Quipper.Algorithms.BF.Main

Author: Alexander S. Green

An implementation of the Boolean Formula algorithm, applied to finding whether a winning strategy exists in a game of Hex.

The algorithm consists of eigenvalue analysis using phase estimation, acting on an oracle that defines a quantum walk over the NAND graph representation of a game of Hex, for a given size of board.

The implementation defines the NAND graph of a game of Hex by adding a few extra nodes to a graph representing pieces being played during a game of Hex. An extra node is added to each leaf node that represents a completed game of Hex, for which the red player has won, as well as two extra nodes being added below the root node.

The general form of the algorithm is described in:

• A. Ambainis, A. M. Childs, B. W. Reichardt, R. Spalek, and S. Zhang. "Any AND-OR formula of size N can be evaluated in time N^1/2+o(1) on a quantum computer." SIAM J. Comput., 39:2513–2530, April 2010. See also http://www.ucw.cz/~robert/papers/andor-siamjc.pdf.
• A. M. Childs, B. W. Reichardt, R. Spalek, and S. Zhang. "Every NAND formula of size N can be evaluated in time N^1/2+o(1) on a quantum computer" 2007. http://arxiv.org/abs/quant-ph/0703015.

The present implementation is based on detailed algorithm and oracle specifications that were provided to us by the IARPA QCS program and written by Patrick Henry.

Modules:

• Quipper.Algorithms.BF.Main: Command line interface.
• Quipper.Algorithms.BF.BooleanFormula: Implementation of the various quantum circuits that make up the boolean formula algorithm.
• Quipper.Algorithms.BF.Hex: Implementation of the circuits for determining which player has won a completed game of Hex.
• Quipper.Algorithms.BF.HexBoard: Code for drawing Hex boards in graphical format.
• Quipper.Algorithms.BF.Testing: Testing facilities for the boolean formula algorithm, and some auxiliary function definitions.

Command line interface

This module provides a command line interface for the Boolean Formula algorithm. This allows the user, for example, to plug in different oracles, show different parts of the circuit, run a demo, and select different output formats.

Option processing

data WhatToDo

An enumeration type for determining the action that should be taken when the executable is run.

Constructors

OutputCircuit

Output the circuit.

Demo

Run a demo of the circuit, which is different for the various parts of the algorithm.

HexBoard

Output a representation of the moves already made for the defined oracle, i.e. a partially filled Hex Board.

Instances

Show WhatToDo

Defined in Quipper.Algorithms.BF.Main

Methods

showsPrec :: Int -> WhatToDo -> ShowS

show :: WhatToDo -> String

showList :: [WhatToDo] -> ShowS

data WhatPart

An enumeration type for determining what the main function should do.

Constructors

WholeCircuit

The whole circuit.

U

Only one iteration of the U from EXP_U circuit.

Oracle

Only the Oracle circuit.

Hex

Only the Hex circuit.

Checkwin_Red

Only the Checkwin_Red circuit, i.e. including Flood_Fill.

Diffuse

Only the Diffuse circuit.

Walk

Only the Walk circuit.

Undo_Oracle

Only the Undo_Oracle circuit.

Instances

Show WhatPart

Defined in Quipper.Algorithms.BF.Main

Methods

showsPrec :: Int -> WhatPart -> ShowS

show :: WhatPart -> String

showList :: [WhatPart] -> ShowS

data OracleSize

An enumeration type for selecting an oracle size.

Constructors

Full

The oracle for a 9 by 7 Hex board, with a 189 qubit phase estimation register.

Small

The oracle for a 5 by 3 Hex board, with a 4 qubit phase estimation register

Custom Int Int Int

A custom oracle.

Instances

Show OracleSize

Defined in Quipper.Algorithms.BF.Main

Methods

showsPrec :: Int -> OracleSize -> ShowS

show :: OracleSize -> String

showList :: [OracleSize] -> ShowS

data Options

A data type to hold values set by command line options.

Constructors

Options

Fields

• what :: WhatToDo

  What we want to do.

• part :: WhatPart

  What part of the algorithm to use.

• format :: Format

  The output format of a circuit.

• oracle_size :: OracleSize

  Which size of oracle to use.

• oracle_init :: [Int]

  A list of moves already made, which is used to define s

• hex :: HexCircuit

  A flag of which HEX circuit to use.

• gatebase :: GateBase

  What kind of gates to decompose into.

defaultOptions :: Options

The default options, which correspond to a Preview of the entire circuit for the small oracle.

options :: [OptDescr (Options -> IO Options)]

The list of command line options, in the format required by getOpt.

part_enum :: [(String, WhatPart)]

An enumeration of available circuit parts and their names.

oracle_enum :: [(String, Maybe OracleSize)]

An enumeration of available oracles and their names.

dopts :: [String] -> IO Options

Process argv-style command line options into an Options structure.

usage :: IO ()

Print usage message to stdout.

The BF main function

main :: IO ()

Main function: read options, then execute the appropriate task.

check_options :: Options -> Options

Check that the given options are valid. This currently is only required to check that the list of moves already made, is valid for the given size of oracle.

getOracle :: OracleSize -> [Int] -> BooleanFormulaOracle

Convert an OracleSize, and a list of played positions, into an actual oracle.

output_part :: WhatPart -> Format -> GateBase -> BooleanFormulaOracle -> IO ()

This function defines what should be output for each part of the circuit.

demo_part :: WhatPart -> Format -> BooleanFormulaOracle -> IO ()

This function defines what should be done for a demo of each part of the circuit.

valid_sizes :: [Int]

An infinite list of all numbers that are one less than an integer power of 2.

valid_size :: Int -> Bool

Return True if the given number is one less than an integer power of 2.

createCustom :: Int -> Int -> Int -> OracleSize

Create a custom sized oracle, by checking the given x,y, and t sizes are valid.

getCustom :: String -> Maybe OracleSize

Parse a string defining a custom oracle size.