docs/examples/notebooks/dense_coding.ipynb

<?xml version="1.0" encoding="utf-8"?>
<notebook>
  <name>dense_coding</name>
  <nbformat>2</nbformat>
  <worksheets>
    <worksheet>
      <cells>
        <markdowncell>
          <source>&lt;h1&gt;Dense Coding
&lt;/h1&gt;</source>
        </markdowncell>
        <codecell>
          <input>%load_ext sympyprinting</input>
          <language>python</language>
          <prompt_number>2</prompt_number>
          <collapsed>1</collapsed>
          <outputs />
        </codecell>
        <codecell>
          <input>from sympy import sqrt, symbols, Rational
from sympy import expand, Eq, Symbol, simplify, exp, sin
from sympy.physics.quantum import *
from sympy.physics.quantum.qubit import *
from sympy.physics.quantum.gate import *
from sympy.physics.quantum.grover import *
from sympy.physics.quantum.qft import QFT, IQFT, Fourier
from sympy.physics.quantum.circuitplot import circuit_plot</input>
          <language>python</language>
          <prompt_number>3</prompt_number>
          <collapsed>1</collapsed>
          <outputs />
        </codecell>
        <codecell>
          <input>psi = Qubit('00')/sqrt(2) + Qubit('11')/sqrt(2); psi
</input>
          <language>python</language>
          <prompt_number>4</prompt_number>
          <collapsed>0</collapsed>
          <outputs>
            <output>
              <output_type>pyout</output_type>
              <text>⎽⎽⎽          ⎽⎽⎽     
╲╱ 2 ⋅❘00⟩   ╲╱ 2 ⋅❘11⟩
────────── + ──────────
    2            2</text>
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</png>
              <latex>$$\frac{1}{2} \sqrt{2} {\left|00\right\rangle } + \frac{1}{2} \sqrt{2} {\left|11\right\rangle }$$</latex>
              <prompt_number>4</prompt_number>
            </output>
          </outputs>
        </codecell>
        <codecell>
          <input>circuits = [H(1)*CNOT(1,0), H(1)*CNOT(1,0)*X(1), H(1)*CNOT(1,0)*Z(1), H(1)*CNOT(1,0)*Z(1)*X(1)]</input>
          <language>python</language>
          <prompt_number>5</prompt_number>
          <collapsed>1</collapsed>
          <outputs />
        </codecell>
        <codecell>
          <input>for circuit in circuits:
    circuit_plot(circuit, nqubits=2)
    display(Eq(circuit*psi,qapply(circuit*psi)))</input>
          <language>python</language>
          <prompt_number>6</prompt_number>
          <collapsed>0</collapsed>
          <outputs>
            <output>
              <output_type>display_data</output_type>
              <text>⎛  ⎽⎽⎽          ⎽⎽⎽     ⎞       
           ⎜╲╱ 2 ⋅❘00⟩   ╲╱ 2 ⋅❘11⟩⎟       
H ⋅CNOT   ⋅⎜────────── + ──────────⎟ = ❘00⟩
 1     1,0 ⎝    2            2     ⎠</text>
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</png>
              <latex>$$H_{1} CNOT_{1,0} \left(\frac{1}{2} \sqrt{2} {\left|00\right\rangle } + \frac{1}{2} \sqrt{2} {\left|11\right\rangle }\right) = {\left|00\right\rangle }$$</latex>
            </output>
            <output>
              <output_type>display_data</output_type>
              <text>⎛  ⎽⎽⎽          ⎽⎽⎽     ⎞       
              ⎜╲╱ 2 ⋅❘00⟩   ╲╱ 2 ⋅❘11⟩⎟       
H ⋅CNOT   ⋅X ⋅⎜────────── + ──────────⎟ = ❘01⟩
 1     1,0  1 ⎝    2            2     ⎠</text>
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</png>
              <latex>$$H_{1} CNOT_{1,0} X_{1} \left(\frac{1}{2} \sqrt{2} {\left|00\right\rangle } + \frac{1}{2} \sqrt{2} {\left|11\right\rangle }\right) = {\left|01\right\rangle }$$</latex>
            </output>
            <output>
              <output_type>display_data</output_type>
              <text>⎛  ⎽⎽⎽          ⎽⎽⎽     ⎞       
              ⎜╲╱ 2 ⋅❘00⟩   ╲╱ 2 ⋅❘11⟩⎟       
H ⋅CNOT   ⋅Z ⋅⎜────────── + ──────────⎟ = ❘10⟩
 1     1,0  1 ⎝    2            2     ⎠</text>
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</png>
              <latex>$$H_{1} CNOT_{1,0} Z_{1} \left(\frac{1}{2} \sqrt{2} {\left|00\right\rangle } + \frac{1}{2} \sqrt{2} {\left|11\right\rangle }\right) = {\left|10\right\rangle }$$</latex>
            </output>
            <output>
              <output_type>display_data</output_type>
              <text>⎛  ⎽⎽⎽          ⎽⎽⎽     ⎞       
                 ⎜╲╱ 2 ⋅❘00⟩   ╲╱ 2 ⋅❘11⟩⎟       
H ⋅CNOT   ⋅Z ⋅X ⋅⎜────────── + ──────────⎟ = ❘11⟩
 1     1,0  1  1 ⎝    2            2     ⎠</text>
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</png>
              <latex>$$H_{1} CNOT_{1,0} Z_{1} X_{1} \left(\frac{1}{2} \sqrt{2} {\left|00\right\rangle } + \frac{1}{2} \sqrt{2} {\left|11\right\rangle }\right) = {\left|11\right\rangle }$$</latex>
            </output>
            <output>
              <output_type>display_data</output_type>
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