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Expand Up @@ -12,32 +12,20 @@ This python package generates QSP phase angles using the code based on two diffe

### QSP conventions

As described in [A Grand Unification of Quantum Algorithms](https://arxiv.org/abs/2105.02859), two important QSP model conventions used in the literature are known as Wx, where the signal W(a) is an X-rotation and QSP signal processing phase shifts are Z-rotations, and Wz, where the signal W(a) is a Z-rotation and QSP signal processing phase shifts are X-rotations.
As described in [A Grand Unification of Quantum Algorithms](https://arxiv.org/abs/2105.02859), two important QSP model conventions used in the literature are known as $W_x$, where the signal W(a) is an X-rotation and QSP signal processing phase shifts are Z-rotations, and $W_z$, where the signal $W(a)$ is a Z-rotation and QSP signal processing phase shifts are X-rotations.

Specifically, in the Wx convention, the QSP operation sequence is:

<!-- U_x = e^{i\phi_0 Z} \prod_{k=1}^L W(a) e^{i\phi_k Z} ~~~~{\rm where} ~~~ W(x)=\left[ \begin{array}{cc} a & i\sqrt{1-a^2} \\ i\sqrt{1-a^2} & a \end{array} \right ] -->
Specifically, in the $W_x$ convention, the QSP operation sequence is:

```math
U_x = e^{i\phi_0 Z} \prod_{k=1}^L W(a) e^{i\phi_k Z} \;\;\;\;\text{here}\;\;\;\; W(x)= \begin{bmatrix} a & i\sqrt{1-a^2} \\ i\sqrt{1-a^2} & a \end{bmatrix}
U_x = e^{i\phi_0 Z} \prod_{k=1}^L W(a) e^{i\phi_k Z} \;\;\;\;\text{where}\;\;\;\; W(x)= \begin{bmatrix} a & i\sqrt{1-a^2} \\ i\sqrt{1-a^2} & a \end{bmatrix}
```

<!-- <center>
<img src="https://latex.codecogs.com/svg.latex?%5Clarge%20U_x%20%3D%20e%5E%7Bi%5Cphi_0%20Z%7D%20%5Cprod_%7Bk%3D1%7D%5EL%20W%28a%29%20e%5E%7Bi%5Cphi_k%20Z%7D%20%7E%7E%7E%7E%7B%5Crm%20where%7D%20%7E%7E%7E%20W%28a%29%3D%5Cleft%5B%20%5Cbegin%7Barray%7D%7Bcc%7D%20a%20%26%20i%5Csqrt%7B1-a%5E2%7D%20%5C%5C%20i%5Csqrt%7B1-a%5E2%7D%20%26%20a%20%5Cend%7Barray%7D%20%5Cright%20%5D" />
</center> -->

And in the $W_z$ convention, the QSP operation sequence is:

```math
U_z = e^{i\phi_0 X} \prod_{k=1}^L W(a) e^{i\phi_k X} \;\;\;\;\text{here}\;\;\;\; W(a)=\begin{bmatrix} a + i\sqrt{1-a^2} & 0 \\ 0 & a - i\sqrt{1-a^2} \end{bmatrix}
U_z = e^{i\phi_0 X} \prod_{k=1}^L W(a) e^{i\phi_k X} \;\;\;\;\text{where}\;\;\;\; W(a)=\begin{bmatrix} a + i\sqrt{1-a^2} & 0 \\ 0 & a - i\sqrt{1-a^2} \end{bmatrix}
```

<!-- U_z = e^{i\phi_0 X} \prod_{k=1}^L W(a) e^{i\phi_k X} ~~~~{\rm where} ~~~ W(a)=\left[ \begin{array}{cc} a + i\sqrt{1-a^2} & 0 \\ 0 & a - i\sqrt{1-a^2} \end{array} \right ] -->

<!-- <center>
<img src="https://latex.codecogs.com/svg.latex?%5Clarge%20U_z%20%3D%20e%5E%7Bi%5Cphi_0%20X%7D%20%5Cprod_%7Bk%3D1%7D%5EL%20W%28a%29%20e%5E%7Bi%5Cphi_k%20X%7D%20%7E%7E%7E%7E%7B%5Crm%20where%7D%20%7E%7E%7E%20W%28a%29%3D%5Cleft%5B%20%5Cbegin%7Barray%7D%7Bcc%7D%20a%20&plus;%20i%5Csqrt%7B1-a%5E2%7D%20%26%200%20%5C%5C%200%20%26%20a%20-%20i%5Csqrt%7B1-a%5E2%7D%20%5Cend%7Barray%7D%20%5Cright%20%5D" />
</center> -->

They are related by a Hadamard transform:

<!-- U_x = H U_z H -->
Expand All @@ -46,43 +34,39 @@ They are related by a Hadamard transform:
U_x = H U_z H
```

<!-- <center>
<img src="https://latex.codecogs.com/svg.latex?%5Clarge%20U_x%20%3D%20H%20U_z%20H" />
</center> -->

The Wz convention is convenient for and employed in [Laurent polynomial formulations of QSP](https://arxiv.org/abs/2003.02831), whereas the Wx convention is more traditional, e.g. as employed in [quantum singular value transform](https://arxiv.org/abs/1806.01838) applications of QSP.
The $W_z$ convention is convenient for and employed in [Laurent polynomial formulations of QSP](https://arxiv.org/abs/2003.02831), whereas the Wx convention is more traditional, e.g. as employed in [quantum singular value transform](https://arxiv.org/abs/1806.01838) applications of QSP.

This package can generate QSP phase angles for both conventions (whereas earlier code only handled the Wz convention). The challenge is that if one wants a certain polynomial $P_x(a) = \langle 0|U_x|0\rangle$<!-- P_x(a) = \langle 0|U_x|0\rangle --> <!-- <img src="https://latex.codecogs.com/svg.latex?%5Clarge%20P_x%28a%29%20%3D%20%5Clangle%200%7CU_x%7C0%5Crangle"/> --> in the Wx convention, one cannot just use the phases generated for this polynomial in the Wz convention. Instead, first the Q_x(a) corresponding to P_x(a) is needed to complete the full U_x. This then gives $P_z(a) = \langle 0|U_z|0\rangle = P_x(a) + Q_x(a)$ <!-- P_z(a) = \langle 0|U_z|0\rangle = P_x(a) + Q_x(a) --> <!-- <img src="https://latex.codecogs.com/svg.latex?%5Clarge%20P_z%28a%29%20%3D%20%5Clangle%200%7CU_z%7C0%5Crangle%20%3D%20P_x%28a%29%20&plus;%20Q_x%28a%29" /> -->. Computing the QSP phases for P_z(a) in the Wz convention then gives the desired QSP phases for P_x(a) in the Wx convention, if suitable care is taken with respect to the Q(a) polynomial.
This package can generate QSP phase angles for both conventions (whereas earlier code only handled the $W_z$ convention). The challenge is that if one wants a certain polynomial $P_x(a) = \langle 0|U_x|0\rangle$ in the $W_x$ convention, one cannot just use the phases generated for this polynomial in the $W_z$ convention. Instead, first the $Q_x(a)$ corresponding to $P_x(a)$ is needed to complete the full $U_x$. This then gives $P_z(a) = \langle 0|U_z|0\rangle = P_x(a) + Q_x(a)$. Computing the QSP phases for $P_z(a)$ in the $W_z$ convention then gives the desired QSP phases for $P_x(a)$ in the $W_x$ convention, if suitable care is taken with respect to the $Q(a)$ polynomial.

In addition to specifying the signal rotation operator W and the signal processing operator phase shifts, the QSP signal basis must also be specified. In this code, the default basis is |+>, but the code also allows the |0> basis to be used (when using tensorflow optimization to generate the phase angles). See the "--measurement" option.
In addition to specifying the signal rotation operator $W$ and the signal processing operator phase shifts, the QSP signal basis must also be specified. In this code, the default basis is $|+\rangle$, but the code also allows the $|0\rangle$ basis to be used (when using tensorflow optimization to generate the phase angles). See the "--measurement" option.

## Examples

This package also plots the QSP response function, and can be run from the command line. In the example below, the blue line shows the target ideal polynomial QSP response function P_x(a); the red line shows the real part of the response achieved by the QSP phases, and the green line shows the imaginary part of the QSP response, with L=20.
This package also plots the QSP response function, and can be run from the command line. In the example below, the blue line shows the target ideal polynomial QSP response function $P_x(a)$; the red line shows the real part of the response achieved by the QSP phases, and the green line shows the imaginary part of the QSP response, with L=20.

![Example QSP response function for 1/a](https://github.com/ichuang/pyqsp/blob/master/docs/IMAGE-sample-qsp-response-for-one-over-x-kappa3.png?raw=true)

This was generated by running `pyqsp --plot invert`, which generated the following text output:

```
b=20, j0=10
[PolyOneOverX] minimum [-2.90002589] is at [-0.25174082]: normalizing
Laurent P poly 0.0002108924030879319 * w ^ (-21) + -0.0006894043260278334 * w ^ (-19) + 0.001994436843136674 * w ^ (-17) + -0.005148265426149545 * w ^ (-15) + 0.011941126989562179 * w ^ (-13) + -0.02504164571900347 * w ^ (-11) + 0.04774921151669176 * w ^ (-9) + -0.08322978307559126 * w ^ (-7) + 0.1333200017468883 * w ^ (-5) + -0.1973241700493915 * w ^ (-3) + 0.2714342596621151 * w ^ (-1) + 0.2714342596621151 * w ^ (1) + -0.1973241700493915 * w ^ (3) + 0.1333200017468883 * w ^ (5) + -0.08322978307559126 * w ^ (7) + 0.04774921151669176 * w ^ (9) + -0.02504164571900347 * w ^ (11) + 0.011941126989562179 * w ^ (13) + -0.005148265426149545 * w ^ (15) + 0.001994436843136674 * w ^ (17) + -0.0006894043260278334 * w ^ (19) + 0.0002108924030879319 * w ^ (21)
Laurent Q poly -2.784691352688532e-05 * w ^ (-21) + -3.467046922689296e-06 * w ^ (-19) + -0.00023447352272530086 * w ^ (-17) + 0.00018731079814912242 * w ^ (-15) + -0.0007159634932856078 * w ^ (-13) + 0.0032821300323626736 * w ^ (-11) + -0.001907488898700853 * w ^ (-9) + 0.012591630667616198 * w ^ (-7) + -0.02371053103573988 * w ^ (-5) + -0.016629567619072996 * w ^ (-3) + -0.18519029619384528 * w ^ (-1) + -0.20476480667058175 * w ^ (1) + -0.5374170003848409 * w ^ (3) + -0.2638332742746844 * w ^ (5) + 0.21212227077358078 * w ^ (7) + 0.27748656462486654 * w ^ (9) + -0.34978762979573563 * w ^ (11) + 0.17888109364117422 * w ^ (13) + -0.07649917245226195 * w ^ (15) + 0.03551004671728361 * w ^ (17) + -0.011926352140796492 * w ^ (19) + 0.001997855069006951 * w ^ (21)
Laurent Pprime poly 0.00018304548956104657 * w ^ (-21) + -0.0006928713729505227 * w ^ (-19) + 0.001759963320411373 * w ^ (-17) + -0.0049609546280004226 * w ^ (-15) + 0.01122516349627657 * w ^ (-13) + -0.021759515686640796 * w ^ (-11) + 0.0458417226179909 * w ^ (-9) + -0.07063815240797505 * w ^ (-7) + 0.10960947071114842 * w ^ (-5) + -0.2139537376684645 * w ^ (-3) + 0.08624396346826982 * w ^ (-1) + 0.06666945299153335 * w ^ (1) + -0.7347411704342324 * w ^ (3) + -0.1305132725277961 * w ^ (5) + 0.12889248769798953 * w ^ (7) + 0.3252357761415583 * w ^ (9) + -0.3748292755147391 * w ^ (11) + 0.1908222206307364 * w ^ (13) + -0.0816474378784115 * w ^ (15) + 0.037504483560420285 * w ^ (17) + -0.012615756466824325 * w ^ (19) + 0.0022087474720948828 * w ^ (21)
[QuantumSignalProcessingWxPhases] Error in reconstruction from QSP angles = 0.4862234440482484
QSP angles = [0.11177647198529908, 0.321286291382733, -2.449109005349844, 0.5475854935639934, -0.23068701778999645, 1.852723853663707, -0.1638598380129409, -0.031620025684050396, -0.3018780297835489, -1.8740083256355018, 0.6314873861312269, -0.036881744534211114, 1.300525766122238, 0.3073180685644643, -0.23974313142278042, 0.09536754986888613, -1.188342341212731, -0.6092537117168764, 0.4850840958947118, 0.8800119831449805, 0.4163653216360098, 2.480415494993986]
b=20, j0=10
[PolyOneOverX] minimum [-2.90002589] is at [-0.25174082]: normalizing
Laurent P poly 0.0002108924030879319 * w ^ (-21) + -0.0006894043260278334 * w ^ (-19) + 0.001994436843136674 * w ^ (-17) + -0.005148265426149545 * w ^ (-15) + 0.011941126989562179 * w ^ (-13) + -0.02504164571900347 * w ^ (-11) + 0.04774921151669176 * w ^ (-9) + -0.08322978307559126 * w ^ (-7) + 0.1333200017468883 * w ^ (-5) + -0.1973241700493915 * w ^ (-3) + 0.2714342596621151 * w ^ (-1) + 0.2714342596621151 * w ^ (1) + -0.1973241700493915 * w ^ (3) + 0.1333200017468883 * w ^ (5) + -0.08322978307559126 * w ^ (7) + 0.04774921151669176 * w ^ (9) + -0.02504164571900347 * w ^ (11) + 0.011941126989562179 * w ^ (13) + -0.005148265426149545 * w ^ (15) + 0.001994436843136674 * w ^ (17) + -0.0006894043260278334 * w ^ (19) + 0.0002108924030879319 * w ^ (21)
Laurent Q poly -2.784691352688532e-05 * w ^ (-21) + -3.467046922689296e-06 * w ^ (-19) + -0.00023447352272530086 * w ^ (-17) + 0.00018731079814912242 * w ^ (-15) + -0.0007159634932856078 * w ^ (-13) + 0.0032821300323626736 * w ^ (-11) + -0.001907488898700853 * w ^ (-9) + 0.012591630667616198 * w ^ (-7) + -0.02371053103573988 * w ^ (-5) + -0.016629567619072996 * w ^ (-3) + -0.18519029619384528 * w ^ (-1) + -0.20476480667058175 * w ^ (1) + -0.5374170003848409 * w ^ (3) + -0.2638332742746844 * w ^ (5) + 0.21212227077358078 * w ^ (7) + 0.27748656462486654 * w ^ (9) + -0.34978762979573563 * w ^ (11) + 0.17888109364117422 * w ^ (13) + -0.07649917245226195 * w ^ (15) + 0.03551004671728361 * w ^ (17) + -0.011926352140796492 * w ^ (19) + 0.001997855069006951 * w ^ (21)
Laurent Pprime poly 0.00018304548956104657 * w ^ (-21) + -0.0006928713729505227 * w ^ (-19) + 0.001759963320411373 * w ^ (-17) + -0.0049609546280004226 * w ^ (-15) + 0.01122516349627657 * w ^ (-13) + -0.021759515686640796 * w ^ (-11) + 0.0458417226179909 * w ^ (-9) + -0.07063815240797505 * w ^ (-7) + 0.10960947071114842 * w ^ (-5) + -0.2139537376684645 * w ^ (-3) + 0.08624396346826982 * w ^ (-1) + 0.06666945299153335 * w ^ (1) + -0.7347411704342324 * w ^ (3) + -0.1305132725277961 * w ^ (5) + 0.12889248769798953 * w ^ (7) + 0.3252357761415583 * w ^ (9) + -0.3748292755147391 * w ^ (11) + 0.1908222206307364 * w ^ (13) + -0.0816474378784115 * w ^ (15) + 0.037504483560420285 * w ^ (17) + -0.012615756466824325 * w ^ (19) + 0.0022087474720948828 * w ^ (21)
[QuantumSignalProcessingWxPhases] Error in reconstruction from QSP angles = 0.4862234440482484
QSP angles = [0.11177647198529908, 0.321286291382733, -2.449109005349844, 0.5475854935639934, -0.23068701778999645, 1.852723853663707, -0.1638598380129409, -0.031620025684050396, -0.3018780297835489, -1.8740083256355018, 0.6314873861312269, -0.036881744534211114, 1.300525766122238, 0.3073180685644643, -0.23974313142278042, 0.09536754986888613, -1.188342341212731, -0.6092537117168764, 0.4850840958947118, 0.8800119831449805, 0.4163653216360098, 2.480415494993986]
```

The sign function is also useful to approximate, e.g. for oblivious amplitude amplification. Running
```
pyqsp --plot-positive-only --plot --polyargs=19,10 --plot-real-only --polyname poly_sign poly
```
produces the QSP phase angles for a degree 19 polynomial approximation, using the error function of kappa * a, where kappa is 10, with a response function as shown in this plot:
produces the QSP phase angles for a degree 19 polynomial approximation, using the error function of `kappa * a`, where kappa is 10, with a response function as shown in this plot:

![Example QSP response function approximating the sign function](https://github.com/ichuang/pyqsp/blob/master/docs/IMAGE-sample-qsp-response-for-sign-kappa-10-degree-19.png?raw=true)

A threshold function is useful, for example, for distinguishing eigenvalues and singular values. Running
A threshold function is useful, for example, for distinguishing eigenvalues and singular values. Running
```
pyqsp --plot-real-only --plot --polyargs=20,20 --polyname poly_thresh poly
```
Expand All @@ -94,7 +78,7 @@ Sine and cosine functions are useful, for example, for Hamiltonian simulation.
```
pyqsp --plot --func "np.cos(3*x)" --polydeg 6 --plot-qsp-model polyfunc
```
produces the QSP phase angles for a degree 6 polynomial approximation of cos(3*x), and produces this plot:
produces the QSP phase angles for a degree 6 polynomial approximation of `cos(3*x)`, and produces this plot:

![Example QSP response function approximating a cosine function](https://github.com/ichuang/pyqsp/blob/master/docs/IMAGE-sample-qsp-response-for-cos-using-tf-order-6.png?raw=true)

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