Folded QCNN state on Quantum Machine Learning using the state vector specification feature of blueqat 0.4.1

Hello everyone, I know you all want to do quantum machine learning and be on the cutting edge of quantum computing. In this article, we'll check how to make quantum machine learning more efficient using a technique called mid-circuit measurement and qubit reuse & we'll try to reduce qubits using a new feature of blueqat.

MCMR, for short, is a technique that involves taking a measurement of only a specific qubit in the middle of a circuit, and then resetting the circuit after the measurement of that qubit. By taking a measurement on a particular qubit and initializing it to $\lvert 0\rangle$, you can sample a more complex quantum state than the state vector alone for a number of qubits.

In this example, we will try to reduce a three-qubit circuit to two qubits. Let's look at the following structure.

|0> --*-----
      |
|0> --U--*--
         |
|0> -----U--

Let's look at the top qubit and the bottom qubit. The circuit will be the same only if the bottom qubit is initialized after the top qubit is measured. If we turn the bottom circuit upside down and rewrite it using MCMR, we can do the following

|0> --*--  U--
      |    |
|0> --U----*--

In mid-circuit measurement and qubit reuse (MCMR), the order of measurement and initialization of qubits is devised in such a way that complex calculations can be performed with fewer qubits. Currently, MCMR is implemented on Honeywell machines and on IBM machines.

To understand how it works, let's look at the transition of the state vector, and check out the circuit of two qubits.

|0> --H--*--
         |
|0> -----X--

This is a quantum entanglement circuit. The state vector is

You can use rxx to write it in one step.

|0> ---*----
       |
|0> ---RXX--

Thus, we have an entangled state of $\frac{\lvert 00\rangle +\lvert 11\rangle}{\sqrt{2}}$. If we measure only the 0th qubit in this state, we get

|0> ---*----測定
       |
|0> ---RXX---------

If $q_0 = 0$ is measured, then $\lvert 00\rangle$ is determined and $\lvert \psi \rangle = \lvert 00\rangle$. The state vector in that case is

If $q_0 = 1$ is measured, then $\lvert 11\rangle$ is confirmed and $\lvert \psi \rangle = \lvert 11\rangle$.

The result is $\lvert 11 \rangle$. Now, if we initialize the 0th qubit to $\lvert 0\rangle$, then $\lvert 11\rangle$ becomes $\lvert 01\rangle$.

In other words, the state vectors are initialized to different quantum states, with the state vectors diverging depending on the measured qubit.

|0> ---*---- |0>--
       |
|0> ---RXX--------

Finally, when we measure the above circuit again, the first time we get $q_0 = 0$, the second time we get $00$, so we get a sample of $000$.

Also, the first time $q_0 = 1$, the second and subsequent measurements will give $01$, giving a sample of $101$. In blueqat, the value of the 0th qubit is on the right.

The result of the first measurement changes the result for subsequent sampling, and by using entanglement many times, the state vector of the quantum state will diverge with each intermediate measurement and reset.

To apply these to quantum machine learning, we can reuse these for convolution and other applications of quantum machine learning. For example, suppose we have a convolutional circuit such as QCNN for image recognition.

|0> --*--
      |
|0> --U--*--
         |
|0> --U--U----
      |  
|0> --*--

This means that the $q_0$ circuit is collapsed into $q_1$, $q_3$ is collapsed into $q_2$, and $q_1$ is collapsed into $q_2$ again. It is a four-qubit circuit, but using MCMR

|0>--U------------*--
     |            |
|0>--*--  |0>--U--U----
               |
|0>------------*--

Using the MPS circuit, the number o qubits can be further reduced. In this way, quantum machine learning can be efficiently used on Honeywell and IBM machines by efficiently using MCMR in quantum circuits that approximate quantum states, such as MPS and TTN/MERA.