In this article, we will be looking into qubit rotation or rotating a |0&gt; state to |1&gt; state using QML optimization where we optimize the parameters of the Rotation operators Rx and Ry in order to achieve the state |1&gt;.We will be using Blueqat SDK on <a href="https://quantumcomputing.com/" rel="noopener noreferrer" target="_blank">quantumcomputing.com</a> (Strangeworks) which is a cloud based quantum computing service.<ul><li>You can start by creating an account on quantumcomputing.com by going to the given link : <a href="https://quantumcomputing.com/beta" rel="noopener noreferrer" target="_blank">https://quantumcomputing.com/beta</a>.</li><li>Next, go to <a href="https://app.quantumcomputing.com/" rel="noopener noreferrer" target="_blank">https://app.quantumcomputing.com/</a> and create a new project there (Empty Python Project).</li></ul>You can also work out this using Blueqat SDK on the local machine. You can install blueqat using python with the pip command :<pre class="ql-syntax" spellcheck="false">pip3 install blueqat
</pre>The task is to rotate |0&gt; state to |1&gt; state. We will be using Rx and Ry as ansatz to explore the state space on the Bloch Sphere.To get started we first import the libraries needed for the program : <pre class="ql-syntax" spellcheck="false">import numpy as np
from blueqat import Circuit
from scipy.optimize import minimize
from strangeworks.blueqat import StrangeworksBackend
</pre> Next, we will be writing the function for the state preparation which is done by using Rx and Ry rotation operators as ansatz. <pre class="ql-syntax" spellcheck="false">def state_prep(state,params):
 state.rx(params[0])[0]
 state.ry(params[1])[0]
 return state
</pre> We use the criterion of expectation value in order to rotate the qubit to |1&gt; state. Thus we take the expectation value as the cost function which needs to be minimized in order to achieve the state |1&gt;. For this, we measure the expectation value of σᴢ operator or do the measurement in the computational basis. We know that the expectation value of σᴢ operator lies in [-1,1] with the minimum expectation value for the state |1&gt; i.e -1.Thus, we try to optimize the parameters of both the ansatz to achieve the minimum expectation value i.e -1 in order to achieve the rotated state. The function for the cost value or the expectation value is: <img src="https://assets.blueqat.com/public/uploads/us-east-2:ced6d487-7663-460c-a3d4-69d55e5600ab/2021/01/11/Screenshot 2021-01-11 at 11.07.39 AM.png"/><pre class="ql-syntax" spellcheck="false">def exp(params):
 state=Circuit()
 state_prep(state,params)
 state.m[:]
 shots=1000
 b=state.run(shots=shots)
 exp_value=(b[&#39;0&#39;]-b[&#39;1&#39;])/shots
 return exp_value
</pre> Next we try to optimize the above parameters using the Scipy.optimize minimum function with initial parameters as [0.1,0.2] and the learning rate being 0.05. <pre class="ql-syntax" spellcheck="false">init_params=np.array([0.1,0.2])
result=minimize(exp,init_params, method=&#34;Powell&#34; , tol=0.05)
print(f&#39;The expectation value after optimization is {result.fun}&#39;)
</pre> Thus the results for the states and params on each iteration are:<pre class="ql-syntax" spellcheck="false">Counter({&#39;0&#39;: 983, &#39;1&#39;: 17})[0.1, 0.2]
Counter({&#39;0&#39;: 992, &#39;1&#39;: 8})[0.1, 0.2]
Counter({&#39;0&#39;: 730, &#39;1&#39;: 270})[1.1, 0.2]
Counter({&#39;1&#39;: 960, &#39;0&#39;: 40})[2.718034, 0.2 ]
Counter({&#39;0&#39;: 790, &#39;1&#39;: 210})[5.33606803, 0.2 ]
Counter({&#39;1&#39;: 949, &#39;0&#39;: 51})[2.718034, 0.2 ]
Counter({&#39;1&#39;: 933, &#39;0&#39;: 67})[2.718034, 0.2 ]
Counter({&#39;1&#39;: 655, &#39;0&#39;: 345})[2.718034, 1.2 ]
Counter({&#39;1&#39;: 553, &#39;0&#39;: 447})[ 2.718034, -1.418034]
Counter({&#39;1&#39;: 954, &#39;0&#39;: 46})[2.718034, 0.2 ]
Counter({&#39;1&#39;: 925, &#39;0&#39;: 75})[ 2.718034, -0.41803397]
Counter({&#39;1&#39;: 890, &#39;0&#39;: 110})[2.718034, 0.581966]
Counter({&#39;1&#39;: 968, &#39;0&#39;: 32})[ 2.71803400e+00, -8.22658533e-04]
Counter({&#39;0&#39;: 788, &#39;1&#39;: 212})[ 5.336068, -0.20164532]
Counter({&#39;1&#39;: 949, &#39;0&#39;: 51})[ 2.71803400e+00, -8.22658533e-04]
Counter({&#39;0&#39;: 783, &#39;1&#39;: 217})[ 5.336068, -0.20164532]
Counter({&#39;0&#39;: 507, &#39;1&#39;: 493})[-1.51803403, 0.32411523]
Counter({&#39;1&#39;: 957, &#39;0&#39;: 43})[ 2.71803400e+00, -8.22658533e-04]
Counter({&#39;0&#39;: 706, &#39;1&#39;: 294})[1.10000004, 0.12329257]
Counter({&#39;1&#39;: 915, &#39;0&#39;: 85})[ 3.71803397, -0.07753009]
Counter({&#39;1&#39;: 999, &#39;0&#39;: 1})[ 3.11343772 -0.03115306]
Counter({&#39;1&#39;: 998, &#39;0&#39;: 2})[ 3.11343772 -0.03115306]
Counter({&#39;1&#39;: 778, &#39;0&#39;: 222})[3.11343772 0.96884694]
Counter({&#39;0&#39;: 531, &#39;1&#39;: 469})[ 3.11343772 -1.64918706]
Counter({&#39;1&#39;: 1000})[ 3.11343772, -0.03115306]
Counter({&#39;1&#39;: 918, &#39;0&#39;: 82})[ 3.11343772, -0.64918704]
Counter({&#39;1&#39;: 975, &#39;0&#39;: 25})[3.11343772 0.35081294]
Counter({&#39;1&#39;: 1000})[ 3.11343772, -0.01112575]
Counter({&#39;1&#39;: 997, &#39;0&#39;: 3})[3.11343772, 0.0890108 ]
Counter({&#39;1&#39;: 1000})[ 3.11343772, -0.01112575]
Counter({&#39;1&#39;: 955, &#39;0&#39;: 45})[ 3.50884144, -0.04145615]
Counter({&#39;1&#39;: 892, &#39;0&#39;: 108})[2.47366106, 0.03794987]
Counter({&#39;1&#39;: 999, &#39;0&#39;: 1})[ 3.11343772, -0.01112575]
Counter({&#39;1&#39;: 978, &#39;0&#39;: 22})[2.86906479, 0.00761947]
Counter({&#39;1&#39;: 999, &#39;0&#39;: 1})[ 3.2644685, -0.02271093]
Counter({&#39;1&#39;: 999, &#39;0&#39;: 1})[ 3.16360143, -0.01497368]
The expectation value after optimization is -1.
</pre><img src="https://assets.blueqat.com/public/uploads/us-east-2:ced6d487-7663-460c-a3d4-69d55e5600ab/2021/01/11/Screenshot 2021-01-11 at 11.10.55 AM.png"/>We can see the state transitioning from |0&gt; to |1&gt; on the histogram in the above diagrams on quantumcomputing.com (Strangeworks) and also see the change in parameters in the circuit diagram above.Thus we see the rotation of a qubit from |0&gt; to |1&gt; using expectation value as the cost function and optimizing the parameters for the ansatz.

Rotating a qubit using QML with Blueqat

Gaurav Singh