUIDDs are analogous to BDDs for Quantum Circuits
  and Quantum Logic Synthesis.
• A canonical and concise representation of quantum
• logic circuits in the form of quantum decision
• diagrams (QDDs),which are amenable to efficient
• manipulation and optimization including recursive
  unitary
• functional bi-decomposition.
• Definition
• A QDD is a directed acyclic graph with three
  types of nodes
• a single terminal node with value Ô, a weighted
  root node,
• and a set of non terminal (internal) nodes. Each
  internal
• node represents a quantum function.

Four-input Toffoli gate
QDD for output s
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Conclusions

Binary Decision Diagrams
Quantum Decision Diagrams
Standard
Can it be applied ??
Satisfiabilty Logic Circuits/Boolean Expressions
Satisfiability Ising Problem
13

• Questions ??
• THANK YOU