Tutorial: Explore quantum entanglement with Q#

Article
11/07/2022

This tutorial shows you how to write a Q# program that manipulates and measures qubits and demonstrates the effects of superposition and entanglement.

Where classical bits hold a single binary value such as a 0 or 1, the state of a qubit can be in a superposition of two quantum states, 0 and 1. Each possible quantum state has an associated probability amplitude.
The act of measuring a qubit produces a binary result, either 0 or 1, with a certain probability, and changes the state of the qubit out of superposition.
Multiple qubits can be entangled such that they cannot be described independently from each other. That is, whatever happens to one qubit in an entangled pair also happens to the other qubit.

In this tutorial, you will prepare two qubits in a specific quantum state, learn how to operate on qubits with Q# to change their state, and demonstrate the effects of superposition and entanglement. You will build your Q# program piece-by-piece to introduce qubit states, operations, and measurements.

Prerequisites

To complete this tutorial, you'll need:

An Azure account with an active subscription. If you don’t have an Azure account, register for free and sign up for a pay-as-you-go subscription.
An Azure Quantum workspace. For more information, see Create an Azure Quantum workspace.

In this tutorial, you'll learn how to

Create Q# operations to measure and initialize a qubit to a desired state.
Create qubits and test your program.
Put a qubit in superposition.
Entangle a pair of qubits.

Create a new Notebook in your workspace

Log in to the Azure portal and select the workspace you created in the previous step.
In the left blade, select Notebooks.
Click My Notebooks and click Add New.
In Kernel Type, select IQ#.
Type a name for the file, for example Entanglement.ipynb, and click Create file.

When your new Notebook opens, it automatically creates the code for the first cell, based on your subscription and workspace information.

%azure.connect "/subscriptions/\<subscription ID>/\<resource group>/providers/Microsoft.Quantum/Workspaces/\<workspace>" \<location>

Note

%azure.connect is an IQ# Magic command, a set of commands that help streamline tasks in Jupyter Notebooks.

If you run this cell, it should authenticate to your subscription and display a list of available providers and their targets.

Initialize a qubit using measurement

The first step is to define a Q# operation that initializes a qubit to a known state. You call this to set a qubit to a classical state, meaning it either returns Zero 100% of the time or returns One 100% of the time. Zero and One are Q# values that represent the only two possible results of a measurement of a qubit.

Click +Code to add a new cell and add the following code:

operation SetQubitState(desired : Result, target : Qubit) : Unit {
    if desired != M(target) {
        X(target);
    }
}

Note

The Microsoft.Quantum.Intrinsic and Microsoft.Quantum.Canon namespaces, which are used by operations in this code, are automatically opened in every cell of an Azure Quantum notebook.

The code example introduces two standard operations, M and X, which transform the state of a qubit.

The SetQubitState operation:

Takes two parameters: a type Result, named desired, that represents the desired state for the qubit to be in (0 or 1), and a type Qubit.
Performs a measurement operation, M, which measures the state of the qubit (Zero or One) and compares the result to the value specified in desired.
If the measurement does not match the compared value, it runs an X operation, which flips the state of the qubit to where the probabilities of a measurement returning Zero and One are reversed. This way, SetQubitState always puts the target qubit in the desired state.

Test the measurement

Next, to demonstrate the effect of the SetQubitState operation, create another operation named TestBellState.

Add another new cell and add the following code:

operation TestBellState(count : Int, initial : Result) : (Int, Int, Int, Int) {
    mutable numOnesQ1 = 0;
    mutable numOnesQ2 = 0;

    // allocate the qubits
    use (q1, q2) = (Qubit(), Qubit());   
    for test in 1..count {
        SetQubitState(initial, q1);
        SetQubitState(Zero, q2);
        
        // measure each qubit
        let resultQ1 = M(q1);            
        let resultQ2 = M(q2);           

        // Count the number of 'Ones':
        if resultQ1 == One {
            set numOnesQ1 += 1;
        }
        if resultQ2 == One {
            set numOnesQ2 += 1;
        }
    }

    // reset the qubits
    SetQubitState(Zero, q1);             
    SetQubitState(Zero, q2);
    

    // Return number of |0> states, number of |1> states
    Message("q1:Zero, One  q2:Zero, One");
    return (count - numOnesQ1, numOnesQ1, count - numOnesQ2, numOnesQ2 );

}

The TestBellStateoperation:

Takes two parameters: count, the number of times to run a measurement, and initial, the desired state to initialize the qubit.
Calls the use statement to initialize two qubits.
Loops for count iterations. For each loop, it
1. Calls SetQubitState to set a specified initial value on the first qubit.
2. Calls SetQubitState again to set the second qubit to a Zero state.
3. Uses the M operation to measure each qubit.
4. Stores the number of measurements for each qubit that return One.
After the loop completes, it calls SetQubitState again to reset the qubits to a known state (Zero) to allow others to allocate the qubits in a known state. This is required by the use statement.
Finally, it uses the Message function to display a message to the console before returning the results.

Test the code

Before moving on to the procedures for superposition and entanglement, test the code up to this point to see the initialization and measurement of the qubits.

To run the TestBellState operation, you use the %simulate magic command to call the Azure Quantum full-state simulator. You need to specify the count and initial arguments, for example, count=1000 and initial=1. This initializes the first qubit to One and measures each qubit 1000 times. Add a new cell with the following command and click Run all:

%simulate TestBellState count=1000 initial=1

and you should observe the following output:

q1:Zero, One  q2:Zero, One
(0, 1000, 1000, 0)

Because the qubits haven't been manipulated yet, they have retained their initial values: the first qubit returns One every time, and the second qubit returns Zero.

If you run the cell again with initial=0, you should observe that the first qubit also returns Zero every time.

%simulate TestBellState count=1000 initial=0

q1:Zero, One q2:Zero, One
(1000, 0, 1000, 0)

Put a qubit in superposition

Currently, the qubits in the program are all in a classical state, that is, they are either 1 or 0. You know this because the program initializes the qubits to a known state, and you haven't added any processes to manipulate them. Before entangling the qubits, you will put the first qubit into a superposition state, where a measurement of the qubit will return Zero 50% of the time and One 50% of the time. Conceptually, the qubit can be thought of as being in a linear combination of all states between the Zero and One.

To put a qubit in superposition, Q# provides the H, or Hadamard, operation. Recall the X operation from the Initialize a qubit using measurement procedure earlier, which flipped a qubit from 0 to 1 (or vice versa); the H operation flips the qubit halfway into a state of equal probabilities of 0 or 1. When measured, a qubit in superposition should return roughly an equal number of Zero and One results.

In the previous cell with the TestBellState, add the H operation inside the for loop:

    for test in 1..count {
        use (q1, q2) = (Qubit(), Qubit());   
        for test in 1..count {
            SetQubitState(initial, q1);
            SetQubitState(Zero, q2);
            
            H(q1);                // Add the H operation after initialization and before measurement

            // measure each qubit
            let resultQ1 = M(q1);            
            let resultQ2 = M(q2);

Initialize the qubit to 1 again in the %simulate command and click Run all, and you can see the results of the first qubit in superposition:

%simulate TestBellState count=1000 initial=1

q1:Zero, One  q2:Zero, One
(523, 477, 1000, 0)      // results will vary

Every time you run the program, the results for the first qubit will vary slightly, but will be close to 50% One and 50% Zero, while the results for the second qubit will remain Zero all the time.

Q1:Zero/One  Q2:Zero/One
(510, 490, 1000, 0)

Initializing the first qubit to Zero returns similar results.

%simulate TestBellState count=1000 initial=0

Q1:Zero/One  Q2:Zero/One
(504, 496, 1000, 0)

Entangle two qubits

As mentioned earlier, entangled qubits are connected such that they cannot be described independently from each other. That is, whatever operation happens to one qubit in an entangled pair, also happens to the other qubit. This allows you to know the resulting state of one qubit without measuring it, just by measuring the state of the other qubit. (This example uses two qubits; however, it is also possible to entangle three or more qubits).

To enable entanglement, Q# provides the CNOT operation, which stands for Controlled-NOT. The result of running this operation on two qubits is to flip the second qubit if the first qubit is One.

Add the CNOT operation to the for loop immediately after the H operation. The TestBellState operation should now look like this:

operation TestBellState(count : Int, initial : Result) : (Int, Int, Int, Int) {
    mutable numOnesQ1 = 0;
    mutable numOnesQ2 = 0;

    // allocate the qubits
    use (q1, q2) = (Qubit(), Qubit());   
    for test in 1..count {
        SetQubitState(initial, q1);
        SetQubitState(Zero, q2);

        H(q1);
        CNOT(q1, q2);                   // added CNOT operation
        
        // measure each qubit
        let resultQ1 = M(q1);            
        let resultQ2 = M(q2);           

        // Count the number of 'Ones':
        if resultQ1 == One {
            set numOnesQ1 += 1;
        }
        if resultQ2 == One {
            set numOnesQ2 += 1;
        }
    }

    // reset the qubits
    SetQubitState(Zero, q1);             
    SetQubitState(Zero, q2);
    

    // Return number of |0> states, number of |1> states
    Message("q1:Zero, One  q2:Zero, One");
    return (count - numOnesQ1, numOnesQ1, count - numOnesQ2, numOnesQ2 );

}

Click Run all to run the updated operation and you should see:

Q1:Zero/One  Q2:Zero/One
(502, 498, 502, 498)      // actual results will vary

The statistics for the first qubit haven't changed (a 50/50 chance of a Zero or a One after measurement), but the measurement results for the second qubit are always the same as the measurement of the first qubit, regardless of what the qubit is initialized to. The CNOT operation has entangled the two qubits, so that whatever happens to one of them, happens to the other.

Prerequisites

Install the Quantum Development Kit (QDK) using your preferred language and development environment.
If you already have the QDK installed, make sure you have updated to the latest version.
Create a Q# project named Bell for either a Q# application or a C# host program. Alternatively, you can run your Q# code directly in Juptyer Notebooks or from a Python host program.

In this tutorial, you'll learn how to

Create Q# operations to measure and initialize a qubit to a desired state.
Create qubits and test your program.
Put a qubit in superposition.
Entangle a pair of qubits.

Initialize a qubit using measurement

The first step is to define a Q# operation that will initialize a qubit to a known state. This can be called to set a qubit to a classical state, meaning it either returns Zero 100% of the time or returns One 100% of the time. Zero and One are Q# values that represent the only two possible results of a measurement of a qubit.

In your project, replace the contents of Program.qs with the following code:

   namespace Bell {
       open Microsoft.Quantum.Intrinsic;
       open Microsoft.Quantum.Canon;

       operation SetQubitState(desired : Result, target : Qubit) : Unit {
           if desired != M(target) {
               X(target);
           }
       }
   }

The code example introduces two standard operations, M and X, which transform the state of a qubit.

The SetQubitState operation:

Takes two parameters: a type Result, named desired, that represents the desired state for the qubit to be in (0 or 1), and a type Qubit.
Performs a measurement operation, M, which measures the state of the qubit (Zero or One) and compares the result to the value specified in desired.
If the measurement does not match the compared value, it runs an X operation, which flips the state of the qubit to where the probabilities of a measurement returning Zero and One are reversed. This way, SetQubitState always puts the target qubit in the desired state.

Test the measurement

Next, to demonstrate the effect of the SetQubitState operation, create another operation named TestBellState.

Add the following operation to your Program.qs file after the SetQubitState operation:

operation TestBellState(count : Int, initial : Result) : (Int, Int, Int, Int) {
    mutable numOnesQ1 = 0;
    mutable numOnesQ2 = 0;

    // allocate the qubits
    use (q1, q2) = (Qubit(), Qubit());   
    for test in 1..count {
        SetQubitState(initial, q1);
        SetQubitState(Zero, q2);
        
        // measure each qubit
        let resultQ1 = M(q1);            
        let resultQ2 = M(q2);           

        // Count the number of 'Ones' we saw:
        if resultQ1 == One {
            set numOnesQ1 += 1;
        }
        if resultQ2 == One {
            set numOnesQ2 += 1;
        }
    }

    // reset the qubits
    SetQubitState(Zero, q1);             
    SetQubitState(Zero, q2);
    

    // Return times we saw |0>, times we saw |1>
    Message("q1:Zero, One  q2:Zero, One");
    return (count - numOnesQ1, numOnesQ1, count - numOnesQ2, numOnesQ2 );

}

The TestBellStateoperation:

Takes two parameters: count, the number of times to run a measurement, and initial, the desired state to initialize the qubit.
Calls the use statement to initialize two qubits.
Loops for count iterations. For each loop, it
1. Calls SetQubitState to set a specified initial value on the first qubit.
2. Calls SetQubitState again to set the second qubit to a Zero state.
3. Uses the M operation to measure each qubit.
4. Stores the number of measurements for each qubit that return One.
After the loop completes, it calls SetQubitState again to reset the qubits to a known state (Zero) to allow others to allocate the qubits in a known state. This is required by the use statement.
Finally, it uses the Message function to print a message to the console before returning the results.

Run the code from the command prompt

Before moving on to the procedures for superposition and entanglement, test the code up to this point to see the initialization and measurement of the qubits.

In order to run the code as a standalone program, the Q# compiler needs to know where to start the program when you run the dotnet run command. This is done in the Q# file by adding an @EntryPoint() directly preceding the operation that you want to run: the TestBellState operation in this case.

Note

@EntryPoint() is only required for standalone Q# programs. When running a Q# program in Jupyter Notebooks, or calling a Q# program from a Python or .NET host file, it is not required and will throw an error if included.

Your program.qs file should now look like this:

namespace Bell {
    open Microsoft.Quantum.Intrinsic;
    open Microsoft.Quantum.Canon;

       operation SetQubitState(desired : Result, target : Qubit) : Unit {
           if desired != M(target) {
               X(target);
           }
       }

    @EntryPoint()
    operation TestBellState(count : Int, initial : Result) : (Int, Int, Int, Int) {
        mutable numOnesQ1 = 0;
        mutable numOnesQ2 = 0;

        // allocate the qubits
        use (q1, q2) = (Qubit(), Qubit());   
        for test in 1..count {
            SetQubitState(initial, q1);
            SetQubitState(Zero, q2);
            
            // measure each qubit
            let resultQ1 = M(q1);            
            let resultQ2 = M(q2);           
    
            // Count the number of 'Ones' we saw:
            if resultQ1 == One {
                set numOnesQ1 += 1;
            }
            if resultQ2 == One {
                set numOnesQ2 += 1;
            }
        }
    
        // reset the qubits
        SetQubitState(Zero, q1);             
        SetQubitState(Zero, q2);
        
    
        // Return times we saw |0>, times we saw |1>
        Message("q1:Zero, One  q2:Zero, One");
        return (count - numOnesQ1, numOnesQ1, count - numOnesQ2, numOnesQ2 );

    }
}

To run the program, you need to specify the count and initial arguments from the command prompt. For example, --count 1000 and --initial One will initialize the first qubit to One and measure each qubit 1000 times. Run the following command:

dotnet run --count 1000 --initial One

and you should observe the following output:

Q1:Zero/One  Q2:Zero/One
(0, 1000, 1000, 0)

Because the qubits haven't been manipulated yet, they have retained their initial values: the first qubit returns One every time, and the second qubit returns Zero.

If you run it with --initial Zero, you should observe that the first qubit also returns Zero every time.

dotnet run --count 1000 --initial Zero

Q1:Zero/One  Q2:Zero/One
(1000, 0, 1000, 0)

Put a qubit in superposition

Currently, the qubits in the program are all in a classical state, that is, they are either 1 or 0. You know this because the program initializes the qubits to a known state, and you haven't added any processes to manipulate them. Before entangling the qubits, you will put the first qubit into a superposition state, where a measurement of the qubit will return Zero 50% of the time and One 50% of the time. Conceptually, the qubit can be thought of as halfway between the Zero and One.

Modify the code in the TestBellState operation to include the H operation:

    for test in 1..count {
        use (q1, q2) = (Qubit(), Qubit());   
        for test in 1..count {
            SetQubitState(initial, q1);
            SetQubitState(Zero, q2);
            
            H(q1);                // Add the H operation after initialization and before measurement

            // measure each qubit
            let resultQ1 = M(q1);            
            let resultQ2 = M(q2);

Now when you run the program, you can see the results of the first qubit in superposition:

dotnet run --count 1000 --initial One

Q1:Zero/One  Q2:Zero/One
(523, 477, 1000, 0)      // results will vary

dotnet run --count 1000 --initial One

Q1:Zero/One  Q2:Zero/One
(510, 490, 1000, 0)

Initializing the first qubit to Zero returns similar results.

dotnet run --count 1000 --initial Zero

Q1:Zero/One  Q2:Zero/One
(504, 496, 1000, 0)

Entangle two qubits

As mentioned earlier, entangled qubits are connected such that they cannot be described independently from each other. That is, whatever operation happens to one qubit, also happens to the entangled qubit. This allows you to know the resulting state of one qubit without measuring it, just by measuring the state of the other qubit. (This example uses two qubits; however, it is also possible to entangle three or more qubits).

Add the CNOT operation to your program immediately after the H operation. Your full program should look like this:

namespace Bell {
    open Microsoft.Quantum.Intrinsic;
    open Microsoft.Quantum.Canon;

       operation SetQubitState(desired : Result, target : Qubit) : Unit {
           if desired != M(target) {
               X(target);
           }
       }

    @EntryPoint()
    operation TestBellState(count : Int, initial : Result) : (Int, Int, Int, Int) {
        mutable numOnesQ1 = 0;
        mutable numOnesQ2 = 0;

        // allocate the qubits
        use (q1, q2) = (Qubit(), Qubit());   
        for test in 1..count {
            SetQubitState(initial, q1);
            SetQubitState(Zero, q2);
        
            H(q1);            
            CNOT(q1, q2);      // Add the CNOT operation after the H operation

            // measure each qubit
            let resultQ1 = M(q1);            
            let resultQ2 = M(q2);           
    
            // Count the number of 'Ones' we saw:
            if resultQ1 == One {
                set numOnesQ1 += 1;
            }
            if resultQ2 == One {
                set numOnesQ2 += 1;
            }
        }
    
        // reset the qubits
        SetQubitState(Zero, q1);             
        SetQubitState(Zero, q2);
        
    
        // Return times we saw |0>, times we saw |1>
        Message("q1:Zero, One  q2:Zero, One");
        return (count - numOnesQ1, numOnesQ1, count - numOnesQ2, numOnesQ2 );

    }
}

Now when you run the program:

dotnet run --count 1000 --initial One

Q1:Zero/One  Q2:Zero/One
(502, 498, 502, 498)

The statistics for the first qubit haven't changed (a 50/50 chance of a Zero or a One after measurement), but the measurement results for the second qubit are always the same as the measurement of the first qubit. The CNOT operation has entangled the two qubits, so that whatever happens to one of them, happens to the other.

Next steps

Continue to explore other quantum algorithms and techniques:

The tutorial Implement Grover’s search algorithm shows how to write a Q# program that uses Grover's search algorithm to solve a graph coloring problem.
The tutorial Write and simulate qubit-level programs in Q# explores how to write a Q# program that directly addresses specific qubits.
The Quantum Katas are Jupyter Notebook-based, self-paced tutorials and programming exercises aimed at teaching the elements of quantum computing and Q# programming at the same time.

Tutorial: Explore quantum entanglement with Q#

Prerequisites

Create a new Notebook in your workspace

Initialize a qubit using measurement

Test the measurement

Test the code

Put a qubit in superposition

Entangle two qubits

Prerequisites

Initialize a qubit using measurement

Test the measurement

Run the code from the command prompt

Put a qubit in superposition

Entangle two qubits

Next steps

Feedback

Additional resources

Additional resources