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FREELessons: 7Length: 36 minutes

• Overview
• Transcript

# 2.1 Targeting Elements by ID and Class Name

In this lesson, I will show you how to use a few JavaScript methods to target elements on the page using their ids and class names.

## 2.All You Need Is JavaScript5 lessons, 35:15

### 2.4JavaScript Event Handling07:52

Hello and welcome back. Over the next few lessons, we're gonna be taking a look at this FAQ page. And we're gonna take away some of the jQuery programming that we've done on this page. And we're going to replace it with plain JavaScript, so that I can prove to you that you can accomplish a lot using plain JavaScript. And it's only a little bit more challenging than using jQuery. So, as you can see, we are using jQuery selectors here to point to all of our items, and we're also using some jQuery functions like hide, show and on. We're also using removeClass and addClass as well as the parent method. These are all jQuery methods here. And we're gonna talk about how we can achieve all of the same functionality here using plain JavaScript instead. So in our FAQ section, you can see that we can click on different areas, or different headers here on these panels, and it will collapse all of the other panels and expand out the panel that we clicked on. If we look in our HTML, we can see that our markup consists of three main sections here. So we have this FAQ section, and in that section we have three divs with a class of panel-question. This first panel also has a class of active, because by default, when the page first loads, this is the one that we want expanded out. So we have this active on first, and inside these panel-question divs, we have a div with a class of panel-heading, which is this section with a dark background. And then a div with a class of panel body, which is where the answer is contained with the white background. So you can find the URL for this particular CodePen and the course note for this lesson. Once you've opened up, click on the fork button to create a new copy of that. And we'll get started with this new copy. So in this pen, again, we've got several jQuery items, there are several objects that we're pointing to using a jQuery. And in this lesson, I wanna focus on pointing to these items using plain JavaScript instead. So let's start with our collapse button. Now, right now there's no functionality for that collapse button, but it's this button in the upper right-hand corner that will collapse everything. And we can create an event for that if we want. It's pretty simple, $btnCollapse.on, and this is gonna be a click event. When we click on that, we want to collapse all of our items. And we already have some code for that. That's this answer variable here, which is pointing to the panel body, which again is the answer section of each of these questions. And if we just point to answer, and then in jQuery do a dot hide, then that will get rid of them. So let's just copy that and paste it here, and that should do it for us. So I'm gonna save that and hit Collapse All. And there we go, now we can see they're all collapsed. And then we can through and click on them individually to open them up again. Okay, so let's talk about how we would point to this collapse button using plain JavaScript instead of jQuery. So I'm gonna go to the next line. I'm gonna create another variable called btnCollapse but without the dollar sign. And the way we point to an element using its ID, which is what we doing here, that's what this hashtag is for. The way we do that in plain JavaScript is we point to document., and then there is a method called getElement, singular, ById, now make sure you get all the right things capitalized. It starts off lowercase and then every word after that starts with a capital letter. So we have document.getElementById, and then inside parentheses and inside quotation marks, we will the type in the name of the ID that we are looking for. Nnow in this case we're not gonna put the hashtag there, this is not a CSS selector, instead we're just typing in the name of the ID. So, btnCollapse, and there we go. So lets take a look at our console to see what these two items look like. So, I'm gonna do a console log, console.log. And first, we are gonna do$btnCollapse with a dollar sign in front of it. And then second, lets just copy this and paste it and get rid of dollar sign. And you'll notice that I put a string here first because I want to see in the console the actual string $btnCollapse and then after that we'll see what that looks like in JavaScript. So, let's save that, and then if we hit F12 to open up our developer tools in Chrome, and then click on the consul tab to make sure it's open, and then well expand this out a little bit, we can see our console logs here. So our first one, here we have$btnCollapse. This is what that looks like, right here. And we can expand that out by clicking on this little triangle to see more information on it. Now the btnCollapse without the dollar sign, you'll notice it looks a little bit different in the console. Here we see the actual HTML for that button. So both of these methods of pointing to an object, a dumb element, both of them will work but you'll notice that both of them work a little bit differently. Now we're not gonna into a lot of detail on how they're different but I wanted to point out that they are different, and we can even do this. Let's get rid of this first one, and instead of putting the string here at the beginning. I'm just going to see, I'm gonna compare this, and see if $btnCollapse is equal to just plain bntCollapse. And when our code refreshes, we see at the bottom that we got a false, which means that these two are not the same thing. And, again, the reason I wanna point that out you is because you need to understand that even though we're pointing to the same element with these two methods, we can't expect everything that we do with this item, this jQuery item, we can't expect to be able to do the same thing to this non-jQuery item. So down here, where we have this$btnCollapse.on, we couldn't do that with the plain JavaScript btnCollapse. If I were to get rid of this dollar sign here and then tried to click on Collapse All, you'll see that it doesn't work. In fact, we can see in our console that btnCollapse.on is not a function. We have to actually have a jQuery object that we're pointing to here. So if we put that dollar sign back then our code will suddenly start working again, and we can collapse everything. So, again, it's important to understand that these two methods of pointing two DOM elements, they're both effective, but we have to do things a different way if we're going to use jQuery versus if we're going to use plain JavaScript. And our whole goal here is to get rid of jQuery anyway. Not that jQuery's a bad thing, but sometimes you don't necessarily need it. So we're gonna find a different way to achieve what we're doing anyways, but I wanted to point that out, these are not the same types of objects. But again, this getElementById is how you would point to an element on the screen or in your HTML using plain JavaScript. Now, before we move on, let me go ahead and point out how we would do that using a class name. Obviously, getElementById is not going to work if we're pointing to a class name. So we have these three items here that are pointing to class names. And let's create versions of them. So we have var question without the dollar sign. And here we're still gonna point to document, but instead of getElementbyId, there's another method, lowercase g here, getElements, plural, ByClassName. And that will get all of the elements in your document that have whatever class name you enter in here. So, inside quotation marks here, we're gonna type in our class name, again, this time without then dot in front of it. And we're gonna point to panel.question. So again, it's very similar to the document.getElementById but it's getElements, plural, don't forget the s at the end of the elementsByClassName. And then we can do the same thing for heading and answer. So var heading = document.getElementsByClassName, and here we're reporting to panel-heading, again, without the dot in front of it, because this is no longer a CSS selector we're using, it's just the class name we're entering. So do the same thing here for answer. And we'll do document.getElementsByClassName, and this time we're pointing to the class of panel-body. So, again, those are two simple ways that you can access elements using plain JavaScript instead of jQuery. As we move forward, we're gonna talk about different ways that we can point to other objects, as well as eventually how to achieve some of these same effects using these JavaScript objects that we've created. So thank you for watching and I'll see you in the next lesson.

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# Equilibrium index of a sequence

Equilibrium index of a sequence is an index such that the sum of elements at lower indexes is equal to the sum of elements at higher indexes. For example, in a sequence A:

A[0]=-7 A[1]=1 A[2]=5 A[3]=2 A[4]=-4 A[5]=3 A[6]=0

3 is an equilibrium index, because:

A[0]+A[1]+A[2]=A[4]+A[5]+A[6]

6 is also an equilibrium index, because:

A[0]+A[1]+A[2]+A[3]+A[4]+A[5]=0

(sum of zero elements is zero) 7 is not an equilibrium index, because it is not a valid index of sequence A.

The idea is to create a program that given a sequence (array), returns its equilibrium index (any) or -1 if no equilibrium indexes exist.

### Golfscript 17 16

Since the form of the input isn't specified, this takes a string in Golfscript array format from stdin.

~0\{1$+.@+\}/])?  So run as e.g. golfscript.ry eqindex.gs <<<"[-7 1 5 2 -4 3 0]"  The idea is very simple: it takes an array of A_i and maps to an array of A_i + 2 SUM_{j<i} A_j and then looks for the first index which is equal to the sum of the whole array. For @mellamokb's challenge I offer: ~0\{1$+.@+\}/:S;]:A,,{A=S=},

for 29 chars.

• Since you easily have the shortest solution, I proclaim you must return all of the indexes, not just the first one :) May 4, 2011 at 17:30
• @mellamokb, with my compliments. May 4, 2011 at 22:29
• Cool! Now I've got some more GolfScript learning to do... May 5, 2011 at 2:27

## Python - 72 chars

A=input()
print[i for i in range(len(A))if sum(A[:i])==sum(A[i+1:])]or-1

Takes comma separated input

• Awesome... this one returns all the equilibrium indexes... really cool. May 4, 2011 at 13:58
• @Christian: Mine does so, too. May 4, 2011 at 18:29
• I see :) I actually don't know how to run haskell code... will have to study. May 5, 2011 at 4:08
• Christian: There is ghc, a compiler and hugs, an interpreter. I'd suggest downloading hugs. It's better then downloading ghc, because hugs is about 7 MiB, while the whole ghc distribution is about 300 MiB. Using hugs, you can just type runhugs FILE.hs to run program FILE.hs. May 6, 2011 at 13:03

e l=[n|n<-[0..length l-1],sum(take n l)==sum(drop(n+1)l)]
main=interact$show.e.read  Reads a list in Haskell style from stdin, eg. [-7,1,5,2,-4,3,0]  and returns a Haskell style list of the indices, eg. [3,6]  The result is [], if there is no index. Please tell me, if your spec wants a different behavior. ### Edits: • (95 → 83): list comprehension is more breve ## C - 96 a[99],*p=a,s;main(){for(;scanf("%d",p)>0;s+=*p++ );for(;p>a;s-=*p)(s-=*--p)||printf("%d\n",p-a);}  Note that this prints the equilibrium indices in reverse order. Sample usage: $ ./equilibrium <<< "-7 1 5 2 -4 3 0"
6
3

## Ruby (83 77)

a=*$<.map(&:to_i) p (0...a.size).select{|x|a[0..x].reduce(:+)==a[x..-1].reduce(:+)}  Edit: Shorter version as suggested by Ventero: a=$<.map &:to_i
p (0...a.size).select{|x|eval"#{a[0..x]*?+}==#{a[x..-1]*?+}"}

Input is one number per line, output is comma separated list of indexes in square brackets.

• You don't need the parentheses in the first line, and you can save a few chars by using join + eval to get the sums: p (0...a.size).select{|x|eval"#{a[0..x]*?+}==#{a[x..-1]*?+}"} (note that this is for Ruby 1.9, since it uses character literals as strings) May 9, 2011 at 17:57
• Great suggestions, thanks! Kind of annoying that Array#sum isn't in Ruby core. May 9, 2011 at 20:48


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en:lesson02

# 02 Lesson. Introducing sensors

## Angular velocity sensor (AVS)

Function of angular velocity sensor Angular velocity sensor is used for measuring of angular speed of rotating object. Angular speed measuring is necessary so that it would be possible to stop the rotating satellite – to stabilize it. Using angular velocity sensor it is also possible to make satellite rotate at a certain speed.

## Operation principle of angular velocity sensor

The main measuring component of angular velocity sensor is a special microelectromechanical gyroscope (MEMS). It is not a conventional gyroscope in which a disk is rotating at a high speed but a miniature vibrating gyroscope. There is a ring inside MEMS vibrating along one plane direction. If such gyroscope is put on a rotating platform the planar surface of which coincides with the plane direction of ring vibration then the Coriolis force will start affecting it in proportion to the speed of platform rotation. Coriolis force is measured using piezoelectric elements that output the voltage in proportion to applied force.

After determining volume of Coriolis force and having data on the speed of vibration it is possible to calculate the angular velocity and its fluctuation (angular acceleration).

### Testing of operational performance of angular velocity sensor

Connect the angular velocity sensor to power-supply unit and to onboard computer. Open Notepad++ and write the following program.

Python code.

hyro_test.py
def control(): # Main program function in which other functions are called

hyro_result = [0,0,0,0] # Initializing hyro_result
num = 1 # Number of angular velocity sensor
print "Enable angular velocity sensor №", num
hyro_turn_on(num) # Turning angular velocity sensor on
sleep(1) # Waiting for 1 second for sensor to turn on
print "Get RAW data from angular velocity sensor"

for i in range(10):  # Reading the indication for 10 times
hyro_result = hyro_request_raw(num) # recording function answer
# hyro_request_raw to variable hyro_result
if not hyro_result[0]: # if sensor does not return error message,
print "state:", hyro_result[0], "x_raw =", hyro_result[1], \
"y_raw =", hyro_result[2], "z_raw =", hyro_result[3]
# Data output
elif hyro_result[0] == 1: # if sensor returns error message 1
print "Fail because of access error, check the connection"
elif hyro_result[0] == 2: # if sensor returns error message 2
print "Fail because of interface error, check your code"

sleep(1) # Indications are read one time per second

print "Disable angular velocity sensor №", num
hyro_turn_off(num)   # Turning angular velocity sensor off

С code.

hyro_test.c
#include "libschsat.h"

int control(){
uint16_t num = 1;  	// Number of angular velocity sensor
int16_t hyro_result[] = {0, 0, 0, 0};
printf("Enable angular velocity sensor № %d", num);
hyro_turn_on(num); 						// Turning angular velocity sensor on
Sleep (1);							// Waiting for 1 second for sensor to turn on
printf("\nGet RAW data from angular velocity sensor\n");

int i;
for (i = 0; i < 10; i++) { 		// Reading the indication for 10 times
hyro_result[0] = hyro_request_raw(num,& hyro_result[1],& hyro_result[2],& hyro_result[3]);

if (!hyro_result[0])
{
printf("state: %d", i);
printf(" x_raw = %d", hyro_result[1]);
printf(" y_raw = %d", hyro_result[2]);
printf(" z_raw = %d\n", hyro_result[3]);

}
else if (hyro_result[0] == 1)
{
printf("Fail because of access error, check the connection");
}
else if (hyro_result[0] == 2)
{
printf("Fail because of interface error, check your code");
}
Sleep(1);

}
printf("\nDisable angular velocity sensor № %d\n", num);
hyro_turn_off(num);
return 0;
}

Run the program and test the operation of angular velocity sensor.

### Code analysis

Please note that after # symbol the comments are written that do not affect the program operation. The program starts running from output of function control(). Then we create the list hyro_result for receiving from sensor the data and variable num storing the sensor number. Then operator print outputs the message that the sensor is turned on. Then function hyro_turn_on(num) turns on the sensor with the specified number. Function sleep(1) suspends the program execution for 1 second. After that in cycle for i in range(10): we read and output the indication of sensor for 10 times. Indicatioin of sensor are read with the help of function hyro_request_raw(num).

## Magnetometer

### Function of magnetometer

Information on orientation is received by the satellite from angular velocity sensor and from magnetometer. Information on angle is needed to rotate the satellite in the necessary direction and information on angular velocity is needed to stabilize the satellite i.e. to stop the angular rotation. The driving moment is created with the help of reaction wheel.

### Operation principle of magnetometer

Operation of magnetometer is based on application of magnetoresistive effect when electrical resistance of a conductor is fluctuating in correspondence with magnetic fields pattern direction. Sensor operation is based on a layer of permalloy (special nickel-iron alloy) possessing a strong magnetoresistive effect. Electrical resistance of permalloy usually fluctuating within the range of ±5 % depending on force and pattern direction of magnetic field.

Therefore measuring the electric current force flowing through the layer of permalloy at the permanent current of +5V it is possible to determine the direction of magnetic fields pattern. To measure the magnetic field pattern direction by all three axes three little sensors are utilized oriented along axes X, Y and Z , installed in one microchip.

### Checking of magnetometer functional capability

Open Notepad++ and write the following program.

Python code.

mag_test.py
def control(): # Main function of program from which other functions are called
mgn_result = [0,0,0,0] # Initializing mgn_result
num = 1  # number of magnetometer
print "Enable magnetometer №", num
magnetometer_turn_on(num)
sleep(1)
print "Get RAW data from magnetometer"
for i in range(10):
mgn_result = magnetometer_request_raw(num)
if not mgn_result[0]: # if sensor returns an error message,
print "state:", mgn_result[0], "x_raw =", mgn_result[1], \
"y_raw =", mgn_result[2], "z_raw =", mgn_result[3]
# Please note the line break symbol!
elif mgn_result[0] == 1:
print "Fail because of access error, check the connection"
elif mgn_result[0] == 2:
print "Fail because of interface error, check your code"
sleep(1)
print "Disable magnetometer №", num
magnetometer_turn_off(num)

С code.

mag_test.c
#define LSS_OK 0
#define LSS_ERROR 1
#define LSS_BREAK 2

#include <stdio.h>
#include <stdint.h>
#include "libschsat.h"
int control() // Main function of program from which other functions are called
{
int16_t mgn_result[] = {0,0,0,0}; // Initializing mgn_result
uint16_t num = 1;	// number of magnetometer
int i;
printf("Rnable magnetometer №%d\n", num);
magnetometer_turn_on(num);
Sleep(1);
printf("Get RAW data from magnetometer\n");
for (i = 0; i < 1000; i++)
{
mgn_result[0] = magnetometer_request_raw(num,& mgn_result[1],& mgn_result[2],& mgn_result[3]);
if (!mgn_result[0])// if sensor returns an error message,
{
printf("\nstate:%d, \nx_raw =%d,\ny_raw =%d, \nz_raw =%d\n",i, mgn_result[1], mgn_result[2], mgn_result[3]); //Please note the line break symbol!
}
else if (mgn_result[0] == 1){
printf("Fail because of access error, check the connection");
}
else if (mgn_result[0] == 2){
printf("Fail because of interdace error, check your code");
}
Sleep(0.1);
}
printf("Disable magnetometer №%d\n", num);
magnetometer_turn_off(num);
return 0;
}

Run the program and check the magnetometer functional capability.

### Code analysis

Program starts from control() function call. Then we create the list mgn_result to get the data from sensor and variable num, storing the number of magnetometer. Then operator print outputs the message on switching on of sensor. After that function magnetometer_turn_on(num) switches on the magnetometer with indicated number. Function sleep(1) suspends the program execution for 1 second. Then in cycle for i in range(10): we read and output the indication of magnetometer for 10 times. Indications of magnetometer are read with the help of function magnetometer_request_raw(num).

## Sun Sensors

### Function of Sun sensors

Function of sun sensors is to determine the location of satellite in relation to the Sun. As the position of the Sun in relation to the Earth at any given moment of time is known fairly precisely, then as it follows from it is possible to determine the position of satellite in relation to the Earth.

### Operation principle of Sun sensors

The basis of Sun sensor is a photodetector measuring the brightness of light. The most popular types of photodetectors are photoresistors cells and photodiodes. Photoresistor consists of material, the resistance of which fluctuates depending on the intensity of impinging light.

Unlike photoresistor, photodiode outputs voltage under the action of light. Photodiode detectors can be of various types — LEP (Latheral Effect Photodiode), QD-photodiode (Quadrant Detector) or matrix type.

LEP photodiode is a single photodiode with large sensitive surface.

QD-photodiode consists of four independent photodiodes located symmetrically in relation to the center of sensitive surface.

Calculation of the spot position on the surface of QD-photodiode фотодиода results from correlation of output currents of photodiodes.

Matrix sensor incorporates a large number of photodiodes and allows to determine the position of the Sun more precisely.

### Checking of operational capability of sun sensors

Start connecting sun sensors in turn to power-supply unit and to onboard computer and check the operational capability.

Open Notepad++ and write the following program.

Python code.

sun_test.py
def control(): # Main function of program from which other functions are called
sun_result = [0,0,0] # Initializing sun_result
num = 1
print "Enable sun sensor №", num
sun_sensor_turn_on(num)
sleep(1)
print "Get RAW data from sun sensor"

for i in range(10):
sun_result = sun_sensor_request_raw(num)
if not sun_result[0]: # if sensor returns error message,
print "state:", sun_result[0], "raw =", sun_result[1], \
sun_result[2]
elif sun_result[0] == 1:
print "Fail because of access error, check the connection"
elif sun_result[0] == 2:
print "Fail because of interface error, check your code"

sleep(1)

print "Disable sun sensor №", num
sun_sensor_turn_off(num)

С code.

sun_test.c
#include <stdio.h>
#include <stdint.h>
#include "libschsat.h"
#define LSS_OK 0
#define LSS_ERROR 1
#define LSS_BREAK 2

int control(){ // Main function
uint16_t sun_result[] = {0, 0, 0}; // Initializing sun_result
uint16_t num = 1; // number of sun sensor
printf("Enable sun sensor №%d\n", num);
sun_sensor_turn_on(num); // turn the sensor on
Sleep(1); // wait 1 second
printf("Get RAW data from sun sensor №%d\n", num);
int i;
for (i = 0; i < 10; i++) // read the sensor 10 times
{
sun_result[0] = sun_sensor_request_raw(num,& sun_result[1],& sun_result[2]);
if (!sun_result[0]){ // if the sensor did not return an error message,
printf("state: %d raw = %d, %d\n", i, sun_result[1], sun_result[2]);
}

else if (sun_result[0] == 1) { // if the sensor did not return an error message 1
printf("Fail because of access error, check the connection\n");
}
else if (sun_result[0] == 2) { // if the sensor did not return an error message 2
printf("Fail because of interface error, check you code\n");
}
Sleep(1); // wait 1 second

}
printf("Disable sun sensor №%d\n", num);
sun_sensor_turn_off(num); // turn sun sensor off
return 0;
}

Run the program and test sun sensor operation.

Test in turn all 4 sun sensors.

### Code analysis

Program starts from declaration of function control().

Then we create the list sun_result to receive data from sensor and variable num, storing the magnetometer number. The operator print returns the message on turning on of sensor. Then function sun_sensor_turn_on(num) turns on the magnetometer with indicated number. Function sleep(1) suspends the program execution for 1 second. Then in cycle for i in range(10): we read and return the value of magnetometer output for 10 times. Indications of magnetometer are read with the help of function sun_sensor_request_raw(num).


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# Compactness Properties of the Space of Genus-$g$ Helicoids

-
Jacob Bernstein, MIT
Fine Hall 314

I will discuss a recent application of the work of Colding and Minicozzi of structure of embedded minimal surfaces in $\Real3$ to the study of compactness properties of the space of genus-$g$ helicoids. I will introduce the theory of Colding and Minicozzi and then show how it can be used to show (among other results) that the space of genus-one helicoids is compact (modulo symmetries). (Joint work with C. Breiner)


text:
# Lesson 14

Multiplying, Dividing, and Estimating with Scientific Notation

Let’s multiply and divide with scientific notation to answer questions about animals, careers, and planets.

### 14.1: True or False: Equations

Is each equation true or false? Explain your reasoning.

1. $$4\times 10^5 \times 4 \times 10^4= 4\times 10^{20}$$
2. $$\dfrac{7 \times 10^6}{2 \times 10^4} = (7\div2) \times 10^{(6-4)}$$
3. $$8.4 \times 10^3 \times 2 = (8.4 \times 2) \times 10^{(3 \times 2)}$$

### 14.2: Biomass

Use the table to answer questions about different creatures on the planet. Be prepared to explain your reasoning.

creature number mass of one individual (kg)
humans $$7.5 \times 10^9$$ $$6.2 \times 10^1$$
cows $$1.3 \times 10^9$$ $$4 \times 10^2$$
sheep $$1.75 \times 10^9$$ $$6 \times 10^1$$
chickens $$2.4 \times 10^{10}$$ $$2 \times 10^0$$
ants $$5 \times 10^{16}$$ $$3 \times 10^{\text -6}$$
blue whales $$4.7 \times 10^3$$ $$1.9 \times 10^5$$
Antarctic krill $$7.8 \times 10^{14}$$ $$4.86 \times 10^{\text -4}$$
zooplankton $$1 \times 10^{20}$$ $$5 \times 10^{\text -8}$$
bacteria $$5 \times 10^{30}$$ $$1 \times 10^{\text -12}$$
1. Which creature is least numerous? Estimate how many times more ants there are.
2. Which creature is the least massive? Estimate how many times more massive a human is.
3. Which is more massive, the total mass of all the humans or the total mass of all the ants? About how many times more massive is it?
4. Which is more massive, the total mass of all the krill or the total mass of all the blue whales? About how many times more massive is it?

### 14.3: Info Gap: Distances in the Solar System

Your teacher will give you either a problem card or a data card. Do not show or read your card to your partner.

If your teacher gives you the problem card:

3. Explain to your partner how you are using the information to solve the problem.

If your teacher gives you the data card:

2. Ask your partner “What specific information do you need?” and wait for your partner to ask for information. Only give information that is on your card. (Do not figure out anything for your partner!)
3. Before telling your partner the information, ask “Why do you need that information?”
4. After your partner solves the problem, ask them to explain their reasoning and listen to their explanation.

Choose two celestial objects and create a scale drawing of them in the applet below.

object distance to Earth (km) diameter (km) mass (kg)
Sun $$1.496 \times 10^8$$ $$1.392 \times 10^6$$ $$1.989 \times 10^{30}$$
Mercury $$7.73 \times 10^7$$ $$4.878 \times 10^3$$ $$3.3 \times 10^{23}$$
Venus $$4 \times 10^7$$ $$1.21 \times 10^4$$ $$4.87 \times 10^{24}$$
Earth N/A $$1.28 \times 10^4$$ $$5.98 \times 10^{24}$$
Mars $$5.46 \times 10^7$$ $$6.785 \times 10^3$$ $$6.4 \times 10^{23}$$
Jupiter $$5.88 \times 10^8$$ $$1.428 \times 10^5$$ $$1.898 \times 10^{27}$$
Saturn $$1.2 \times 10^9$$ $$1.199 \times 10^5$$ $$5.685 \times 10^{26}$$
Uranus $$2.57 \times 10^9$$ $$5.149 \times 10^4$$ $$8.68 \times 10^{25}$$
Neptune $$4.3 \times 10^9$$ $$4.949 \times 10^4$$ $$1.024 \times 10^{26}$$

Plot a point  for the center of each circle. Select the Circle with Center and Radius tool  and click on a point. When the dialog box opens, enter the radius.

GeoGebra Applet esEdKmyB

### 14.4: Professions in the United States

Use the table to answer questions about professions in the United States as of 2012.

profession number typical annual salary (U.S. dollars)
architect $$1.074 \times 10^5$$ $$7.3 \times 10^4$$
artist $$5.14 \times 10^4$$ $$4.4 \times 10^4$$
programmer $$1.36 \times 10^6$$ $$8.85 \times 10^4$$
doctor $$6.9 \times 10^5$$ $$1.87 \times 10^5$$
engineer $$6.17 \times 10^5$$ $$8.6 \times 10^4$$
firefighter $$3.07 \times 10^5$$ $$4.5 \times 10^4$$
military—enlisted $$1.16 \times 10^6$$ $$4.38 \times 10^4$$
military—officer $$2.5 \times 10^5$$ $$1 \times 10^5$$
nurse $$3.45 \times 10^6$$ $$6.03 \times 10^4$$
police officer $$7.8 \times 10^5$$ $$5.7 \times 10^4$$
college professor $$1.27 \times 10^6$$ $$6.9 \times 10^4$$
retail sales $$4.67 \times 10^6$$ $$2.14 \times 10^4$$
truck driver $$1.7 \times 10^6$$ $$3.82\times 10^4$$

1. Estimate how many times more nurses there are than doctors.
2. Estimate how much money all doctors make put together.
3. Estimate how much money all police officers make put together.
4. Who makes more money, all enlisted military put together or all military officers put together? Estimate how many times more.

### Summary

Multiplying numbers in scientific notation extends what we do when we multiply regular decimal numbers. For example, one way to find $$(80)(60)$$ is to view 80 as 8 tens and to view 60 as 6 tens. The product $$(80)(60)$$ is 48 hundreds or 4,800. Using scientific notation, we can write this calculation as $$\displaystyle (8 \times 10^1) (6 \times 10^1) = 48 \times 10^2.$$ To express the product in scientific notation, we would rewrite it as $$4.8 \times 10^3$$.

Calculating using scientific notation is especially useful when dealing with very large or very small numbers. For example, there are about 39 million or $$3.9 \times 10^7$$ residents in California. Each Californian uses about 180 gallons of water a day. To find how many gallons of water Californians use in a day, we can find the product $$(180) (3.9 \times 10^7) = 702 \times 10^7$$, which is equal to $$7.02 \times 10^9$$. That’s about 7 billion gallons of water each day!

Comparing very large or very small numbers by estimation also becomes easier with scientific notation. For example, how many ants are there for every human? There are $$5 \times 10^{16}$$ ants and $$7 \times 10^9$$ humans. To find the number of ants per human, look at $$\frac{5 \times 10^{16}}{7 \times 10^9}$$. Rewriting the numerator to have the number 50 instead of 5, we get $$\frac{50 \times 10^{15}}{7 \times 10^9}$$. This gives us $$\frac{50}{7} \times 10^6$$. Since $$\frac{50}{7}$$ is roughly equal to 7, there are about $$7 \times 10^6$$ or 7 million ants per person!

### Glossary Entries

• scientific notation

Scientific notation is a way to write very large or very small numbers. We write these numbers by multiplying a number between 1 and 10 by a power of 10.

For example, the number 425,000,000 in scientific notation is $$4.25 \times 10^8$$. The number 0.0000000000783 in scientific notation is $$7.83 \times 10^{\text-11}$$.


text:
Page updated: March 20, 2021
Author: Curtis Mobley
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# Wave Variance Spectra: Examples

This page gives two examples of wave elevation variance spectra. The Pierson-Moskowitz omnidirectional spectrum describes gravity waves in a “fully developed” sea. A fully developed sea is an idealization of the statistically steady-state wave ﬁeld resulting from a steady wind blowing for an inﬁnitely long time over an inﬁnite fetch. (In practice, the duration and fetch required to achieve something close to a fully developed sea depend on the wind speed. A steady wind of $5\phantom{\rule{2.6108pt}{0ex}}m\phantom{\rule{2.6108pt}{0ex}}{s}^{-1}$ blowing for 10 hours over a fetch of 60 km might be adequate; for hurricane winds of $35\phantom{\rule{2.6108pt}{0ex}}m\phantom{\rule{2.6108pt}{0ex}}{s}^{-1}$, a fetch of a few thousand kilometers and a duration of several days would be required. Thus it is much easier to approach a fully developed sea at low wind speeds than at high.)

The Elfouhaily et al. directional spectrum includes both gravity and capillary waves scales. Moreover, it has a parameter that describes the wave age, so than any sea state from young (the wind has just started blowing) to fully developed can be simulated. The Elfouhaily et al. spectrum will be used to generate the two-dimensional sea surface examples on the following pages.

### The Pierson-Moskowitz Omnidirectional Gravity Wave Spectrum

The omnidirectional Pierson and Moskowitz (1964) spectrum, formulated in terms of angular spatial frequency $k$, is

 ${\mathsc{𝒮}}_{PM}\left(k\right)=\frac{\alpha }{2{k}^{3}}exp\left[-\beta {\left(\frac{g}{k}\right)}^{2}\frac{1}{{U}_{19}^{4}}\right]\phantom{\rule{2em}{0ex}}\left[{m}^{2}∕\left(rad∕m\right)\right]\phantom{\rule{0.3em}{0ex}},$ (1)

where

$\alpha =0.0081$,
$\beta =0.74$,
$g=9.82\phantom{\rule{2.6108pt}{0ex}}m\phantom{\rule{2.6108pt}{0ex}}{s}^{-2}$ is the acceleration of gravity,
${U}_{19}$ is the wind speed in $m\phantom{\rule{2.6108pt}{0ex}}{s}^{-1}$ at 19.5 m above the sea surface, and
$k$ is the angular spatial frequency in $rad\phantom{\rule{2.6108pt}{0ex}}{m}^{-1}$.

The wind speed at 19.5 m can be obtained from the more commonly used wind at 10 m above the sea surface by the approximate formula

 ${U}_{19}\approx 1.026{U}_{10}\phantom{\rule{0.3em}{0ex}}.$

As has already been noted, it is often of interest to express a variance spectrum in terms of other variables, such as the wavenumber $\nu$ or the temporal angular frequency $\omega$. To change variables in a spectral density function, the key is to recall that a variance density function by deﬁnition expresses the variance per unit frequency interval. The variance contained in some interval $dk$ of the spatial angular frequency equals the variance contained in the corresponding interval $d\nu$ of the wavenumber or the interval $d\omega$ of the temporal frequency. Thus we have

 ${\mathsc{𝒮}}_{PM}\left(k\right)dk={\mathsc{𝒮}}_{PM}\left(\nu \right)d\nu ={\mathsc{𝒮}}_{PM}\left(\omega \right)d\omega \phantom{\rule{0.3em}{0ex}}.$

To change the variable from $k=2\pi \nu$ to $\nu$, the previous equation gives

 ${\mathsc{𝒮}}_{PM}\left(\nu \right)={\mathsc{𝒮}}_{PM}\left(k\right)\frac{dk}{d\nu }={\mathsc{𝒮}}_{PM}\left(k=2\pi \nu \right)2\pi \phantom{\rule{0.3em}{0ex}},$

which results in

 ${\mathsc{𝒮}}_{PM}\left(\nu \right)=\frac{\alpha }{8{\pi }^{2}{\nu }^{3}}exp\left[-\beta {\left(\frac{g}{2\pi \nu }\right)}^{2}\frac{1}{{U}_{19}^{4}}\right]\phantom{\rule{2em}{0ex}}\left[{m}^{2}∕\left(1∕m\right)\right]\phantom{\rule{0.3em}{0ex}}.$ (2)

To change variables from spatial angular frequency $k$ to temporal angular frequency $\omega$, we use the dispersion relation for gravity waves in deep water,

 ${\omega }^{2}=gk\phantom{\rule{0.3em}{0ex}},$

to evaluate

 $\frac{dk}{d\omega }=\frac{2\omega }{g}\phantom{\rule{0.3em}{0ex}},$

 ${\mathsc{𝒮}}_{PM}\left(\omega \right)=\frac{\alpha {g}^{2}}{{\omega }^{5}}exp\left[-\beta {\left(\frac{g}{\omega {U}_{19}}\right)}^{4}\right]\phantom{\rule{2em}{0ex}}\left[{m}^{2}∕\left(rad∕s\right)\right]\phantom{\rule{0.3em}{0ex}}.$ (3)

All of these formulas have units of meters squared over the appropriate frequency. (The quantities $dk∕d\nu$ and $dk∕d\omega$ seen in the conversions are the Jacobians for the one-dimensional changes of variables.) Figure 1 shows the Pierson-Moskowitz spectrum as functions of $k$ and $\omega$ for wind speeds of ${U}_{10}=5,10$, and $15\phantom{\rule{2.6108pt}{0ex}}m\phantom{\rule{2.6108pt}{0ex}}{s}^{-1}$.

This spectrum has withstood the test of time fairly well (Alves and Banner (2003)) as a description of gravity waves in fully developed seas. However, it should not be used for high-frequency (short-wavelength) gravity waves, and certainly not for capillary waves. Likewise, it does not describe young seas, which have not had the time or fetch needed to approach the state of a well developed sea.

Figure 2 shows the Pierson-Moskowitz slope spectra for three wind speeds. Note that the slope spectrum falls oﬀ much more slowly for high frequencies than does the elevation spectrum. That means that the higher frequencies contribute much more to the total slope variance than they do to the total elevation variance.

### The Elfouhaily et al. Directional Gravity-Capillary Wave Spectrum

In order to model two-dimensional sea surfaces $z\left(x,y\right)$, we need a 2-D wave elevation variance spectrum that describes the distribution of wave variance for waves propagating in diﬀerence directions. The numerical examples of 2-D sea surfaces to be seen on the following pages use the 2-D elevation variance spectrum of Elfouhaily et al. (1997), which is described here for later reference. For brevity, this paper and their model are denoted here by “ECKV,” taken from the initials of the authors’ last names.

The ECKV spectrum has an omnidirectional variance spectrum that explicitly includes both the gravity and capillary wave scales. The boundary between gravity and capillary waves is taken to be $k=\sqrt{\rho g∕\tau }=370\phantom{\rule{2.6108pt}{0ex}}rad∕m$, the angular spatial frequency at which the restoring forces (which tend to bring a wave surface back to an unperturbed level surface) of gravity and surface tension are equal. The corresponding wavelength is $\Lambda =2\pi ∕370=0.017\phantom{\rule{2.6108pt}{0ex}}m$. ECKV then combine their omnidirectional spectrum with a spreading function to obtain their one-sided, directional variance spectrum. Using their notation, the 2-D ECKV spectrum has the form

 $\Psi \left(k,\phi \right)=\frac{1}{k}\mathsc{𝒮}\left(k\right)\Phi \left(k,\phi \right)\phantom{\rule{2em}{0ex}}\left[ECKV\phantom{\rule{1em}{0ex}}45\right]\phantom{\rule{0.3em}{0ex}}.$ (4)

Here $\mathsc{𝒮}\left(k\right)$ is the 1-D omnidirectional spectrum with units of ${m}^{2}∕\left(rad∕m\right)$, and $\Phi \left(k,\phi \right)$ is a non-dimensional spreading function. $\Psi \left(k,\phi \right)$ thus has units of ${m}^{2}∕{\left(rad∕m\right)}^{2}$. Equation labels such as [ECKV 45] give for reference the corresponding equation in the ECKV paper.

The ECKV omnidirectional spectrum is

 $\mathsc{𝒮}\left(k\right)=\frac{{B}_{l}+{B}_{h}}{{k}^{3}}\phantom{\rule{2em}{0ex}}\left[ECKV\phantom{\rule{1em}{0ex}}30\right]\phantom{\rule{0.3em}{0ex}},$ (5)

where ${B}_{l}$ is the low-frequency (long gravity wave) contribution to the variance, and ${B}_{h}$ is the high-frequency (short gravity wave to capillary wave) contribution. (The quantity ${k}^{3}\mathsc{𝒮}\left(k\right)$ is called the curvature or saturation spectrum and is of interest in physical oceanography because it is related to the rate of variance dissipation of the waves. Thus ECKV refer to ${B}_{l}$ and ${B}_{h}$ as the low and high frequency curvature spectra.) The components of the omnidirectional spectrum are given by

$\begin{array}{llll}\hfill {L}_{PM}=& exp\left[-1.25{\left({k}_{p}∕k\right)}^{2}\right]\phantom{\rule{2em}{0ex}}\left[ECKV\phantom{\rule{1em}{0ex}}2\right]\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill \Gamma =& exp\left\{-\frac{1}{2{\sigma }^{2}}\left[{\left(\sqrt{k∕{k}_{p}}-1\right)}^{2}\right]\right\}\phantom{\rule{2em}{0ex}}\left[below\phantom{\rule{1em}{0ex}}ECKV\phantom{\rule{1em}{0ex}}3\right]\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill {J}_{p}=& {\gamma }^{\Gamma }\phantom{\rule{2em}{0ex}}\left[ECKV\phantom{\rule{1em}{0ex}}3\right]\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill {F}_{p}=& {L}_{PM}{J}_{p}exp\left\{-0.3162{\Omega }_{c}\left(\sqrt{k∕{k}_{p}}-1\right)\right\}\phantom{\rule{2em}{0ex}}\left[ECKV\phantom{\rule{1em}{0ex}}32\right]\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill {F}_{m}=& {L}_{PM}{J}_{p}exp\left[-0.25{\left(k∕{k}_{m}-1\right)}^{2}\right)\right]\phantom{\rule{2em}{0ex}}\left[ECKV\phantom{\rule{1em}{0ex}}41\right]\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & \left(Note:A\phantom{\rule{2.6108pt}{0ex}}typo\phantom{\rule{2.6108pt}{0ex}}in\phantom{\rule{2.6108pt}{0ex}}ECKV\phantom{\rule{2.6108pt}{0ex}}Eq.\phantom{\rule{2.6108pt}{0ex}}41\phantom{\rule{2.6108pt}{0ex}}omitted\phantom{\rule{2.6108pt}{0ex}}the\phantom{\rule{2.6108pt}{0ex}}{L}_{PM}{J}_{p}\phantom{\rule{1em}{0ex}}factor\phantom{\rule{2.6108pt}{0ex}}in\phantom{\rule{2.6108pt}{0ex}}{F}_{m}\right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill {B}_{l}=& 0.5{\alpha }_{p}\left({c}_{p}∕c\right){F}_{p}\phantom{\rule{2em}{0ex}}\left[ECKV\phantom{\rule{1em}{0ex}}31\right]\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill {B}_{h}=& 0.5{\alpha }_{m}\left({c}_{m}∕c\right){F}_{m}\phantom{\rule{2em}{0ex}}\left[ECKV\phantom{\rule{1em}{0ex}}40\right]\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

where

$\alpha =0.0081$,
$\beta =1.25$,
$g=9.82\phantom{\rule{1em}{0ex}}m\phantom{\rule{2.6108pt}{0ex}}{s}^{-2}$ is the acceleration of gravity,
${U}_{10}$ is the wind speed in $m\phantom{\rule{2.6108pt}{0ex}}{s}^{-1}$ at 10 m above the sea surface
$k$ is the angular spatial frequency in $rad\phantom{\rule{2.6108pt}{0ex}}{m}^{-1}$
${\Omega }_{c}$ is deﬁnes the age of the waves for the given wind speed:
= 0.84 for a fully developed sea (corresponds to Pierson-Moskowitz)
= 1 for a “mature” sea [used in ECKV Fig 8a]
= 2 to 5 for a “young” sea; the maximum allowed value is 5
$C{d}_{10N}=0.00144$ is a drag coeﬃcient [value deduced from ECKV Fig 11]
${u}^{\ast }=\sqrt{C{d}_{10N}}{U}_{10}$ is the friction velocity [using ECKV 61]
${a}_{o}=0.1733$ (ECKV 59)
${a}_{p}=4.0$
${k}_{m}=370.0\phantom{\rule{2.6108pt}{0ex}}rad∕m$
${c}_{m}=0.23\phantom{\rule{2.6108pt}{0ex}}m∕s$ is the phase speed of the wave with spatial frequency ${k}_{m}$
${a}_{m}=0.13{u}^{\ast }∕{c}_{m}$ [ECKV 59]
$\gamma =1.7\phantom{\rule{1em}{0ex}}if\phantom{\rule{1em}{0ex}}{\Omega }_{c}\le 1\phantom{\rule{1em}{0ex}}else\phantom{\rule{1em}{0ex}}\gamma =1.7+6{log}_{10}\left({\Omega }_{c}\right)$
$\sigma =0.08\left(1+4{\Omega }_{c}^{-3}\right)$
${\alpha }_{p}=0.006{\Omega }_{c}^{0.55}$ [ECKV Eq. 34]
${\alpha }_{m}=0.01\left[1+ln\left({u}^{\ast }∕{c}_{m}\right)\right]\phantom{\rule{1em}{0ex}}if\phantom{\rule{1em}{0ex}}{u}^{\ast }\le {c}_{m}\phantom{\rule{1em}{0ex}}else\phantom{\rule{1em}{0ex}}{\alpha }_{m}=0.01\left[1+3ln\left({u}^{\ast }∕{c}_{m}\right)\right]$ [ECKV 44]
${k}_{o}=g∕{U}_{10}^{2}$
${k}_{p}={k}_{o}{\Omega }_{c}^{2}$ is the spatial frequency of the maximum of the spectrum
${c}_{p}=\sqrt{g∕{k}_{p}}$ is the phase speed of the wave with spatial frequency ${k}_{p}$
$c=\sqrt{\left(g∕k\right)\left(1+{\left(k∕{k}_{m}\right)}^{2}\right)}$ is the phase speed of the wave

At the lower frequencies, the ECKV spectrum is essentially the Pierson-Moskowitz spectrum (the ${L}_{PM}$ term above) with an enhancement (the ${J}_{p}$ term) that adds more energy to the lower frequencies. The highest frequencies have a cutoﬀ due to viscous damping of the smallest capillary waves. The ECKV omnidirectional elevation and slope spectra are illustrated in Fig. 3 for the case of a fully developed sea and three wind speeds. Figure 4 shows the spectra as a function of wave age for a wind speed of ${U}_{10}=10\phantom{\rule{1em}{0ex}}m\phantom{\rule{1em}{0ex}}{s}^{-1}$.

The ECKV spreading function is given by

$\begin{array}{llll}\hfill \Phi \left(k,\phi \right)=& \frac{1}{2\pi }\left[1+\Delta \left(k\right)cos\left(2\phi \right)\right]\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill =& \frac{1}{2\pi }\left\{1+tanh\left[{a}_{o}+{a}_{p}{\left(c∕{c}_{p}\right)}^{2.5}+{a}_{m}{\left({c}_{m}∕c\right)}^{2.5}\right]cos\left(2\phi \right)\right\}\phantom{\rule{2em}{0ex}}& \hfill \text{(6)}\end{array}$

Note that this function is symmetric about $\phi =\pi ∕2$; i.e., the function has symmetric spreading downwind and upwind. This is consistent with a symmetric variance spectrum $\Psi \left(-k\right)=\Psi \left(k\right)$ as would be obtained from the Fourier transform of a snapshot of a sea surface. This symmetry will be explained on the following pages.

A commonly used family of alternate spreading functions is given by the “cosine-2s” functions of Longuet-Higgins et al. 1963, which have the form

 $\Phi \left(k,\phi \right)={C}_{s}{cos}^{2s}\left(\phi ∕2\right)\phantom{\rule{0.3em}{0ex}},$ (7)

where the normalizing coeﬃcient is

 ${C}_{s}=\frac{1}{2\sqrt{\pi }}\frac{\Gamma \left(s+1\right)}{\Gamma \left(s+1∕2\right)}\phantom{\rule{0.3em}{0ex}},$

and $s$ is a spreading parameter that in general depends on $k$, ${U}_{10}$, and wave age. In this equation $\Gamma$ is the customary gamma function deﬁned by $\Gamma \left(p\right)\equiv {\int }_{0}^{\infty }{x}^{p-1}{e}^{-x}dx$ where $p>0$. The cosine-2s functions are asymmetric, with much stronger downwind than upwind propagation.

The ECKV and cosine-2s spreading functions are illustrated in Fig. 5. Both of these functions satisfy the normalization condition (??). Both spreading functions transition from strongly forward peaked at low spatial frequencies (long gravity waves; the red curves) to curves with signiﬁcant propagation at right angles to the wind at high frequencies (capillary waves; the blue curves). The cosine-2s curves are asymmetric in $±k$ and have at least a small amount of upwind propagation at all frequencies (except at exactly upwind, $\phi =\pi$). Not surprisingly, the real ocean is more complicated than either of these models. In particular, observations of long-wave gravity waves tend to show a bimodal spreading about the downwind direction, which transitions to a more isotropic, unimodal spreading at shorter wavelengths (Heron (2006). However, the simple models of Eqs. (6) and (7) are adequate for the present purpose of illustrating surface-generation techniques. The eﬀect of the choice of spreading function on the generated waves will be illustrated on the Spreading Function Eﬀects page.

On the following pages we will learn the important distinction between “one-sided” or “folded” spectra and the associated “two-sided” spectra. The ECKV spectrum as given above is a one-sided spectrum.


text:
# 2014 Talk Abstracts

## 1. Towards an Efficient Decoder for Quantum LDPC Codes

Jonas Anderson, Université de Sherbrooke

(Session 9a : Friday from 5:30pm - 6:00pm)

Quantum low-density parity-checking (LDPC) codes can greatly reduce the overhead associated with fault-tolerant quantum computation (FTQC) by providing a nonzero-rate code family with low-weight stabilizer generators. In principle this means that as the code distance grows so does the number of encoded qubits thus allowing FTQC with constant overhead [1]. Exact decoding of classical LDPC codes is computationally difficult, but approximate decoders such as the belief propagation (BP) decoder are known to work well. For quantum LDPC codes much less is known and BP without modifications is plagued with issues due to degeneracy and short cycles in the Tanner graph. Here we improve upon the work of Poulin and Chung [2] by modifying BP to correct for some of the effects of message passing on a Tanner graph with cycles. Our technique uses nonlinear message weights to offset the additional correlations picked up due to cycles. For physical error rates an order of magnitude below pseudothreshold, arguably the most important regime for FTQC, we improve upon the best-known decoding schemes by an order of magnitude. We will also discuss ideas to further improve upon these schemes. [1] Daniel Gottesman, “What is the Overhead Required for Fault-Tolerant Quantum Computation?” arxiv.org/1310.2984. [2] David Poulin and Yeojin Chung, “On the iterative decoding of sparse quantum codes” Quantum Information and Computation, Vol. 8, No. 10 (2008) 0987–1000.

## 2. Towards practical quantum simulators for quantum chemistry

Alan Aspuru-Guzik, Harvard University

(Session 8 : Friday from 1:45pm - 2:30pm)

My first talk about quantum computing for chemistry was at SQUINT 2005. Back then, I presented a gate-model approach for the simulation of quantum chemistry. In this talk, almost a decade later, I will present two approaches that are much less demanding on the requirements of the quantum device, yet are able to simulate Fermionic Hamiltonians such as those of molecular quantum chemistry. First, I will talk about the variational quantum eigensolver approach for solving chemistry problems in an arbitrary {\sl hardware ansatz}. I will follow by describing an approach for the simulation of quantum chemistry using adiabatic quantum computers. Both approaches are scalable and good candidates for an early implementation of quantum devices that could carry out a simulation of practical relevance to medical or industrial applications.

## 3. Robust and high-sensitivity nonorthogonal coherent state discrimination

Francisco Elohim Becerra, University of New Mexico

(Session 3 : Thursday from 2:15pm - 2:45pm)

Measurements for accessing the information contained in quantum states are limited by the inherent noise of these states. Strategies for nonorthogonal state discrimination for optimally extracting information from these states have fundamental interest in quantum mechanics and can allow for communications approaching the quantum limits. Conventional measurements for nonorthogonal state discrimination of coherent states with different phases implement a direct phase-sensitive detection, and can ideally reach the standard quantum limit (SQL). However, measurement strategies based on the quantum properties of these states can allow for better measurements which surpass the SQL and approach the ultimate measurement limits allowed by quantum mechanics. We present the demonstration of a receiver based on adaptive measurements and single-photon counting that unconditionally discriminates multiple nonorthogonal coherent states below the SQL. We also discuss the potential of photon-number-resolving detection to provide robustness under realistic conditions for an adaptive coherent receiver with detectors with finite photon-number resolution.

## 4. Quantum assisted sensing with diamond spins

Ania Bleszynski-Jayich, University of California Santa Barbara

(Session 13 : Saturday from 4:15pm - 5:00pm)

Nitrogen-vacancy (NV) centers in diamond are atomic-scale spin systems with remarkable quantum properties that persist to room temperature. They are highly sensitive to a wide variety of fields (magnetic, electric, thermal) and are easy to initialize, read-out, and manipulate on the individual spin level; thus they make excellent nanoscale sensors. The NV’s sensitivity is a double-edged sword however; environmental fluctuating fields are also a source of decoherence. We use the NV to probe these fluctuating fields, both their frequency spectrum and spatial character, and we mitigate their induced decoherence through engineered CVD diamond growth and quantum control of the NV. I will also present my group’s work on quantum assisted sensing of strain fields on the nanoscale. We demonstrate strain coupling of a single NV spin to a high quality factor mechanical mode of a single-crystal diamond mechanical resonator. This hybrid system has exciting prospects for a phonon-based approach to integrating NVs into quantum networks.

## 5. Simulating Hamiltonian evolution on a quantum computer

Richard Cleve, University of Waterloo

(Session 2 : Thursday from 10:45am - 11:30am)

I will explain various quantum algorithms that have been proposed for simulating the evolution of a quantum state under a Hamiltonian, including my recent joint work (with Dominic Berry, Andrew Childs, Robin Kothari, and Rolando Somma) that dramatically improves the running time as a function of the precision of the output data.

## 6. Probabilistic protocols in quantum information? Probably not.

Joshua Combes, University of New Mexico

(Session 13 : Saturday from 5:00pm - 5:30pm)

Probabilistic protocols in quantum information are an attempt to improve performance by occasionally reporting a better result than could be expected from a deterministic protocol. Here we show that probabilistic protocols can never improve performance beyond the quantum limits on the corresponding deterministic protocol. To illustrate this result we examine three common probabilistic protocols: probabilistic amplification, weak value amplification, and probabilistic metrology. In each of these protocols we show explicitly that the optimal deterministic protocol is better than the corresponding probabilistic protocol when the probabilistic nature of the protocol is correctly accounted for.

## 7. Shortcuts to adiabaticity in many-body systems

Adolfo del Campo, Los Alamos National Laboratory

(Session 9b : Friday from 5:00pm - 5:30pm)

The evolution of a system induced by counter-diabatic driving mimics the adiabatic dynamics without the requirement of slow driving. Engineering it involves diagonalizing the instantaneous Hamiltonian of the system and results in the need of auxiliary non-local interactions for matter-waves. Here experimentally realizable driving protocols are presented for a large class of single-particle, many-body, and non-linear systems without demanding the spectral properties as an input. The method is applied to the fast decompression of quantum fluids realizing a dynamical quantum microscope, as well as to the fast transport of ion chains.

## 8. Trapped-ion quantum information processing experiments at NIST

John Gaebler, National Institute of Standards and Technology

(Session 1 : Thursday from 9:15am - 9:45am)

We report experiments towards scalable quantum information processing with laser-cooled trapped ions. Quantum information is stored in internal (hyperfine ground) states of ions and gate operations are performed with laser and microwave fields. We describe the current status of quantum information experiments using multi-zone trap arrays to investigate the basic tasks of a quantum information processor including transport of ions between zones and sympathetic cooling. In one recent experiment we created an entangled steady state of two trapped ions using dissipation. The steady state can be maintained for a duration of several times the entanglement generation duration and the entanglement fidelity is currently limited by identified technical issues. We also briefly describe recent progress with other quantum-information-focused experiments in our group including the generation of entanglement between two ions held in distinct but coupled trap zones, efforts to reduce the electric field noise from trap surfaces, superconducting photon detectors and junctions for switching the transport pathway of ions in multi-zone traps structures. *This work is supported by IARPA, ONR, and the NIST Quantum Information Program.

## 9. Quantum gate set tomography

John Gamble, Sandia National Laboratories

(Session 11 : Saturday from 10:15am - 10:45am)

In this talk, I will discuss a recently-proposed framework called "gate set tomography" (GST) for self-consistently characterizing an entire set of quantum logic gates on a black-box quantum device. Until recently, protocols for quantum tomography relied on a pre-existing and perfectly calibrated reference frame for the measurements used to characterize a device. GST eschews this artificial separation entirely, instead characterizing quantum processes, preparations, and measurements concurrently. I will then describe an explicit closed-form protocol for linear-inversion GST, whose reliability is independent of pathologies such as local maxima of the likelihood function. This initial estimate can then be refined using standard likelihood maximization techniques. Finally, I discuss recent experimental implementations of GST for single qubits in both ion traps and electrostatically defined quantum dot systems. This work was supported in part by the Laboratory Directed Research and Development program at Sandia National Laboratories. Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000.

## 10. Understanding the effects of leakage in superconducting quantum error detection circuits

Joydip Ghosh, University of Calgary

(Session 9a : Friday from 4:30pm - 5:00pm)

The majority of quantum error detection and correction protocols assume that the population in a qubit does not leak outside of its computational subspace. For many existing approaches, however, the physical qubits do possess more than two energy levels and consequently are prone to such leakage events. Analyzing the effects of leakage is therefore essential to devise optimal protocols for quantum gates, measurement, and error correction. In this talk, I discuss the role of leakage in a two-qubit superconducting quantum error detection circuit. We simulate the repeated ancilla-assisted measurement of a single Z operator for a data qubit, record the outcome at the end of each measurement cycle, and explore the signature of leakage events in the obtained readout statistics. An analytic model is also developed that closely approximates the results of our numerical simulations. We find that leakage leads to destructive features in the quantum error detection scheme, making additional hardware and software protocols necessary.

## 11. Practical and Fast Gaussian State Estimation

Scott Glancy, National Institute of Standards and Technology

(Session 11 : Saturday from 11:45am - 12:15pm)

Many experiments on quantum systems involve the preparation and measurement of Gaussian states of a multi-system continuous variable Hilbert space. Examples include optical and microwave systems involving squeezing and linear interactions and nanomechanical resonators described with second order Hamiltonians. The state space that these systems access is much smaller than the full Hilbert space and can be fully characterized with a 2Nx2N covariance matrix and 2N means vector, where N is the number of individual modes or resonators. We describe here a very simple and fast method for estimating the covariance matrix and means vector from homodyne (or quadrature) measurement data collected at arbitrary phases. The method computes observed means of simple functions of the homodyne (phase, quadrature) pairs, which are easily related to the covariance matrix and means vector. We characterize uncertainty through a parametric bootstrap strategy. Our method is particularly useful for the analysis of large data sets.

## 12. The effect of realistic noise models on quantum error correction thresholds

Mauricio Gutierrez, Georgia Institute of Technology

(Session 9a : Friday from 5:00pm - 5:30pm)

Classical simulations of noisy stabilizer circuits are often used to estimate the threshold of a quantum error-correcting code (QECC). In this context, it is common to model the noise as a depolarizing channel by inserting Pauli gates randomly throughout the circuit [1]. However, it is not clear how sensitive a code's threshold is to the noise model, and whether or not a depolarizing channel is a good approximation for realistic non-stabilizer errors. Within the stabilizer formalism, we have shown that for a single qubit more accurate approximations can be obtained by including in the noise model Clifford operators and Pauli operators conditional on measurement [2]. Independent work by Magesan et al. has also shown the utility of adding Clifford operators to error models [3]. We now examine the feasibility of employing these error approximations at the single-qubit level to obtain better estimates of a QECC's threshold. For several codes and various noise models, we simulate an error-correction step and compute the pseudo-threshold by determining the noise strength above which encoding reduces the qubit fidelity. We compare the pseudo-threshold values for the real noise with its Pauli and expanded Pauli approximations. In most cases, the expanded Pauli channel provides a significantly better approximation to the real pseudo-threshold suggesting that our expanded error models will lead to more accurate stabilizer-based threshold estimations for realistic noise models. [1] A.M. Steane, Phys. Rev. A 68, 042322 (2003) [2] M. Gutiérrez, L. Svec, A. Vargo, and K. R. Brown, Phys. Rev. A. 87, 030302(R) (2013) [3] E. Magesan, D. Puzzuoli, C. E. Granade, D. G. Cory, Phys. Rev. A 87, 012324 (2013)

## 13. Spacetime, quantum cloning and black holes

Patrick Hayden, Stanford University

(Session 7 : Friday from 10:30am - 11:15am)

Reconciling black hole evaporation with the unitarity of quantum mechanics is an endeavour frought with conceptual difficulties. Not least among them is the apparent need for quantum cloning or, equivalently, violations of the monogamy of entanglement. The most recent and confusing incarnation of this problem is the so-called firewall paradox, which interprets monogamy violations as an indication that black holes may not have interiors. This talk will begin with a more pedestrian question: understanding all the ways in which quantum information can be replicated in Minkowski spacetime. It turns out that there is an amazing variety, perhaps an indication that we should not be so worried about apparent violations of no-cloning in situations in which the causal structure of spacetime is itself in doubt. Towards the end, I will return to the black hole firewall problem and sketch some of the quantum information theoretic ideas that have been proposed as possible resolutions.

## 14. Period Finding with Adiabatic Quantum Computation

Itay Hen, University of Southern California

(Session 9a : Friday from 4:00pm - 4:30pm)

We outline an efficient quantum-adiabatic algorithm that solves Simon's problem, in which one has to determine the period', or xor-mask, of a given black-box function. We show that the proposed algorithm is exponentially faster than its classical counterpart and has the same complexity as the corresponding circuit-based algorithm. Together with other related studies, this result supports a conjecture that the complexity of adiabatic quantum computation is equivalent to the circuit-based computational model in a stronger sense than the well-known, proven polynomial equivalence between the two paradigms. We also discuss the importance of the algorithm and its implications for the existence of an optimal-complexity adiabatic version of Shor's integer factorization algorithm and the experimental implementation of the latter.

## 15. Single charged impurities inside a Bose-Einstein condensate

Sebastian Hofferberth, Universität Stuttgart

(Session 1 : Thursday from 8:30am - 9:15am)

We investigate the interaction of a single electron as well as a single ion with a Bose-Einstein condensate (BEC). The charge impurities are produced by exciting exactly one atom from the BEC to a Rydberg state. Since the ionic core and the Rydberg electron have vastly different mass and interaction range with the surrounding ground state, we can consider both parts separately. Firstly, for low-L Rydberg states, the electron wavefunction is fully immersed in the BEC, and we observe electron-phonon coupling. We observe that single electron excite collective modes of the whole condensate. Alternatively, for high-L states the electron can be moved completely outside of the BEC, enabling us to study the interaction of the ionic core with the BEC. We are currently studying ion-ground state Feshbach resonances and investige the possibility of trapping the ion inside the BEC without any external electric fields.

## 16. Quantum process tomography of near-unitary maps

Amir Kalev, University of New Mexico

(Session 11 : Saturday from 11:15am - 11:45am)

We study the problem of quantum process tomography given the prior information that the implemented map is near to a unitary map on a d-dimensional Hilbert space. In particular, we show that a perfect unitary map is completely characterized by a minimum of d^2 + d measurement outcomes. This contrasts with the d^4 measurement outcomes required in general. To achieve this lower bound, one must probe the system with a particular set of d states in a particular order. This order exploits unitarity but does not assume any other structure of the map. We further numerically study the behaviors of two of compressed sensing estimators based on correct or faulty prior information caused by noise. The results show two important features: (1) When we have accurate prior information, one can drastically reduce the required data needed; (2) Different estimators applied to the same data are sensitive to different types of noise. The estimators could, therefore, be used as indicators of particular error models and to validate the use of prior assumptions for compressed sensing quantum process tomography. Finally, we consider the more general case of noisy quantum maps, with a low level of noise. Our study indicates that transforming to the interaction picture, where the noiseless map is represented by a diagonal operator, can provide a useful tool to identify the noise structure. This, in turn, can lead to a substantial reduction in the numerical resources needed to estimate the noisy map.

## 17. Certifying violations of local realism

Emanuel Knill, University of Colorado at Boulder

(Session 12 : Saturday from 1:45am - 2:30pm)

Many applications of quantum systems require measurements that verify the presence of sufficiently strong quantum correlations. The probability of the following unwanted event must be extremely small: The event where the correlations are not sufficiently strong but one is nevertheless convinced that they are strong enough. Important examples of quantum correlation occur in experiments showing violations of Bell's inequalities, which are thought to invalidate local realism. This is a review of how such violations are quantified and robustly certified, with or without predetermined Bell's inequalities.

## 18. True Quantum Precision and Unique Optimal Probes in presence of Decoherence.

Sergey Knysh, NASA Ames Research Center

(Session 13 : Saturday from 5:30pm - 6:00pm)

Quantum instruments derived from composite systems allow greater measurement precision than their classical counterparts due to coherences maintained between the N component elements; spins, atoms or photons. Typical decoherence that plagues real-world devices can be dephasing, particle loss, thermal excitation and relaxation. All these adversely affect precision (mean squared error), whether one is measuring time or phase, or even the noise amplitude itself. We develop a novel technique that uncovers the uniquely optimal probe states of the N qubits' alongside new tight bounds on precision under local and collective mechanisms of these noise types above.   For large quantum ensembles (where numerical techniques fail), the problem reduces by analogy  to finding the ground state of a 1-D particle in a potential well, with the shape of the well dictated by the type and strength of decoherence. Under collective dephasing alone we find that optimal estimation of phase and noise parameter can be effected simultaneously, utilizing the same optimal probe and measurement scheme. The formalism is applied to real-world devices such as the Mach-Zehnder interferometer and atom clocks.

## 19. Optimal quantum-enhanced interferometry using a laser power source

Matthias Lang, University of New Mexico

(Session 9c : Friday from 5:00pm - 5:30pm)

We consider an interferometer powered by laser light (a coherent state) into one input port and ask the following question: what is the best state to inject into the second input port, given a constraint on the mean number of photons this state can carry, in order to optimize the interferometer’s phase sensitivity? This question is the practical question for high-sensitivity interferometry. We answer the question by considering the quantum Cramer-Rao bound for such a setup. The answer is squeezed vacuum, if there are no photon losses in the interferometer. For a lossy interferometer, the squeezed vacuum is the best choice for the practical case where the laser power is much bigger than the power put into the squeezing.

## 20. Symmetry-Protected Topological Entanglement

Iman Marvian, University of Southern California

(Session 9b : Friday from 5:30pm - 6:00pm)

We propose an order parameter for the Symmetry-Protected Topological (SPT) phases which are protected by an Abelian on-site symmetry. This order parameter, called the SPT entanglement, is defined as the entanglement between A and B, two distant regions of the system, given that the total charge (associated with the symmetry) in a third region C is measured and known, where C is a connected region surrounded by A and B and the boundaries of the system. In the case of 1-dimensional systems we prove that at the limit where A and B are large and far from each other compared to the correlation length, the SPT entanglement remains constant throughout a SPT phase, and furthermore, it is zero for the trivial phase while it is nonzero for all the non-trivial phases. Moreover, we show that the SPT entanglement is invariant under the low-depth local quantum circuits which respect the symmetry, and hence it remains constant throughout a SPT phase in the higher dimensions as well. Finally, we show that the concept of SPT entanglement leads us to a new interpretation of the string order parameters and based on this interpretation we propose an algorithm for extracting the relevant information about the SPT phase of the system from the string order parameters.

## 21. An on-chip toolset for surface-electrode trap based quantum processors

True Merrill, Georgia Tech Research Institute

(Session 1 : Thursday from 9:45am - 10:15am)

Increasing the size and complexity of ion-trap quantum computing experiments requires improvements in automation, hardware, and control.  We report on several technologies which incorporate control electronics, diffractive ion imaging optics, and quantum control techniques for microwave gates in microfabricated surface-traps.  We demonstrate a compact, in-vacuum 80 channel digital-to-analog converter (DAC) system controlling a surface-electrode trap.  The DAC system transports  40Ca+ ions for over 70 m at 1 m/s without cooling, and the measured 0.8 quanta/ms  ion-heating rate is comparable to external DAC systems.  A second project incorporates diffractive Fresnel mirrors onto a trap surface for enhanced ion imaging and state detection.  Optics for light collimation and refocusing are demonstrated, achieving a ~8.3 x enhancement in the total fluorescence signal.  We comment on possible limits to asynchronous ion-qubit readout and strategies to mitigate decoherence from stray photons during measurement processes. Finally, we discuss composite pulse techniques for gates on 171Yb+ qubits that yield accurate quantum control despite classical control errors.

## 22. Symmetry-protected topological ordered phases and their use for quantum computation

Akimasa Miyake, University of New Mexico

(Session 4 : Thursday from 4:30pm - 5:00pm)

Collective phenomena, like superconductivity and magnetism, are usually robust features of quantum many-body systems. They only depend on a few key parameters of a system Hamiltonian, and often symmetries are sufficient enough to characterize different (quantum) phases associated with different collective behaviors. Through remarkable progress at the crossover between quantum information and quantum many-body physics, it gets more and more clear that certain strongly-correlated ground states could be harnessed for quantum information processing, based on their underlying entanglement structure and thus inherent complexity. For example, some concrete models of topological orders are most known by the application to quantum error correction. Here we address a question: "to what extent computational usefulness of quantum many-body ground states would be determined ubiquitously by symmetries, without the system Hamiltonian specified in detail?" Our approach may be also applicable to how quantum state preparation and verification can be made without detailed knowledge about the system, in the context of quantum simulation. This is a joint work with Jacob Miller.

## 23. Topological defect formation and dynamics in ion Coulomb crystals

Heather Partner, Physikalisch-Technische Bundesanstalt

(Session 8 : Friday from 3:00pm - 3:30pm)

Topological defects (kinks) in laboratory systems have attracted recent interest because of their universal nature. In our system, kinks form during nonlinear quenches from the linear to zigzag phase in Coulomb crystals of about 30 172Yb+ ions in a segmented linear trap. I will present our experimental study of probabilistic kink creation in the context of the Kibble-Zurek mechanism, which predicts a scaling for defect creation as a function of quench rate, and discuss the effects of inhomogeneity and finite size in such systems. In addition, I will describe the dynamics of such defects and how they can be controllably modified. These methods provide a toolbox for using kinks to study phase transitions and soliton physics, and as a potential carrier of quantum information.

## 24. Optimal dissipative encoding and state preparation for topological order

Fernando Pastawski, California Institute of Technology

(Session 9b : Friday from 4:30pm - 5:00pm)

We study the suitability of dissipative (non-unitary) processes for (a) encoding logical information into a topologically ordered ground space and (b) preparing an (arbitrary) topologically ordered state. We give a construction achieving (a) in time O(L) for the LxL-toric code by evolution under a geometrically local, time-independent Liouvillian. We show that this scaling is optimal: even the easier problem (b) takes at least O(L) time when allowing arbitrary (possibly time-dependent) dissipative evolution. For more general topological codes, we obtain similar lower bounds on the required time for (a) and (b). These bounds involve the code distance and the dimensionality of the lattice. The proof involves Lieb-Robinson bounds, recent cleaning-lemma-type arguments for topological codes, as well as uncertainty relations between complementary observables. By allowing general locality-preserving evolutions (including, e.g., circuits of CPTPMs), our results extend earlier work characterizing unitary state preparation.

## 25. An atomic superfluid Bose-Einstein condensate in a ring

William Phillips, Joint Quantum Institute/National Institute of Standards and Technology

(Session 6 : Friday from 8:30am - 9:15am)

A Bose-Einstein condensate extending around a ring-shaped trap that is interrupted by a repulsive barrier can exhibit behavior similar to that of a superconducting loop interrupted by a weak link or Josephson junction. [ See, for example, K. Wright, R. Blakestad, C. Lobb, W. Phillips, and G. Campbell, Phys. Rev. Lett 110, 025302 (2013). DOI: 10.1103/PhysRevLett.110.025302]. We observe controllable, discrete phase slips (jumps in the phase winding number around the ring) and hysteretic behavior. Analogies to the superconducting circuit can describe much of the behavior of this atomtronic circuit. This work was partially supported by the ONR, the NSF, and the ARO.

## 26. Spectrally Entangled Photon Pairs for Ultrafast Probing of Molecules

Michael Raymer, University of Oregon

(Session 3 : Thursday from 2:45pm - 3:15pm)

We introduce a new method, called entangled photon-pair two-dimensional fluorescence spectroscopy (EPP-2DFS), to sensitively probe the nonlinear electronic response of complex molecular systems. The method incorporates a separated two-photon (Franson) interferometer, which generates time-frequency-entangled photon pairs, into the framework of a fluorescence-detected 2D optical spectroscopic experiment. The EPR-like correlations in time and frequency allow circumventing the usual restrictions imposed by the time-frequency uncertainty principle.

## 27. Classical command of quantum systems

Ben Reichardt, University of Southern California

(Session 12 : Saturday from 2:30pm - 3:15pm)

Can a classical experimentalist command an untrusted quantum system to realize arbitrary quantum dynamics, aborting if it misbehaves?  We give a way for a classical system to certify the joint, entangled state of a bipartite quantum system, as well as command the application of specific operators on each subsystem. This is accomplished by showing a strong converse to Tsirelson's optimality result for the CHSH game: the only way to win many games is if the bipartite state is close to the tensor product of EPR states, and the measurements are the optimal CHSH measurements on successive qubits. This leads directly to a scheme for device-independent quantum key distribution. Control over the state and operators can also be leveraged to create more elaborate protocols for reliably realizing general quantum circuits. Joint work with Falk Unger and Umesh Vazirani.

## 28. Quantum optics experiments at Earth-orbital scales and beyond

David Rideout, University of California, San Diego

(Session 9c : Friday from 4:30pm - 5:00pm)

A number of national and international entities are racing to set up satellite-based quantum communication infrastructure, which would allow the construction of a global network for quantum key distribution. The construction of such a network poses numerous technological challenges, considering that quantum entanglement has not yet been demonstrated at such scales. I will provide some details of a Canadian Space Agency funded mission to demonstrate quantum entanglement between the Earth's surface and Low Earth Orbit, and sketch some tests of fundamental physics which could be enabled by such satellites, ranging from verification of quantum mechanics to tests of spacetime discreteness from quantum gravity.

## 29. Tomography of Quantum Fields

Carlos Riofrio, Freie Universität Berlin

(Session 11 : Saturday from 10:45am - 11:15am)

Understanding the fundamental interactions in many-body physical systems is of great interest in current theoretical and experimental efforts. In particular, continuous many-body systems or fields are exciting because they offer the tools for performing quantum simulations of processes of non-equilibrium, equilibration and thermalization. In this context, the problem of developing tools for identifying and reconstructing the state or some aspect of such systems is needed on a practical level. In this talk, I will present a first approach in that direction and possible applications for reconstructing quantum fields from low order correlation functions readily measurable in current experiments. We concentrate on one dimensional systems with spatially limited entanglement which are well described by the continuous matrix product state (cMPS) formalism.

## 30. Mapping the topological phase diagram of superconducting qubit systems

Pedram Roushan, UCSB

(Session 10 : Saturday from 9:15am - 9:45am)

Building a practical quantum simulator requires a scalable architecture suitable for large numbers of qubits. By combining the high coherence Xmon qubits with an adjustable inductance, we have developed a new qubit architecture called g-mon, which has a tunable qubit-qubit interaction. To demonstrate this tunability, we have performed high fidelity single and two-qubit gates. Turning on the qubit-qubit interaction allows for fast multi-qubit operations implemented in less than 30 ns, achieving multi-qubit gate times approaching that of single qubit gates. Furthermore, we show the versatility of this system by mapping the topological phase diagram of interacting Hamiltonians. So far, experimental studies of topological invariants in condensed matter systems have been limited to transport measurements in non-interacting systems. Recently, it was proposed [1] that the topological properties of Hamiltonians can be inferred from quantum dynamics. Using superconducting g-mon qubits, we experimentally measure the Berry curvature, a quantity that reflects the geometrical properties of the eigenstates, for various eigenstates of the Hamiltonian of the system. We will discuss the phase diagram of various topological phases and the robustness of the measured Chern numbers. [1] Gritsev and Polkovnikov, PNAS, 109, 6457 (2012).

## 31. Quantum Behavior of Electro-mechanical Structures

Keith Schwab, Caltech

(Session 6 : Friday from 9:15am - 10:00am)

Electro-mechanical structures composed of a radio-frequency mechanical resonator parametrically coupled to a superconducting microwave frequency electrical resonator, offer an opportunity to study quantum behavior of both the electronic and mechanical degrees of freedom. We have recently demonstrated detection of a single motional quadrature with imprecision less than x_zp, and avoidance of the backaction due to the shot noise of the microwave field. We have also demonstrated the measurement of the imbalance between up and down conversion and will discuss the interpretation of these measurements. Finally, we have also recently demonstrated quantum squeezing of the motion using a reservoir engineering technique described by Aash Clerk et al and will present this new data.

## 32. Autonomously stabilized entanglement between two superconducting qubits

Shyam Shankar, Yale University

(Session 10 : Saturday from 8:30am - 9:15am)

Quantum error-correction codes are designed to protect an arbitrary state of a multi-qubit register against decoherence-induced errors, but their implementation is an outstanding challenge for the development of large-scale quantum computers. A first step is to stabilize a non-equilibrium state of a simple quantum system such as a qubit or a cavity mode, in the presence of decoherence. Several groups have recently accomplished this goal using measurement-based feedback schemes. A next step is to prepare and stabilize a state of a composite system. Here we demonstrate the stabilization of an entangled Bell state of a quantum register of two superconducting qubits for an arbitrary time. Our result[1] is achieved by an autonomous feedback scheme which combines continuous drives along with a specifically engineered coupling between the two-qubit register and a dissipative bath. Similar bath engineering techniques have recently been used for qubit reset, single qubit state stabilization, as well as for the creation and stabilization of states of multipartite quantum systems. Unlike conventional, measurement-based schemes, an autonomous approach which uses engineered dissipation to counteract decoherence, obviates the need for a complicated external feedback loop to correct errors. Instead the feedback loop is built into the Hamiltonian such that the steady state of the system in the presence of drives and dissipation is a Bell state, an essential building-block for quantum information processing. Such autonomous schemes, which are broadly applicable to a variety of physical systems, will be an essential tool for the implementation of quantum-error correction. [1] http://dx.doi.org/10.1038/nature12802

## 33. Tensor networks and Symmetries

Sukhbinder Singh, Macquarie University, Sydney

(Session 9b : Friday from 4:00pm - 4:30pm)

Tensors networks methods, which are based on ideas of entanglement and renormalization group, have significantly progressed our understanding of strongly correlated quantum lattice systems in recent years. Examples of popular tensor network states include the matrix product state (MPS)[1], which results naturally from both Wilson’s numerical renormalization group[2] and White’s density matrix renormalization group(DMRG)[3], and the multi-scale entanglement renormalization ansatz (MERA) [4] which is based a specific RG scheme known as entanglement renormalization [5]. These tensor networks have been applied to the exploration of frustrated antiferromagnets, interacting fermions, quantum criticality, topological order and symmetry protected order, and more recently, the MERA has been used to explore[6] the holographic correspondence[7] of string theory. The careful incorporation[8] of lattice symmetries (both spacetime and internal symmetries) in tensor networks is playing an increasingly important role in these applications. In this talk I will outline some aspects of this role in the context of the MPS and the MERA where symmetries have been exploited to (i) target specific quantum number sectors of the Hilbert space, which subsequently also allows for the efficient simulation of lattice systems of anyons[9], (ii) identification of the quantum order of a ground state from its tensor network description[10], and (iii) to realize certain symmetry features of the holographic correspondence in the MERA. References [1] S. Ostlund and S. Rommer, Phys. Rev. Lett. 75, 3537 (1995). [2] K.G. Wilson, Rev. Mod. Phys. 47, 4, 773 (1975). [3] S.R. White, Phys. Rev. Lett. 69, 2863 (1992). [4] G. Vidal, Phys. Rev. Lett. 101, 110501 (2008). [5] G. Vidal, Phys. Rev. Lett. 99, 220405 (2007). [6] B. Swingle, Phys. Rev. D 86, 065007 (2012). [7] J. Maldacena, Adv. Theor. Math. Phys. 2, 231 (1998). [8] S. Singh, R. N. C. Pfeifer, and G. Vidal, Phys. Rev. A 82, 050301 (2010). [9] S. Singh, R. N. C. Pfeifer, G. Vidal, and G. K. Brennen. arXiv:1311.0967 (2013). [10] S. Singh and G. Vidal. Phys. Rev. B 88, 121108(R) (2013).

## 34. A quantum fractional Fourier transform

Rolando Somma, Los Alamos National Laboratory

(Session 2 : Thursday from 11:30am - 12:00pm)

The Fourier transform (FT) is ubiquitous in signal processing, as it can be used to filter noise. The digital version, often named the discrete Fourier transform, when formulated on a basis of quantum states, is the quantum Fourier transform (QFT). The efficiency in the implementation of the QFT is the main reason for several quantum speedups, including the one for factoring and the one in phase estimation at the Heisenberg limit. The fractional FT (frFT) is a generalization of the FT. The frFT has recently gained attention in signal analysis as it can filter noise in scenarios where the FT is not useful. Quantum frFTs (QfrFTs), however, have never been analyzed or applied; We propose a QfrFT and show that a good approximation of this transformation can be implemented on a quantum computer with exponentially less resources than those required for its conventional implementation. We then analyze some problems in signal analysis (parameter estimation) for which our defined QfrFT is useful. Applications of the QfrFT for the simulation of continuous-variable quantum mechanics will also be considered.

## 35. New Tools for Unitary Control of Cold Atom Qudits

Hector Sosa Martinez, University of Arizona

(Session 9c : Friday from 4:00pm - 4:30pm)

Accurate and robust quantum control of single or coupled qubit systems is a key element of quantum information science. In practice, the actual physical building blocks (atoms, ions, superconducting devices) are often qudits with state space dimension d>2, and the available auxiliary levels have proven useful for information processing tasks such as implementing Toffoli gates with two-body interactions. More generally, large internal state spaces may prove a useful resource if good laboratory tools for qudit manipulation can be developed. As a laboratory test bed for such development, we have implemented a protocol to perform arbitrary unitary transformations in the 16 dimensional ground hyperfine manifold of individual 133Cs atoms, by driving this system with phase modulated rf and microwave magnetic fields and using the tools of optimal control to find appropriate control waveforms. Similar to what can be achieved for qubits, we show that accurate unitary control can be achieved in the presence of simultaneous static and dynamical perturbations and imperfections in the control fields, simply by optimizing with respect to the appropriate cost function when designing control waveforms. We anticipate this approach to prove helpful for control in less than ideal environments, such as atoms moving around in the light shift potential of a dipole trap. We are currently exploring the prospects for inhomogeneous quantum control, with the goal of performing different unitary transformations on qudits that see different light shifts from an optical addressing field. Ultimately this may lead to addressable unitary control similar to what has been demonstrated for atomic qubits in optical lattices.

## 36. Repeat-Until-Success: Non-deterministic decomposition of single-qubit unitaries

Krysta Svore, Microsoft Research

(Session 8 : Friday from 2:30pm - 3:00pm)

We present a non-deterministic circuit decomposition technique for approximating an arbitrary single-qubit unitary to within distance epsilon that requires significantly fewer non-Clifford gates than deterministic decomposition techniques. We develop Repeat-Until-Success" (RUS) circuits and characterize unitaries that can be exactly represented as an RUS circuit. Our RUS circuits operate by conditioning on a given measurement outcome and using only a small number of non-Clifford gates and ancilla qubits. We construct an algorithm based on RUS circuits that approximates an arbitrary single-qubit Z-axis rotation to within distance epsilon, where the number of T gates scales as 1.26*log_2(1/\epsilon) - 3.53, an improvement of roughly three-fold over state-of-the-art techniques. We then extend our algorithm and show that a scaling of 2.4 * log_2(1/\epsilon) - 3.28 can be achieved for arbitrary unitaries and a small range of epsilon, which is roughly twice as good as optimal deterministic decomposition methods.

## 37. On generating macroscopic superpositions via nonlinear dynamics of stopped light in a two-component Bose-Einstein condensate

Collin Trail, University of Calgary

(Session 9c : Friday from 5:30pm - 6:00pm)

We investigate a method for generating nonlinear phase shifts on superpositions of photon number states. The light is stored in a Bose-Einstein condensate via electromagnetically-induced transparency memory techniques. The atomic collisions are exploited to generate a nonlinear evolution for the stored state. The stored light is then revived with the nonlinear phase shift imprinted upon it. For the special case of a coherent state input we find that this method can be used to generate an optical cat state. We investigate the validity of using the Thomas-Fermi and mean-field approximations.

## 38. Mismatched quantum filtering and entropic information

Mankei Tsang, National University of Singapore

(Session 7 : Friday from 11:15am - 11:45am)

Quantum filtering is a signal processing technique that estimates the posterior state of a quantum system under continuous measurements and has become a standard tool in quantum information processing, with applications in quantum state preparation, quantum metrology, and quantum control. If the filter assumes a wrong model due to assumptions or approximations, however, the estimation accuracy is bound to suffer. I shall present formulas that relate the error penalty caused by quantum filter mismatch to the relative entropy between the true model and the nominal model, with one formula for Gaussian measurements, such as homodyne detection, and another for Poissonian measurements, such as photon counting. These formulas generalize recent seminal results in classical information theory and provide new operational meanings to relative entropy, mutual information, and channel capacity in the context of quantum experiments. See http://arxiv.org/abs/1310.0291 for details.

## 39. All-Optical Switching and Router via the Direct Quantum Control of Coupling between Cavity Modes

Jason Twamley, Macquarie University

(Session 12 : Saturday from 3:15pm - 3:45pm)

We describe a scheme to execute all-optical routing of photonic information by optically controlling the internal quantum state of a individual scatterer coupled to two independent cavity modes. We show that through this quantum control one can dynamically and rapidly modulate the cavity coupling. This allows all-optical modulation of intercavity couplings via ac Stark or shuffle (stimulated Raman adiabatic passage) control of the scatterer’s internal states, and from this modulation, we show that we can perform all-optical switching and all-optical routing with near-unit switching contrast and with high bandwidth. [1] K. Xia and J. Twamley, Phys Rev X 3, 031013 (2013)

## 40. A polynomial-time algorithm for the ground state of 1D gapped local Hamiltonians

Thomas Vidick, UC Berkeley

(Session 4 : Thursday from 3:45pm - 4:30pm)

Computing ground states of local Hamiltonians is a fundamental problem in condensed matter physics. We give the first randomized polynomial-time algorithm for finding ground states of gapped one-dimensional Hamiltonians: it outputs an (inverse-polynomial) approximation, expressed as a matrix product state (MPS) of polynomial bond dimension. The algorithm combines many ingredients, including recently discovered structural features of gapped 1D systems, convex programming, insights from classical algorithms for 1D satisfiability, and new techniques for manipulating and bounding the complexity of MPS. Our result provides one of the first major classes of Hamiltonians for which computing ground states is provably tractable despite the exponential nature of the objects involved.


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# Generate domain DCV random value

POST https://www.digicert.com/services/v2/domain/{{domain_id}}/dcv/token

Use this endpoint to generate a new domain control validation (DCV) random value for the domain. Use this random value to validate the domain using the DCV method set for the domain.

To use this endpoint, the DCV method for the domain must be set to dns-txt-token, dns-cname-token, or http-token. To change the domain's DCV method, use the Change domain DCV method API endpoint

To get the existing domain random value instead of generating a new one, use the Domain info endpoint.

## DCV random values for domains vs. orders

This endpoint generates a new random value for the domain. This is the domain random value. You can only use the domain random value to complete the DCV check for the given domain.

### Danger

Generating a new domain random value invalidates the domain random value previously generated for the given domain.

When you place an OV or EV TLS order using a DCV method that requires a random value, DigiCert generates a single random value for the entire order. This is the order random value. You can use the order random value to complete the DCV check for any domain submitted with the order.

### Tip

Order random values remain valid even when you generate a new domain random value.

When requesting a DCV check, you can use either check DCV endpoint: Check order DCV or Check domain DCV. If DigiCert finds a valid random value on the domain, the DCV check is completed, regardless of whether the random value is for the domain or the order.

## Request parameters

Name

Opt/Req

Type

Description

validations Deprecated*

ignored

array

List of validation types for the domain.

*Note: DigiCert has deprecated the validations request parameter.

.. type

ignored

string

Validation type identifier.

## Response parameters

Name

Type

Description

token

string

Random value used to validate domain control for domains using the dns-txt-token, http-token, or dns-cname-token DCV methods.

expiration_date

string

Date and time when the random value expires.

Format: UTC timezone and ISO 8601 date

http_token_url

string

For domains using the http-token DCV method, the URL path where the fileauth.txt file should be placed on the web server. Not returned for domains using other DCV methods.

verification_value

string

For domains using the dns-cname-token DCV method, the domain to use in your CNAME record. For more information, see Use the DNS CNAME validation method to verify domain control.

For domains using the http-token DCV method, the URL path where the fileauth.txt file should be placed on the web server.

Not returned for domains using the dns-txt-token DCV method.


text:
What does the $[(0 : 3 : 1)]$ means in Sage.

I tried to solve the integer points of $y(y+2)=x^3+(x+3)(x+5)$ by using Sage's command E.integral_points(). Its output was $[(0 : 3 : 1)]$. I tried that $(x,y)=(0,3)$ satisfies my equation. But what does the third number $1$ means in Sage's notation?

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