text:
Q: How do I use Assert to verify that an exception has been thrown with MSTest? How do I use Assert (or other Test class) to verify that an exception has been thrown when using MSTest/Microsoft.VisualStudio.TestTools.UnitTesting?

A: If you're using MSTest, which originally didn't have an ExpectedException attribute, you could do this:
try
{
    SomeExceptionThrowingMethod()
    Assert.Fail("no exception thrown");
}
catch (Exception ex)
{
    Assert.IsTrue(ex is SpecificExceptionType);
}

A: In VS built-in unit testing if you simply want to verify that "any exception" is thrown, but you don't know the type, you can use a catch all:
[TestMethod]
[ExpectedException(typeof(Exception), AllowDerivedTypes = true)]
public void ThrowExceptionTest()
{
    //...
}

A: MSTest (v2) now has an Assert.ThrowsException function which can be used like this:
Assert.ThrowsException<System.FormatException>(() =>
            {
                Story actual = PersonalSite.Services.Content.ExtractHeader(String.Empty);
            });

You can install it with nuget: Install-Package MSTest.TestFramework

A: I do not recommend using the ExpectedException attribute (since it's too constraining and error-prone) or to write a try/catch block in each test (since it's too complicated and error-prone).  Use a well-designed assert method -- either provided by your test framework or write your own.  Here's what I wrote and use.
public static class ExceptionAssert
{
    private static T GetException<T>(Action action, string message="") where T : Exception
    {
        try
        {
            action();
        }
        catch (T exception)
        {
            return exception;
        }
        throw new AssertFailedException("Expected exception " + typeof(T).FullName + ", but none was propagated.  " + message);
    }

    public static void Propagates<T>(Action action) where T : Exception
    {
        Propagates<T>(action, "");
    }

    public static void Propagates<T>(Action action, string message) where T : Exception
    {
        GetException<T>(action, message);
    }

    public static void Propagates<T>(Action action, Action<T> validation) where T : Exception
    {
        Propagates(action, validation, "");
    }

    public static void Propagates<T>(Action action, Action<T> validation, string message) where T : Exception
    {
        validation(GetException<T>(action, message));
    }
}

Example uses:
    [TestMethod]
    public void Run_PropagatesWin32Exception_ForInvalidExeFile()
    {
        (test setup that might propagate Win32Exception)
        ExceptionAssert.Propagates<Win32Exception>(
            () => CommandExecutionUtil.Run(Assembly.GetExecutingAssembly().Location, new string[0]));
        (more asserts or something)
    }

    [TestMethod]
    public void Run_PropagatesFileNotFoundException_ForExecutableNotFound()
    {
        (test setup that might propagate FileNotFoundException)
        ExceptionAssert.Propagates<FileNotFoundException>(
            () => CommandExecutionUtil.Run("NotThere.exe", new string[0]),
            e => StringAssert.Contains(e.Message, "NotThere.exe"));
        (more asserts or something)
    }

NOTES
Returning the exception instead of supporting a validation callback is a reasonable idea except that doing so makes the calling syntax of this assert very different than other asserts I use.
Unlike others, I use 'propagates' instead of 'throws' since we can only test whether an exception propagates from a call. We can't test directly that an exception is thrown.  But I suppose you could image throws to mean: thrown and not caught.
FINAL THOUGHT
Before switching to this sort of approach I considered using the ExpectedException attribute when a test only verified the exception type and using a try/catch block if more validation was required.  But, not only would I have to think about which technique to use for each test, but changing the code from one technique to the other as needs changed was not trivial effort.  Using one consistent approach saves mental effort.
So in summary, this approach sports: ease-of-use, flexibility and robustness (hard to do it wrong).
UPDATE
My approach is no longer valuable with mstest V2 which seems to have come out in 2018 or something. Use Assert.ThrowsException.
Unless you are stuck using an old version of mstest. Then, my approach still applies.

A: Well i'll pretty much sum up what everyone else here said before...Anyways, here's the code i built according to the good answers :) All is left to do is copy and use...
/// <summary>
/// Checks to make sure that the input delegate throws a exception of type TException.
/// </summary>
/// <typeparam name="TException">The type of exception expected.</typeparam>
/// <param name="methodToExecute">The method to execute to generate the exception.</param>
public static void AssertRaises<TException>(Action methodToExecute) where TException : System.Exception
{
    try
    {
        methodToExecute();
    }
    catch (TException) {
        return;
    }
    catch (System.Exception ex)
    {
        Assert.Fail("Expected exception of type " + typeof(TException) + " but type of " + ex.GetType() + " was thrown instead.");
    }
    Assert.Fail("Expected exception of type " + typeof(TException) + " but no exception was thrown.");
}

A: The helper provided by @Richiban above works great except it doesn't handle the situation where an exception is thrown, but not the type expected. The following addresses that:
using System;
using Microsoft.VisualStudio.TestTools.UnitTesting;

namespace YourProject.Tests
{
    public static class MyAssert
    {
        /// <summary>
        /// Helper for Asserting that a function throws an exception of a particular type.
        /// </summary>
        public static void Throws<T>( Action func ) where T : Exception
        {
            Exception exceptionOther = null;
            var exceptionThrown = false;
            try
            {
                func.Invoke();
            }
            catch ( T )
            {
                exceptionThrown = true;
            }
            catch (Exception e) {
                exceptionOther = e;
            }

            if ( !exceptionThrown )
            {
                if (exceptionOther != null) {
                    throw new AssertFailedException(
                        String.Format("An exception of type {0} was expected, but not thrown. Instead, an exception of type {1} was thrown.", typeof(T), exceptionOther.GetType()),
                        exceptionOther
                        );
                }

                throw new AssertFailedException(
                    String.Format("An exception of type {0} was expected, but no exception was thrown.", typeof(T))
                    );
            }
        }
    }
}

A: Since you mention using other test classes, a better option than the ExpectedException attribute is to use Shoudly's Should.Throw.
Should.Throw<DivideByZeroException>(() => { MyDivideMethod(1, 0); });

Let's say we have a requirement that the customer must have an address to create an order. If not, the CreateOrderForCustomer method should result in an ArgumentException. Then we could write:
[TestMethod]
public void NullUserIdInConstructor()
{
  var customer = new Customer(name := "Justin", address := null};

  Should.Throw<ArgumentException>(() => {
    var order = CreateOrderForCustomer(customer) });
}

This is better than using an ExpectedException attribute because we are being specific about what should throw the error. This makes requirements in our tests clearer and also makes diagnosis easier when the test fails.
Note there is also a Should.ThrowAsync for asynchronous method testing.

A: As an alternative you can try testing exceptions are in fact being thrown with the next 2 lines in your test.
var testDelegate = () => MyService.Method(params);
Assert.Throws<Exception>(testDelegate);

A: I know this thread is old and has many good answers but maybe worth mentioning that local function can help in a very simple way.
//Arrange

//Act
void LocalFunction() => mr.ActualMethod(params);

//Assert
Assert.Throws<Exception>(LocalFunction);

A: Be wary of using ExpectedException, as it can lead to several pitfalls as demonstrated here:
Link
And here:
http://xunit.github.io/docs/comparisons.html
If you need to test for exceptions, there are less frowned upon ways. You can use the try{act/fail}catch{assert} method, which can be useful for frameworks that don't have direct support for exception tests other than ExpectedException.
A better alternative is to use xUnit.NET, which is a very modern, forward looking, and extensible unit testing framework that has learned from all the others mistakes, and improved. One such improvement is Assert.Throws, which provides a much better syntax for asserting exceptions.
You can find xUnit.NET at github: http://xunit.github.io/

A: In case of using NUnit, try this:
Assert.That(() =>
        {
            Your_Method_To_Test();
        }, Throws.TypeOf<Your_Specific_Exception>().With.Message.EqualTo("Your_Specific_Message"));

A: There is an awesome library called NFluent which speeds up and eases the way you write your assertions.
It is pretty straightforward to write an assertion for throwing an exception:
    [Test]
    public void given_when_then()
    {
        Check.ThatCode(() => MethodToTest())
            .Throws<Exception>()
            .WithMessage("Process has been failed");
    }

A: Usually your testing framework will have an answer for this. But if it's not flexible enough, you can always do this:
try {
    somethingThatShouldThrowAnException();
    Assert.Fail(); // If it gets to this line, no exception was thrown
} catch (GoodException) { }

As @Jonas points out, this DOES NOT work for catching a base Exception:
try {
    somethingThatShouldThrowAnException();
    Assert.Fail(); // raises AssertionException
} catch (Exception) {
    // Catches the assertion exception, and the test passes
}

If you absolutely must catch Exception, you need to rethrow the Assert.Fail(). But really, this is a sign you shouldn't be hand-writing this; check your test framework for options, or see if you can throw a more meaningful exception to test for.
catch (AssertionException) { throw; }

You should be able to adapt this approach to whatever you like -- including specifying what kinds of exceptions to catch. If you only expect certain types, finish the catch blocks off with:
} catch (GoodException) {
} catch (Exception) {
    // not the right kind of exception
    Assert.Fail();
}

A: In a project i´m working on we have another solution doing this.
First I don´t like the ExpectedExceptionAttribute becuase it does take in consideration which method call that caused the Exception.
I do this with a helpermethod instead.
Test
[TestMethod]
public void AccountRepository_ThrowsExceptionIfFileisCorrupt()
{
     var file = File.Create("Accounts.bin");
     file.WriteByte(1);
     file.Close();

     IAccountRepository repo = new FileAccountRepository();
     TestHelpers.AssertThrows<SerializationException>(()=>repo.GetAll());
}

HelperMethod
public static TException AssertThrows<TException>(Action action) where TException : Exception
    {
        try
        {
            action();
        }
        catch (TException ex)
        {
            return ex;
        }
        Assert.Fail("Expected exception was not thrown");

        return null;
    }

Neat, isn´t it;)

A: This is going to depend on what test framework are you using?
In MbUnit, for example, you can specify the expected exception with an attribute to ensure that you are getting the exception you really expect.
[ExpectedException(typeof(ArgumentException))]

A: Check out nUnit Docs for examples about:
[ExpectedException( typeof( ArgumentException ) )]

A: It is an attribute on the test method... you don't use Assert. Looks like this:
[ExpectedException(typeof(ExceptionType))]
public void YourMethod_should_throw_exception()

A: You can achieve this with a simple one-line.
If your operation foo.bar() is async:
await Assert.ThrowsExceptionAsync<Exception>(() => foo.bar());

If foo.bar() is not async
Assert.ThrowsException<Exception>(() => foo.bar());

A: You can download a package from Nuget using: PM> Install-Package MSTestExtensions that adds Assert.Throws() syntax in the style of nUnit/xUnit to MsTest.
High level instructions: download the assembly and inherit from BaseTest and you can use the Assert.Throws() syntax.
The main method for the Throws implementation looks as follows:
public static void Throws<T>(Action task, string expectedMessage, ExceptionMessageCompareOptions options) where T : Exception
{
    try
    {
        task();
    }
    catch (Exception ex)
    {
        AssertExceptionType<T>(ex);
        AssertExceptionMessage(ex, expectedMessage, options);
        return;
    }

    if (typeof(T).Equals(new Exception().GetType()))
    {
        Assert.Fail("Expected exception but no exception was thrown.");
    }
    else
    {
        Assert.Fail(string.Format("Expected exception of type {0} but no exception was thrown.", typeof(T)));
    }
}

Disclosure: I put together this package.
More Info: http://www.bradoncode.com/blog/2012/01/asserting-exceptions-in-mstest-with.html

A: My preferred method for implementing this is to write a method called Throws, and use it just like any other Assert method. Unfortunately, .NET doesn't allow you to write a static extension method, so you can't use this method as if it actually belongs to the build in Assert class; just make another called MyAssert or something similar. The class looks like this:
using System;
using Microsoft.VisualStudio.TestTools.UnitTesting;

namespace YourProject.Tests
{
    public static class MyAssert
    {
        public static void Throws<T>( Action func ) where T : Exception
        {
            var exceptionThrown = false;
            try
            {
                func.Invoke();
            }
            catch ( T )
            {
                exceptionThrown = true;
            }

            if ( !exceptionThrown )
            {
                throw new AssertFailedException(
                    String.Format("An exception of type {0} was expected, but not thrown", typeof(T))
                    );
            }
        }
    }
}

That means that your unit test looks like this:
[TestMethod()]
public void ExceptionTest()
{
    String testStr = null;
    MyAssert.Throws<NullReferenceException>(() => testStr.ToUpper());
}

Which looks and behaves much more like the rest of your unit test syntaxes.

A: if you use NUNIT, you can do something like this:
Assert.Throws<ExpectedException>(() => methodToTest());

It is also possible to store the thrown exception in order to validate it further:
ExpectedException ex = Assert.Throws<ExpectedException>(() => methodToTest());
Assert.AreEqual( "Expected message text.", ex.Message );
Assert.AreEqual( 5, ex.SomeNumber);

See: http://nunit.org/docs/2.5/exceptionAsserts.html

A: For "Visual Studio Team Test" it appears you apply the ExpectedException attribute to the test's method.
Sample from the documentation here: A Unit Testing Walkthrough with Visual Studio Team Test
[TestMethod]
[ExpectedException(typeof(ArgumentException),
    "A userId of null was inappropriately allowed.")]
public void NullUserIdInConstructor()
{
   LogonInfo logonInfo = new LogonInfo(null, "P@ss0word");
}

A: Even though this is an old question, I would like to add a new thought to the discussion. I have extended the Arrange, Act, Assert pattern to be Expected, Arrange, Act, Assert. You can make an expected exception pointer, then assert it was assigned to. This feels cleaner than doing your Asserts in a catch block, leaving your Act section mostly just for the one line of code to call the method under test. You also don't have to Assert.Fail(); or return from multiple points in the code. Any other exception thrown will cause the test to fail, because it won't be caught, and if an exception of your expected type is thrown, but the it wasn't the one you were expecting, Asserting against the message or other properties of the exception help make sure your test won't pass inadvertently.
[TestMethod]
public void Bar_InvalidDependency_ThrowsInvalidOperationException()
{
    // Expectations
    InvalidOperationException expectedException = null;
    string expectedExceptionMessage = "Bar did something invalid.";

    // Arrange
    IDependency dependency = DependencyMocks.Create();
    Foo foo = new Foo(dependency);

    // Act
    try
    {
        foo.Bar();
    }
    catch (InvalidOperationException ex)
    {
        expectedException = ex;
    }

    // Assert
    Assert.IsNotNull(expectedException);
    Assert.AreEqual(expectedExceptionMessage, expectedException.Message);
}

A: This works for Visual Studio Team Test (a.k.a MSTest)
While dealing with databases or http transaction. System should throw an exception somewhere, using Assert.ThrowExceptionAsync<>() will catch the your Throw event. (In these cases, Assert.ThrowException<>() does not catch the exception).
   [TestMethod]
   public void Invalid_Input_UserName_Should_Throw_Exception()
   {
       await Assert.ThrowExceptionAsync<ExpectedExceptionType>(()=> new LogonInfo(InvalidInputInUserNameFormat,"P@ssword"));
   }

A: FluentAssertions Examples
Adding an example using FluentAssertions for those using that library.
// act
Action result = () => {
    sut.DoSomething();
};

// assert
result.Should().Throw<Exception>();

Async Example
// act
Func<Task> result = async () => {
    await sut.DoSomethingAsync();
};

// assert
await result.Should().ThrowAsync<Exception>();


text:
Hebrews Verse by Verse
: Verse by Verse
by:Grant R Osborne
The letter to the Hebrews is unique in the New Testament for its focus on the priesthood of Jesus and its interaction with the Old Testament. But beyond this deep theology, Hebrews is a practical book that addresses the very real challenges believers face when life gets difficult. In Hebrews Verse by Verse, the late Grant R. Osborne, with George H. Guthrie, shows readers how this beautifully crafted letter encourages believers to endure in faithfulness... Show More
The letter to the Hebrews is unique in the New Testament for its focus on the priesthood of Jesus and its interaction with the Old Testament. But beyond this deep theology, Hebrews is a practical book that addresses the very real challenges believers face when life gets difficult. In Hebrews Verse by Verse, the late Grant R. Osborne, with George H. Guthrie, shows readers how this beautifully crafted letter encourages believers to endure in faithfulness to Jesus. By using Scripture and theology to lay the foundation for these exhortations, the central message of Hebrews continues to be relevant for the church today. Osborne's commentary delves into the grand implications of Christ's identity and its importance for our spiritual lives. Show Less
Publish Date Sep. 28th, 2021
Publisher Zlexham Press
About Grant R Osborne
b>Grant R. Osborne is Professor of New Testament at Trinity Evangelical Divinity School. He is the author of numerous books, including The Hermeneutical Spiral: A Comprehensive Introduction to Biblical Interpretation, and commentaries on Revelation (Baker Exegetical Commentary on the New Testament), Romans (IVP New Testament Commentary), and Matthew (Zondervan Exegetical Commentary on the New Testament). He has also taught at Winnipeg Theological Seminary and the University of Aberdeen, and has pastored churches in Ohio and... Show More
b>Grant R. Osborne is Professor of New Testament at Trinity Evangelical Divinity School. He is the author of numerous books, including The Hermeneutical Spiral: A Comprehensive Introduction to Biblical Interpretation, and commentaries on Revelation (Baker Exegetical Commentary on the New Testament), Romans (IVP New Testament Commentary), and Matthew (Zondervan Exegetical Commentary on the New Testament). He has also taught at Winnipeg Theological Seminary and the University of Aberdeen, and has pastored churches in Ohio and Illinois. Show Less

redpajama_set_name: RedPajamaCommonCrawl


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text:
Shutdown Delaying Lakeland Housing Authority Construction
By Stephanie Claytor Polk County
PUBLISHED 7:30 AM ET Jan. 18, 2019 PUBLISHED 7:30 AM EST Jan. 18, 2019 UPDATED 7:20 AM ET Jan. 20, 2019
LAKELAND, Fla. — Housing Authorities across the nation are worried about the government shutdown.
Agency constructing new 100-unit senior public housing building
The United States Department of Housing and Urban Development funds everything from section 8 subsidies to public housing units.
Lakeland Housing Authority Executive Director Ben Stevenson said the shutdown was terrible timing for his agency.
It's constructing a new 100-unit senior public housing building. The goal was to open in April, but if the shutdown continues, that date will have to be pushed back due to federal subsidies funding much of the rent.
"The longer the shutdown goes on the worse it is for everybody," said Ben Stevenson.
Stevenson said the agency also planned to tear down standing public housing units from the 1940s, and build 132 multi-family units. The shutdown has that on hold too.
"We were wanting to do the closing so we could start construction but we need HUD approvals and HUD is not there," Stevenson said.
What to Do? Bills Stacking Up For Furloughed Workers
Florida Lawmakers Push Bill to Pay Coast Guard During the Shutdown
Crisis Center of Tampa Bay Feeling Government Shutdown Pinch
A delay that could be costly.
"So the price, cost of the construction could potentially go up as well. A million, two million dollars," Stevenson said.
Stevenson said the delay is ultimately hurting the people.
"Here in Lakeland, for every 10 families that need affordable housing, there's only 3 units of affordable housing available," Stevenson said.
Not only that, he said Section 8 landlords are in for a big surprise if the government stays shut down until March.
The housing authority wouldn't be able to pay them the Section 8 subsidies.
Section 8 landlord Don Brown said if that happened, he'd make his tenant pay all of the rent and workout a payment plan.
"I normally will allow them to go maybe a month or two then something has to happen. I'm going to get my money," said Don Brown.
Ultimately Stevenson said it'd be up to landlords to decide whether to let the tenant stay.
Stevenson said the landlords would receive back rent when the government reopened.
If the shutdown continues into March, Stevenson said the housing authority would have to dip into cash reserves to pay for operational expenses and may have to consider furloughing employees.


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GEDANKEN TOTEN LEBENS – Gedanken Toten Lebens
Gedanken Toten Lebens is a new atmospheric dark black metal band from Germany, the personal project of Solveig, who is responsible for the whole concept, lyrics and vocals. She formed the band in 2012 and has cooperated with other bands and musicians for her project since then. In the summer of 2015 the band self-released its debut full-length album "Gedanken Toten Lebens", its only official release so far. For this album Solveig collaborated with Freigeist, who played all Read more…
DAWN OF A DARK AGE – The Six Elements, Vol.4 Air
Dawn of a Dark Age is a very interesting band that has managed to release 4 quality albums in a very short period of time. It is the personal project of the multi-instrumentalist Vittorio Sabelli from Italy, who has been active like composer, saxophonist and clarinetist in the classical, jazz and avant-garde scene for over 15 years. In 2010 he formed the Pink Floyd anti-tribute prog jazz metal group R-Evolution Band. In 2014 he decided to share his personal eclectic Read more…
AUÐN – Auðn
Auðn is a new atmospheric black metal band from Hafnarfjörður of Iceland, a country with many interesting releases lately. The band was formed in 2010 and released its eponymous debut full-length album "Auðn" in the end of November 2014, via the US labels Black Plague Records and Metallic Media. It was the first official release of the band, without any demos before it, thus "Auðn" was my only contact with their music. Read more…
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DE PROFUNDIS – This Winter in My Heart
De profundis is a Hungarian atmospheric black metal band from Szombathely, formed back in 2000, by Yzokras and Zoltan, the band's only members since then. Unluckily, due to personal reasons, the band split up in 2001. They decided to reunite 7 years later, in 2008 and released the 4-track demo "Haunted By Ill Angels Only" in 2010. In 2013 the demo plus one song was re-released as a split tape with the demo of another Hungarian black metal Read more…
TIVAZ – Under Nordic Stars
Tivaz is a dark ambient / atmospheric epic black metal project from Moscow, Russia. There is not much info about the band's bio or line-up, but they have already released 3 full-length albums since 2010. "Under Nordic Stars" is the second full-length album for Tivaz and it is also their first work ever that came out in physical format. It was released in CD edition via the American label Razed Soul Productions in July 2014. In May 2015 Read more…
SVIATIBOR – La Foi des Ancêtres
Sviatibor is the personal project of Thomas Hornstein (also in Nervengeist) from Albi, France. He formed this band just in 2013, but he has been quite over-productive since then, with several demos, EPs and 3 full-length albums already released! Actually he has almost finished his next album, which will be released this Spring. All Sviatibor's full-length albums were released in 2014, beginning with "From Yav to Nav" in February, "L'Oeuvre de Rod" in April and finally Read more…
ENISUM – Samoht Nara
Enisum is an Italian atmospheric black metal band from Susa Valley (Val di Susa), the longest valley of Italian Alps, near Turin. They took their name from a local mountain, since Enisum is the anagram Musinè mountain, spelled backwards. The band was formed in 2006 by Lys as his personal project and self-released several demos, without any proper distribution. Finally in 2014 they signed to Dusktone label and in October 2014 their first professional full-length album entitled "Samoht Nara Read more…
IMMORIOR – Herbstmär
www.youtube.com/watch?v=Cw0DpMx7Zno"
Immorior is a new atmospheric post black metal band from Saarland of Germany. They were formed in 2013 by Nordmann and Sarghas, who remain their only members since then. They've been together in other bands too (Nelandhir, Seuchenzug, Yarr and Stardust), but with Immorior they wanted to explore new musical paths. On October 17th, 2014, their debut full-length album "Herbstmär" was finally released by the German labels Narbentage Produktionen and Black Blood Records. Read more…


text:
Title: So. Yes. A post.
discuss: food, tv, work
Date: Jul 11th, 2007 8:25 pm
Whokay - long over due for a post here....
As I said to Lissa today, there are some times I kind of wish I was back in Minneapolis so I could hurl myself off the Foshay tower. She laughed and agreed - it would be so romantic. Chef Eric asked about the Golden Gate Bridge and Lissa brushed him off - it's just too cliche. But the Foshay Tower? It would be a short flight but ever so art deco.
I have *another* interview with THE SAK on Friday and, as much as I love my Salty boys and the bohemian life of a reservationists, after today and Monday (the story of Monday includes, but is not limited to, me doing nothing to stop A-Rod from canceling his reservation for that night. Yes. A-Rod. The man whom we would have slaughtered babies for if he wanted to feast upon their tender flesh. Bald eagles? His. Sea Otters? Done) I kind of want a job that doesn't involve me having to work the job of two people and sounds impressive when I tell people what I do. And a real pay check? And real hours? Yes. Yes I think I can do this. So hopefully they think I can do this too (they like me well enough that, while I didn't have experience for the last position, called me right up last week when this one opened up).
Tomorrow I will be going to Davis and I'll have my body comp test done. I've started to eat more real food. Right now it's just salad and fruit, but I am now incorporating some meat too. In fact, after I write this, I am going to grill up 3oz of pork and have some asparagus (I cuss, you cuss, we all cuss for asparagus!) and a salad. I am so hard core.
Kevin and I finished Season 1 of New Who. I am wondering who Billy Piper slept with so that she could be OMG STAR! of the companion world. Not that I really hate her or anything, it's just that I'm slightly annoyed with her and the whole lovey dovey thing with her and the Doctor. I might have to bring back some Old Who that I know mom and dad have to compare Companion reltaionships with the Doctor. Because damn. She sucks. But not really. It's just...hmm.
Date: Jul. 12th, 2007 3:38 am (UTC)
Billy Piper kinda grew on me.... and it was funny when Madam Hooch took over her body and made her Uber-Cockney in the second season :-)
and good luck with the SAK!
Author: amaresu
Date: Jul. 12th, 2007 10:51 am (UTC)
You're annoyed with her now? Oh just wait S2 is even worse. *runs back to her wonderful audios where the companions do stuff*
Author: fantom07
Date: Jul. 12th, 2007 2:39 pm (UTC)
Yeah, but the few episodes of season 2 that I have seen I didn't hate her all that much. Even when she had to switch universes because alternate earth blew up or something. I just don't like the relationship with the doctor. Like, when they aren't talking about it (or, rather, not talking about it but the subtext is hitting you over the head with a brick) she's just fine.
~ Thread ~ Parent ~ Reply
Date: Jul. 13th, 2007 10:45 pm (UTC)
S2 is when they went full force with the Doctor/Rose ship. I remained in deep denial up until the last three minutes of Doomsday myself, but it's there. And they are really annoying about it. But there are several episodes of S2 that I love. Tooth and Claw and all of the episodes with Cybermen. But you know of my love for Cybermen.


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text:
Odilon Redon, French (Bordeaux, France 1840 - 1916 Paris, France)
Alternate Title: Mystical Personality of Saint Sebastian; Mystical Saint Sebastian
Watercolor and gouache on off-white wove paper
actual: 23.4 x 15.3 cm (9 3/16 x 6 in.)
(?) Bauchy (Wildenstein 1992-98, vol. 1 [1992], no. 534); his sale, Hôtel Drouot, Paris, March 20, 1923, no. 52; acquired at that sale through Martin Birnbaum by Grenville L. Winthrop, April 1930; his bequest to the Fogg Art Museum, 1943.
Klaus Berger, Odilon Redon: Fantasy and Colour, McGraw-Hill Book Company (New York, Toronto, London, 1965), no. 534, pl. 24
Alec Wildenstein, Odilon Redon: Catalogue raisonné de l'oeuvre peint et dessiné, Wildenstein Institute (Paris, 1992-1998), vol. 1, no. 534, repr.
Douglas Druick, Odilon Redon: Prince of Dreams, 1840-1916, exh. cat., Harry N. Abrams, Inc. (New York, 1994), pp. 349 (fig. 116), 350
Stephan Wolohojian, ed., A Private Passion: 19th-Century Paintings and Drawings from the Grenville L. Winthrop Collection, Harvard University, exh. cat., The Metropolitan Museum of Art and Yale University Press (U.S.) (New York, 2003), cat. no. 109, repr. (color)
Franklin Einspruch, Fuse Visual Art Review: A Pair of Drawing Shows at the Harvard Art Museums, The Arts Fuse ([e-journal], June 9, 2016), http://artsfuse.org/146319/fuse-visual-arts-review-a-pair-of-drawing-shows-at-the-harvard-art-museums/, accessed June 9, 2016
Paysages d'après nature: Aquarelles et dessins par Odilon Redon, Bernheim-Jeune, Paris, 04/18/1917 - 04/28/1917
A Private Passion: 19th-Century Paintings and Drawings from the Grenville L. Winthrop Collection, Harvard University, The Metropolitan Museum of Art, New York, 10/23/2003 - 01/25/2004
Flowers of Evil: Symbolist Drawings, 1870–1910, Harvard Art Museums, Cambridge, 05/21/2016 - 08/14/2016
"Saint Sebastian (Odilon Redon) , 1943.396," Harvard Art Museums collections online, Jan 20, 2022, https://hvrd.art/o/298160.


There are around 7,830 certified nursing assistants (CNAs) in the Orlando area. Although a CNA holds an entry-level position, the job nevertheless requires a certain degree of responsibility and calls for excellent interpersonal skills. Students who wish to attend a CNA program but are tight in cash can apply to public higher education institutions such as Orlando Tech.
Becoming certified as a licensed practical nurse (LPN) can open the door to an array of employment opportunities, such as hospitals and correctional facilities. Those interested in attending an LPN program may be eligible for Florida's Nursing Student Loan Forgiveness Program, created to encourage qualified nurses to seek employment in state areas where there is a critical shortage of nursing employees. LPNs working in Orlando have a median income of $39,630.
A career as a registered nurse (RN) can allow students to work under a variety of employers, both within and beyond the traditional healthcare industry. Current LPNs may wish to enroll in an associate of science in nursing degree program, which qualifies them to work as RNs. Valencia College also offers a generic track, from which students can enroll in one of Florida's public universities to complete a four-year bachelor's degree in nursing.
Below is a full list of nursing schools in Orlando, Florida, to help you choose the right nursing program.


text:
\section{\label{sec:intro} Introduction}
The core tenet of the kernel method in machine learning is that it allows one to apply linear statistical methods to data sets that are more complex and non-linear in nature. The kernel function corresponds to an inner product of vectors in a (potentially) high-dimensional euclidean space referred to as the feature space \cite{boser1992training}. The data is mapped to this high-dimensional space by means of a non-linear feature map. This feature map has to be chosen in such a way that data, initially not tractable by linear methods, can be separated by linear means in the higher-dimensional space. The choice of the non-linear map is therefore central to this approach. The use of the kernel trick allows one to process the data in the high-dimensional space without explicitly computing the feature vector. This trick has found its way into many machine learning tasks such as classification~\cite{boser1992training}, regression~\cite{vapnik1996regression, smola2004tutorial}, clustering~\cite{benhur2002support, girolami2002mercer}, correlation analysis~\cite{lai2000kernel} and filtering~\cite{liu2011kernel}. The most prominent application of the kernel method is arguably for binary classification in the use of the support vector machine (SVM)~\cite{boser1992training, vapnik2013nature}.

A quantum computer can be used to perform the feature mapping into a quantum-enhanced feature space and estimate the kernel matrix for the training data \cite{havlivcek2019supervised, schuld2019quantum}. This quantum kernel can then be used in most machine learning algorithms that use the kernel method. As such, the quantum kernel approach is a general method for constructing learning algorithms that utilize quantum resources. In fact, it was observed in \cite{havlivcek2019supervised} that many of the recently introduced variational quantum algorithms ~\cite{mitarai2018quantum, farhi2018classification, grant2018hierarchical, schuld2020circuit, benedetti2019parameterized} reduce to a quantum kernel method, since these algorithms are only linear methods in the quantum feature space. This observation immediately implies that most of these quantum learning algorithms only have the potential of a quantum advantage over classical approaches if the corresponding quantum kernel is hard to estimate classically. The hardness of estimating the kernel with classical resources is of course only a necessary and not always sufficient condition to obtain a quantum advantage. However, it was proven recently \cite{liu2020rigorous} that learning problems exist for which learners with access to quantum kernel methods have a quantum advantage over all classical learners.

 Here, we experimentally implement a binary SVM classifier that utilizes the quantum kernel estimation (QKE) subroutine for a new class of kernels. For binary classification we consider a data set $T = \{\myVec{x}_i,y_i\}_{i=1\ldots m} \subset {\cal X} \times \mathbbm{Z}_2$ for a data space ${\cal X}$. In the following, this space will correspond to some group with binary labels. The goal is to predict correct labels $y$ for previously unseen data $\myVec{z} \in {\cal X}$ with high probability relative to the unknown probability distribution that produced the data samples.

The quantum kernel can be estimated as a transition amplitude of a quantum circuit. Following \cite{havlivcek2019supervised}, the feature mapping for a datum $\myVec{x} \in {\cal X}$ through some unitary circuit $U(\myVec{x})$  produces the feature state $\Phi(\myVec{x}) = U(\myVec{x})\proj{0^n}U ^\dagger(\myVec{x})$. Here, $U(\myVec{x})$ is assumed to depend non-linearly on the datum and it is assumed that it can be implemented efficiently.

Importantly, different choices of this circuit family will lead to different quantum kernels. The kernel entry for two points $\myVec{x},\tilde{\myVec{x}}$ is obtained as the Hilbert-Schmidt inner product $K(\myVec{x},\tilde{\myVec{x}}) = \tr{\Phi^\dagger(\myVec{x}) \Phi(\tilde{\myVec{x}})}$ of the quantum states. This inner product evaluates to the fidelity between the two quantum states and can be understood as the transition amplitude
\begin{equation}
   K(\myVec{x},\tilde{\myVec{x}}) = |\bra{0^n} U^\dagger(\myVec{x}) U(\tilde{\myVec{x}}) \ket{0^n}|^2,
\end{equation}
which can be estimated on a quantum computer by evolving the initial state $\ket{0^n}$ with  $U^\dagger(\myVec{x}) U(\tilde{\myVec{x}})$ and recording the frequency of the all-zero outcome $0^n$.
While it may appear to be only a detail, it is important to define  the feature state as the density matrix $\Phi(\myVec{x})$. Formally using only $\ket{\phi(x)} = U(\myVec{x})\ket{0^n}$ and the canonical inner product $\langle \phi(x)|\phi(y)\rangle$ \cite{schuld2019quantum} would lead to a kernel and classifiers that are sensitive to the global phase of the state.\\

The quantum kernel can now be used in lieu of a classical kernel in conventional machine learning algorithms. For the SVM classifier, with training set $T = \{(\myVec{x}_i,y_i)\}_{i=1\ldots m}$, the kernel is used to construct the threshold function to compute the label $y = f(\myVec{z})$ for a new datum $\myVec{z} \notin T$ as
\begin{equation}
\label{eqn:svm-decision-function}
    f(\myVec{z}) = \mbox{sign}\left(\sum_{i=1}^m y_i \alpha_i K(\myVec{x}_i,\myVec{z})\right).
\end{equation}
The $\myVec{\alpha}$ parameters are constrained as $0 \leq \alpha_i \leq C$ and $\sum_i y_i \alpha_i = 0$ and will be optimized during the training. Here $C$ can be chosen at will and is referred to as the {\it box parameter}. It determines how strongly data points that are not linearly separable contribute to the cost function. The classifier is trained by solving the quadratic program  $\myVec{\alpha}$ with the cost function
\begin{equation}
\label{eqn:svm-dual}
     F(\myVec{\alpha}) = \sum_{i} \alpha_i - \frac12 \sum_{i,j} \alpha_i \alpha_j y_i y_j K(\myVec{x}_i,\myVec{x}_j).
\end{equation}
The quantum computer is utilized to estimate kernel matrices in training as well as in classification. \\

A core challenge for the development of learning algorithms with QKE is to find {\it good} quantum kernels for practically relevant problems. While a formal quantum advantage for learning with QKE has been proven \cite{liu2020rigorous}, it is necessary to find both learning problems as well as the corresponding kernels that can benefit from the quantum advantage in a practically relevant setting. The problem is therefore to identify an explicit circuit family $U(\myVec{x})$ that can provide an advantage for learning on some data ${\cal X}$ that is of wide relevance. An observation that appears to make this endeavour  challenging is that most quantum algorithms that provide a significant advantage over their classical counterparts exploit a particular structural insight \cite{aaronson2009need} into the problem. For an ad hoc quantum feature map circuit that is to be applied to a generic data set there appears to be little structural information to exploit. Therefore, one does not expect a quantum algorithm to outperform classical learners in this scenario \cite{wu2021application,peters2021machine,huang2020power}. It is important to identify a broad class of kernels that relate to learning problems that have a particular structural aspect that can lead to an advantage for the quantum kernel.

\begin{figure*}[t!]
	\centering
	\includegraphics[width=\textwidth]{FIG_1.pdf}
    \caption{(a), (b) Two covariant feature maps for a single qubit example of the \emph{labeling cosets} learning problem introduced in sec. \ref{sec:experiment}. We consider $S = \{\mathds{1}, A, A^2\}$ as subgroup of $G = SU(2)$, where $A = \exp(i (2\pi/3) Z)$. If we choose two elements $\myVec{c}_+, \myVec{c}_- \in SU(2)$, with their representation $D_{\myVec{c}_+}=H$, $D_{\myVec{x}_-}=\mathds{1}$, we can form two cosets: $C^+ = D_{\myVec{c}_+} S = \{H, HA, HA^2\}$ and $C^- = D_{\myVec{c}_-} S = \{\mathds{1}, A, A^2\}$. (a) Suppose we select the fiducial state $|\psi\rangle = |+\rangle$. Data points $\myVec{x} \in C^{\pm}$ for which $\myVec{x} = \myVec{c}_{\pm} \myVec{s}$ get mapped to either $D_{\myVec{c}_+} D_{\myVec{s}} |+\rangle$ or $D_{\myVec{c}_-} D_{\myVec{s}} |+\rangle$ for all $\myVec{s} \in S$ respectively. The states correspond to the points on the Bloch sphere marked by three red stars and three blue dots, respectively. We see that the data from the different cosets lives on orthogonal planes and cannot be linearly separated. (b) However, suppose we choose the fiducial state $|\psi\rangle = |0\rangle$. This is a subgroup invariant state, which means that the data from a given coset gets mapped to a single state: $D_{\myVec{c}_+} D_{\myVec{s}} |0\rangle = D_{\myVec{c}_+}|0\rangle = |+\rangle$ and $D_{\myVec{c}_-} D_{\myVec{s}} |0\rangle = D_{\myVec{c}_-}|0\rangle = |0\rangle$ for all $\myVec{s} \in S$. Now, we only need to distinguish between two points $|+\rangle$ and $|0\rangle$, which are linearly separable by an optimal hyperplane tilted at 45$^{\circ}$ from the $zy$ plane.
    (c) Quantum circuit for calculating the matrix elements of a covariant quantum kernel. The feature map $U(\myVec{x})$ is defined in terms of a unitary representation $D_{\myVec{x}}$ for a group $G$ with $\myVec{x} \in G$ and a circuit $V_{\myVec{\lambda}}$ that prepares the fiducial state $|\psi_{\myVec{\lambda}}\rangle = V_{\myVec{\lambda}} |0^n\rangle$. The frequency of measuring all zeros in the computational basis is an estimate for the kernel matrix element $K_{\myVec{\lambda}}(\myVec{x}, \myVec{\tilde{x}})$.}
    \label{fig:kernel}
\end{figure*}

\section{\label{cov} Covariant quantum kernels and feature maps}
We consider a class of feature map circuits that we call {\it covariant feature maps}. The feature states are intimately related to covariant measurements \cite{Decker2005,holevo1979covariant} and the circuits that implement them. The feature states are defined relative to a unitary representation $D_{\myVec{x}}$ for some group $G$ with $\myVec{x} \in G$. Let us consider here the case of a classification for group elements and their labels $\{(\myVec{x}_i,y_i)\}_{i = 1 \ldots m} \subset G \times {\mathbbm Z}_2$. The data space ${\cal X} = G$ is now identified with a group and the definitions given here are meant to apply different candidate groups that have relevance for the learning task at hand. Examples could constitute learning permutations or classifying translations or rotations. Exploiting group structure and learning problems for group data have already been investigated for classical kernels \cite{kondor2008group}.\\

After the group has been determined, the covariant feature map is defined in terms of the representation $D_{\myVec{x}}$ and a fiducial state $\ket{\psi} \in \mathbbm{C}^{2^n}$ on $n$ qubits as
\begin{equation}
    \Phi(\myVec{x}) = D_{\myVec{x}} \proj{\psi} D_{\myVec{x}}^\dagger \;\;\;\; \forall \, \myVec{x} \in G.
\end{equation}
The fiducial state is chosen as some reference state from the representing Hilbert space. In principle, any state that can be prepared efficiently may be chosen. However, the choice will greatly affect the performance of the classifier as we will discuss shortly.\\

The {\it covariant quantum kernel} $K(\myVec{x},\tilde{\myVec{x}}) = \tr{\Phi^\dagger(\myVec{x}) \Phi(\tilde{\myVec{x}})}$ is then estimated as the fidelity
\begin{eqnarray}\label{cov_kern}
K(\myVec{x},\tilde{\myVec{x}})  = |\bra{\psi} D^\dagger_{\myVec{x}} D_{\tilde{\myVec{x}}} \ket{\psi}|^2.
\end{eqnarray}
Note, that the circuit can be simplified if $\myVec{x}^{-1} \tilde{\myVec{x}}$ can be computed efficiently classically since $D^\dagger_{\myVec{x}} D_{\tilde{\myVec{x}}} = D_{\myVec{x}^{-1} \tilde{\myVec{x}}}$ for unitary representations.

We assume the fiducial state $\ket{\psi} = V \ket{0^n}$ can be prepared by applying an efficient quantum circuit $V$. Likewise, it is important to also assume that the representation $D$ of $G$ can be implemented on a quantum computer. In this case, the QKE routine reduces to estimating the transition amplitude
\begin{equation}
\label{eqn:kernel}
K(\myVec{x}, \tilde{\myVec{x}}) = |\bra{0^n} V^\dagger D_{\myVec{x}}^\dagger D_{\tilde{\myVec{x}}}V\ket{0^n}|^2,
\end{equation}
and the generic feature map circuit assumes the form $U(\myVec{x}) = D_{\myVec{x}} V$ c.f. FIG ~\ref{fig:kernel}(c). For the learning problem, the group $G$ and the representation $D_{\myVec{x}}$ will be assumed as predetermined. This means that the only degree of freedom in the kernel will be the choice of the fiducial state, $\ket{\psi} = V\ket{0^n}$. We will now see an example demonstrating that the choice of this state can impact the performance of the classifier significantly. The example we use is the recently introduced learning problem \cite{liu2020rigorous} for the the discrete logarithm problem (DLOG)\cite{rosen2011elementary}.\\

\paragraph*{The DLOG feature map:}
In this example the group is given by $\mathbbm{Z}^*_p$, i.e. the integers with multiplication modulo $p$. We consider a fixed generator $g \in \mathbbm{Z}^*_p$ that generates the full group through modular exponentiation. Given the generator $g$ we will always be able to write any $\myVec{x}  = g^{\myVec{v}}$ with ${\myVec{v}} \in \{0,1,\ldots,p-1\}$. The data to be classified is drawn uniformly at random from $\mathbbm{Z}^*_p \subset \{0,1\}^n$ with $n=\lceil\log_2 p\rceil$ and labels are assigned according to the following rule: To generate the particular data set to be learned, we first draw an element ${\myVec{s}} \in \mathbbm{Z}^*_p$ uniformly at random. Then, every sampled element from the training set $\myVec{x} \in T \subset \mathbbm{Z}^*_p$ is assigned a label $y$ according to
\begin{eqnarray}
y=\begin{cases}+1,&\text{if }\log_{g} {\myVec{x}}\in[{\myVec{s}},{\myVec{s}}+\frac{p-3}{2}],\\ -1,
                  &\text{else.}\end{cases}
\end{eqnarray}
Here, addition is also taken modulo $p$. It was shown \cite{liu2020rigorous} that no classical classifier can assign the correct labels of this decision rule with a probability more than $\frac{1}{2} + \mbox{poly}(n)^{-1}$, relative to the hardness of DLOG by utilizing a result by Blum and Micali \cite{blum1984generate}. On the other hand, in the same paper it was also proven that near-perfect classification accuracy could be obtained with polynomial effort when the appropriate quantum kernel function was used.

The feature states for this kernel are constructed as follows. Let $D_{\myVec{x}}$ denote the regular representation of $\mathbbm{Z}^*_p \subset \{0,1\}^n$ defined on $n$ qubits that acts on a computational basis state $\ket{\myVec{z}} \in \mathbbm{C}^{2^n}$ through
\begin{equation}
D_{\myVec{x}} \ket{\myVec{z}} = \ket{{\myVec{x}} \circ {\myVec{z}} \mbox{ mod } p}.
\end{equation}
Given the generator $g$ we then define the following subset state on $\{0,1\}^k \subset \mathbbm{Z}^*_{p}$ as the fiducial state
\begin{equation}
\ket{\psi} = \frac{1}{2^{\frac{k}{2}}}\sum_{{\myVec{v}} \in \{0,1\}^k} \ket{g^{\myVec{v}}}.
\end{equation}
The feature state $\Phi({\myVec{x}}) = \proj{\phi({\myVec{x}})}$ for which the quantum speedup was proven is then given by $\ket{\phi({\myVec{x}})} = D_{\myVec{x}}\ket{\psi} = 2^{-\frac{k}{2}}\sum_{{\myVec{v}} \in \{0,1\}^k} \ket{{\myVec{x}} g^{\myVec{v}}}$. The kernel is constructed in the canonical form, c.f. eqn.~(\ref{cov_kern}).\\

\paragraph*{The importance of the fiducial state:} We can now understand the relevance of choosing a good reference state $\ket{\psi}$. If one were to modify the kernel and select the computational basis state $\ket{\psi} = \ket{0^n}$ as the fiducial reference state, the action of the representation is simply $D_{\myVec{x}} \ket{\psi} = \ket{\myVec{x}}$ and the kernel matrix $K({\myVec{x}},\tilde{\myVec{x}}) = \delta_{{\myVec{x}},\tilde{\myVec{x}}}$ reduces to the identity matrix. This kernel is trivial and will lead to a very poor classifier \cite{vapnik2013nature}. This example stresses the importance of the choice of $\ket{\psi}$ and demonstrates that an advantage can only be obtained when a suitable state is chosen. Moreover, the example highlights that the choice of $\ket{\psi}$ may strongly depend on the particular form of the learning problem and the ground truth distribution. This poses a challenge we seek to address in the next section.

\section{Kernel alignment for the fiducial state}
If sufficient structural knowledge about the problem is present a suitable fiducial can be chosen \emph{a priori}. However, in general it may a challenging problem to construct a reliable fiducial state with little prior knowledge. One could argue that learning the structure is in fact the task of the learning algorithm in the first place. We want a method that, subject to the available data, will search for potential fiducial states and choose the state that will lead to the best classification performance. For other quantum kernel families recent proposals have been put forth that optimize different metrics \cite{lloyd2020quantum,otten2020quantum} motivated by quantum information theoretic intuition. Here we will follow a method commonly used in the classical literature referred to as {\it kernel alignment} \cite{cristianini2001kernel,cortes2012algorithms}. This approach can be seen to optimize the SVM upper bound on the generalization error directly.

We define a family of fiducial states by applying a parameterized quantum circuit $V_{\myVec{\lambda}}$ with  parameters $\myVec{\lambda} \in \Omega \subset \mathbbm{R}^q$ to
\begin{equation}
\ket{\psi_{\myVec{\lambda}}} = V_{\myVec{\lambda}} \ket{0^n}.
\end{equation}
This will lead to a set of quantum kernels $K_{\myVec{\lambda}}(\myVec{x},\myVec{\tilde{x}}) = |\bra{0^n }V_{\myVec{\lambda}}D_{\myVec{x}}D^\dagger_{\myVec{\tilde{x}}}V^\dagger_{\myVec{\lambda}} \ket{0^n}|^2$  that can be  depicted as in Fig.~\ref{fig:kernel}(c). The parameters $\myVec{\lambda}$ are to be optimized in the kernel alignment procedure. \\

The initial definition \cite{cristianini2001kernel} of the alignment $\hat{A}$ of two kernels $K_1$ and $K_2$ is made in terms of the training data  $T = \{\myVec{x}_{1},\myVec{x}_{2},\ldots,\myVec{x}_{m}\}$. The alignment of $K_1,K_2$ is given in terms of the empirical Hilbert-Schmidt inner product $\hat{A} \propto \sum_{j,i=1}^m K_1(\myVec{x}_{i},\myVec{x}_{j})K_2(\myVec{x}_{i},\myVec{x}_{j})$. It can be interpreted as a similarity measure of the kernels relative to the training data $T$. In \cite{cristianini2001kernel} it is shown that when choosing $K_1(\myVec{x}_{i},\myVec{x}_{j}) = y_i y_j$ and $K_2(\myVec{x}_{i},\myVec{x}_{j}) = K_{\myVec{\lambda}}(\myVec{x}_{i},\myVec{x}_{j})$ the resulting alignment can be used to give an upper bound on the generalization error of the Parzen-window classifier.\\

For threshold functions of general SVM classifiers, a ``weighted'' version of the alignment is optimized \cite{cortes2012algorithms,bullins2017not}. The intuition here is that only the data points that constitute the support vectors should determine the kernel parameters. The following cost function, c.f. eqn. (\ref{eqn:svm-dual}) for comparison,
\begin{equation}
\label{eqn:svm-objective}
	F(\myVec{\alpha}, \myVec{\lambda}) = \sum_i \alpha_i - \frac12 \sum_{i,j} \alpha_i \alpha_j y_i y_j K_{\myVec{\lambda}}(\myVec{x}_i, \myVec{x}_j),
\end{equation}
is optimized in both $\myVec{\alpha}$ and $\myVec{\lambda}$. The kernel alignment procedure solves the following optimization problem
\begin{equation}
\label{eqn:minmaxgame}
    \min_{\myVec{\lambda}} \max_{\myVec{\alpha}} F(\myVec{\alpha}, \myVec{\lambda}),
\end{equation}
subject to the constraints of the feasible set $0 \leq \alpha_i \leq C$, where $C$ is the box parameter, and $\sum_i y_i \alpha_i = 0$. Here $\myVec{\lambda} \in \Omega$ is determined by the particular form of the kernel family.\\

This min-max problem has an interesting interpretation as choosing a kernel among all the available $K_{\myVec{\lambda}}$ that minimizes the SVM bound on the classification error. The classification error $\mathbbm{P}(y \neq f(\myVec{z}))$ of $f$ is the probability that the classifier fails to predict the correct label $y$ subject to $(\myVec{z},y)$ being drawn from the true data distribution. This error was upper bounded by Shawe-Taylor and Cristianini~\cite{shawe2002generalization} in terms of the fat-shattering dimension for linear threshold functions. The fat-shattering bound is data dependent and encodes the generalization error of the classifier. The bound is, up to polylogarithmic factors, given by the primal cost function \cite{vapnik2013nature} of the SVM classifier.
The optimization of the primal SVM cost function or it's Wolfe dual in kernel space, c.f. eqn. (\ref{eqn:svm-dual}) considered here can be interpreted as minimizing this upper bound over all admissible threshold functions. For all $\myVec{\lambda}$ the maximum $F^*(\myVec{\lambda}) = \max_{\myVec{\alpha}} F(\myVec{\alpha}, \myVec{\lambda})$ yields the upper bound to the classification error $\mathbbm{P}(y \neq f(\myVec{z})) \leq \tilde{\cal O}(m^{-1} F^*(\myVec{\lambda}))$. The notation $\tilde{\cal O}$ suppresses polylogarithmic factors. The optimization problem in eqn. (\ref{eqn:minmaxgame}) thus optimizes the parameters in  $K_{\myVec{\lambda}}$ to yield the smallest upper bound on the classification error. The kernel alignment procedure for fiducial states $\ket{\psi_{\myVec{\lambda}}}$ can be interpreted as the search for the fiducial state that gives rise to the best bound on the data-dependent generalization as measured in terms of the fat-shattering dimension.\\

\paragraph*{Stochastic Algorithm:}
Our approach to solving the optimization problem with a dataset $T$ as stated in eqn.~(\ref{eqn:minmaxgame}) is described in Algorithm~\ref{algo:quantum-kernel-alignment}. This iterative classical-quantum algorithm evaluates kernel matrices on a quantum processor and updates the parameters $\myVec{\lambda}$ and $\myVec{\alpha}$ with classical optimization routines. At each iteration, kernel matrices are computed using the QKE routine~\cite{havlivcek2019supervised} on quantum circuits parametrized by kernel parameters $\myVec{\lambda}$. The optimal values $\myVec{\lambda}^{*}$ and $\myVec{\alpha}^{*}$ obtained can then be used in a standard SVM program to predict labels for a test set.

The parametrization of quantum circuits for the kernel matrices leads, in general, to a highly non-convex objective function~\eqref{eqn:svm-objective} in the kernel parameters. We use simultaneous perturbation stochastic approximation (SPSA)~\cite{spall1992multivariate} for the minimization over the kernel parameters $\myVec{\lambda}$ in~\eqref{eqn:minmaxgame}. This method approximates the gradient of~\eqref{eqn:svm-objective} with respect to $\myVec{\lambda}$ using only two objective function evaluations $F(\myVec{\alpha}_\pm, \myVec{\lambda}_\pm)$ independent of the dimension of $\myVec{\lambda}$ and is suitable when measurements of the objective function are subject to stochastic fluctuations, which is the case for kernels evaluated on noisy quantum hardware. For the concave optimization over $\myVec{\alpha}$ in~\eqref{eqn:minmaxgame}, we use a standard classical solver \texttt{CVXOPT}~\cite{cvxopt2021}, which yields a unique solution for $\myVec{\alpha}$.

\onecolumngrid

\begin{algorithm}[H]
\centering
\caption{\label{algo:quantum-kernel-alignment} Quantum kernel alignment: learning the maximum-margin kernel}
\begin{algorithmic}[1]
\State  {\bf Input} Training set $T = \{ \myVec{x}_i\in \mathbbm{R}^n\}_{i=1}^m$ with labels $y \in \{-1, 1\}^m$, quadratic program solver \texttt{qp} (e.g., \texttt{CVXOPT}).
\State  {\bf Parameters} Number of measurement shots $R$, box constraint $C>0$ (the SVM regularization parameter), initial kernel parameters $\myVec{\lambda}_0 \in \mathbbm{R}^q$, and SPSA steps $P$.
\State  Calibrate the quantum hardware to generate short depth quantum kernel circuits.
\State  Set initial values of the kernel parameters $\myVec{\lambda} = \myVec{\lambda}_0$ for the quantum kernel circuits.
	\For{\texttt{$i=0$ to $P$ }}

		\State Generate random vector $\Delta \in \{-1,1\}^q$.
		\State Evaluate $\myVec{\lambda}_{\pm,i} = \myVec{\lambda}_i \pm c_i \, \Delta$, where $c_i = c / (i+1)^\gamma$ for constants $c$, $\gamma$.
		\State Evaluate the kernel matrices $K_\pm = K(\myVec{\lambda}_{\pm,i}, T)$ on quantum device with $R$ measurement shots per~\cite{havlivcek2019supervised}.
        \State Maximize the SVM objective~\eqref{eqn:svm-objective} over $\myVec{\alpha}$ via \texttt{qp} solver $F(\myVec{\alpha}_{\pm,i}, \myVec{\lambda}_{\pm,i}) \leftarrow \texttt{qp}(K_\pm, y, C)$ subject to $0\le \myVec{\alpha}_\pm \le C$, $y^T \myVec{\alpha}_\pm = 0$.
		\State Update $\myVec{\lambda}_{i+1} \leftarrow \myVec{\lambda}_i - \frac{a_i}{2 c_i} \left[F(\myVec{\alpha}_{+,i}, \myVec{\lambda}_{+,i}) - F(\myVec{\alpha}_{-,i}, \myVec{\lambda}_{-,i}) \right]$ via SPSA, where $a_i = a / (i + 1 + A)^\sigma$ for constants $A$, $\sigma$.
	\EndFor
\State {\Return the final kernel parameters $\myVec{\lambda}^*$.}
\State Evaluate aligned kernel matrix $K(\myVec{\lambda}^*, T)$ on the quantum device with $R$ measurement shots per~\cite{havlivcek2019supervised}.
\end{algorithmic}
\end{algorithm}
\twocolumngrid

\section{Experiment}
\label{sec:experiment}
In the experimental implementation we  consider a special case of the following learning problem. We refer to this problem as {\it labeling cosets with error}. Given a group $G$ and a subgroup $S$ we can define two distinct left-cosets $C_\pm = {\myVec{c}}_\pm S$ by choosing two ${\myVec{c}}_+,{\myVec{c}}_- \in G$ at random. The ${\myVec{c}}_{\pm}$ can be drawn according to the Haar measure of $G$, however other measures could also be considered. The $m$ data samples for training and testing are then generated from the two cosets as follows: For each of the $i=1 \ldots m$ pick a label $y_i = \pm 1$ and chose $\myVec{q}_i \in C_\pm$ uniformly at random. Then, we draw $e_i \in G$ according to the error distribution $P_{\epsilon} : G \rightarrow \mathbbm{R}^+$. The specific choice for $P_{\epsilon}$ will depend on the group $G$. However, we assume that most of the measure will be close to the identity element ${\myVec{1}} \in G$. The group elements $\myVec{e}$ drawn from $P_{\epsilon}$ are treated as being $\epsilon$-close to the identity. Each data sample is then generated by setting  $\myVec{x}_i = \myVec{q}_i \myVec{e}_i$. The resulting data is of the form $T = \{(\myVec{x}_1,y_1),\ldots,(\myVec{x}_m,y_m)\}$. Note that this problem is related to the problem of deciding membership in a subgroup \cite{watrous2000succinct} in the error free limit $\epsilon = 0$. This connection could be used to investigate the hardness of this learning problem, but will not be explored further here.\\

\begin{figure*}[t]
	\centering
	\includegraphics[width=\textwidth]{FIG_2.pdf}
	\caption{\label{Fig:Chip_Circuit}
	(a) The connectivity of the 27-qubit device ibmq kolkata.
	(b) The quantum circuit used to evaluate the kernel matrix elements~\eqref{eqn:kernel-coset} for the learning problem \emph{labeling cosets with errors}. Here, we define the single-qubit rotations as $R_{P}(\phi) = \exp(i \phi P)$ for $P \in \{X,Y,Z\}$. We choose the representation of the group $G=SU(2)^{\otimes 27}$ to be $D(\mathbf{\theta}) = \bigotimes_{k=1}^{27} \exp(i \theta^{2k-1}_{\pm} X) \exp(i \theta^{2k}_{\pm} Z)$. The fiducial state~\eqref{eqn:fiducial-state} is prepared by the circuit $V_\lambda$ for a single parameter $\lambda$. The entanglers $\mathrm{CZ}$, which are the controlled phase gates diag$(1,1,1,-1)$, match the connectivity of the quantum device in (a). The frequency of measuring all zeros in the computational basis is an estimate for a matrix element.}
\end{figure*}

Let us briefly consider the error free version of this problem to motivate the ansatz for the fiducial state family $\ket{\psi_{\myVec{\lambda}}}$. A natural candidate for the reference state, if it can be prepared efficiently, would be a state that remains invariant under the action of the subgroup $S$. We would like to have that
\begin{equation}
 \forall \;\; \myVec{s} \in S \;\;\;\; \mbox{the state satisfies} \;\;\;\; D_{\myVec{s}} \ket{\psi} = \ket{\psi}.
\end{equation}
For such a state, the data gets mapped to a unique representing state for each coset, since  all elements $\myVec{x} \in C_{\pm}$ are of the form $\myVec{x} = \myVec{x}_{\pm}\myVec{s}$ with $\myVec{s} \in S$. We therefore have that
\begin{equation}
\label{eqn:subgroup-invariant}
 D_{\myVec{c}_{\pm}\myVec{s}}\ket{\psi} = D_{\myVec{c}_{\pm}} D_{\myVec{s}} \ket{\psi} = D_{\myVec{c}_{\pm}} \ket{\psi}.
\end{equation}
In this case, the classifier effectively only needs to distinguish between the two states $D_{\myVec{c}_{\pm}}\proj{\psi} D_{\myVec{c}_{\pm}}^\dagger$. This situation changes, however, when we learn cosets with error. In this case the added error is expected to act as a perturbation on each of the two states. If the perturbation is small, e.g. $\epsilon \ll 1$, one would expect that the states will still be classified correctly.\\

\begin{figure*}[t]
	\centering
	\includegraphics[width=\textwidth]{FIG_3.pdf}
	\caption{\label{FigExp}  (a) Kernel alignment cost function and fiducial state parameter $\lambda$ for each SPSA trial step, for the error mitigated (blue symbols) and unmitigated (red) experimental realizations. The ideal $\lambda$ is $\pi/2$. The training data set during this optimization contains 10 data points per label. (b) Accuracy tests results at odd trial steps. For these tests, a training set kernel is estimated with the $\lambda$ parameter at that step, from which a separating hyperplane is obtained and used to classify a test set consisting of 50 data points per label. The training set kernels used for each classification are shown in (c) for both the mitigated and the unmitigated cases. The identity matrix has been subtracted from these kernels for clarity.}
\end{figure*}

In our experiment we consider a special case of this problem with a group $G$ motivated by our physical hardware. We implement our kernel alignment, training, and classification test experiments on a $n = 27$-qubit superconducting quantum processor with heavy-hexagon topology, c.f. Fig~\ref{Fig:Chip_Circuit}(a)~\cite{chamberland2020topo, jurcevic2021demonstration}. We choose $G = \mathsf{SU}(2)^{\otimes n}$ with the natural representation $D$ of $\mathsf{SU}(2)$ for each tensor component. We choose a Pauli-stabilizer group $S = \avr{s_1,\ldots,s_n}$ as subgroup of $G$. The stabilizers fix a stabilizer state as the joint eigenvector for unit eigenvalue $D_{\myVec{s}} \ket{\psi} = \ket{\psi}$. We choose the graph-stablizer
\begin{equation}
    S_{\mbox{graph}} = \avr{\{X_i \bigotimes_{k : (k,i) \in E} Z_k\}_{i \in V}}
\end{equation}
associated with the coupling graph $(E,V)$ given by the device connectivity. This graph also determines the stabilizer state \cite{hein2006entanglement}. Subgroup membership for Abelian subgroups is efficient classically. We don't know whether this is still the case when considering the problem with errors as done here. However, for the specific instance we implement the circuit Fig~\ref{Fig:Chip_Circuit}(b) is simple enough that it can still be estimated classically~\cite{bravyi2021classical}.
To generate the data we represent the rotations $D(\theta_1,\theta_2,\theta_3) \in  \mathsf{SU}(2)$ by their Euler angles $D(\theta_1,\theta_2,\theta_3) = \exp(i \theta_1 X) \exp(i \theta_2 Z) \exp(i \theta_3 X)$. For simplicity we set all $\theta_3 = 0$ and randomly draw two $\myVec{c}_{\pm} \in [-\pi/4,\pi/4]^{2n}$ and define each class by the cosets $C_{\pm} = \myVec{c}_{\pm} S_{\mbox{graph}}$. The rotations with the representative angles $\myVec{c}_{\pm}$ can be combined with the elements from the stabilizer group so that each datum in the coset is expressed as the rotation $D(\myVec{\theta}_{\pm}) = \bigotimes_{k=1}^{n} D(\theta^{2k-1}_{\pm},\theta^{2k}_{\pm},0)$. We express every element in the coset in terms of the Euler angles $\myVec{\theta}_{\pm}$. To generate data for training and testing we uniformly sample elements from $C_{\pm}$ and perturb each set of Euler angles with a normal random error of variance $\epsilon = 0.01$. The family of fiducial states we optimize over during the kernel alignment procedure is given by
\begin{equation}
\label{eqn:fiducial-state}
    \ket{\psi_\lambda} = \prod_{(k,t) \in E} \mbox{CZ}_{k,t} \prod_{k \in V} R_{Y_k}(\lambda)\ket{0^n},
\end{equation}
where $R_{Y}(\lambda) = \exp(i \lambda Y)$ for a uniform $\lambda \in [0,\pi]$. This state interpolates between a trivial initial state $\ket{\psi_0} = \ket{0^n}$ to a graph state stabilized by the subgroup. The resulting covariant quantum kernel, c.f. Fig.~\ref{Fig:Chip_Circuit}(b), is given as
\begin{equation}
\label{eqn:kernel-coset}
    K_{\lambda} (\myVec{\theta} ,\; \myVec{\theta}') = |\bra{\psi_\lambda} D(\myVec{\theta}) D^\dagger(\myVec{\theta}') \ket{\psi_\lambda}|^2.
\end{equation}

The device (\textit{ibmq\_kolkata}, with the same topology as the device in Ref.~\cite{jurcevic2021demonstration}), consists of 27 fixed-frequency transmons with fundamental transition frequencies near 5 GHz and anharmonicities around -340 MHz, coupled by co-planar waveguide bus resonators on a top chip die, bump-bonded to a bottom die for readout and signal delivery. Single qubit gates are implemented via microwave pulses with Gaussian envelop with variance $\sigma$ equal to a quarter of the total pulse length and  with DRAG correction~\cite{motzoi2009simple} and two-qubit gates use the cross-resonance interaction ~\cite{Chow2011} with target rotary pulses ~\cite{Sundaresan2020}. The experiments described in this work take place over four consecutive days, and use a unique calibration for all the gates and readouts. The median qubit lifetime $T_1$ for the entire device is 132 $\mu$s, the median coherence time $T_2^{\mathrm{echo}}$ is 148 $\mu$s, and the median readout error (defined as $[P(0|1) + P(1|0)]/2$, where $P(i|j)$ represents the probability of measuring $i$ when the state is $j$) across all 27 qubits is 96e-3. During idle times of length $T_{\mathrm{idle}}$ for any qubit in our system we use a dynamical decoupling protocol whenever that idle time is longer than twice our single qubit gate. This dynamical decoupling consists of the following sequence: $\tau/2 - X_p - \tau - X_m - \tau/2$ with $\tau = (T_{\mathrm{idle}} - 2T_{X_{p/m}})/2$.
We exploit error mitigation techniques~\cite{temme2017error,li2017efficient, kandala2019error} in our experiments, using stretches $c=1$ and $c=1.3$, from which we apply a first-order Richardson extrapolation to estimate the kernel entries $K_{\lambda} (\myVec{\theta} ,\; \myVec{\theta}')$. For stretch $c=1$ the single qubit gates are 35.6 ns long and the median CNOT length across the device is 476.4 ns, with standard deviation of 134 ns. The cross-resonance pulses have Gaussian turn-on and -off envelopes with a variance of 28.4 ns for all the stretches. The median CNOT error across the device is 7.33e-3 and the median single qubit gate error is 2.97e-4 as measured by randomized benchmarking ~\cite{Magesan2011}.
We run kernel alignment with and without error mitigation, using Simultaneous Perturbation Stochastic Approximation (SPSA) as our optimizer in both cases. We cover 21 trial steps, starting from $\lambda = 0.1$ in both experimental realizations, using the same training set consisting of 10 data points per label. As we can see in Fig.~\ref{FigExp}(a), the cost flattens after around 11 SPSA trials both for the mitigated an unmitigated cases, with a lower cost for the mitigated experiments. The kernel parameter $\lambda$ approaches its ideal value of $\pi/2$ at a faster rate in the mitigated case. Once the training is completed, accuracy tests can be run on different sets of data points. We run these classification tests at each odd SPSA step for the mitigated and the unmitigated protocols, targeting a test set with 50 data points per label. As shown in Fig. ~\ref{FigExp}(b), the mitigated experiments reach high classification success percentages faster than the unmitigated version, although both approaches reach 100\% success after SPSA step 13. We show the training set Gram matrices at odd steps of SPSA for the mitigated and unmitigated approaches in Fig.~\ref{FigExp}(c). Each circuit in all these experiments is sampled 8,192 times.

\section{Outlook and Conclusions}
The class of covariant kernels introduced here can be taken as a blueprint to construct explicit kernels for machine learning problems where the data satisfies a group structure. This construction can directly be applied to many groups, beyond the example considered here, such as the general permutation group $S_n$, the Heisenberg-Weyl group, and dihedral group~\cite{terras1999fourier, jacobson2012basic, Decker2005}, and representations of the braid group~\cite{aharonov2009polynomial, jones1985polynomial}. A particular aspect of the {\it covariant quantum kernel} family is that it already contains an example that has a provable quantum advantage, as given by the DLOG kernel \cite{liu2020rigorous}, over classical learners. However, the fact that it can be applied to more data sets motivates the statement that it is actually a kernel family with great potential for hard learning problems of practical relevance. Interesting examples of learning with group data have already been introduced in the classical literature \cite{kondor2008group}, for example, kernels for permutations that can be used to learn rank~\cite{kondor2010ranking, jiao2015kendall}, and we expect that the covariant kernels will find their application in this area of learning theory. Potentially interesting examples, where one would expect data with group structure to be relevant, are reinforcement learning \cite{jerbiquantum2021}, regression~\cite{vapnik1996regression, smola2004tutorial}, clustering~\cite{benhur2002support, girolami2002mercer}, correlation analysis~\cite{lai2000kernel}, and nonlinear adaptive filtering \cite{liu2011kernel}, which has applications in control theory \cite{bouvrie2010balanced}. A further interesting class of kernels that exploit the group structure present in the data have been discussed in \cite{kondor2008group} where they have been referred to as left- and right-invariant kernels. It would be very interesting to construct group-invariant quantum kernels with a provable quantum advantage.

The core element of the QKE routine reduces to the estimation of an expectation value. If the circuit remains within the coherence limit set by the device error mitigation techniques \cite{temme2017error,li2017efficient, kandala2019error} can be applied. This has been demonstrated in the $27$ qubit experiment presented here. The experiment shows that even for a $27$ qubit experiment the kernel matrix can be significantly improved by using error mitigation techniques. Note, that no readout error mitigation~\cite{bravyi2021mitigating} has been applied here, giving rise to the hope that this is approach is indeed robust against small experimental imperfections. The considered circuit, although supported on a larger number of qubits, is still sufficiently shallow so that classical simulation methods can be applied \cite{bravyi2021classical}. These methods exploit the explicit two-dimensional structure, an advantage that disappears for deeper circuits or other coupling topologies.

The kernel alignment algorithm presented requires the construction of a full kernel matrix for every updated step in the parameters $\myVec{\lambda}$. This is one of the most costly aspects when accounting for the  experimental time. We used a direct application of SPSA \cite{spall1992multivariate} due to its sample robustness to shot noise. However, more advanced algorithms \cite{rafique2018non,thekumparampil2019efficient} to solve min-max optimization problems are available. To make this approach more practically viable for larger alignment problems it is important to explore the application of other optimizers to the kernel alignment in the quantum setting. Methods are known \cite{scholkopf2002sampling} that reduce the overhead of reconstructing the full kernel matrix by subsampling the kernel entries and only evaluating a few kernels at a time. It speed up the alignment in the quantum setting it would be important to understand whether such methods can be applied to QKE without jeopardizing a potential quantum advantage the kernel can offer.

redpajama_set_name: RedPajamaArXiv


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Just play the song and get the lyrics, it is only this simple with Lyrics app. The music player with lyrics is lightweight in size and simple to use. It does implements with your favorite music player without replacing it. Customize the look of the app with light and dark themes.
Get recently viewed lyrics to access the last seen lyrics instantly. The app comes with a safe mode to give the user a clear lyrics option. Quickly launch your favorite player through the app without leaving it. More than a dozen music players come with the support of app to deliver lyrics to every user in the world.
Download: PlayStore
Lyrics Plus
Get lyrics of almost any song with Lyrics Plus app. The app allows you to have instant access to the lyrics of a particular song. Lyrics database of the app is huge and expands with each passing day. The translucency skin of the app makes it a visual treat for the eyes.
We can follow all the artists and catch almost every word of the song. The lyrics search is automatic in the app for the user. You can also save the displayed lyrics to the device and view them on any other iOS running device. Furthermore, you can search manually for the song if you are not satisfied with the automated results of the app.
Download: iTunes
Music Player with Lyrics
Music serves as an all-time remedy whether you are a great day or a bad one. While listening to them, most of us get an overflow of emotions. We all are hooked to a particular song due to its tone and lyrics. There are times when we do forget the lyrics of the playing song and are unable to follow it as it plays on. However, you can avoid this entire situation through a music player with lyrics.
The mobile technology has made it easy for all of us to use these music players with lyrics to satisfy your demand to hum the song lyrics. Brighten your entire day by signing your heart out by seeing the lyrics as the song plays on. Make the song lyrics search easy with these apps allowing you to play that song lyrics.
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I just wrote several thoughtful paragraphs about my experience with something of a creative drought through most of 2006, then lost it with an accidental page-back or page-forward thingy I couldn't recover from in OmniWeb.
OK. Here's the file I was going to post: Sam Hall. The first ukulele song I've recorded, and very nearly the first I've played, in many many months. From the chords on Alligator Boogaloo's uke songbook.
Gotta get to bed. Maybe I'll write up what I can remember tomorrow.
If you're not familiar with my glorious one-take-wonder uke songs from back when I used to post them here, they're not being posted because they're "good" but because I enjoy making them and like to share. Download if you want the experience of sitting in the living room with a not particularly skilled ukulele player playing while he reads chords and lyrics off a web site. Don't download if you don't.
AFTERTHOUGHT: I have had the song "Harris and the Mare" by Stan Rogers in my head tonight. I'd love to play that. Anyone know where I can find chords for it?
Thanks Ed, for "Sam Hall." I never heard of it before but just found a version done by Cash. I'll be working on this.
Hey, please remember that post – it sounds interesting and I know I'd like to read it. Damn yer eyes.
I'll try, jbm, I'll try!


text:
Every Day Sun Rise
'Every Day Sun Rise' is the first song from my second album, which will be released very soon. I am sharing this song with you today (15th October 2019) to celebrate three years since my debut album 'The Pass Of Storms' was released.
Thank you to everyone who has supported me over the years.
'Every Day Sun Rise' is a song of hope and resilience inspired by Greek philosopher, scientist and mathematician, Hypatia of Alexandria. The song features William Fletcher on drums. It was recorded, mixed and mastered at Woolpack Studios in Otley by Richard Sabey. I hope you enjoy it x


Today 118 Swainson's Hawks migrated from Borrego Springs. Tonight a large kettle of over 100 birds descended into the Elis farm on Borrego Valley Rd. Tomorrow with light winds we expect the hawks to first go out to feed before kettling up between 8 and 9:30am. A large number of caterpillars are now in several locations throughout the valley. We do not know where the hawks will feed. I suggest you go to the day count site 2.6 miles north of Palm Canyon Rd on DiGiorgio Rd to find the best location to view the hawks in the air.