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The quantum Sine and Cosine transforms functions are the quantum analog for the discrete Sine and Cosine transforms.
The unitary versions of the type I and type II transforms are defined as follows:
D C T j k ( 1 ) ( N ) = α j k 2 N − 1 cos ( π j k N − 1 ) , α j k = { 1 2 j = 0 , N − 1 , 1 2 k = 0 , N − 1 , 1 else , j , k = 0 … , N − 1 {\rm DCT}^{(1)}_{jk}(N) = \alpha_{jk}\sqrt{\frac{2}{N-1}} \cos\left(\frac{\pi j k}{N-1}\right), \qquad
\alpha_{jk} = \left\{ \begin{array}{l l}
\frac{1}{\sqrt{2}} & j = 0,N-1 ,\\
\frac{1}{\sqrt{2}} & k = 0,N-1 ,\\
1 & \text{else}
\end{array}
\right.,
\qquad j,k = 0\dots,N-1 DCT jk ( 1 ) ( N ) = α jk N − 1 2 cos ( N − 1 πjk ) , α jk = ⎩ ⎨ ⎧ 2 1 2 1 1 j = 0 , N − 1 , k = 0 , N − 1 , else , j , k = 0 … , N − 1 D S T j k ( 1 ) ( N ) = 2 N + 1 sin ( π j k N + 1 ) , j , k = 0 … , N − 1 {\rm DST}^{(1)}_{jk}(N) = \sqrt{\frac{2}{N+1}} \sin\left(\frac{\pi j k}{N+1}\right), \qquad j,k = 0\dots,N-1 DST jk ( 1 ) ( N ) = N + 1 2 sin ( N + 1 πjk ) , j , k = 0 … , N − 1 D C T j k ( 2 ) ( N ) = α j k 2 N cos ( π ( j + 1 / 2 ) k N ) , α j k = { 1 2 k = 0 , 1 else , j , k = 0 … , N − 1 {\rm DCT}^{(2)}_{jk}(N) = \alpha_{jk}\sqrt{\frac{2}{N}} \cos\left(\frac{\pi (j+1/2) k }{N}\right), \qquad
\alpha_{jk} = \left\{ \begin{array}{l l}
\frac{1}{\sqrt{2}} & k = 0 ,\\
1 & \text{else}
\end{array}
\right.,
\qquad j,k = 0\dots,N-1 DCT jk ( 2 ) ( N ) = α jk N 2 cos ( N π ( j + 1/2 ) k ) , α jk = { 2 1 1 k = 0 , else , j , k = 0 … , N − 1 D S T j k ( 2 ) ( N ) = α j k 2 N sin ( π ( j + 1 / 2 ) ( k + 1 ) N ) , α j k = { 1 2 k = N − 1 , 1 else , j , k = 0 … , N − 1 {\rm DST}^{(2)}_{jk}(N) = \alpha_{jk} \sqrt{\frac{2}{N}} \sin\left(\frac{\pi (j+1/2) (k+1)}{N}\right), \qquad
\alpha_{jk} = \left\{ \begin{array}{l l}
\frac{1}{\sqrt{2}} & k = N-1 ,\\
1 & \text{else}
\end{array}
\right.,
\qquad j,k = 0\dots,N-1 DST jk ( 2 ) ( N ) = α jk N 2 sin ( N π ( j + 1/2 ) ( k + 1 ) ) , α jk = { 2 1 1 k = N − 1 , else , j , k = 0 … , N − 1 The open library includes four functions, following the implementation in Ref. [1 ]:
QCT and QST of type I Function: qct_qst_type1
Arguments:
The x quantum argument is the quantum state on which we apply the transforms, according to the following unitary on n ≡ n\equiv n ≡ x.len qubits:
( D C T ( 1 ) ( 2 n − 1 + 1 ) 0 0 i D S T ( 1 ) ( 2 n − 1 − 1 ) ) \left(
\begin{array}{ccc|c}
{} &{} &{} \\
{}&{\rm DCT}^{(1)}(2^{n-1}+1) & {}& 0\\
{} &{} &{} \\
\hline
{} & 0 & {} & i{\rm DST}^{(1)}(2^{n-1}-1)
\end{array}
\right) DCT ( 1 ) ( 2 n − 1 + 1 ) 0 0 i DST ( 1 ) ( 2 n − 1 − 1 ) Example import numpy as np from classiq import * NUM_QUBITS = 4 execution_preferences = ExecutionPreferences( num_shots = 1 , backend_preferences = ClassiqBackendPreferences( backend_name = ClassiqSimulatorBackendNames. SIMULATOR_STATEVECTOR ), ) np.random.seed( 123 ) cos_data = np.random.rand( 2 ** ( NUM_QUBITS - 1 ) + 1 ) cos_data = cos_data / np.linalg.norm(cos_data) sin_data = np.random.rand( 2 ** ( NUM_QUBITS - 1 ) - 1 ) sin_data = sin_data / np.linalg.norm(sin_data) combined_data = np.append(cos_data / np.sqrt( 2 ), sin_data / np.sqrt( 2 ))
@qfunc def main ( x : Output[QNum]): prepare_amplitudes(combined_data.tolist(), 0.0 , x) qct_qst_type1(x) qmod = create_model(main, execution_preferences = execution_preferences)
result = execute(qprog).result_value()
qct_data = np.zeros( 2 ** ( NUM_QUBITS - 1 ) + 1 ).astype( complex ) qst_data = np.zeros( 2 ** ( NUM_QUBITS - 1 ) - 1 ).astype( complex ) for sample in result.parsed_state_vector: value = int (sample.state[ "x" ]) if value < 2 ** ( NUM_QUBITS - 1 ) + 1 : qct_data[value] += sample.amplitude else : qst_data[ int (value - 2 ** ( NUM_QUBITS - 1 ) - 1 )] += sample.amplitude
def dct1 ( n ): dct = np.array( [ [ np.cos(np.pi * j * k / (n - 1 )) * (np.sqrt( 1 / 2 ) if j == 0 or j == n - 1 else 1 ) * (np.sqrt( 1 / 2 ) if k == 0 or k == n - 1 else 1 ) for j in range (n) ] for k in range (n) ] ) / np.sqrt((n - 1 ) / 2 ) return dct def dst1 ( n ): dst = np.array( [ [np.sin(np.pi * (j + 1 ) * (k + 1 ) / (n + 1 )) for j in range (n)] for k in range (n) ] ) / np.sqrt((n + 1 ) / 2 ) return dst
global_phase = np.exp( 1 j * np.angle(qct_data[ 0 ])) measured_cos_res = np.real(qct_data / global_phase) expected_cos_res = (dct1( 2 ** ( NUM_QUBITS - 1 ) + 1 ) @ cos_data) / np.sqrt( 2 ) print ( "measured result:" , measured_cos_res) print ( "expected result:" , expected_cos_res) assert np.allclose(measured_cos_res, expected_cos_res, atol = 0.01 )
Output: measured result: [ 0.64936034 -0.13662084 0.02159456 0.08089956 0.06722088 0.18669631 0.02254746 -0.01198615 0.1123771 ] expected result: [ 0.64936034 -0.13662084 0.02159456 0.08089956 0.06722088 0.18669631 0.02254746 -0.01198615 0.1123771 ]
global_phase = np.exp( 1 j * np.angle(qst_data[ 0 ])) measured_sin_res = np.real(qst_data / global_phase) expected_sin_res = (dst1( 2 ** ( NUM_QUBITS - 1 ) - 1 ) @ sin_data) / np.sqrt( 2 ) print ( "measured result:" , measured_sin_res) print ( "expected result:" , expected_sin_res) assert np.allclose(measured_sin_res, expected_sin_res, atol = 0.01 )
Output: measured result: [ 0.57556991 0.04712138 0.22433696 -0.27513447 0.17796799 0.07685908 0.05378552] expected result: [ 0.57556991 0.04712138 0.22433696 -0.27513447 0.17796799 0.07685908 0.05378552]
QCT and QST of type II Function: qct_qst_type2
Arguments:
The x quantum argument is the quantum state on which we apply the transforms, whereas the single q qubit indicates the block, according to the following unitary on n + 1 ≡ n+1\equiv n + 1 ≡ x.len + 1 +1 + 1
( D C T ( 2 ) ( 2 n − 1 ) 0 0 − D S T ( 2 ) ( 2 n − 1 ) ) \left(
\begin{array}{c|c}
{\rm DCT}^{(2)}(2^{n-1}) & 0\\
\hline
0 & -{\rm DST}^{(2)}(2^{n-1})
\end{array}
\right) ( DCT ( 2 ) ( 2 n − 1 ) 0 0 − DST ( 2 ) ( 2 n − 1 ) ) Function: qct_type2
Arguments:
x: QArray[QBit]: the quantum state on which we apply D C T ( 2 ) {\rm DCT}^{(2)} DCT ( 2 )
Function: qst_type2
Arguments:
x: QArray[QBit]: the quantum state on which we apply D S T ( 2 ) {\rm DST}^{(2)} DST ( 2 )
Example NUM_QUBITS = 4 cos_sin_data = np.random.rand( 2 ** ( NUM_QUBITS - 1 )) cos_sin_data = cos_sin_data / np.linalg.norm(cos_sin_data)
@qfunc def main ( x : Output[QNum], q : Output[QBit]): prepare_amplitudes(cos_sin_data.tolist(), 0.0 , x) allocate(q) H(q) qct_qst_type2(x, q) qmod = create_model(main, execution_preferences = execution_preferences)
result = execute(qprog).result_value()
qct_data = np.zeros( 2 ** ( NUM_QUBITS - 1 )).astype( complex ) qst_data = np.zeros( 2 ** ( NUM_QUBITS - 1 )).astype( complex ) for sample in result.parsed_state_vector: if sample.state[ "q" ] == 0 : qct_data[ int (sample.state[ "x" ])] += sample.amplitude else : qst_data[ int (sample.state[ "x" ])] += sample.amplitude
def dct2 ( n ): dct = np.array( [ [ np.cos(np.pi * j * (k + 1 / 2 ) / n) * (np.sqrt( 1 / 2 ) if j == 0 else 1 ) for j in range (n) ] for k in range (n) ] ) / np.sqrt(n / 2 ) return dct.T def dst2 ( n ): dst = np.array( [ [ np.sin(np.pi * (j + 1 ) * (k + 1 / 2 ) / n) * (np.sqrt( 1 / 2 ) if j == n - 1 else 1 ) for j in range (n) ] for k in range (n) ] ) / np.sqrt(n / 2 ) return dst.T
global_phase = np.exp( 1 j * np.angle(qct_data[ 0 ])) measured_cos_res = np.real(qct_data / global_phase) expected_cos_res = (dct2( 2 ** ( NUM_QUBITS - 1 )) @ cos_sin_data) / np.sqrt( 2 ) print ( "measured result:" , measured_cos_res) print ( "expected result:" , expected_cos_res) assert np.allclose(measured_cos_res, expected_cos_res, atol = 0.01 )
Output: measured result: [ 0.65104654 -0.23305325 -0.11473439 0.02595719 -0.04930425 0.03323495 0.06553173 0.01252819] expected result: [ 0.65104654 -0.23305325 -0.11473439 0.02595719 -0.04930425 0.03323495 0.06553173 0.01252819]
global_phase = np.exp( 1 j * np.angle(qst_data[ 0 ])) measured_sin_res = np.real(qst_data / global_phase) expected_sin_res = (dst2( 2 ** ( NUM_QUBITS - 1 )) @ cos_sin_data) / np.sqrt( 2 ) print ( "measured result:" , measured_sin_res) print ( "expected result:" , expected_sin_res) assert np.allclose(measured_sin_res, expected_sin_res, atol = 0.01 )
Output: measured result: [ 0.63981132 -0.2180174 0.13530848 -0.08552637 0.02796936 -0.03450769 0.12370004 -0.01455965] expected result: [ 0.63981132 -0.2180174 0.13530848 -0.08552637 0.02796936 -0.03450769 0.12370004 -0.01455965]
@qfunc def main ( x : Output[QNum]): prepare_amplitudes(cos_sin_data.tolist(), 0.0 , x) qct_type2(x) qmod = create_model(main, execution_preferences = execution_preferences) qprog = synthesize(qmod) result = execute(qprog).result_value() qct_data = np.zeros( 2 ** ( NUM_QUBITS - 1 )).astype( complex ) for sample in result.parsed_state_vector: qct_data[ int (sample.state[ "x" ])] += sample.amplitude global_phase = np.exp( 1 j * np.angle(qct_data[ 0 ])) measured_cos_res = np.real(qct_data / global_phase) expected_cos_res = dct2( 2 ** ( NUM_QUBITS - 1 )) @ cos_sin_data print ( "measured result:" , measured_cos_res) print ( "expected result:" , expected_cos_res) assert np.allclose(measured_cos_res, expected_cos_res, atol = 0.01 )
Output: measured result: [ 0.92071884 -0.32958707 -0.16225893 0.03670901 -0.06972674 0.04700132 0.09267587 0.01771754] expected result: [ 0.92071884 -0.32958707 -0.16225893 0.03670901 -0.06972674 0.04700132 0.09267587 0.01771754]
@qfunc def main ( x : Output[QNum]): prepare_amplitudes(cos_sin_data.tolist(), 0.0 , x) qst_type2(x) qmod = create_model(main, execution_preferences = execution_preferences) qprog = synthesize(qmod) result = execute(qprog).result_value() qst_data = np.zeros( 2 ** ( NUM_QUBITS - 1 )).astype( complex ) for sample in result.parsed_state_vector: qst_data[ int (sample.state[ "x" ])] += sample.amplitude global_phase = np.exp( 1 j * np.angle(qst_data[ 0 ])) measured_sin_res = np.real(qst_data / global_phase) expected_sin_res = dst2( 2 ** ( NUM_QUBITS - 1 )) @ cos_sin_data print ( "measured result:" , measured_sin_res) print ( "expected result:" , expected_sin_res) assert np.allclose(measured_sin_res, expected_sin_res, atol = 0.01 )
Output: measured result: [ 0.90482984 -0.30832317 0.19135509 -0.12095255 0.03955464 -0.04880124 0.17493827 -0.02059046] expected result: [ 0.90482984 -0.30832317 0.19135509 -0.12095255 0.03955464 -0.04880124 0.17493827 -0.02059046]
References [1] : Klappenecker, A., & Rotteler M., “Discrete Cosine Transforms on Quantum Computers”. 
