Tutorial: Explore entanglement with Q#

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

You will write an application called Bell to demonstrate quantum entanglement. The name Bell is in reference to Bell states, which are specific quantum states of two qubits that are used to represent the simplest examples of superposition and quantum entanglement.

Pre-requisites

You can also follow along with the narrative without installing the QDK, reviewing the overviews of the Q# programming language and the first concepts of quantum computing.

In this tutorial, you'll learn how to:

  • Create and combine operations in Q#.
  • Create operations to put qubits in superposition, entangle and measure them.
  • Demonstrate quantum entanglement with a Q# program run in a simulator.

Demonstrating qubit behavior with the QDK

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 0 and 1. Conceptually, the state of a qubit can be thought of as a direction in an abstract space (also known as a vector). A qubit state can be in any of the possible directions. The two classical states are the two directions; representing 100% chance of measuring 0 and 100% chance of measuring 1.

The act of measurement produces a binary result and changes a qubit state. Measurement produces a binary value, either 0 or 1. The qubit goes from being in superposition (any direction) to one of the classical states. Thereafter, repeating the same measurement without any intervening operations produces the same binary result.

Multiple qubits can be entangled. When we make a measurement of one entangled qubit, our knowledge of the state of the other(s) is updated as well.

Write the Q# application

The goal is to prepare two qubits in a specific quantum state, demonstrating how to operate on qubits with Q# to change their state and demonstrate the effects of superposition and entanglement. You will build this up piece by piece to introduce qubit states, operations, and measurement.

Initialize qubit using measurement

The code snippet below shows how to work with qubits in Q#. The code introduces two operations, M and X that transform the state of a qubit.

An operation SetQubitState is defined that takes as a parameter a qubit and another parameter, desired, representing the desired state for the qubit to be in. The operation SetQubitState performs a measurement on the qubit using the operation M. In Q#, a qubit measurement always returns either Zero or One. If the measurement returns a value not equal to the desired value, SetQubitState “flips” the qubit; that is, it runs an X operation, which changes the qubit state to a new state in which the probabilities of a measurement returning Zero and One are reversed. This way, SetQubitState always puts the target qubit in the desired state.

Replace the contents of Program.qs with the following code:

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

       operation SetQubitState(desired : Result, q1 : Qubit) : Unit {
           if desired != M(q1) {
               X(q1);
           }
       }
   }

This operation may now be called to set a qubit to a classical state, either returning Zero 100% of the time or returning One 100% of the time. Zero and One are constants that represent the only two possible results of a measurement of a qubit.

The operation SetQubitState measures the qubit. If the qubit is in the desired state, SetQubitState leaves it alone; otherwise, by running the X operation, the qubit state changes to the desired state.

About Q# operations

A Q# operation is a quantum subroutine. That is, it is a callable routine that contains calls to other quantum operations.

The arguments to an operation are specified as a tuple, within parentheses.

The return type of the operation is specified after a colon. In this case, the SetQubitState operation has no return type, so it is marked as returning Unit. This is the Q# equivalent of unit in F#, which is roughly analogous to void in C#, and an empty tuple in Python ((), represented by the type hint Tuple[()]).

You have used two quantum operations in your first Q# operation:

  • The M operation, which measures the state of the qubit
  • The X operation, which flips the state of a qubit

A quantum operation transforms the state of a qubit. Sometime people talk about quantum gates instead of operations, in analogy to classical logic gates. This is rooted in the early days of quantum computing when algorithms were merely a theoretical construct and visualized as diagrams similarly to circuit diagrams in classical computing.

Counting measurement outcomes

To demonstrate the effect of the SetQubitState operation, a TestBellState operation is then added. This operation takes as input a Zero or One, and calls the SetQubitState operation some number of times with that input, and counts the number of times that Zero was returned from the measurement of the qubit and the number of times that One was returned. Of course, in this first simulation of the TestBellState operation, one expects that the output will show that all measurements of the qubit set with Zero as the parameter input will return Zero, and all measurements of a qubit set with One as the parameter input will return One. Further on, code will be added to TestBellState to demonstrate superposition and entanglement.

Add the following operation to the Program.qs file, inside the namespace, after the end of the SetQubitState operation:

   operation TestBellState(count : Int, initial : Result) : (Int, Int) {

       mutable numOnes = 0;
       use qubit = Qubit();
       for test in 1..count {
           SetQubitState(initial, qubit);
           let res = M(qubit);

           // Count the number of ones we saw:
           if res == One {       
               set numOnes += 1;
           }
       }

       SetQubitState(Zero, qubit);

       

       // Return number of times we saw a |0> and number of times we saw a |1>
       Message("Test results (# of 0s, # of 1s): ");
       return (count - numOnes, numOnes);
   }

Note that there is a line before the return to print an explanatory message in the console with the function (Message)[microsoft.quantum.intrinsic.message]

This operation (TestBellState) will loop for count iterations, set a specified initial value on a qubit and then measure (M) the result. It will gather statistics on how many zeros and ones we've measured and return them to the caller. It performs one other necessary operation. It resets the qubit to a known state (Zero) before returning it allowing others to allocate this qubit in a known state. This is required by the use statement.

About variables in Q#

By default, variables in Q# are immutable; their value may not be changed after they are bound. The let keyword is used to indicate the binding of an immutable variable. Operation arguments are always immutable.

If you need a variable whose value can change, such as numOnes in the example, you can declare the variable with the mutable keyword. A mutable variable's value may be changed using a set statement.

In both cases, the type of a variable is inferred by the compiler. Q# doesn't require any type annotations for variables.

About use statements in Q#

The use statement is also special to Q#. It is used to allocate qubits for use in a block of code. In Q#, all qubits are dynamically allocated and released, rather than being fixed resources that are there for the entire lifetime of a complex algorithm. A use statement allocates a set of qubits at the start, and releases those qubits at the end of the block.

Run the code from the command prompt

In order to run the code we need to tell the compiler which callable to run when we provide the dotnet run command. This is done with a simple change in the Q# file by adding a line with @EntryPoint() directly preceding the callable: the TestBellState operation in this case. The full code should be:

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

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

    @EntryPoint()
    operation TestBellState(count : Int, initial : Result) : (Int, Int) {

        mutable numOnes = 0;
        use qubit = Qubit();
        for test in 1..count {
            SetQubitState(initial, qubit);
            let res = M(qubit);
            

            // Count the number of ones we saw:
            if res == One {
                  set numOnes += 1;
            }
        }

        SetQubitState(Zero, qubit);
        

    // Return number of times we saw a |0> and number of times we saw a |1>
    Message("Test results (# of 0s, # of 1s): ");
    return (count - numOnes, numOnes);
    }
}

To run the program we need to specify count and initial arguments from the command prompt. Let's choose for example count = 1000 and initial = One. Enter the following command:

dotnet run --count 1000 --initial One

And you should observe the following output:

Test results (# of 0s, # of 1s):
(0, 1000)

If you try with initial = Zero you should observe:

dotnet run --count 1000 --initial Zero
Test results (# of 0s, # of 1s):
(1000, 0)

Prepare superposition

Now let’s look at how Q# expresses ways to put qubits in superposition. Recall that the state of a qubit can be in a superposition of 0 and 1. This can be accomplish by using Hadamard operation. If the qubit is in either of the classical states (where a measurement returns Zero always or One always), then the Hadamard or H operation will put the qubit in a state where a measurement of the qubit will return Zero 50% of the time and return One 50% of the time. Conceptually, the qubit can be thought of as halfway between the Zero and One. Now, when in the simulation of the TestBellState operation, the results will return roughly an equal number of Zero and One after measurement.

X flips qubit state

First, let's just try to flip the qubit (if the qubit is in Zero state it will flip to One and vice versa). This is accomplished by performing an X operation before we measure it in TestBellState:

X(qubit);
let res = M(qubit);

Now the results are reversed:

dotnet run --count 1000 --initial One
Test results (# of 0s, # of 1s):
(1000, 0)
dotnet run --count 1000 --initial Zero
Test results (# of 0s, # of 1s):
(0, 1000)

Now let's explore the quantum properties of the qubits.

H prepares superposition

All you need to do is replace the X operation in the previous run with an H or Hadamard operation. Instead of flipping the qubit all the way from 0 to 1, this will only flip it halfway. The replaced lines in TestBellState now look like:

H(qubit);
let res = M(qubit);

Now the results get more interesting:

dotnet run --count 1000 --initial One
Test results (# of 0s, # of 1s):
(496, 504)
dotnet run --count 1000 --initial Zero
Test results (# of 0s, # of 1s):
(506, 494)

Every measures asks for a classical value, but the qubit is halfway between 0 and 1, so we get (statistically) 0 half the time and 1 half the time. This is known as superposition and gives us our first real view into a quantum state.

Prepare entanglement

Now let’s look at how Q# expresses ways to entangle qubits. First, set the first qubit to the initial state and then use the H operation to put it into superposition. Then, before measuring the first qubit, use a new operation (CNOT), 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. Now, the two qubits are entangled. The statistics for the first qubit haven't changed (50-50 chance of a Zero or a One after measurement), but now when measured the second qubit, it is always the same as what we measured for the first qubit. Our CNOT has entangled the two qubits, so that whatever happens to one of them, happens to the other. If you reversed the measurements (did the second qubit before the first), the same thing would happen. The first measurement would be random and the second would be in lock step with whatever was discovered for the first.

To prepare entanglement, first allocate two qubits instead of one in TestBellState:

use (q0, q1) = (Qubit(), Qubit());

This will allow us to add a new operation (CNOT) before measuring (M) in TestBellState:

SetQubitState(initial, q0);
SetQubitState(Zero, q1);

H(q0);
CNOT(q0, q1);
let res = M(q0);

Add another SetQubitState operation to initialize the first qubit to make sure that it's always in the Zero state.

And finally, reset the second qubit before releasing it.

SetQubitState(Zero, q0);
SetQubitState(Zero, q1);

The full routine now looks like this:

    operation TestBellState(count : Int, initial : Result) : (Int, Int) {

        mutable numOnes = 0;
        use (q0, q1) = (Qubit(), Qubit());
        for test in 1..count {
            SetQubitState(initial, q0);
            SetQubitState(Zero, q1);

            H(q0);
            CNOT(q0,q1);
            let res = M(q0);

            // Count the number of ones we saw:
            if res == One {
                set numOnes += 1;
            }
        }

        SetQubitState(Zero, q0);
        SetQubitState(Zero, q1);
        

        // Return number of times we saw a |0> and number of times we saw a |1>
        return (count-numOnes, numOnes);
    }

If you run this, you will get exactly the same 50-50 result we got before. However, what the interesting part is how the second qubit reacts to the first being measured. This statistic is added with a new version of the TestBellState operation:

    operation TestBellState(count : Int, initial : Result) : (Int, Int, Int) {
        mutable numOnes = 0;
        mutable agree = 0;
        use (q0, q1) = (Qubit(), Qubit());
        for test in 1..count {
            SetQubitState(initial, q0);
            SetQubitState(Zero, q1);

            H(q0);
            CNOT(q0, q1);
            let res = M(q0);

            if M(q1) == res {
                set agree += 1;
            }

            // Count the number of ones we saw:
            if res == One {
                set numOnes += 1;
            }
        }

        SetQubitState(Zero, q0);
        SetQubitState(Zero, q1);
        

        // Return times we saw |0>, times we saw |1>, and times measurements agreed
        Message("Test results (# of 0s, # of 1s, # of agreements)");
        return (count-numOnes, numOnes, agree);
    }

The new return value (agree) keeps track of every time the measurement from the first qubit matches the measurement of the second qubit.

Running the code results in:

dotnet run --count 1000 --initial One
(505, 495, 1000)
dotnet run --count 1000 --initial Zero
Test results (# of 0s, # of 1s, # of agreements)
(507, 493, 1000)

As stated in the overview, the statistics for the first qubit haven't changed (50-50 chance of a 0 or a 1), but now when measuring the second qubit, it is always the same as what for the first qubit, because they are entangled!

Next steps

The tutorial Grover’s search shows you how to build and run Grover search, one of the most popular quantum computing algorithms and offers a nice example of a Q# program that can be used to solve real problems with quantum computing.

Set up Azure Quantum recommends more ways to learn Q# and quantum programming.