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<a name="Functions-and-Variables-for-Quantum_005fComputing"></a>
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Previous: <a href="maxima_347.html#Package-quantum_005fcomputing" accesskey="p" rel="previous">Package quantum_computing</a>, Up: <a href="maxima_346.html#quantum_005fcomputing_002dpkg" accesskey="u" rel="up">quantum_computing-pkg</a> [<a href="maxima_toc.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="maxima_423.html#Function-and-Variable-Index" title="Index" rel="index">Index</a>]</p>
</div>
<a name="Functions-and-Variables-for-Quantum_005fComputing-1"></a>
<h3 class="section">83.2 Functions and Variables for Quantum_Computing</h3>
<a name="binlist"></a><a name="Item_003a-quantum_005fcomputing_002fdeffn_002fbinlist"></a><dl>
<dt><a name="index-binlist"></a>Function: <strong>binlist</strong> <em><br> <tt>binlist</tt> (<var>k</var>) <br> <tt>binlist</tt> (<var>k</var>, <var>n</var>)</em></dt>
<dd>
<p><code>binlist</code>(<var>k</var>), where <var>k</var> must be a natural number,
returns a list of binary digits 0 or 1 corresponding to the digits of
<var>k</var> in binary representation. <code>binlist</code>(<var>k</var>, <var>n</var>) does
the same but returns a list of length <var>n</var>, with leading zeros as
necessary. Notice that for the result to represent a possible state of
<var>m</var> qubits, <var>n</var> should be equal to 2^<var>m</var> and <var>k</var> should
be between 0 and 2^<var>m</var>-1.
</p>
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</dd></dl>
<a name="binlist2dec"></a><a name="Item_003a-quantum_005fcomputing_002fdeffn_002fbinlist2dec"></a><dl>
<dt><a name="index-binlist2dec"></a>Function: <strong>binlist2dec</strong> <em>(<var>lst</var>)</em></dt>
<dd>
<p>Given a list <var>lst</var> with <var>n</var> binary digits, it returns the decimal
number it represents.
</p>
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</dd></dl>
<a name="CNOT"></a><a name="Item_003a-quantum_005fcomputing_002fdeffn_002fCNOT"></a><dl>
<dt><a name="index-CNOT"></a>Function: <strong>CNOT</strong> <em>(<var>q</var>, <var>i</var>, <var>j</var>)</em></dt>
<dd>
<p>Changes the value of the <var>j</var>’th qubit, in a state <var>q</var> of <var>m</var>
qubits, when the value of the <var>i</var>’th qubit equals 1. It modifies the
list <var>q</var> and returns its modified value.
</p>
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</dd></dl>
<a name="controlled"></a><a name="Item_003a-quantum_005fcomputing_002fdeffn_002fcontrolled"></a><dl>
<dt><a name="index-controlled"></a>Function: <strong>controlled</strong> <em>(<var>U</var>, <var>q</var>, <var>c</var>, <var>i</var>)</em></dt>
<dd>
<p>Applies a matrix <var>U</var>, acting on <var>m</var> qubits, on qubits <var>i</var>
through <var>i</var>+<var>m</var>-1 of the state <var>q</var> of <var>n</var> qubits
(<var>n</var> > <var>m</var>), when the value of the <var>c</var>’th qubit in <var>q</var>
equals 1. <var>i</var> should be an integer between 1 and <var>n</var>+1-<var>m</var>
and <var>c</var> should be an integer between 1 and <var>n</var>, excluding the
qubits to be modified (<var>i</var> through <var>i</var>+<var>m</var>-1).
</p>
<p><var>U</var> can be one of the indices of the array of common matrices
<var>qmatrix</var> (see <code><a href="#qmatrix">qmatrix</a></code>). The state <var>q</var> is modified and
shown in the output.
</p>
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</dd></dl>
<a name="gate"></a><a name="Item_003a-quantum_005fcomputing_002fdeffn_002fgate"></a><dl>
<dt><a name="index-gate"></a>Function: <strong>gate</strong> <em><br> <tt>gate</tt> (<var>U</var>, <var>q</var>) <br> <tt>gate</tt> (<var>U</var>, <var>q</var>, <var>i</var>) <br> <tt>gate</tt> (<var>U</var>, <var>q</var>, <var>i1</var>, …, <var>im</var>)</em></dt>
<dd>
<p><var>U</var> must be a matrix acting on states of <var>m</var> qubits; <var>q</var> a
list corresponding to a state of <var>n</var> qubits (<var>n</var> >= <var>m</var>);
<var>i</var> and the <var>m</var> numbers <var>i1</var>, …, <var>im</var> must be
different integers between 1 and <var>n</var>.
</p>
<p><code>gate</code>(<var>U</var>, <var>q</var>) applies matrix <var>U</var> to each qubit of
<var>q</var>, when <var>m</var> equals 1, or to the first <var>m</var> qubits of
<var>q</var> when <var>m</var> is bigger than 1.
</p>
<p><code>gate</code>(<var>U</var>, <var>q</var>, <var>i</var>) applies matrix <var>U</var> to the
qubits <var>i</var> through <var>i</var>+<var>m</var>-1 of <var>q</var>.
</p>
<p><code>gate</code>(<var>U</var>, <var>q</var>, <var>i1</var>, …, <var>in</var>) applies
matrix <var>U</var> to the in the positions <var>i1</var>, …, <var>im</var>.
</p>
<p><var>U</var> can be one of the indices of the array of common matrices
<var>qmatrix</var> (see <code><a href="#qmatrix">qmatrix</a></code>). The state <var>q</var> is modified and
shown in the output.
</p>
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</dd></dl>
<a name="gate_005fmatrix"></a><a name="Item_003a-quantum_005fcomputing_002fdeffn_002fgate_005fmatrix"></a><dl>
<dt><a name="index-gate_005fmatrix"></a>Function: <strong>gate_matrix</strong> <em><br> <tt>gate_matrix</tt> (<var>U</var>, <var>n</var>) <br> <tt>gate_matrix</tt> (<var>U</var>, <var>n</var>, <var>i1</var>, …, <var>im</var>)</em></dt>
<dd>
<p><var>U</var> must be a 2 by 2 matrix or one of the indices of the array of
common matrices <var>qmatrix</var> (see <code><a href="#qmatrix">qmatrix</a></code>).
<code>gate_matrix</code>(<var>U</var>, <var>n</var>) returns the matrix corresponding to
the action of <var>U</var> on each qubit in a state of <var>n</var> qubits.
</p>
<p><code>gate_matrix</code> (<var>U</var>, <var>n</var>, <var>i1</var>, …, <var>im</var>)
returns the matrix corresponding to the action of <var>U</var> on qubits
<var>i1</var>, …, <var>im</var> of a state of <var>n</var> qubits, where
<var>i1</var>, …, <var>im</var> are different integers between 1 and
<var>n</var>.
</p>
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</dd></dl>
<a name="linsert"></a><a name="Item_003a-quantum_005fcomputing_002fdeffn_002flinsert"></a><dl>
<dt><a name="index-linsert"></a>Function: <strong>linsert</strong> <em>(<var>e</var>, <var>lst</var>, <var>p</var>)</em></dt>
<dd>
<p>Inserts the expression or list <var>e</var> into the list <var>lst</var> at position
<var>p</var>. The list can be empty and <var>p</var> must be an integer between 1 and
the length of <var>lst</var> plus 1.
</p>
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</dd></dl>
<a name="lreplace"></a><a name="Item_003a-quantum_005fcomputing_002fdeffn_002flreplace"></a><dl>
<dt><a name="index-lreplace"></a>Function: <strong>lreplace</strong> <em>(<var>e</var>, <var>lst</var>, <var>p</var>)</em></dt>
<dd>
<p>If <var>e</var> is a list of length <var>n</var>, the elements in the positions
<var>p</var>, <var>p</var>+1, …, <var>p</var>+<var>n</var>-1 of the list <var>lst</var> are
replaced by <var>e</var>, or the first elements of <var>e</var> if the end of
<var>lst</var> is reached. If <var>e</var> is an expression, the element in
position <var>p</var> of list <var>lst</var> is replaced by that expression.
<var>p</var> must be an integer between 1 and the length of <var>lst</var>.
</p>
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</dd></dl>
<a name="normalize"></a><a name="Item_003a-quantum_005fcomputing_002fdeffn_002fnormalize"></a><dl>
<dt><a name="index-normalize"></a>Function: <strong>normalize</strong> <em>(<var>q</var>)</em></dt>
<dd>
<p>Returns the normalized version of a quantum state given as a list <var>q</var>.
</p>
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</dd></dl>
<a name="qdisplay"></a><a name="Item_003a-quantum_005fcomputing_002fdeffn_002fqdisplay"></a><dl>
<dt><a name="index-qdisplay"></a>Function: <strong>qdisplay</strong> <em>(<var>q</var>)</em></dt>
<dd>
<p>Represents the state <var>q</var> of a system of <var>n</var> qubits as a linear
combination of the computational states with <var>n</var> binary digits. It
returns an expression including strings and symbols.
</p>
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</dd></dl>
<a name="qmatrix"></a><a name="Item_003a-quantum_005fcomputing_002fdefvr_002fqmatrix"></a><dl>
<dt><a name="index-qmatrix"></a>System variable: <strong>qmatrix</strong></dt>
<dd>
<p>This variable is a predefined hash array of two by two matrices with the
standard matrices: identity, Pauli matrices, Hadamard matrix and the
phase matrix. The six possible indices are I, X, Y, Z, H,
S. <var>qmatrix</var>[I] is the identity matrix, <var>qmatrix</var>[X] the Pauli x
matrix, <var>qmatrix</var>[Y] the Pauli y matrix, <var>qmatrix</var>[Z] the Pauli
z matrix, <var>qmatrix</var>[H] the Hadamard matrix and <var>qmatrix</var>[S] the
phase matrix.
</p>
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</dd></dl>
<a name="qmeasure"></a><a name="Item_003a-quantum_005fcomputing_002fdeffn_002fqmeasure"></a><dl>
<dt><a name="index-qmeasure"></a>Function: <strong>qmeasure</strong> <em><br> <tt>qmeasure</tt> (<var>q</var>) <br> <tt>qmeasure</tt> (<var>q</var>, <var>i1</var>, …, <var>im</var>)</em></dt>
<dd>
<p>Measures the value of one or more qubits in a system of <var>n</var> qubits
with state <var>q</var>. The <var>m</var> positive integers <var>i1</var>, …,
<var>im</var> are the positions of the qubits to be measured It requires 1 or
more arguments. The first argument must be the state q. If the only
argument given is <var>q</var>, all the n qubits will be measured.
</p>
<p>It returns a list with the values of the qubits measured (either 0 or
1), in the same order they were requested or in ascending order if the
only argument given was <var>q</var>. It modifies the list <var>q</var>,
reflecting the collapse of the quantum state after the measurement.
</p>
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</dd></dl>
<a name="qubits"></a><a name="Item_003a-quantum_005fcomputing_002fdeffn_002fqubits"></a><dl>
<dt><a name="index-qubits"></a>Function: <strong>qubits</strong> <em><br> <tt>qubits</tt> (<var>n</var>) <br> <tt>qubits</tt> (<var>i1</var>, …, <var>in</var>)</em></dt>
<dd>
<p><code>qubits</code>(<var>n</var>) returns a list representing the ground state of a
system of <var>n</var> qubits.
</p>
<p><code>qubits</code>(<var>i1</var>, …, <var>in</var>) returns a list with
representing the state of <var>n</var> qubits with values <var>i1</var>, …,
<var>in</var>.
</p>
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</dd></dl>
<a name="qswap"></a><a name="Item_003a-quantum_005fcomputing_002fdeffn_002fqswap"></a><dl>
<dt><a name="index-qswap"></a>Function: <strong>qswap</strong> <em>(<var>q</var>, <var>i</var>, <var>j</var>)</em></dt>
<dd>
<p>Interchanges the states of qubits <var>i</var> and <var>j</var> in the state
<var>q</var> of a system of several qubits. It modifies the list <var>q</var> and
returns its modified value.
</p>
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</dd></dl>
<a name="Rx"></a><a name="Item_003a-quantum_005fcomputing_002fdeffn_002fRx"></a><dl>
<dt><a name="index-Rx"></a>Function: <strong>Rx</strong> <em>(<var>a</var>)</em></dt>
<dd>
<p>Returns the 2 by two matrix (acting on one qubit) corresponding to a
rotation of with an angle of <var>a</var> radians around the x axis.
</p>
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</dd></dl>
<a name="Ry"></a><a name="Item_003a-quantum_005fcomputing_002fdeffn_002fRy"></a><dl>
<dt><a name="index-Ry"></a>Function: <strong>Ry</strong> <em>(<var>a</var>)</em></dt>
<dd>
<p>Returns the 2 by two matrix (acting on one qubit) corresponding to a
rotation of with an angle of <var>a</var> radians around the y axis.
</p>
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</dd></dl>
<a name="Rz"></a><a name="Item_003a-quantum_005fcomputing_002fdeffn_002fRz"></a><dl>
<dt><a name="index-Rz"></a>Function: <strong>Rz</strong> <em>(<var>a</var>)</em></dt>
<dd>
<p>Returns the 2 by two matrix (acting on one qubit) corresponding to a
rotation of with an angle of <var>a</var> radians around the z axis.
</p>
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</dd></dl>
<a name="tprod"></a><a name="Item_003a-quantum_005fcomputing_002fdeffn_002ftprod"></a><dl>
<dt><a name="index-tprod"></a>Function: <strong>tprod</strong> <em>(<var>o1</var>, …, <var>on</var>)</em></dt>
<dd>
<p>Returns the tensor product of the <var>n</var> matrices or lists <var>o1</var>,
…, <var>on</var>.
</p>
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<a name="toffoli"></a><a name="Item_003a-quantum_005fcomputing_002fdeffn_002ftoffoli"></a><dl>
<dt><a name="index-toffoli"></a>Function: <strong>toffoli</strong> <em>(<var>q</var>, (<var>i</var>, (<var>j</var>, (<var>k</var>)</em></dt>
<dd>
<p>Changes the value of the <var>k</var>’th qubit, in the state <var>q</var> of
<var>n</var> qubits, if the values of the <var>i</var>’th anf <var>j</var>’th qubits
are equal to 1. It modifies the list <var>q</var> and returns its new value.
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Previous: <a href="maxima_347.html#Package-quantum_005fcomputing" accesskey="p" rel="previous">Package quantum_computing</a>, Up: <a href="maxima_346.html#quantum_005fcomputing_002dpkg" accesskey="u" rel="up">quantum_computing-pkg</a> [<a href="maxima_toc.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="maxima_423.html#Function-and-Variable-Index" title="Index" rel="index">Index</a>]</p>
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