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Variational Quantum Classifier: Parity
This demonstration is a translation of the tutorial “” in PennyLane by Maria Schuld. It demonstrates variational quantum classifiers; that is, quantum circuits that can be trained from labelled data to classify new data samples.
In particular, we will see that the variational quantum classifier can reproduce the parity function , where if there are even number of ones in and otherwise. This optimization example demonstrates how to encode binary inputs into the initial state of the variational circuit, which is simply a computational basis state.
f:→{1,-1}
n
{0,1}
f(x)=1
x
-1
Encodes data into amplitudes of a quantum state | |
Find the minimum of a function |
Frequently used function in variational classification.
Make sure to load the Q3 package.
In[10]:=
Needs["QuantumMob`Q3`"]
Setup
The qubits to be modeled are S[,$] for .
i
i=1,…,n
In[5]:=
Let[Qubit,S]
In[9]:=
$n=4;$N=Power[2,$n];kk=Range[$n];SS=S[kk,$]
Out[12]=
{,,,}
S
1
S
2
S
3
S
4
Data
Load and preprocess the data.
In[17]:=
bits=Tuples[{0,1},$n];parity=2*Mod[Count[#,1]&/@bits,2]-1;data=AssociationThread[bitsparity];Normal[data]//TableForm
Out[20]//TableForm=
{0,0,0,0}-1 |
{0,0,0,1}1 |
{0,0,1,0}1 |
{0,0,1,1}-1 |
{0,1,0,0}1 |
{0,1,0,1}-1 |
{0,1,1,0}-1 |
{0,1,1,1}1 |
{1,0,0,0}1 |
{1,0,0,1}-1 |
{1,0,1,0}-1 |
{1,0,1,1}1 |
{1,1,0,0}-1 |
{1,1,0,1}1 |
{1,1,1,0}1 |
{1,1,1,1}-1 |
Embed the input data to quantum states.
In[47]:=
$in=BasisEmbedding[SS]/@Keys[data];$in=Matrix/@$in;MatrixForm/@$in
Out[49]=
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Quantum Circuit Layer
The parameters to be optimized are a[] for and .
layer,i,j
i=1,…,n
j=1,…,3
In[50]:=
Let[Real,a]
Define a quantum circuit layer.
In[51]:=
layer[aa_?MatrixQ,ss:{__?QubitQ}]:=QuantumCircuit[Table[EulerRotation[aa〚k〛,ss〚k〛],{k,kk}],Apply[Sequence,CNOT@@@Transpose@{ss,RotateLeft@ss}]]/;Length[aa]Length[ss]layer[aa_?MatrixQ]:=layer[aa,SS]layer[n_Integer]:=QuantumCircuit@@Table[layer[Array[a[k],{$n,3}]],{k,n}]
For example, the following quantum circuit contains 24 real parameters, with three parameters associated with each qubit on layer .
a[k,i,1],a[k,i,2],a[k,i,3]
i
k
In[54]:=
ll=layer[2]
Out[54]=
Optimization
Using a single layer
Using a single layer
In[55]:=
$L=1;
Pick a quantum circuit layer to apply on the input states.
In[56]:=
op=layer[$L]
Out[56]=
In[63]:=
mat=Normal@Matrix[op];
In[64]:=
Z1=Matrix[S[1,3],SS];
In[65]:=
w0=Flatten@RandomReal[{0,1}Pi,{$L,$n,3}]
Out[65]=
{1.41694,0.769011,1.86193,2.64199,2.90684,1.28007,2.76019,2.58545,2.36391,1.93902,3.01061,1.32815}
The parameters to use.
In[66]:=
aa=a[Range@$L,Range@$n,{1,2,3}];
The parameters are angles and the following constraints are imposed.
In[68]:=
constraints=Thread[0≤aa<Pi];
In[72]:=
optimizer[ww_]:=Module[{ii=RandomChoice[Range@Length@data,4],yy,vv,avg,sol,cost},yy=Part[Values@data,ii];vv=Transpose@Part[$in,ii];vv=Transpose[mat.vv];avg=Map[Conjugate[#].Z1.#&,vv];cost=Mean@Abs[avg-yy]^2;EchoTiming[{cost,sol}=FindMinimum[{cost,constraints},Transpose@{aa,ww},StepMonitorPrintTemporary["Cost: ",cost],Method"IPOPT"];];Echo[{Chop[avg/.sol],yy,cost},"{avg, yy, cost}"];Return[aa/.sol]]
Test the optimizer.
Find the optimal values of the parameters iteratively.
From the optimized parameters, make predictions.
Verdict
Verdict
◼
A single-layer is not sufficient to predict the parity values properly.
Using two layers without imposing the constraints
Using two layers without imposing the constraints
Pick a quantum circuit layer to apply on the input states.
Check which parameters are actually involved in the above quantum circuit.
The set of parameters.
We prepare the natural constraints, but in this example, we will not impose them.
Test the optimizer.
Find the optimal values of the parameters iteratively.
From the optimized parameters, make predictions.
Verdict
Verdict
◼
Two layers is sufficient to properly predict the parity values.
Using two layers imposing the constraints
Using two layers imposing the constraints
Pick a quantum circuit layer to apply on the input states.
Check which parameters are actually involved in the above quantum circuit.
The set of parameters.
Prepare the natural constraints for angles.
Test the optimizer.
Find the optimal values of the parameters iteratively.
From the optimized parameters, make predictions.
Verdict
Verdict
◼
Two layers is sufficient to properly predict the parity values.