Those wishing to have a more profound understanding of this exceptional & Unique Cultural Ceremony should click the link included herewith for a fascinating insight into the event.
After days of persistent rain the sudden glorious sunset over the shimmering waters directly across to Faanui Temple augured well for the night that lay ahead.
All roads led to the Temple; on every pathway strolling towards the Temple, Tahitians dressed in colourful traditional mumus (long Tahitain dress), their men in shirts with equally colourful Tahitian motifs & all adorned in simply magnificent tiare Tahiti shoulder leis & floral head leis to match. Some carried simply their parahiraa – a small hand-made wooden seat on which the singers of himene. The singing tonight would be uniquely beautiful voices with not a musical instrument in sight, for tonight would mark the start of the Taurua Varua, a celebratory competition of himene & orero unique to Bora Bora & running for 3 consecutive Sunday nights – right through to 6am on Monday!
After delving into the history of this fascinating celebration in my report last year (see below) I'll cover the 3 nights for this year with a photographic report for each of the 3 gatherings with a final overall coverage following the last night of competition.
FIRST DAY OF COMPETITION – Faanui Temple, 10 January, 2016.
I found the himene to be of great quality & appreciated the 3 Aitos (oreros). I am greatly looking forward to the ensuing competition ………………… see you there!
Tonight we would gather in the annex to the Protestant Temple at Anau, a district of Bora Bora made famous for, amongst other things, being the birthplace of Tarita Teriipaia who played opposite Marlon Brando in "Mutiny On The Bounty" before she in turn married the famous actor & together they had 2 children.
Tonight's celebrations would start just after 5pm & end at 6am the following morning! A traditional dinner accompanied by Ori Tahiti – the famous Tahitian tamure (dance) – for over 400 in the annex stunningly decorated with plaited niau (coconut leaves) & adorned in tropical flowers of every description & colour – there is no limit to the imagination & creativity of the Polynesians.
Shortly thereafter participants would gather in the large hall alongside the Temple – itself another marvel of Polynesian decoration with the walls adorned with magnificent hand-made tifaifai (Tahitian quilt) & an impressive array of hand-made hats fashioned from local foliage. The judges & dignitaries would assemble here before the himenes (local Tahitian chants) would start. I was struck by the fact that this was a tradition that has endured for hundreds of years.
You could feel Anau's pride in stagging tonight's celebrations & it was justly rewarded by a greater number of participants putting great joy as well as polish into tonight's presentations. A night of great quality rounded off by two Aitos (oreros).
Although I eagerly awaited the final night & the announcement of this year's winners on the following Sunday in Vaitape/Nunue, I regretably agitated an old sciatic problem whist gardening on Saturday & was immobile for Sunday's final – BOO!!! I was probably not missed as none other than the President of French Polynesia flew in specifically to witness the final of this unique event.
And the winners are……….Ierutalema – those in the black & white patterned attire with yellow leis won (& justifiably so).
Well worth catching the event next year if you are in Bora Bora one January.


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text:
Q: Filtering a TableView with sections I try to make a little program to learn more about searching a tableview about a searchbar. Meanwhile i'm in the third week trying to do this. I found many examples in different ways in the internet. The most with a searchdisplaycontroller but this way is only for a viewcontroller. i prefer a way that also works with an uiview and so i use the searchbar delegatemethod to filter my data dictionary. Normally i try as long i get handle such things by myself. But this is my nemesis. I really can't find a way to get this problem solved. Can anyone help me with my code?
Here it is.
#import "FilterDemoTableViewController.h"

@implementation FilterDemoTableViewController

@synthesize filteredTableData;
@synthesize searchBar;
@synthesize isFiltered;

@synthesize tableContents;
@synthesize Keys;

- (void)viewDidLoad
{
    [super viewDidLoad];

    searchBar.delegate = (id)self;

    //-----------------------My TableView Data ------------------------------

    NSArray *array1 = [[NSArray alloc]initWithObjects:@"Berlin",@"München",@"Stuttgart",nil];
    NSArray *array2 = [[NSArray alloc]initWithObjects:@"Paris",@"Bordeaux",@"Marseille",@"Toulouse",nil];
    NSArray *array3 = [[NSArray alloc]initWithObjects:@"London",@"Portsmouth",@"Oxford",@"York",@"Dover",nil];
    NSArray *array4 = [[NSArray alloc]initWithObjects:@"Rom" ,@"Genua",@"Mailand",@"Florenz",nil];
    NSArray *array5 = [[NSArray alloc]initWithObjects:@"Madrid",@"Barcelona",@"Toledo",@"Saragossa",@"Pamplona",nil];
    NSDictionary *dictionary =[[NSDictionary alloc]initWithObjectsAndKeys:array1,@"Deutschland",array2,@"Frankreich",array3,@"Großbritannien",array4,@"Italien",array5,@"Spanien",nil];

    self.tableContents = dictionary;
    self.Keys = [self.tableContents allKeys];

    //--------------------------------------------------------------------------
}

- (NSInteger)numberOfSectionsInTableView:(UITableView *)tableView
{
    if (self.isFiltered) {
        return [filteredTableData count];
    } else {
        return [Keys count];}
}

- (NSInteger)tableView:(UITableView *)tableView numberOfRowsInSection:(NSInteger)section
{

    NSArray *listData =[self.tableContents objectForKey:[self.Keys objectAtIndex:section]];

    int rowCount;
    if(self.isFiltered)
        rowCount = filteredTableData.count;
    else
        rowCount = [listData count];

    return rowCount;

}

- (UITableViewCell *)tableView:(UITableView *)tableView cellForRowAtIndexPath:(NSIndexPath *)indexPath
{

    static NSString *CellIdentifier = @"Cell";

    UITableViewCell *cell = [tableView dequeueReusableCellWithIdentifier:CellIdentifier];
    if (cell == nil)
        cell = [[UITableViewCell alloc] initWithStyle:UITableViewCellStyleSubtitle reuseIdentifier:CellIdentifier];

    NSDictionary* sectionDictionary;

    if (isFiltered) {
        sectionDictionary = [filteredTableData objectAtIndex:indexPath.section];
    } else {
        sectionDictionary = [self.tableContents objectForKey:[self.Keys objectAtIndex:indexPath.section]];
    }

    NSArray* sectionEntries = [self.tableContents objectForKey:[self.Keys objectAtIndex:indexPath.section]];

    cell.textLabel.text = [sectionEntries objectAtIndex:indexPath.row];
    cell.accessoryType = UITableViewCellAccessoryDisclosureIndicator;

    return cell;
}

-(void)searchBar:(UISearchBar*)searchBar textDidChange:(NSString*)text
{
    if(text.length == 0)
    {
        isFiltered = FALSE;
    }
    else
    {
     //I think here is something wrong but i don't know what

        isFiltered = true;

        filteredTableData = [[NSMutableArray alloc] init];

        NSMutableArray *searchArray = [[NSMutableArray alloc] init];

        for (NSDictionary *dictionary in tableContents)    //dictionary read
        {
            NSArray *array = [dictionary objectForKey:Keys];  //section of dictionary read
            [searchArray addObjectsFromArray:array];
        }

        for (NSString *sTemp in searchArray)
        {
            NSRange titleResultsRange = [sTemp rangeOfString:text options:NSCaseInsensitiveSearch];

            if (titleResultsRange.length != 0)
                [filteredTableData addObject:sTemp];
        }
    }
    [self.tableView reloadData];
}

- (void)viewDidUnload{
    [self setSearchBar:nil];
    [super viewDidUnload];
}

- (BOOL)shouldAutorotateToInterfaceOrientation:(UIInterfaceOrientation)interfaceOrientation
{
    // Return YES for supported orientations
    return (interfaceOrientation == UIInterfaceOrientationPortrait);
}

@end

A: Now i post a complete refreshed code. I tested each component of the code step by step. It works and the search is also fine (in a UITableViewController). But in this example the hole code is in a UIView. The reason, on this way it is possible to create non full-size tableviews and instanzes of tableviews. Much better for a clear Viewcontroller. I know that an UIView has no method like this reloadData but it is necessary.
Near the code end you can see the problem line. And for this last step i have no idea how i can solve this. [self.tableView reloadData];
#import "TableView.h"
#import <QuartzCore/QuartzCore.h>

@interface TableView ()

@end

@implementation TableView
@synthesize delegate;
@synthesize dropDownHeight;
@synthesize labelText;
@synthesize enabled;

@synthesize tableContents;
@synthesize Keys;

@synthesize searchBar;
@synthesize isFiltered;
@synthesize filteredTableData;

- (void)__show {
   viewControl.alpha = 0.0f;
   UIWindow *mainWindow = [[UIApplication sharedApplication] keyWindow];
   [mainWindow addSubview:viewControl];
   [UIView animateWithDuration:0.3f
                 animations:^{
                     viewControl.alpha = 1.0f;
                 }
                 completion:^(BOOL finished) {}];
}
- (void)__hide {
   [UIView animateWithDuration:0.2f
                 animations:^{
                     viewControl.alpha = 0.0f;
                 }
                 completion:^(BOOL finished) {
                     [viewControl removeFromSuperview];
                 }];
}
- (void) setLabelText:(NSString *)_labelText{
   [button setTitle:labelText forState:UIControlStateNormal];
}
- (void) setEnable:(BOOL)_enabled{
   enabled = _enabled;
   [button setEnabled:_enabled];
}
- (void) setArrayData:(NSArray *)_arrayData{
   [table reloadData];
}
- (void) buttonPressed{
   [self __show];
}
- (void) controlPressed{
 //[viewControl removeFromSuperview];
   [self __hide];
}

- (id) initWithFrame:(CGRect)frame
{

self = [super initWithFrame:frame];
if (self) {
    // Initialization code
    button = [UIButton buttonWithType:UIButtonTypeCustom];
    [button setFrame:CGRectMake(10, 0, 280, 30)];
    [button setTitle:@"--Auswahl--" forState:UIControlStateNormal];
    [button setBackgroundImage:[UIImage imageNamed:@"combo_bg.png"] forState:UIControlStateNormal];
    [button setTitleColor:[UIColor blackColor] forState:UIControlStateNormal];
    [button addTarget:self action:@selector(buttonPressed) forControlEvents:UIControlEventTouchUpInside];
    [button.titleLabel setFont:[UIFont fontWithName:@"Helvetica" size:14]];
    [self addSubview:button];
    dropDownHeight = 706;

    viewControl = [[UIControl alloc] initWithFrame:CGRectMake(0, 0, [UIScreen mainScreen].bounds.size.width, [UIScreen mainScreen].bounds.size.height)];
    [viewControl addTarget:self action:@selector(controlPressed) forControlEvents:UIControlEventTouchUpInside];

    CGFloat x = self.frame.origin.x;
    CGFloat y = (viewControl.frame.size.height - dropDownHeight)/2;

    table = [[UITableView alloc] initWithFrame:CGRectMake(x, y, frame.size.width, dropDownHeight) style:UITableViewStyleGrouped];
    table.dataSource = self;
    table.delegate = self;

    searchBar = [[UISearchBar alloc] init];
    [searchBar sizeToFit];
    searchBar.delegate = (id)self;
    table.tableHeaderView = searchBar;

    CALayer *layer = table.layer;
    layer.masksToBounds = YES;
    layer.cornerRadius = 2.0f;
    layer.borderWidth = 1.0f;
    [layer setBorderColor:[UIColor darkGrayColor].CGColor];
    [viewControl addSubview:table];

    CGAffineTransform rotateTable = CGAffineTransformMakeRotation(M_PI_2);
    table.transform = rotateTable;
    table.frame = CGRectMake(-2, -1, table.frame.size.width, table.frame.size.height);

    UIImageView *tempImageView = [[UIImageView alloc] initWithImage:[UIImage imageNamed:@"gradientBackground.png"]];
    [tempImageView setFrame:self->table.frame];
    table.backgroundView = tempImageView;

    //----------------------- TableView Daten ------------------------------

    NSArray *array1 = [[NSArray alloc]initWithObjects:@"Berlin",@"München",@"Stuttgart",@"Hamburg",nil];
    NSArray *array2 = [[NSArray alloc]initWithObjects:@"Paris",@"Bordeaux",@"Marseille",@"Toulouse",nil];
    NSArray *array3 = [[NSArray alloc]initWithObjects:@"London",@"Portsmouth",@"Oxford",@"York",@"Dover",nil];
    NSArray *array4 = [[NSArray alloc]initWithObjects:@"Rom" ,@"Genua",@"Mailand",@"Florenz",nil];
    NSArray *array5 = [[NSArray alloc]initWithObjects:@"Madrid",@"Barcelona",@"Toledo",@"Saragossa",@"Pamplona",nil];
    NSDictionary *dictionary =[[NSDictionary alloc]initWithObjectsAndKeys:array1,@"Deutschland",array2,@"Frankreich",array3,@"Großbritannien",array4,@"Italien",array5,@"Spanien",nil];

    self.tableContents = dictionary;
    self.Keys = [self.tableContents allKeys];

    // ---------------------------------------------------------------------
  }
return self;
}

- (CGFloat)tableView:(UITableView *)tableView heightForHeaderInSection:(NSInteger)section {
if ([self tableView:tableView titleForHeaderInSection:section] != nil) {
    return 40;
}
else {
    return 0;
}
}

- (UIView *)tableView:(UITableView *)tableView viewForHeaderInSection:(NSInteger)section      {
NSString *sectionTitle = [self tableView:tableView titleForHeaderInSection:section];
if (sectionTitle == nil) {
    return nil;
}

// Create label with section title
UILabel *label = [[UILabel alloc] init];
label.frame = CGRectMake(20, 6, 300, 30);
label.backgroundColor = [UIColor clearColor];
label.textColor = [UIColor colorWithRed:1.0 green:1.0 blue:1.0 alpha:1.0];
label.shadowColor = [UIColor grayColor];
label.shadowOffset = CGSizeMake(0.0, 1.0);
label.font = [UIFont boldSystemFontOfSize:16];
label.text = sectionTitle;

// Create header view and add label as a subview
UIView *view = [[UIView alloc] initWithFrame:CGRectMake(0, 0, 320, 40)];
[view addSubview:label];

return view;
}

- (CGFloat)tableView:(UITableView *)tableView heightForRowAtIndexPath:(NSIndexPath    *)indexPath {
return 60.0;
}

- (NSString *)tableView:(UITableView *)tableView titleForHeaderInSection: (NSInteger)section {

return [self.Keys objectAtIndex:section];

}

- (NSInteger)numberOfSectionsInTableView:(UITableView *)tableView
{
if (self.isFiltered)
{
    return 1;
} else {
    return [Keys count];}
}

- (NSInteger)tableView:(UITableView *)tableView numberOfRowsInSection:(NSInteger)section
{
int rowCount;
if(self.isFiltered)
{
    rowCount = [filteredTableData count];
}
else
{
    NSArray *listData =[self.tableContents objectForKey:[self.Keys objectAtIndex:section]];
    rowCount = [listData count];
}
return rowCount;

}

- (UITableViewCell *)tableView:(UITableView *)tableView cellForRowAtIndexPath:(NSIndexPath *)indexPath {

static NSString *SimpleTableIdentifier = @"SimpleTableIdentifier";
UITableViewCell * cell = [tableView dequeueReusableCellWithIdentifier: SimpleTableIdentifier];

if(cell == nil) {

    cell = [[UITableViewCell alloc]initWithStyle:UITableViewCellStyleDefault reuseIdentifier:SimpleTableIdentifier];
}

if (isFiltered)
{
    NSString * stadt  = [filteredTableData objectAtIndex:indexPath.row];
    cell.textLabel.text = stadt;
}
else
{
    NSDictionary* sectionDictionary;
    sectionDictionary = [self.tableContents objectForKey:[self.Keys objectAtIndex:indexPath.section]];
    NSArray* sectionEntries = [self.tableContents objectForKey:[self.Keys objectAtIndex:indexPath.section]];
    cell.textLabel.text = [sectionEntries objectAtIndex:indexPath.row];
}

cell.accessoryType = UITableViewCellAccessoryDisclosureIndicator;

return cell;
}

- (void)tableView:(UITableView *)tableView didSelectRowAtIndexPath:(NSIndexPath *)indexPath {

NSArray *listData =[self.tableContents objectForKey:[self.Keys objectAtIndex:[indexPath section]]];
NSUInteger row = [indexPath row];
NSString *rowValue = [listData objectAtIndex:row];
[tableView deselectRowAtIndexPath:indexPath animated:YES];

selectedIndex = [indexPath row];

[self __hide];
[button setTitle:[[NSString alloc] initWithFormat:rowValue] forState:UIControlStateNormal];

}

- (NSInteger) selectedIndex {

return selectedIndex;
}

-(void)searchBar:(UISearchBar*)searchBar textDidChange:(NSString*)text
{
if(text.length == 0)
{
    isFiltered = FALSE;
}
else
{
    isFiltered = true;
    [filteredTableData removeAllObjects];

    for (NSString* key in self.tableContents)
    {

        NSArray *staedte = [self.tableContents objectForKey:key];

        for (NSString *stadt in staedte)
        {

            NSRange titleResultsRange = [stadt rangeOfString:text options:NSCaseInsensitiveSearch];

            if (titleResultsRange.length != 0)
            {
                [filteredTableData addObject:stadt];
            }
        }
    }
}
[self.tableView reloadData];  //Here is the error
}

-(void)didChangeTableViewValue:(TableView *)TableView selectedIndex:(NSInteger)selectedIndex {

}

@end

redpajama_set_name: RedPajamaStackExchange


As an asbestos attorney who has been handling mesothelioma, asbestos lung cancer, and asbestosis cases for more than 20 years, I urge you to call or write your elected official in the U.S. House of Representatives in Washington, DC and tell them to oppose H.R. 982, the Furthering Asbestos Claim Transparency (FACT) Act of 2013.
In effect, this House bill is a one-sided attempt to hurt asbestos victims by allowing the asbestos industry to delay and deny accountability so victims die before they are able to receive justice.
To learn how H.R. 982, the FACT Act of 2013, is a misguided piece of legislation, read this insightful piece: "What "Lies" Beneath the Wall Street Journal Asbestos Article, Myths and Facts Exposed".
The dangers associated with asbestos exposure have been known since the early 1930s.
Still the asbestos industry hid the facts, exposed workers and families, and now nearly 10,000 Americans are killed by asbestos diseases every year. This bill does not focus on banning asbestos nor does it look at ways to expedite asbestos cases.
This bill, H.R. 982, instead would require private asbestos bankruptcy trusts to publicly release extensive individual information about asbestos victims and slow down asbestos cases by allowing asbestos defendants to bury the trusts in information requests, no matter how unnecessary or irrelevant. This denies and delays justice for asbestos victims who do not have time on their side.
Congress should focus on keeping Americans safe from asbestos exposure and not protecting corporations that deliberately exposed workers and consumers for decades. Oppose H.R. 982!


text:
\section*{Abstract}

Classical simulation of quantum computation is necessary for studying the numerical behavior of quantum algorithms, as there does not yet exist a large viable quantum computer on which to perform numerical tests. Tensor network (TN) contraction is an algorithmic method that can efficiently simulate some quantum circuits, often greatly reducing the computational cost over methods that simulate the full Hilbert space. In this study we implement a tensor network contraction program for simulating quantum circuits using multi-core compute nodes. We show simulation results for the Max-Cut problem on 3- through 7-regular graphs using the quantum approximate optimization algorithm (QAOA), successfully simulating up to 100 qubits. We test two different methods for generating the ordering of tensor index contractions: one is based on the tree decomposition of the line graph, while the other generates ordering using a straight-forward stochastic scheme. Through studying instances of QAOA circuits, we show the expected result that as the treewidth of the quantum circuit's line graph decreases, TN contraction becomes significantly more efficient than simulating the whole Hilbert space. The results in this work suggest that tensor contraction methods are superior only when simulating Max-Cut/QAOA with graphs of regularities approximately five and below. Insight into this point of equal computational cost helps one determine which simulation method will be more efficient for a given quantum circuit. The stochastic contraction method outperforms the line graph based method only when the time to calculate a reasonable tree decomposition is prohibitively expensive. Finally, we release our software package, qTorch (Quantum TensOR Contraction Handler), intended for general quantum circuit simulation. For a nontrivial subset of these quantum circuits, 50 to 100 qubits can easily be simulated on a single compute node.

\section{Introduction}

Experimental hardware for quantum computing has been steadily improving in the past twenty years, indicating that a useful quantum computer that outperforms a classical computer may eventually be built. However, until a large-scale and viable quantum computer has been realized, numerically simulating quantum circuits on a classical computer will be necessary for predicting the behavior of quantum computers.

Such simulations can play an important role in the development of quantum computing by (1) numerically verifying the correctness and characterizing the performance of quantum algorithms \cite{Liquid,qhipster,qcmpi,qcwave,1012.6035}, (2) simulating error and decoherence due to the interaction between the quantum computer and its environment \cite{qhip-jctc,silva2008,geller2013,tomita2014}, and (3) improving our understanding of the boundary between classical and quantum computing in terms of computational power, for which recent efforts for characterizing the advantage of quantum computers over classical computers \cite{boixo18a,farhi16,boixo18b,chen18a,childs17,bouland18,pednault17,chen18b} serve as an example of this direction.

In this work, we consider the problem of quantum circuit simulation as one where we are given a quantum circuit and an initial state, with the goal of determining the probability of a given output state. Several approaches are possible for such simulation tasks. The most general method is to represent the state vector of an $N$-qubit state by a complex unit vector of dimension $2^N$ and apply the quantum gates by performing matrix-vector multiplications. This is essentially the approach adopted in, for example, \cite{Liquid,qcmpi,qhipster,projq,haner-steiger2017}. This method has the advantage that full information of the quantum state is represented at any point during the circuit propagation. However, the exponential cost of storing and updating the state vector renders it prohibitive for simulating circuits larger than $\sim$45 qubits. On the other hand, for a wide class of circuits with restricted gate sets and input states\cite{Gottesman_1998,quant-ph/0406196,Valiant:2001:QCS:380752.380785,quant-ph/0108010,2016PhRvL.116y0501B}, efficient classical simulation algorithms are available. For example, the numerical package Quipu \cite{6657072,garcia17} has been developed for taking advantage of prior results \cite{Gottesman_1998,quant-ph/0406196,2016PhRvL.116y0501B} on the stabilizer formalism to speed up general quantum circuit simulation. Finally, path integral-based methods \cite{gould-PI-2006} have also been proposed---though they do not improve the simulation cost, they lead to reduced memory storage requirements.

Other than considering the gate sets involved, an alternative perspective of viewing a quantum circuit is through its geometry or topology\cite{quant-ph/0403114,quant-ph/0511069}. Under this view, a quantum circuit is simulated via tensor network contractions. An advantage of viewing quantum circuits as tensor networks is that one can afford to ignore the particular kinds of quantum gates used in a circuit, and instead only focus on the graph theoretic properties. While general quantum circuits involving universal sets of elementary gates are likely hard to simulate on a classical computer \cite{mikeike11}, this geometric perspective sometimes allows for the efficient simulation of a quantum circuit with a universal gate set, provided that it satisfies certain graph theoretic properties. We note that at least one open source implementation of tensor network simulation for quantum circuits already exists, called TNQVM\cite{McCaskey_2016}, which can simulate tensor networks but also focuses on integrating algorithms with real quantum hardware. Aside from the field of quantum computation, tensor networks and related methods are an important and active area of research in the simulation of quantum mechanical problems in theoretical physics \cite{orus13,orus14,ran17}.

Among others, \emph{treewidth} is an important graph theoretic parameter that determines the efficiency of contracting a tensor network of quantum gates. A property of graphs that is actively studied in the graph theory literature \cite{Robertson1991153,Bodlaender2010259,BODLAENDER20111103,qbb}, the treewidth provides important structural information about a quantum circuit. Namely, if the circuit's underlying tensor network has treewidth $T$, it is shown in \cite{quant-ph/0511069} that the cost of simulating the circuit is $O(\text{exp}(T))$. In \cite{boixo18a} treewidth is also used for estimating the classical resource needed for simulating certain quantum circuits.

Motivated by the importance of tensor networks in quantum circuit simulation in general (and for example quantum computational supremacy tests in particular), it is useful to have a circuit simulation platform singularly dedicated to tensor network contraction. One immediate challenge in contracting tensor networks is to find an efficient contraction ordering, which relies on explicitly or implicitly finding a reasonable \textit{tree decomposition} of the underlying graph (definitions are further discussed in Section \ref{sec:prelim}). However, finding the optimal contraction ordering (or equivalently finding the minimum-size tree decomposition, or finding the treewidth of a graph) is NP-complete \cite{Arnborg:1987:CFE:37170.37183}. Therefore one must typically resort to heuristic methods when finding this decomposition \cite{amir10}.

For this study, we implemented a plug-and-play tensor network (TN) contraction code with two contraction schemes. Other schemes were attempted, but were significantly inferior to those that became part of the software package. However, there are likely other heuristic schemes that outperform our stochastic algorithm, and this is an avenue worth pursuing. For a large set of quantum circuits, our tensor network based methods are shown to be less costly than simulation of the full Hilbert space, by comparing to simulations using the LIQ$Ui|>$ \space software package \cite{Liquid}. We emphasize again that the tests in this report give timing data for finding the expectation value of a measurement performed after implementing a quantum circuit, not for fully characterizing a circuit's final state.

The remainder of the paper is organized as the follows. Section \ref{sec:prelim} sets up the definitions and notations used in the paper. Section \ref{sec:contract} describes the heuristic methods used for contracting the quantum circuit tensor networks, along with other relevant details of the code implementation. Section \ref{sec:examples} presents the example quantum circuits used as benchmarks for demonstrating the performance of our contraction algorithms. Section \ref{sec:res} gives results of comparisons between the qTorch contraction methods, and between qTorch simulations and LIQ$Ui|>$'s Hilbert space simulations.

\section{Preliminaries}\label{sec:prelim}

In this section, we provide an overview of relevant concepts and definitions. All graphs that we consider in this paper are undirected. We denote a graph as $G(V,E)$, consisting of the set of nodes $V=\{v_1, v_2,\cdots,v_n\}$ and edges $E\subseteq V\times V$.

Two relevant concepts are a graph's tree decomposition and treewidth \cite{Robertson1991153,amir10}. A \textit{tree decomposition} of a graph $G(V,E)$ is a pair $(S,T(I,F))$, where  $S=\{X_i|i\in I\}$ is a collection of subsets $X_i\subseteq V$ and $T$ is a tree (with edge set $F$ and node set $I$), such that $\cup_{i\in I}X_i=V$. Two nodes $X_i$ and $X_j$ are connected by an edge only if the intersection between $X_i$ and $X_j$ is not null. The \textit{width} of a tree decomposition  $(S,T)$ is $\max_{i\in I}|X_i|-1$. The \textit{treewidth} of a graph $G$ is the minimum width among all tree decompositions of $G$.

Another important concept in tensor network methods is the \textit{linegraph} of graph $G$, denoted by $L(G)$. $L(G)$ is itself an undirected graph, with each edge in $G$ corresponding to a node in $L(G)$. Two nodes in $L(G)$ are connected if and only if these two nodes' corresponding edges in $G$ are connected to the same node in $G$. There exists an optimal tree decomposition of $L(G)$ that provides the optimal contraction ordering of $G$ \cite{quant-ph/0511069}.

In the context of this work, a \textit{tensor} is defined as a data structure with rank $k$ and dimension $m$. More specifically, each tensor is a multidimensional array with $m^{k}$ complex entries. Each index may have a different dimension, though in this work each index has the same dimension $m=4$. A tensor $A_{i_{1},i_{2},i_{3},...i_{k}}$ has $k$ indices, which take values from $0$ to $m-1$.

A \textit{tensor contraction} is a generalized tensor-tensor multiplication. Here a rank $(x+y)$ dimension $m$ tensor $A$ and a rank $(y+z)$ dimension $m$ tensor $B$ are contracted into $C$, a rank $(x+z)$ dimension $m$ tensor.
\begin{center}
\begin{equation}\label{eq:contr}
C_{i_{1},i_{2}...,i_{x},k_{1},k_{2}...,k_{z}} = \sum_{j_{1},j_{2}...,j_{y} \in \{0,..,m-1\}}{A_{i_{1},i_{2}...,i_{x},j_{1},...,j_{y}} B_{j_{1},j_{2}...,j_{y},k_{1},...,k_{z}}}
\end{equation}
\end{center}

Note that the number of floating point operations performed is $m^{x+y+z}$, exponential in the number of indices contracted on $y$ and the rank of the resulting tensor ($x+z$). It is important to point out that pairwise contractions are always optimal \cite{pfeifer14}. In other words, a function that contracts three or more nodes at a time will not achieve an improvement in scaling.

A \textit{tensor network} is a graph $G = (V,E)$ with tensors as vertices, and edges labeled by a set of indices. The rank of each tensor is given by the number of edges connected to it. An edge from one tensor to another indicates a contraction between the two tensors, and multiple connected edges indicate a contraction on multiple indices. Fig \ref{fig:tn} shows an example of a tensor network. Note that a tensor may have open edges, \textit{i.e.} edges that do not connect to any other tensor, though this possibility is not allowed in the current version of qTorch.

\begin{figure}[h]
\centering
\includegraphics[width=0.7\textwidth]{img/Fig1.png}
\caption{An example of a tensor network. The number of edges (or wires) connecting to a tensor is equal to that tensor's rank. When an index (edge) is contracted by combining two tensors according to Equation \ref{eq:contr}, the two tensor are replaced by a new one. The number of scalar entries in the tensor scales exponentially in the number of edges to which it connects. In general it is not trivial to choose an efficient contraction ordering that minimizes the total number of floating point operations.}
\label{fig:tn}
\end{figure}

A contraction ordering or contraction scheme determines the order in which the tensor network is contracted. The ordering chosen for the contraction will greatly affect the computation and memory requirements, because some contraction orderings can result in much larger intermediate tensors than others. Although in this work we focus on contracting the tensor network to a scalar that equals the expectation value of the quantum circuit's measurement, the goal of a tensor network algorithm is often \textit{not} to contract it to a scalar\cite{orus13,ran17}. An example of this is the infinite tensor networks used to study periodic systems in condensed matter physics.

An important goal is to avoid tensors of large intermediate rank when contracting the network, as floating point operations grow exponentially with tensor rank. However, it is often the case that increasing the tensor rank is unavoidable. A simple example of this issue is a tetrahedral graph of rank 3 tensors (Fig \ref{fig:tetr}), which cannot be contracted without forming intermediate tensors of rank greater than 3. The larger the treewidth of $L(G)$ is, the more one will be forced to create new tensors of higher rank as the network is contracted, greatly increasing the computational cost \cite{quant-ph/0511069}.

\begin{figure}[h]
\centering
\includegraphics[width=0.3\textwidth]{img/Fig2.png}
\caption{A tetrahedral graph illustrates why it is often unavoidable to form tensors of higher rank while contracting a tensor network. In this example, contracting any of the six edges produces a tensor of rank 4, even though all of the original tensors were of rank 3.}
\label{fig:tetr}
\end{figure}

We note that tensor network methods are commonly used to efficiently find approximate solutions---indeed this is often the main source of a TN's utility. In approximate TN methods, the space of the smaller values of the eigenspectrum are removed, after e.g. performing a singular value decomposition on the tensor\cite{orus13,ran17}. This allows one to contract to form a larger tensor, then renormalize its size before continuing to contract the network. Though this strategy is useful in the context of quantum circuits as well, the current version of qTorch is limited to numerically exact contractions of the tensor network.

Before contracting, the tensor network graph must first be created from a quantum circuit, a procedure that has been summarized in previous work \cite{quant-ph/0511069}. Each node on the graph represents one of the following: An initial state of the qubit (usually $|0 \rangle \langle 0|$), a gate operation, or a measurement. The initial density matrix is represented as a rank 1 dimension 4 tensor (i.e. a vector), $[\rho_{|0 \rangle \langle 0|};\rho_{|0 \rangle \langle 1|};\rho_{|1 \rangle \langle 0|};\rho_{|1 \rangle \langle 1|}]$. Measurement nodes are rank 1 as well. All indices in the graph are dimension 4, regardless of rank. Nodes corresponding to quantum gates are represented in superoperator form. Hence a gate $U$ which acts on the quantum state as $\rho \rightarrow U \rho U^\dag$ is represented by the superoperator $\tilde{U}$. The same operation can be expressed as $\tilde{\rho} \rightarrow \tilde{U}\tilde{\rho}$, where $\tilde{\rho}$ is the Lindblad representation of the density operator. Single qubit gates correspond to rank 2 tensors and two-qubit gates correspond to rank 4 tensors. The graph's connectivity is identical to the connectivity of the original quantum circuit.

We end this section with explicit examples of tensors for standard quantum circuit components. Tensors for other circuit components can be viewed in the source code for qTorch.

The initial state $|0 \rangle\langle 0|$ corresponds to the tensor

\begin{equation}
\tilde{\rho}_0 = [1,0,0,0].
\end{equation}

Superoperator tensors for the Pauli matrices are

\begin{equation}
\tilde{X} =
\begin{bmatrix}
   0 & 0 & 0 & 1 \\
   0 & 0 & 1 & 0 \\
   0 & 1 & 0 & 0 \\
   1 & 0 & 0 & 0 \\
\end{bmatrix};
\tilde{Y} =
\begin{bmatrix}
   0 & 0 & 0 & 1 \\
   0 & 0 & -1 & 0 \\
   0 & -1 & 0 & 0 \\
   1 & 0 & 0 & 0 \\
\end{bmatrix};
\tilde{Z} =
\begin{bmatrix}
   1 & 0 & 0 & 0 \\
   0 & 1 & 0 & 0 \\
   0 & 0 & 1 & 0 \\
   0 & 0 & 0 & 1 \\
\end{bmatrix}.
\end{equation}

The CNOT gate is represented as a sparse rank 4 tensor for which only the following entries are nonzero:

\begin{align}
\tilde{U}_{CNOT}:
\tilde{U}_{0000} =
\tilde{U}_{0101} =
\tilde{U}_{0202} =
\tilde{U}_{0303} =
\tilde{U}_{1011} =
\tilde{U}_{1110} =
\tilde{U}_{1213} =
\tilde{U}_{1312} = \nonumber \\
\tilde{U}_{2022} =
\tilde{U}_{2123} =
\tilde{U}_{2220} =
\tilde{U}_{2321} =
\tilde{U}_{3033} =
\tilde{U}_{3132} =
\tilde{U}_{3231} =
\tilde{U}_{3330} = 1
\end{align}

Finally, the nodes for measurement are rank 1 tensors. $\tilde{M}_X$, $\tilde{M}_Y$, and $\tilde{M}_Z$ correspond respectively to determining expectation values for measurements in the $X$, $Y$, and $Z$ bases. Note that using $\tilde{M}_X$, $\tilde{M}_Y$, or $\tilde{M}_Z$ is equivalent to inserting a Pauli gate at the end of the circuit before implementing $\tilde{M}_{Trace}$.
\begin{equation}
\tilde{M}_{Trace} =
\begin{bmatrix}
1 \\ 0 \\ 0 \\ 1
\end{bmatrix};
\tilde{M}_{X} =
\begin{bmatrix}
0 \\ 1 \\ 1 \\ 0
\end{bmatrix};
\tilde{M}_{Y} =
\begin{bmatrix}
0 \\ i \\ -i \\ 0
\end{bmatrix};
\tilde{M}_{Z} =
\begin{bmatrix}
1 \\ 0 \\ 0 \\ -1
\end{bmatrix}
\end{equation}

\section{Contraction schemes and implementation details}\label{sec:contract}
For many problems in quantum physics to which matrix product states (MPS) or other tensor network methods have been applied, an efficient contraction scheme is often obvious from the underlying structure of the Hamiltonian \cite{orus14}. However, efficient contraction schemes are not available for arbitrary tensor networks. A general heuristic contraction scheme is important for the simulation of general quantum circuits, when one does not know a priori the topological properties of the underlying tensor network problem.

\subsection{Contraction schemes}

qTorch implements two algorithms for determining the contraction ordering. For what we call the line graph (\textit{LG}) method, outlined in Algorithm 1, we first create the line graph of the quantum circuit's graph. Then, the software package QuickBB \cite{qbb} is used to determine an approximately optimal tree decomposition of this linegraph. The resulting tree decomposition is used to define the order of contraction. This linegraph-based approach was first described by Markov and Shi \cite{quant-ph/0511069}.

QuickBB uses a so-called anytime algorithm, meaning that it can be run for an arbitrary amount of time, such that when the program is stopped it provides the best solution found thus far. The algorithm is based on the branch and bound (B\&B) method, though the authors use several techniques based on modern graph theory to improve efficiency in the pruning and propagation steps, making QuickBB faster at finding low-width tree decompositions than vanilla B\&B.

The second contraction scheme is a simple stochastic procedure we refer to as \textit{Stoch} (Algorithm 2). First, a wire is randomly chosen. If the rank of the contracted tensor is higher than the highest rank of the two nodes, plus a given threshold, the contraction is rejected. After a fixed number of rejected contraction attempts, the threshold is relaxed.

\begin{algorithm}
\caption{Contraction via TD of L(G)}\label{td}
\begin{algorithmic}[1]

\State Create line graph L(G) of graph G
\State $\pi \gets$ (Calculate approx. optimal elimination ordering of L(G))
\State Eliminate wires of G in order $\pi$

\end{algorithmic}
\end{algorithm}

\begin{algorithm}
\caption{Simple stochastic contraction}\label{simple}
\begin{algorithmic}[1]
\State Define $G \gets$ The tensor network Graph
\State $Threshold \gets -1$
\State Define $MaxRejections$ $\gets$ Maximum Number of Rejections

\Repeat
  \State Choose a random wire $w$
  \State ($A,B$) $\gets$ (Nodes of $w$)
  \State Cost $\gets$ rank($C$) $-$ max(rank($A$),rank($B$))

  \If{Cost $\leq$ Threshold}
      \State Contract $w$ to form node $C$
      \State $Rejections$ $\gets$ 0
      \State $Threshold$ $\gets$ -1
      \State Update $G$
  \Else
      \State $Rejections \gets Rejections + 1$
      \If{$Rejections > MaxRejections$}
      	\State $Threshold \gets Threshold + 1$
        \State $Rejections \gets 0$
        \State Continue
      \EndIf
  \EndIf
\Until{Graph completely contracted}

\end{algorithmic}
\end{algorithm}

\subsection{Threading}
The tensor contractions are parallelized using the C++ standard library's \texttt{std::thread} class. A particular tensor-tensor contraction is parallelized if the cost of contracting a pair is higher than a provided threshold. We implement other parallelization schemes at the network level, i.e. splitting up the nodes into separate groups to compute on different threads, but the vast majority of the parallelization speedup comes from threading the tensor--tensor contractions. Currently, qTorch does not support parallelization across multiple compute nodes within a cluster, but it allows the user to specify the number of threads (default of 8).

\subsection{Estimating the answer string}\label{ssec:ansstr}

qTorch computes expectation values of the form $\langle \psi | M | \psi \rangle$, where $M$ is a measurement operator such as a Pauli string, and $|\psi\rangle$ is the quantum state produced by the circuit. If instead one wishes to capture all the information of this final state of $n$ qubits, it generally requires $O(2^n)$ repetitions of the algorithm. However, many quantities of interest may be calculated efficiently. For instance, the probability that one measurement operator (e.g. a Pauli string) will provide a particular outcome can be estimated in just one contraction of the tensor network, a result essential to simulating the variational quantum eigensolver (VQE) \cite{Peruzzo.NC.5.4213.2014,weckervqe15,Yung.SR.4.3589.2014,Mcclean.NJP.18.023023.2016}.

qTorch provides a heuristic scheme to output a high-probability answer string from the circuit, which we summarize here. Though this scheme is not used for the results presented in Section \ref{sec:res}, it may be useful in the future for simulating algorithms (like QAOA) where the goal is to estimate a most likely bit string.

The scheme is implemented as follows. First we run one simulation, and measure in the computational basis to project the first qubit into 0 or 1. Based on the resulting expectation value from the simulation, we choose the value for the first qubit that has the greater probability. If the 0 and 1 are equally likely, one is chosen randomly. In the next full contraction, we set the resulting binary value as the measurement outcome for the first qubit in the next simulation, and repeat with a projective measurement on the second qubit. We continue this process for the rest of the qubits. As we show below, this method often gives a good approximation of the most likely final computational basis state. In original tests on 3-regular graphs of 30 vertices, the scheme (used on Quantum Approximate Optimization Algorithm [QAOA] circuits) gave bit strings that provided good estimates to the solution of the Max-Cut problem (average approximation ratio of 94\% compared to the exact brute force solution).

As a way to test the general applicability of this scheme, we performed some tests on more general circuits than the QAOA problem. These tests are meant to provide some early insight into how useful this heuristic would be for estimating the most likely bit string of a quantum algorithm, for the users who are interested in running this string estimation subroutine. We note that it is abundantly clear that in many cases this scheme does not produce a string closest to the most likely string---indeed, if it was a generally accurate scheme then we would have no need for a quantum computer.

In the remainder of this section, we consider the most likely bit string of the final state $|\psi\rangle=\sum_i\psi_i|i\rangle$, which we define as $\text{argmax}_i|\psi_i|^2$, where the vectors $\{|i\rangle\}$ are in the computational basis. We apply a unitary of the form
\begin{equation} \label{eq:unitary}
U_p({\boldsymbol\beta},{\boldsymbol\gamma})=
\prod_{j=1}^p
\text{exp}\left(i\beta_j\sum_{i=1}^n X_i\right)
\text{exp}\left(i\gamma_j{D}_j\right)
\end{equation}
where the matrix ${D}_j$ is a diagonal matrix with entries chosen randomly from the integers $\{1,\cdots,nm\}$. Here $m$ is a parameter that could be interpreted as the number of clauses, if this were a QAOA problem. The elements of the $p$-dimensional vectors $\boldsymbol\beta$ and $\boldsymbol\gamma$ are drawn uniformly from $[0,\pi]$ and $[0,2\pi]$ respectively. We use the construction of $U_p({\boldsymbol\beta},{\boldsymbol\gamma})$ to emulate the form of parametrized unitary operations used in QAOA with the same $p$. Starting from the uniform superposition over all $2^n$ bit strings $|s\rangle$, we apply $U_p$ to compute the final state $|\Psi\rangle=U_p|s\rangle=\sum_{i=0}^{2^n-1}\psi_i|i\rangle$. Let $p_i=|\psi_i|^2$ denote the probability distribution associated to the QAOA-like output state $|\Psi\rangle$. We ran 10,000 trials (with n=6, m=10, and p=2) using equation \ref{eq:unitary}, and ranked the result by how many bit strings in the true state were more likely than our outputted bit string.

Conceptually, our likely string estimation algorithm can be thought of as falsely assuming that the output state is a product state. Suppose we apply our algorithm to the state $|\Psi\rangle$. The product state then reads
\begin{equation}\label{eq:psi_}
|\Psi'\rangle=(\alpha_1|0\rangle+\beta_1|1\rangle)\otimes
(\alpha_2|0\rangle+\beta_2|1\rangle)\otimes\cdots
(\alpha_n|0\rangle+\beta_n|1\rangle)
\end{equation}
where $|\alpha_k|^2$ is the probability of $|0\rangle$ that the algorithm obtains at the $k^\text{th}$ step, with an analogous definition for $\beta_k$. With this conceptual framing, we also numerically study the 1-norm distance $\|{\bf p}'-{\bf p}\|_1$ between the approximate distribution ${\bf p}'$ which the algorithm effectively assumes and the actual distribution ${\bf p}$.

The results are shown in Fig \ref{fig:prodstate}. Here we use the number of qubits $n=6$, with parameters $m=10$ and $p=2$. Fig \ref{fig:prodstate} (a) shows that most of the time our algorithm produces a high ranking bit string---roughly 90\% of the time the output of the algorithm is among the top 10\% most likely bit strings. Fig \ref{fig:prodstate} (b) shows that the 1-norm distance between the approximate and exact distributions is less than 0.1 for nearly all of the data points. These results suggest that our heuristic for estimating an output bit string will produce acceptable estimates for some circumstances---in QAOA for instance, where one might be interested in a good approximate (as opposed to exact) solution to the constraint satisfaction problem.

\begin{figure}[H]
	\centering
 	\includegraphics[width=1.0\textwidth]{img/Fig3.png}
\caption{Results from implementing 10,000 trials of Equation \ref{eq:unitary}. We use $n=6$ qubits, $m=10$, and $p=2$. \textbf{(a)} The histogram plots how close the method's output string is to the actual most likely string. qTorch's procedure for the ``estimate'' is given in the text. The horizontal axis \emph{Ranking} is the number of computational basis states in $|\Psi\rangle$ with higher probability than the estimated string---a lower number ranking indicates a better estimate. \textbf{(b)} Distribution of the 1-norm distance between the approximate distribution ${\bf p}'$ arising from the product state approximation $|\Psi'\rangle$ in and the distribution  ${\bf p}$ arising from the exact state $|\Psi\rangle$. }
\label{fig:prodstate}
\end{figure}

\subsection{A note on noise}
It is possible to simulate noise within the quantum circuit model, by mapping a noise model onto a set of one-qubit or multi-qubit operators \cite{mikeike11,kraus83}.

Note that any quantum operation can be expressed in terms of Kraus operators $\{E_j\}$
\begin{equation}\label{eq:kraus}
\rho \rightarrow \sum_j E_j \rho E_j^\dag
\end{equation}

where $\{E_j\}$ are called Kraus operators \cite{kraus83}, and $\sum_j E_j E_j^\dag=1$ because for our purposes the noise process is assumed to be trace-preserving. A noise model can be expressed in terms of such Kraus operators, which can in turn be expressed as superoperators for insertion into the quantum circuit's tensor network.

The most commonly used approximations assume that noise on different qubits is uncorrelated, which allows for single-qubit ``noise gates'' to be used. In this case, because rank 2 tensors can always be contracted without increasing the rank of the resulting tensors, the cost of simulating the resulting ``noisy'' quantum circuit would not substantially increase. One common and easily implementable approximation is the Pauli twirl approximation, which approximates a noise process purely in terms of Pauli rotations \cite{silva2008,geller2013,tomita2014,qhip-jctc}, and therefore can be implemented with the built-in quantum gates of qTorch.

A more physically realistic noise model would assume correlated noise \cite{aharonov06,preskill13}, which necessitates the insertion of noise gates that operate on at least two qubits. In this case, the tree width of the circuit's underlying line graph, and hence the complexity of the problem, would increase in all but the most trivial cases.

qTorch does not incorporate built-in noise gates. Instead, we include functionality that allows for user-defined gates.

\section{Circuit simulations} \label{sec:examples}

Here we describe the classes of quantum circuits that were simulated for this work.

\subsection{QAOA / Max-Cut}\label{ssec:main-qaoa}

The quantum approximate optimization algorithm (QAOA) was recently developed \cite{farhi14a}, for the purpose of demonstrating quantum speedup for combinatorial problems on low-depth quantum circuits. Given a constraint satisfaction problem (CSP), a QAOA quantum circuit produces an output that provides approximate solutions. Several aspects of QAOA have been studied since its introduction, including its application to different classes of CSP, implementations of different classical optimization routines, and numerical and analytical comparisons to classical algorithms \cite{farhi14a,farhi14b,farhi16,wecker16,lin16,gg17}.

We use qTorch to simulate QAOA for the Max-Cut problem, a combinatorial problem that has been a focus of QAOA \cite{farhi14a}. Given an arbitrary undirected graph, the goal of Max-Cut is to assign one of two colors to each node, so as to maximize the number of cuts. A cut is any edge that connects two nodes of different color

In QAOA, a set of constraints is mapped to a an objective function represented by a set of operators. Specifically for the Max-Cut problem, the object function is

\begin{equation} \label{eq:maxcutobj}
C = \sum_{\langle ij\rangle }{C_{\langle ij \rangle}},
\end{equation}

with

\begin{equation} \label{eq:pairwise}
C_{\langle ij\rangle } = \frac{1}{2}(1-\sigma^{z}_{i}\sigma^{z}_{j}),
\end{equation}
where $\langle ij\rangle$ represents the edge between nodes $i$ and $j$, $\sigma_k^z$ is the Pauli-Z operator on qubit $k$, and each node in the underlying Max-Cut graph (which is related to but not the same as the quantum circuit's graph) corresponds to one qubit in the quantum circuit.

Define two operators $U(C,\gamma)$ and $U(B, \beta)$ as
\begin{equation}
U(C,\gamma) = e^{-i\gamma C} = \prod_{m=1}^{n}{e^{-i\gamma C_{m}}}
\end{equation}

and

\begin{equation}
U(B, \beta) = e^{-i\beta B} = \prod_{k=1}^{q}{e^{-i\beta \sigma_k^x}}.
\end{equation}

where $B=\sum_{k=1}^q \sigma_k^x$, $\sigma_x$ is the Pauli-X operator, $q$ is the number of qubits, and $n$ is the number of clauses (for Max-Cut this is the number of edges). These two operators are applied $p$ times (with different $\gamma$ and $\beta$ allowed for each step), with a larger $p$ providing a better approximation. The $\gamma$ and $\beta$ parameters are modified with a classical optimization routine to maximize the cost function. The cost function is evaluated after each measurement, with the bit string that resulted from the measurement.

To generate the graphs for the underlying Max-Cut problem, we made random $k$-regular graphs by placing edges randomly throughout a given vertex set to satisfy a given regularity, before ensuring that disconnected graphs are rejected. QAOA/Max-Cut Quantum circuits based on these graphs are then constructed.

In the numerical results of this paper, we report only the timing for a single contraction of each quantum circuit. A full analysis of QAOA is beyond the scope of this work. However, we note that once the graphs have been created, qTorch currently has the functionality to optimize the QAOA angles using the classical optimization library NLopt \cite{nlopt}. Finally, one can use qTorch to estimate a Max-Cut for the randomly-generated graph, using the most-likely bit string estimation method described above.

\subsection{Hubbard Model}

Quantum simulation of fermionic systems is one of the most relevant applications of quantum computers, with direct impact on chemistry and materials science, including for the design of new drugs and materials. Among all the algorithms proposed for quantum simulation of fermions, the quantum variational algorithm (VQE) and related approaches \cite{Peruzzo.NC.5.4213.2014,weckervqe15,Yung.SR.4.3589.2014,Mcclean.NJP.18.023023.2016} are arguably the most promising for near-term hardware because they have lower circuit depth requirements \cite{mcclean2016hybrid,Omalley.PRX.6.031007.2016}. We note that many types of chemistry-related circuits can be prepared with the software package OpenFermion \cite{openfermion}.

In the VQE algorithm, a quantum computer is employed to prepare and measure the energy of quantum states associated with a parameterized quantum circuit. The approximate ground state of a Hamiltonian is obtained by variationally minimizing the cost function (corresponding to e.g. the molecular energy) with respect to the circuit parameters using a classical optimization routine. This hybrid quantum-classical approach offers a good compromise between classical and quantum resources. Classical simulations of the VQE algorithm for tens of qubits could provide insights into the complexity of the circuits used for state preparation and help design better ansatzes for the quantum simulation of fermions.

As an example of a VQE simulation, we used qTorch to classically simulate variational circuits employed for the quantum simulation of 1D Hubbard lattices. We consider half-filled Hubbard models on $N$ sites, with periodic boundary conditions.

To construct variational circuits for these systems, we considered the variational ansatz introduced by Wecker et al \cite{weckervqe15}. In this case, the Hubbard Hamiltonian is divided as $H=h_h+h_U$, where $h_h$ is the sum of hopping terms in the horizontal dimension and $h_U$ is the repulsion term. The variational circuit is constructed as a sequence of unitary rotations by terms in the Hamiltonian with different variational parameters, with the sequence being repeated $S$ times. In each step, there are two variational parameters, $\theta^{b}_{U}$ and $\theta^{b}_{h}$, where $b=1,\cdots,N$ such that

\begin{equation}\label{variationalC}
U = \prod^{S}_{b=1} \left[ U_U \left(\frac{\sigma^{b}_U}{2}\right) U_h(\theta^{b}_{h})  U_U\left(\frac{\theta^{b}_U}{2}\right) \right]
\end{equation}

where $U_X(\theta_X)$ denotes a Trotter approximation to $\exp(i\theta_X h_X)$, and $X$ can be $U$ or $h$. For our numerical simulations, we employed the variational circuit of Eq. \ref{variationalC} with $S=1$ using a 1-step Trotter formula for all the $U_X(\theta_X)$ terms. Notice that this is approximate for the $h_h$ term, which comprises a sum of non-commuting terms. We also assigned the value of 1 to all variational amplitudes. The corresponding unitary was mapped to a quantum circuit using the Jordan-Wigner transformation and the circuit was generated using a decomposition into CNOT gates and single-qubit rotations \cite{mikeike11,Whitfield.MP.109.735.2011}.

\section{Results}\label{sec:res}

Simulations were performed on the NERSC Cori supercomputer, using one "Knights Landing" (KNL) node per simulation, each of which contains 68 cores and 96 GB of memory. Each LIQ$Ui|>$ \space simulation was run on a full node as well, using Docker \cite{docker2014}
The free version of LIQ$Ui|>$ \space allows for the simulation of 24 qubits. Because full Hilbert space simulation scales exponentially regardless of the quantum algorithm's complexity, we would not have been able to simulate more than $\sim$31 qubits on one of these compute nodes. For each set of parameters (regularity and number of vertices/qubits) 50 instances of Max-Cut/QAOA circuit were created. For higher qubit counts and higher regularities, only a subset of these circuits were completed, since many simulations exceeded memory capacity. In this section, \textit{LG} or \textit{qTorch-LG} refer to the use of qTorch with the linegraph-based contraction, \textit{Stoch} or \textit{qTorch-Stoch} refer to qTorch with stochastic contraction. To determine a qTorch-LG contraction ordering, QuickBB simulations were run for an arbitrary time of 3000 seconds for each quantum circuit. The plotted qTorch results do not include the QuickBB run time.

We note that LIQ$Ui|>$ \space implements a thorough set of important optimizations, which makes it a fair benchmark against which to compare qTorch. For example, LIQ$Ui|>$ \space fuses many gates together before acting on the state vector, and uses sparse operations. qTorch, on the other hand, does not yet use sparsity at all (even when the circuit consists primarily of sparse CNOT gates), which is one of several optimizations that we expect would further improve performance.

LIQ$Ui|>$ \space is the fastest simulation method to use for the Hubbard simulations, as shown in Fig \ref{fig:hubb}. This is because the treewidth of the circuit's graph increases substantially with the number of qubits, even for these short-depth circuits. The result is not surprising---if the algorithm were easy to simulate with a tensor network on a classical computer, then it would not have been worth proposing as a candidate for a quantum computer.

\begin{figure}[H]
	\centering
 	\includegraphics[width=0.8\textwidth]{img/Fig4.png}
	\caption{Time results for simulating quantum circuits of the Hubbard model. LG, Stoch, and LIQ$Ui|>$ \space denote linegraph-based tensor contraction, stochastic tensor contraction, and LIQ$Ui|>$, respectively. LIQ$Ui|>$'s full Hilbert simulation method is substantially faster than either tensor contraction method. Missing data points resulted from exceeding memory capacity.}
    \label{fig:hubb}
\end{figure}

Simulation timing results for 3-, 4-, and 5-regular Max-Cut/QAOA circuits are shown using Tukey boxplots in Figs \ref{fig:t-nq-few} and \ref{fig:t-nq-many}. Stoch and LG simulation times are of similar order of magnitude for these circuits, though LG is generally faster. The exception is the 3-regular graph problems, where Stoch often appears to find a more efficient contraction than the 3000-second run of QuickBB does. We note that if the QuickBB algorithm were run for infinite time before beginning the contraction, then qTorch-LG should always (except in very simple graphs) contract the circuit faster than qTorch-Stoch. This is because, while the Stoch search is purely local (considering only individual wires), the tree decomposition approach of QuickBB implicitly considers the effects of multiple contraction steps. Note that actual search time of Stoch is negligible compared to the tensor contraction time. Note that LIQ$Ui|>$ \space begins to outperform tensor contraction methods once the algorithm is run on 5-regular graphs, because the increased circuit complexity leads to larger intermediate tensors in qTorch.

Note that in principle, Hilbert space simulation can be considered a subset of TN contraction, where the state vector is simply a large tensor. Hence one might expect that there would not be a crossover point at all, \textit{i.e.} that in the worst case TN contraction would not ever be slower than Hilbert space simulation. However, because our implementation considers density matrices instead of state vectors, one would in fact expect this crossover point to exist. The largest tensor in qTorch would have $4^{N}$ entries, while the state vector has just $2^N$ entries. The various choices made in software implementations for qTorch and LIQ$Ui|>$ \space would also affect the position of this crossover point.

Using a single Cori NERSC node, we were able to contract quantum circuits of 90 qubits for a very small subset of the simulated graphs, though not on enough graphs to report statistics. Full Hilbert space methods would be limited to $\sim$30 qubits on these nodes, and indeed previous simulation packages have not yet surpassed 46 qubits \cite{qhipster,haner-steiger2017}, using thousands of nodes.

Interesting trends appear when the simulation time is plotted against regularity of the Max-Cut problem's graph (Fig \ref{fig:t-reg}). It is notable that the LG method runs out of memory before the Stoch method does. As previously mentioned, the LG method contracts more efficiently the longer QuickBB has been run, and we chose 3000 seconds as an arbitrary QuickBB limit for all circuits. There is a trade-off between running a longer QuickBB simulation and instead immediately using the Stoch method. Even with few qubits, at higher regularities the full Hilbert space simulation (using LIQ$Ui|>$) performs better. This is expected, since as the complexity of the quantum circuit increases, higher-rank intermediate tensors appear.

Fig \ref{fig:t-tw} shows simulation time as the estimated upper bound for the treewidth increases, for Max-Cut/QAOA circuits of 18 qubits. These include 3- through 7-regular graphs. This treewidth upper bound is simply the treewidth of the tree decomposition that defines the contraction ordering. The plot shows the expected general trend of an increase simulation time with increased treewidth, regardless of contraction scheme.

Finally, we note that we were easily able to perform simulations of 100 qubits for less complex graphs. To report one such example, we produced a random 3-regular graph with a slightly different procedure from that given in of Section \ref{ssec:main-qaoa}. Beginning with a 2-regular graph (i.e. a ring) of 100 vertices, we added edges between random pairs of vertices until all vertices were of 3 degrees. Contracting this graph's Max-Cut/QAOA circuit took $\sim$150 seconds.

\begin{figure}[H]
	\centering
 	\includegraphics[width=\textwidth]{img/Fig5.png}
	\caption{Simulation time plotted against number of qubits for Max-Cut/QAOA circuits. LG, Stoch, and LIQ$Ui|>$ \space denote linegraph-based tensor contraction, stochastic tensor contraction, and the LIQ$Ui|>$ \space software package, respectively. Tree decompositions for the LG method were determined by running the QuickBB simulation for 3000 seconds. For lower regularities, the tensor contraction methods outperform LIQ$Ui|>$, since LIQ$Ui|>$ \space simulates the full Hilbert space. However, as the regularity of the Max-Cut graphs (and hence the treewidth of the quantum circuits' line graphs) increase, full Hilbert space simulation using LIQ$Ui|>$ \space becomes more efficient.}
    \label{fig:t-nq-few}
\end{figure}

\begin{figure}[H]
	\centering
 	\includegraphics[width=\textwidth]{img/Fig6.png}
	\caption{Simulation time plotted against number of qubits for 3-regular Max-Cut/QAOA circuits. LG and Stoch denote linegraph-based tensor contraction and stochastic tensor contraction respectively. For 3-regular Max-Cut/QAOA circuits, we were able to simulate a small subset of the 100-qubit circuits we created, not shown here.}
    \label{fig:t-nq-many}
\end{figure}

\begin{figure}[H]
	\centering
 	\includegraphics[width=0.8\textwidth]{img/Fig7.png}
	\caption{Simulation time plotted against the regularity of the underlying Max-Cut graph, for Max-Cut/QAOA circuits. LG, Stoch, and LIQ$Ui|>$ \space denote linegraph-based tensor contraction, stochastic tensor contraction, and LIQ$Ui|>$, respectively. As regularity increases, full Hilbert space simulation (using LIQ$Ui|>$ \space) becomes a more competitive simulation method. Missing data points resulted from running out of memory..}
    \label{fig:t-reg}
\end{figure}

\begin{figure}[H]

    \hspace{-1in}
 	\includegraphics[scale=0.43]{img/Fig8.png}
	\caption{Simulation time plotted against approximate treewidth, for all simulated Max-Cut/QAOA quantum circuits of 18 qubits. The plot demonstrates the general trend of increased simulation time with the quantum circuit's line graphs's treewidth, despite a constant number of qubits.}
    \label{fig:t-tw}
\end{figure}

\section{Conclusions}
\label{sec:concl}
We have implemented a tensor contraction code for the efficient simulation of quantum circuits. We compared a stochastic contraction scheme to one based on the line graph of the quantum circuit's graph, showing that the latter is more efficient in most circuits simulated herein. However, there is a subset of cases for which calculating an efficient approximate optimal tree decomposition of the line graph takes longer than contracting the circuit stochastically, in which case the stochastic scheme is superior. For the circuits studied in this work, our simulations suggest that the point at which qTorch is no longer faster than LIQ$Ui|>$ \space occurs in QAOA/Max-Cut approximately when the Max-Cut graph has a regularity of five. In the future, qTorch may be used to estimate these points of equivalent computational cost in other classes of circuits, which may help to determine which simulation method to use in simulations.

Several immediate algorithmic improvements are possible for this software. The use of sparse tensors would reduce the number of floating point operations for some relevant circuits. Tensor decompositions (such as the singular value decomposition) along with trimming can be added as intermediate steps, as has been done in tensor network based simulations of physical systems\cite{orus13,ran17}. Additionally, more advanced parallelization methods would allow for faster calculation of a tree decomposition as well as faster contractions.

\section*{Supporting information}

\section*{Acknowledgements}
We are grateful to Edward Farhi and Aram Harrow for discussions about QAOA, to Salvatore Mandr\`a for general discussions, and to Dave Wecker for helpful advice on using LIQ$Ui|>$. N.S. acknowledges the use of resources of the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231. A.A.-G. and E.F. were supported by the Office of Naval Research under grant N00014-16-1-2008 (Vannevar Bush Faculty Fellowship). A. A.-G. and Y.C. were supported by NSF grant CHE-1655187.

redpajama_set_name: RedPajamaArXiv


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I got the skirt at a little store in Ogudu and I got the top from New Look in London last summer. The heels were gotten from Mr price in South Africa and my hair is styled in a simple Side Part Afro.
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I plan to do a lot more top notch style posts this year so if you are a designer or you have any friends or family members who run a fashion brand, send them my way so we can collaborate.
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Rest: An FCA Devotional For Competitors - This 5-day reading plan, adapted from content within FCA Magazine, focuses on "Rest" and explores the benefits of rest, renewal, and honoring the divinely created Sabbath. It offers perspective and encouragement for the coach or athlete struggling to make time for rest – both physically and spiritually – in their busy schedule. To learn more about FCA Magazine and the Fellowship of Christian Athletes, visit fca.org.
This 5-day reading plan, adapted from content within FCA Magazine, focuses on "Rest" and explores the benefits of rest, renewal, and honoring the divinely created Sabbath. It offers perspective and encouragement for the coach or athlete struggling to make time for rest – both physically and spiritually – in their busy schedule. To learn more about FCA Magazine and the Fellowship of Christian Athletes, visit fca.org.


text:
Brazil and Poland set up title rematch after eight years
Poland played the very first match at this world championship in Warsaw and they will play the very last match against Brazil in Katowice
Lausanne, Switzerland, September 21, 2014 – Brazil and Poland have played each other in the FIVB Volleyball Men's World Championship final in 2006, now they will have a chance to repeat this encounter to determine the winners of the 2014 edition.
Brazil have previously reached the World Championship final four times, including the last three editions, which they eventually won. Their other gold medal match appearance was in 1982, which they lost to the Soviet Union. Poland, on the other hand, have previously reached the World Championship final two times, including the 1974 edition, which they won; and the 2006 encounter with Brazil which they lost in straight sets.
Meanwhile, France will square-off with Germany for the bronze medal match. France can win their second bronze medal and equal their best ranking at the World Championship after their 3-0 win against Yugoslavia in 2002. Germany will finish in the top four for the first time since 1974.
Teams with 15-1 win-records meet in final – Poland have won 15 of their 16 matches, with their only loss coming against USA in the second round. Brazil, on the other hand, have the same record, their only loss coming against Poland in the third round. These teams have last played against each other in the final of the World Championship in 2006, when Brazil won in straight sets. A win by Brazil will make them the first team to win four consecutive World Championship titles. If Poland will win, they will earn their third medal at the World Championship.
France and Germany's exchange in wins – France and Germany have exchanged wins in their last six encounters at all major tournaments: W-L-W-L-W-L-W from France's perspective. France need one more win to reach 10 victories – their best team record at a single edition. Germany have already jumped four places from their eight place finish in 2010. If France will win, they can get their second bronze medal in history. If Germany will win, they will move five places up from their previous standing, and will improve on their last top four appearance in 1974.

redpajama_set_name: RedPajamaCommonCrawl


text:
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To get started with Tentrr BackCountry, the first step is to tell us a little bit about your property. We're looking for information about the acres you have, the privacy and seclusion of the spot you have in mind, and the awesome local activities that make your piece of paradise unforgettable!
The first step to becoming a BackCountry CampKeeper is to complete the Tentrr Self Scout Report. Be sure to send in all information required, and feel free to send us more photos and info about your land as well.
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text:
This track shows the CpG annotations on the 09 Jul 2014 Opisthorchis viverrini/GCF_000715545.1_OpiViv1.0 genome assembly.
CpG islands are associated with genes, particularly housekeeping genes, in vertebrates. CpG islands are typically common near transcription start sites and may be associated with promoter regions. Normally a C (cytosine) base followed immediately by a G (guanine) base (a CpG) is rare in vertebrate DNA because the Cs in such an arrangement tend to be methylated. This methylation helps distinguish the newly synthesized DNA strand from the parent strand, which aids in the final stages of DNA proofreading after duplication. However, over evolutionary time, methylated Cs tend to turn into Ts because of spontaneous deamination. The result is that CpGs are relatively rare unless there is selective pressure to keep them or a region is not methylated for some other reason, perhaps having to do with the regulation of gene expression. CpG islands are regions where CpGs are present at significantly higher levels than is typical for the genome as a whole.
The unmasked version of the track displays potential CpG islands that exist in repeat regions and would otherwise not be visible in the repeat masked version.
CpG islands were predicted by searching the sequence one base at a time, scoring each dinucleotide (+17 for CG and -1 for others) and identifying maximally scoring segments. Each segment was then evaluated for the following criteria:
GC content of 50% or greater
length greater than 200 bp
ratio greater than 0.6 of observed number of CG dinucleotides to the expected number on the basis of the number of Gs and Cs in the segment
The entire genome sequence, masking areas included, was used for the construction of the track Unmasked CpG. The track CpG Islands is constructed on the sequence after all masked sequence is removed.
The CpG count is the number of CG dinucleotides in the island. The Percentage CpG is the ratio of CpG nucleotide bases (twice the CpG count) to the length. The ratio of observed to expected CpG is calculated according to the formula (cited in Gardiner-Garden et al. (1987)):
Obs/Exp CpG = Number of CpG * N / (Number of C * Number of G)
where N = length of sequence.
CpG item counts
masked sequence: item count: 21,299 bases covered: 7,051,228
unmasked sequence: item count: 59,620 bases covered: 25,727,867


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text:
Resonate Activate
The Event Is Coming
Urgent! Daily meditation for India and Pakistan at 4:15 PM UTC
Tension is escalating rapidly between nuclear-armed neighbors India and Pakistan after air combat between the rivals ended with the reported loss of aircraft by both nations.
The ongoing flare-up started on February 14 with the suicide bombing of Indian police troops in Kashmir, a divided region in the north that has been contested by Islamabad and New Delhi since 1947.
https://www.rt.com/news/451532-india-isolate-pakistan-kashmir-attack/
https://www.dw.com/en/kashmir-suicide-car-bomb-kills-dozens-of-indian-troops/a-47523394
In light of that incident, we organized a special daily Buddhic Column Meditation at 4 PM UTC for Kashmir during the previous week.
https://www.welovemassmeditation.com/2019/02/meditation-update.html
Indian raid targeted an area near the town of Balakot in the Khyber-Pakhtunkhwa province of Pakistan, which is located some 50km (32 miles) from the UN-mandated Line of Control (LoC) in Kashmir. This Line of Control serves as the de facto border in the disputed region. Source: https://www.rt.com/news/452540-india-pakistan-hostilities-escalate/
Tensions between India and Pakistan flared up dramatically on Tuesday, after New Delhi sent a dozen of fighter jets to strike what it said were terrorist camps on Pakistani soil.
Then on Wednesday, both countries said they shot down each other's fighter jets, with Pakistan capturing an Indian pilot a day after Indian warplanes struck inside Pakistan for the first time since a 1971 war.
There are also reports that ground forces from both countries have exchanged fire in more than a dozen locations.
https://www.rt.com/news/452540-india-pakistan-hostilities-escalate/
https://www.thestar.com.my/news/world/2019/02/27/india-pakistan-trade-fire-in-kashmir-us-urges-calm-after-indian-airstrike/
Although both governments publicly stated they want to defuse the tensions, the situation may still escalate further and become a full-scale war between two nations with nuclear weapons.
https://www.rt.com/news/452498-india-pakistan-nuclear-power-conflict/
Therefore, we are going to organize a daily meditation at 4:15 PM UTC for India and Pakistan to harmonize the situation. The time of this meditation is chosen so that more people in India, Pakistan and the rest of Asia can participate. We will be doing this meditation until the sitatuion is resolved peacefully.
You can convert UTC time zone to your local time zone here:
https://www.thetimezoneconverter.com/
Here is the Facebook event for this meditation:
Instructions of this meditation: (Suggested time for this meditation is 15 minutes)
Use your own technique to bring you to a relaxed state of consciousness
State your intent to bring peace and healing to the situation in Inadia and Pakistan and especially, to the disputed region in Kashmir.
Visualize a pillar of pink Light, emanating from the Galactic Central Sun and then descending down through your body.
Visualize that Light flowing through your heart and then through your hands directly to everybody involved in the conflict between India and Pakistan , harmonizing everyone and bringing peace. Let that Light then flow onwards and heal all other conflicts worldwide.
Ask the Goddess presence (divine feminine) to come through your body.
Ask that Goddess presence to direct energies of peace and harmony to all unresolved situations regarding the Kashmir conflict and everyone involved in them. Let these energies then flow onwards and heal all other conflicts worldwide. Allow these energies to flow through your heart and then through your hands into this situation in Kashmir for a few minutes.
Goddess wants peace and peace it will be.
You can also create a Buddhic column above Kashmir during the special Buddhic Column Meditation at 4 PM UTC if you feel so guided. You can find the instructions and guided audio for the Buddhic Column Meditation in the link below:
https://www.welovemassmeditation.com/2018/10/daily-buddhic-columns-meditation-at-330.html
This is the link to the Facebook event for this special Buddhic Column Meditation:
We also suggest to include this latest armed escalation in the daily Emergency Meditation at 2 PM UTC. Visualize all tensions completely dissolving and that a peaceful solution is achieved as soon as possible.
https://www.welovemassmeditation.com/2018/02/breakthrough-meditation-at-2-pm-utc-and.html
Source: https://www.welovemassmeditation.com/2019/02/urgent-daily-meditation-for-india-and.html
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