P^{13}_{\theta\theta}(k)=k^{3}\frac{f(k)}{f_{0}}P_{L}(k)\left(\frac{f(k)}{f_{0}}\displaystyle\sum\limits_{m_{1}}c_{m_{1}}k^{-2\nu_{1}}M^{f_{\bm{k}}}_{13,\theta\theta}(\nu_{1})+\displaystyle\sum\limits_{m_{1}}c^{f}_{m_{1}}k^{-2\nu_{1}}M^{f_{\bm{p}}}_{13,\theta\theta}(\nu_{1})\right),$
(B.12) $\displaystyle
M^{f_{\bm{k}}}_{13,\theta\theta}(\nu_{1})=-\frac{\tan(\nu_{1}\pi)}{14\pi(\nu_{1}+1)\nu_{1}(\nu_{1}-1)(\nu_{1}-2)(\nu_{1}-3)},$
(B.13) $\displaystyle
M^{f_{\bm{p}}}_{13,\theta\theta}(\nu_{1})=M^{f_{\bm{k}}}_{13,\delta\theta}(\nu_{1}).$
(B.14)

Since we use $\nu=-0.1$, the only corrections that must be added are

$\displaystyle P^{13,\text{UV}}_{\delta\delta}(k)$
$\displaystyle=-\frac{61}{105}P_{L}(k)k^{2}\sigma^{2}_{\Psi},$ (B.15)
$\displaystyle P^{13,\text{UV}}_{\delta\theta}(k)$
$\displaystyle=-\left(\frac{23}{21}\frac{f(k)}{f_{0}}\sigma^{2}_{\Psi}+\frac{2}{21}\sigma^{2}_{v}\right)k^{2}P_{L}(k),$
(B.16) $\displaystyle P^{13,\text{UV}}_{\theta\theta}(k)$
$\displaystyle=-\left(\frac{169}{105}\frac{f(k)}{f_{0}}\sigma^{2}_{\Psi}+\frac{4}{21}\sigma^{2}_{v}\right)k^{2}P^{L}_{\delta\theta}(k),$
(B.17)

where $\sigma^{2}_{\Psi}$ and $\sigma^{2}_{v}$ are given by eq. (4.32).

### Biasing power spectra

The integrals of eqs. (3.49) – (3.53) can be approximated by

$P(k)=k^{3}\sum_{m_{1},m_{2}}c_{m_{1}}k^{-2\nu_{1}}M_{P}(\nu_{1},\nu_{2})\,c_{m_{2}}k^{-2\nu_{2}},$
(B.18)

where “$P$” refers to $P_{{b_{1}}b_{2}}$, $P_{b_{1}b_{s^{2}}}$,
$P_{b^{2}_{2}}$, $P_{b_{2}b_{s^{2}}}$ and $P_{b^{2}_{s^{2}}}$, while the
matrices involved are

$\displaystyle M_{P_{b_{1}b_{2}}}\left(\nu_{1},\nu_{2}\right)$
$\displaystyle=\frac{(2\nu_{12}-3)(7\nu_{12}-4)}{28\nu_{1}\nu_{2}}I(\nu_{1},\nu_{2}),$
(B.19) $\displaystyle M_{P_{b_{1}b_{s^{2}}}}\left(\nu_{1},\nu_{2}\right)$
$\displaystyle=\Big{[}14\nu_{1}^{2}(2\nu_{2}-1)+\nu_{1}\big{(}4\nu_{2}(7\nu_{2}-11)-3\big{)}-\nu_{2}(14\nu_{2}+3)+2\Big{]}$
$\displaystyle\quad\times\frac{(2\nu_{12}-3)}{168\nu_{1}(\nu_{1}+1)\nu_{2}(\nu_{2}+1)}I(\nu_{1},\nu_{2}),$
(B.20) $\displaystyle M_{P_{b^{2}_{2}}}(\nu_{1},\nu_{2})$
$\displaystyle=\frac{1}{2}I(\nu_{1},\nu_{2}),$ (B.21) $\displaystyle
M_{P_{b_{2}b_{s^{2}}}}\left(\nu_{1},\nu_{2}\right)$
$\displaystyle=\frac{(2\nu_{1}-3)(2\nu_{2}-3)}{12\nu_{1}\nu_{2}}I(\nu_{1},\nu_{2}),$
(B.22) $\displaystyle M_{P_{b_{s^{2}}^{2}}}\left(\nu_{1},\nu_{2}\right)$
$\displaystyle=\Big{[}4\Big{(}\big{(}3+2(\nu_{1}-2)\nu_{1}\big{)}\nu_{2}^{2}+(17-4\nu_{1})\nu_{1}\nu_{2}+3(\nu_{1}-5)\nu_{1}\Big{)}-60\nu_{2}+63\Big{]}$
$\displaystyle\quad\times\frac{1}{36\nu_{1}(\nu_{1}+1)\nu_{2}(\nu_{2}+1)}I(\nu_{1},\nu_{2}).$
(B.23)

Notice that for eqs. (3.51) – (3.53), we have to subtract their large scale
constant contributions, e.g. $P_{b_{2}^{2}}(k)\rightarrow
P_{b_{2}^{2}}(k)-P_{b_{2}^{2}}(k\rightarrow 0)$.

The integrals (3.54) and (3.55) can be approximated by

$\displaystyle
P(k)=k^{3}\sum_{m_{1},m_{2}}c^{f}_{m_{1}}k^{-2\nu_{1}}M_{P}(\nu_{1},\nu_{2})\,c_{m_{2}}k^{-2\nu_{2}},$
(B.24)

here “$P$” stands for $P_{b_{2},\theta}$, $P_{b_{s^{2}},\theta}$, while

$\displaystyle M_{P_{b_{2},\theta}}\left(\nu_{1},\nu_{2}\right)$
$\displaystyle=\frac{(7\nu_{1}-4)(2\nu_{12}-3)}{14\nu_{1}\nu_{2}}I(\nu_{1},\nu_{2}),$
(B.25) $\displaystyle M_{P_{b_{s^{2}},\theta}}\left(\nu_{1},\nu_{2}\right)$
$\displaystyle=\frac{(2\nu_{12}-3)\Big{[}\nu_{1}\big{(}14\nu_{1}(2\nu_{2}-1)-30\nu_{2}+39\big{)}-10\nu_{2}-19\Big{]}}{84\nu_{1}(\nu_{1}+1)\nu_{2}(\nu_{2}+1)}I(\nu_{1},\nu_{2}),$
(B.26)

and eq. (3.57) takes the form

$\displaystyle\sigma_{3}^{2}(k)=k^{3}\sum_{m_{1}}c_{m_{1}}k^{-2\nu_{1}}M_{\sigma_{3}^{2}}(\nu_{1}),$
(B.27) $\displaystyle
M_{\sigma_{3}^{2}}(\nu_{1})=\frac{45\tan(\nu_{1}\pi)}{128\pi(\nu_{1}-3)(\nu_{1}-2)(\nu_{1}-1)\nu_{1}(\nu_{1}+1)}.$
(B.28)

For $\nu=-1.51$, the biasing power spectra do not require UV or IR
corrections.

### A function

We move our attention to eq. (A.3), which can be computed through

$\displaystyle I^{1,udd}_{1,a}(k)$
$\displaystyle=k^{3}P_{L}(k)\left[\frac{f(k)}{f_{0}}\sum_{m_{1}}c_{m_{1}}k^{-2\nu_{1}}M^{f_{\bm{k}}}_{{\text{\bf
a}}_{11}}(\nu_{1})+\sum_{m_{1}}c^{f}_{m_{1}}k^{-2\nu_{1}}M^{f_{\bm{p}}}_{{\text{\bf
a}}_{11}}(\nu_{1})\right],$ (B.29) $\displaystyle M^{f_{\bm{k}}}_{{\text{\bf
a}}_{11}}(\nu_{1})$
$\displaystyle=\frac{(15-7\nu_{1})\tan(\nu_{1}\pi)}{56\pi(\nu_{1}-3)(\nu_{1}-2)(\nu_{1}-1)\nu_{1}},$
(B.30) $\displaystyle M^{f_{\bm{p}}}_{{\text{\bf a}}_{11}}(\nu_{1})$
$\displaystyle=\frac{(7\nu_{1}-6)\tan(\nu_{1}\pi)}{56\pi(\nu_{1}-3)(\nu_{1}-2)(\nu_{1}-1)\nu_{1}}.$
(B.31)

$\displaystyle I^{2,uud}_{1,a}(k)$
$\displaystyle=k^{3}P_{L}(k)\left[\frac{f(k)}{f_{0}}\sum_{m_{1}}c^{f}_{m_{1}}k^{-2\nu_{1}}M^{f_{\bm{k}}f_{\bm{p}}}_{{\text{\bf
a}}_{12}}(\nu_{1})+\sum_{m_{1}}c^{ff}_{m_{1}}k^{-2\nu_{1}}M^{f_{\bm{p}}f_{\bm{p}}}_{{\text{\bf
a}}_{12}}(\nu_{1})\right],$ (B.32) $\displaystyle
M^{f_{\bm{k}}f_{\bm{p}}}_{{\text{\bf a}}_{12}}(\nu_{1})$
$\displaystyle=\frac{3(7\nu_{1}-13)\tan(\nu_{1}\pi)}{224\pi(\nu_{1}-3)(\nu_{1}-2)(\nu_{1}-1)\nu_{1}(\nu_{1}+1)},$
(B.33) $\displaystyle M^{f_{\bm{p}}f_{\bm{p}}}_{{\text{\bf a}}_{12}}(\nu_{1})$
$\displaystyle=\frac{3(1-7\nu_{1})\tan(\nu_{1}\pi)}{224\pi(\nu_{1}-3)(\nu_{1}-2)(\nu_{1}-1)\nu_{1}(\nu_{1}+1)}.$
(B.34)

$\displaystyle I^{3,uuu}_{3,a}(k)$
$\displaystyle=k^{3}\frac{f(k)}{f_{0}}P_{L}(k)\Bigg{[}\frac{f(k)}{f_{0}}\sum_{m_{1}}c^{f}_{m_{1}}k^{-2\nu_{1}}M^{f_{\bm{k}}f_{\bm{k}}f_{\bm{p}}}_{{\text{\bf
a}}_{33}}(\nu_{1})$
$\displaystyle\quad+\sum_{m_{1}}c^{ff}_{m_{1}}k^{-2\nu_{1}}M^{f_{\bm{k}}f_{\bm{p}}f_{\bm{p}}}_{{\text{\bf
a}}_{33}}(\nu_{1})\Bigg{]},$ (B.35) $\displaystyle
M^{f_{\bm{k}}f_{\bm{k}}f_{\bm{p}}}_{{\text{\bf a}}_{33}}(\nu_{1})$
$\displaystyle=\frac{\big{(}\nu_{1}(53-28\nu_{1})+21\big{)}\tan(\nu_{1}\pi)}{224\pi(\nu_{1}-3)(\nu_{1}-2)(\nu_{1}-1)\nu_{1}(\nu_{1}+1)},$
(B.36) $\displaystyle M^{f_{\bm{k}}f_{\bm{p}}f_{\bm{p}}}_{{\text{\bf
a}}_{33}}(\nu_{1})$
$\displaystyle=\frac{\big{(}\nu_{1}(28\nu_{1}-17)-21\big{)}\tan(\nu_{1}\pi)}{224\pi(\nu_{1}-3)(\nu_{1}-2)(\nu_{1}-1)\nu_{1}(\nu_{1}+1)}.$
(B.37) $\displaystyle I^{2,uud}_{2,a}(k)$
$\displaystyle=\frac{f_{0}}{f(k)}I^{3,uuu}_{3,a}(k)+\frac{f(k)}{f_{0}}I^{1,udd}_{1,a}(k),$
(B.38) $\displaystyle I^{3,uuu}_{2,a}(k)$
$\displaystyle=\frac{f(k)}{f_{0}}I^{2,uud}_{1,a}(k),$ (B.39)

where (B.38) and (B.39) reduce to trivial computations.

The rest of the functions $I^{m}_{n}$ are

$\displaystyle
I^{1,udd}_{1,b}(k)=k^{3}\sum_{m_{1},m_{2}}c^{f}_{m_{1}}k^{-2\nu_{1}}M^{f_{\bm{p}}}_{\tilde{\mathcal{A}}_{11}}(\nu_{1},\nu_{2})\,c_{m_{2}}k^{-2\nu_{2}},$
(B.40) $\displaystyle
M^{f_{\bm{p}}}_{\tilde{\mathcal{A}}_{11}}(\nu_{1},\nu_{2})=\frac{(2\nu_{12}-3)(2\nu_{12}-1)\big{(}\nu_{1}(7\nu_{12}-4)-5\big{)}}{7\nu_{1}(\nu_{1}+1)(2\nu_{1}-1)\nu_{2}}I(\nu_{1},\nu_{2}).$
(B.41)

$\displaystyle
I^{2,uud}_{1,b}(k)=k^{3}\sum_{m_{1},m_{2}}c^{f}_{m_{1}}k^{-2\nu_{1}}M^{f_{\bm{k}-\bm{p}}f_{\bm{p}}}_{\tilde{\mathcal{A}}_{12}}(\nu_{1},\nu_{2})\,c^{f}_{m_{2}}k^{-2\nu_{2}},$
(B.42) $\displaystyle
M^{f_{\bm{k}-\bm{p}}f_{\bm{p}}}_{\tilde{\mathcal{A}}_{12}}(\nu_{1},\nu_{2})=-\frac{(2\nu_{12}-3)(2\nu_{12}-1)(7\nu_{12}+6)}{56\nu_{1}(\nu_{1}+1)\nu_{2}(\nu_{2}+1)}I(\nu_{1},\nu_{2}).$
(B.43)

$\displaystyle I^{2,uud}_{2,b}(k)$
$\displaystyle=k^{3}\sum_{m_{1},m_{2}}c^{f}_{m_{1}}k^{-2\nu_{1}}M^{f_{\bm{k}-\bm{p}}f_{\bm{p}}}_{\tilde{\mathcal{A}}_{22}}(\nu_{1},\nu_{2})\,c^{f}_{m_{2}}k^{-2\nu_{2}}$
$\displaystyle\quad+k^{3}\sum_{m_{1},m_{2}}c^{ff}_{m_{1}}k^{-2\nu_{1}}M^{f_{\bm{p}}f_{\bm{p}}}_{\tilde{\mathcal{A}}_{22}}(\nu_{1},\nu_{2})\,c_{m_{2}}k^{-2\nu_{2}},$
(B.44) $\displaystyle
M^{f_{\bm{k}-\bm{p}}f_{\bm{p}}}_{\tilde{\mathcal{A}}_{22}}(\nu_{1},\nu_{2})$
$\displaystyle=\Big{[}336\nu_{1}\nu_{2}^{4}+4\big{(}4\nu_{1}(35\nu_{1}-19)-39\big{)}\nu_{2}^{3}+48\nu_{1}\big{(}\nu_{1}(7\nu_{1}-3)-9\big{)}\nu_{2}^{2}$
$\displaystyle\quad+8\nu_{1}\Big{(}2\nu_{1}\big{(}\nu_{1}(7\nu_{1}-4)-28\big{)}+41\Big{)}\nu_{2}+3\nu_{1}\big{(}4\nu_{1}(10-9\nu_{1})+1\big{)}$
$\displaystyle\quad+75\nu_{2}-18\Big{]}\times\frac{2\nu_{12}-3}{56\nu_{1}(\nu_{1}+1)(2\nu_{1}-1)\nu_{2}(\nu_{2}+1)(2\nu_{2}-1)}I(\nu_{1},\nu_{2}),$
(B.45) $\displaystyle
M^{f_{\bm{p}}f_{\bm{p}}}_{\tilde{\mathcal{A}}_{22}}(\nu_{1},\nu_{2})$
$\displaystyle=\frac{(2\nu_{12}-3)\left[\nu_{1}(7\nu_{12}-4)+3\nu_{2}-5\right]}{7\nu_{1}(\nu_{1}+1)\nu_{2}}I(\nu_{1},\nu_{2}).$
(B.46)

$\displaystyle
I^{3,uuu}_{2,b}(k)=k^{3}\sum_{m_{1},m_{2}}c^{ff}_{m_{1}}k^{-2\nu_{1}}M^{f_{\bm{k}-\bm{p}}f_{\bm{p}}f_{\bm{p}}}_{\tilde{\mathcal{A}}_{23}}(\nu_{1},\nu_{2})\,c^{f}_{m_{2}}k^{-2\nu_{2}},$
(B.47) $\displaystyle
M^{f_{\bm{k}-\bm{p}}f_{\bm{p}}f_{\bm{p}}}_{\tilde{\mathcal{A}}_{23}}(\nu_{1},\nu_{2})=-\frac{(7\nu_{1}-1)(2\nu_{12}-3)(2\nu_{12}-1)}{28\nu_{1}(\nu_{1}+1)\nu_{2}(\nu_{2}+1)}I(\nu_{1},\nu_{2}).$
(B.48)

$\displaystyle
I^{3,uuu}_{3,b}(k)=k^{3}\sum_{m_{1},m_{2}}c^{ff}_{m_{1}}k^{-2\nu_{1}}M^{f_{\bm{k}-\bm{p}}f_{\bm{p}}f_{\bm{p}}}_{\tilde{\mathcal{A}}_{33}}(\nu_{1},\nu_{2})\,c^{f}_{m_{2}}k^{-2\nu_{2}},$
(B.49) $\displaystyle
M^{f_{\bm{k}-\bm{p}}f_{\bm{p}}f_{\bm{p}}}_{\tilde{\mathcal{A}}_{33}}(\nu_{1},\nu_{2})=\Big{[}4(7\nu_{1}+3)\nu_{2}^{2}+2(\nu_{1}(14\nu_{1}-1)-11)\nu_{2}-13(\nu_{1}+1)\Big{]}$
$\displaystyle\quad\qquad\qquad\qquad\qquad\times\frac{(2\nu_{12}-3)(2\nu_{12}-1)}{28\nu_{1}(\nu_{1}+1)\nu_{2}(\nu_{2}+1)(2\nu_{2}-1)}I(\nu_{1},\nu_{2}).$
(B.50)

Finally, for the $A$ functions, we have to add the following corrections

$\displaystyle I^{1,udd,\,\text{UV}}_{1,a}(k)$
$\displaystyle=\left(\frac{92}{35}\frac{f(k)}{f_{0}}\sigma^{2}_{\Psi}-\frac{18}{7}\sigma^{2}_{v}\right)k^{2}P_{L}(k),$
(B.51) $\displaystyle I^{2,uud,\,\text{UV}}_{1,a}(k)$
$\displaystyle=-\left(\frac{38}{35}\frac{f(k)}{f_{0}}\sigma^{2}_{v}+\frac{2}{7}\sigma^{2}_{vv}\right)k^{2}P_{L}(k),$
(B.52) $\displaystyle I^{3,uuu,\,\text{UV}}_{3,a}(k)$
$\displaystyle=-\left(\frac{16}{35}\frac{f(k)}{f_{0}}\sigma^{2}_{v}+\frac{6}{7}\sigma^{2}_{vv}\right)k^{2}P^{L}_{\delta\theta}(k),$
(B.53)

with $\sigma^{2}_{\Psi}$ and $\sigma^{2}_{v}$ given by eq. (4.32), and
$\sigma^{2}_{vv}$ by eq. (A.18).

### D function

Finally, we display the FFTLog expressions for the $D(k,\mu)$ function of eq.
(A).

For $I^{2,uudd}_{n,D}=I^{2,uudd}_{n,B}+I^{2,uudd}_{n,C}$ with $n=1,2$,
corresponding to eqs. (A.19) and (A.22):

$\displaystyle
I^{2,uudd}_{n,B}(k)=k^{3}\sum_{m_{1},m_{2}}c^{f}_{m_{1}}k^{-2\nu_{1}}M_{\mathcal{B}^{n}_{11}}(\nu_{1},\nu_{2})\,c^{f}_{m_{2}}k^{-2\nu_{2}},$
(B.54) $\displaystyle
M_{\mathcal{B}^{1}_{11}}(\nu_{1},\nu_{2})=\frac{3-2\nu_{12}}{4\nu_{1}\nu_{2}}I(\nu_{1},\nu_{2}),$
(B.55) $\displaystyle
M_{\mathcal{B}^{2}_{11}}(\nu_{1},\nu_{2})=\frac{(2\nu_{12}-3)(2\nu_{12}-1)}{4\nu_{1}\nu_{2}}I(\nu_{1},\nu_{2}).$
(B.56)

and

$\displaystyle
I^{2,uudd}_{n,C}(k)=k^{3}\sum_{m_{1},m_{2}}c_{m_{1}}k^{-2\nu_{1}}M_{\mathcal{C}^{n}_{11}}(\nu_{1},\nu_{2})\,c^{ff}_{m_{2}}k^{-2\nu_{2}},$
(B.57) $\displaystyle
M_{\mathcal{C}^{1}_{11}}(\nu_{1},\nu_{2})=\frac{(2\nu_{1}-3)(2\nu_{12}-3)}{4\nu_{2}(\nu_{2}+1)(2\nu_{2}-1)}I(\nu_{1},\nu_{2}),$
(B.58) $\displaystyle
M_{\mathcal{C}^{2}_{11}}(\nu_{1},\nu_{2})=\frac{(2\nu_{12}-3)(2\nu_{12}-1)}{4\nu_{2}(\nu_{2}+1)}I(\nu_{1},\nu_{2}).$
(B.59)

For $I^{3,uuud}_{n,D}(k)$ with $n=2,3$, corresponding to eqs. (A.25):

$\displaystyle I^{3,uuud}_{n,D}(k)$
$\displaystyle=k^{3}\sum_{m_{1},m_{2}}c^{f}_{m_{1}}k^{-2\nu_{1}}M_{\mathcal{D}^{n}_{21}}(\nu_{1},\nu_{2})\,c^{ff}_{m_{2}}k^{-2\nu_{2}},$
(B.60) $\displaystyle M_{\mathcal{D}^{2}_{21}}(\nu_{1},\nu_{2})$
$\displaystyle=\frac{(2\nu_{1}-4\nu_{2}-1)(2\nu_{12}-3)(2\nu_{12}-1)}{4\nu_{1}\nu_{2}(\nu_{2}+1)(2\nu_{2}-1)}I(\nu_{1},\nu_{2}),$
(B.61) $\displaystyle M_{\mathcal{D}^{3}_{21}}(\nu_{1},\nu_{2})$
$\displaystyle=\frac{(3-2\nu_{12})\left(1-4\nu^{2}_{12}\right)}{4\nu_{1}\nu_{2}(\nu_{2}+1)}I(\nu_{1},\nu_{2}).$
(B.62)

Finally for $I^{4,uuuu}_{n,D}(k)$ with $n=2,3,4$, corresponding to eqs.
(A.28):

$\displaystyle
I^{4,uuuu}_{n,D}(k)=k^{3}\sum_{m_{1},m_{2}}c^{ff}_{m_{1}}k^{-2\nu_{1}}M_{\mathcal{D}^{n}_{22}}(\nu_{1},\nu_{2})\,c^{ff}_{m_{2}}k^{-2\nu_{2}},$
(B.63) $\displaystyle
M_{\mathcal{D}^{2}_{22}}(\nu_{1},\nu_{2})=\frac{3(3-2\nu_{12})(1-2\nu_{12})}{32\nu_{1}(\nu_{1}+1)\nu_{2}(\nu_{2}+1)}I(\nu_{1},\nu_{2}),$
(B.64) $\displaystyle
M_{\mathcal{D}^{3}_{22}}(\nu_{1},\nu_{2})=\frac{(3-2\nu_{12})\left(1-4\nu^{2}_{12}\right)\left[1+2\left(\nu^{2}_{1}-4\nu_{1}\nu_{2}+\nu^{2}_{2}\right)\right]}{16\nu_{1}(\nu_{1}+1)(2\nu_{1}-1)\nu_{2}(\nu_{2}+1)(2\nu_{2}-1)}I(\nu_{1},\nu_{2}),$
(B.65) $\displaystyle
M_{\mathcal{D}^{4}_{22}}(\nu_{1},\nu_{2})=\frac{\left(9-4\nu^{2}_{12}\right)\left(1-4\nu^{2}_{12}\right)}{32\nu_{1}(\nu_{1}+1)\nu_{2}(\nu_{2}+1)}I(\nu_{1},\nu_{2}).$
(B.66)

Functions $D$ do not require IR or UV corrections with our choice of bias
$\nu=-0.1$.

##  Appendix C Complementary plots

In this appendix we show complementary plots to §5 that we decided to not show
in the main text to avoid too much cluttering.

Figure 10: Contour plots for the posterior distributions at 68 and 95%
confidence level computed with different covariance matrices, corresponding to
$V_{1}$, $V_{5}$, $V_{25}$ and $V_{100}$ effective volume data sets. This is
the case of mass $M_{\nu}=0.4\,\text{eV}$ at redshift $z=0.5$. The fittings
are performed using the monopole and quadrupole of the power spectrum up to
$k_{\text{max}}=0.2\,\,h\,\text{Mpc}^{-1}$. Vertical and horizontal dashed
lines show the values of the simulations. This is a complementary figure to 2
and table 2 Figure 11: Comparison on the constraints of the cosmological
parameters when using fk\- and EdS-kernels for the volume $V_{100}$. This is
the case of $M_{\nu}=0.4\,\text{eV}$ at redshift $z=0.5$ when fitting up to
$k_{\text{max}}=0.2\,\,h\,\text{Mpc}^{-1}$ Figure 12: Comparison on the
constraints of the cosmological parameters when using fk\- and EdS-kernels for
the volume $V_{25}$. This is the case of $M_{\nu}=0.4\,\text{eV}$ at redshift
$z=0.5$ when fitting up to $k_{\text{max}}=0.2\,\,h\,\text{Mpc}^{-1}$ Figure
13: Contour plots for the posterior distributions at 68 and 95% confidence
level computed for different values of the maximum wave-number
$k_{\text{max}}=0.15,\,0.2,\,0.25\,h\,\text{Mpc}^{-1}$. This is the case of
mass $M_{\nu}=0.4\,\text{eV}$ at redshift $z=0.5$ using the DESI-like volume
$V_{25}$. Vertical and horizontal dashed lines show the values of the
simulations. This is a complementary figure to 4.

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Dual-Parameterized Quantum Circuit GAN Model in High Energy Physics

Su Yeon<EMAIL_ADDRESS>Steven Herbert3<EMAIL_ADDRESS>Sofia<EMAIL_ADDRESS>Elías F. Combarro5 Ross Duncan3,6,7

OpenLab, CERN, Geneva, Switzerland
Department of Physics, EPFL, Lausanne, Switzerland
Cambridge Quantum Computing Ltd, Cambridge, UK
Department of Computer Science and Technology, University of Cambridge, Cambridge, UK
Department of Computer Science, University of Oviedo, Oviedo, Spain
Department of Computer and Information Sciences, University of Strathclyde, Glasgow, UK
Department of Physics, University College London, London, UK

Generative models, and Generative Adversarial Networks (GAN) in
particular, are being studied as possible alternatives to Monte
Carlo simulations.  It has been proposed that, in
certain circumstances, simulation using GANs can be sped-up
by using quantum GANs (qGANs).

We present a new design of qGAN, the dual-Parameterized Quantum
Circuit (PQC) GAN, which consists of a classical discriminator and two
quantum generators which take the form of PQCs. The first PQC learns
a probability distribution over $N$-pixel images, while
the second generates normalized pixel intensities of an individual
image for each PQC input.

With a view to HEP applications, we evaluated the dual-PQC
architecture on the task of imitating calorimeter outputs,
translated into pixelated images.  The results demonstrate that the
model can reproduce a fixed number of images with a reduced size as
well as their probability distribution and we anticipate it should
allow us to scale up to real calorimeter outputs.

§ INTRODUCTION

The next High Luminosity Large Hadron Collider (HL-LHC) phase will
collect an overwhelming amount of data, with complex physics and small
statistical error.  To analyse this data, high precision methods which
use only limited resources are needed.  Traditional Monte Carlo based
simulation, such as Geant4 <cit.> and
the GeantV prototype <cit.> for full
simulation of particle transport, is however very time-consuming,
therefore new approaches using deep neural networks have been studied
for fast simulations.

Generative Adversarial Networks (GAN) are a strong candidate for such fast simulations. Based on two neural networks, generator and discriminator, trained alternatively, GANs have been widely explored thanks to their ability to generate images with complex structures at much high speed.  In HEP, the variations of GAN, such as CaloGAN <cit.> and 3DGAN <cit.>, have achieved similar performance as full Monte Carlo based simulation, but with reduced time taken.

At the same time, quantum computing has emerged as another important pillar in modern research attracting the attention of many researchers due to its potential to execute certain tasks with an exponentially reduced amount of resources both in time and space compared to classical processors <cit.>. It has already shown promising results in various fields, such as optimization <cit.> and  cryptography <cit.>.

Advances in both deep learning and quantum computing suggest to merge them to benefit their advantages at once, leading to a new field of study, so-called Quantum Machine Learning (QML). Quantum Generative Adversarial Networks, which are the quantum version of GANs, are one of its examples. Several quantum GAN models have been investigated in the last few years, but the scientific community still confronts a need to further explore in order to apply the model to more realistic use-cases.

In this paper, we propose for the first time a Dual Parameterized Quantum Circuit GAN model (dual-PQC GAN model) as one of the improvements to overcome the remaining limitations of quantum GANs. This model uses two parametrized quantum circuits, which share the role of a single quantum generator: the first PQC learns the distribution over image samples, while the second PQC determines the amplitude distribution over pixels on a single image. Thanks to this separation, it is possible to exploit the continuous nature of probability distributions over output states in quantum circuits to represent continuous variables.

This paper is organized as follows. sec:GAN_in_HEP summarizes the application of Generative Adversarial Networks in HEP. We then present a short overview of  a quantum version of GAN in sec:QGAN. In sec:dual_PQC and sec:dual_PQC_training, the first prototype of a dual-PQC GAN model is proposed, with the results of its simulation. This paper concludes with sec:conclusion, which summarizes and gives an outlook of future works.

§ APPLICATIONS OF GANS IN HEP

GANs, designed by I. Goodfellow et al. in 2014 <cit.>, are
deep generative models which aim to reproduce new data from a given
original training set. They are characterized by two deep neural
networks, Generator $G$ and Discriminator $D$, which
are trained alternatively. During the training, $G$ progressively
generates data similar to the real one, while $D$ increases the
probability of assigning the correct labels to both real and fake
data. Numerous improvements have been made since the initial proposal
and, in particular, GANs have achieved remarkable success in image
processing and generation, via variations such as Deep Convolutional
GAN (DCGAN) <cit.>, Auxiliary Classifier GAN (ACGAN)
<cit.>, Progressive GAN <cit.>, etc.

The evolution of GANs has attracted strong interest in the high energy physics domain. In HEP, the detectors can be described as 3D cameras, recording pictures of particle collisions. Calorimeters, in particular, measure the energies deposited by a shower of the particles that traverse them. They generally consist of alternate arrays of active sensor material and passive dense layers to ensure that the incoming (primary) particle will deposit most of its energy inside their volume. These energy depositions can be compared to the monochromatic pixel intensities of a 3D image. Because of their high granularity, the detailed simulation of a calorimeter is particularly time-consuming. As a result, GANs come into the limelight to allow fast simulation of particle showers with high fidelity.

One possible application of GAN in HEP is 3DGAN <cit.>, which is a 3D extension of GAN, using 3D (de-)convolutional layers to capture the whole 3D energy profile. It simultaneously performs two additional tasks of estimating the incoming particle energy and measuring the total deposited energy to enhance the stability and convergence of networks. Details on 3DGAN architecture and its performance validation  are available in  <cit.>.

§ QUANTUM GENERATIVE ADVERSARIAL NETWORKS

The possibility of combining machine learning and quantum computing also led to the generalization of GAN to quantum systems by S. Lloyd and C. Weedbrook <cit.>. The main mechanism of the model, the adversarial training,  is reproduced, but different scenarios are possible:  the input data can be either quantum data or classical data embedded in quantum states and the discriminator/the generator can also be either classical or quantum.

Since its initial proposal, several quantum GAN (qGAN) variations have
been suggested to generate either classical data <cit.> or quantum data <cit.>.
Zoufal et al <cit.> proposed a hybrid qGAN model composed
of a quantum generator and a classical discriminator to train on
classical data. During the training, the generator learns an
arbitrary probability distribution over discrete variables, which is
encoded in the amplitudes of the final quantum state. This model was
applied of quantum finance, and demonstrated using the real quantum
hardware, IBM Q Boeblingen. Anand et al <cit.> also present similar results, but simulated on the real Rigetti quantum hardware, Aspen-4-2Q-A.
Unlike the aforementioned models treating classical data, Situ et al <cit.> propose a qGAN model which aims to approximate an unknown pure quantum state with a quantum generator and a quantum discriminator. One problem with quantum machine learning models is the apparent difficulty of training PQCs, captured by the vanishing gradient and barren plateau problems. Fortunately, there have been also studies on the methods to improve the performance of qGAN, for instance, Quantum Multiplicative Matrix Weight (QMMW), which helps to avoid mode collapse or vanishing gradient problem in qGAN <cit.>.

The results from the previous research are impressive, showing the potential of qGAN for the near-term quantum hardware. However, additional investigations are needed in order to fully understand the quantum advantages of qGAN and generate not only simple probability distributions but also  more complex image samples.

§ DUAL-PQC GAN MODEL

rounded corners,fill=yellow!20, inner xsep=1pt, inner ysep = 1pt,background, label style=label position=below,anchor=north,yshift=-0.3cm ]Initialization         1[3,steps=3,style=dashed,
rounded corners,fill=blue!20, inner xsep=1pt, inner ysep = 1pt,background, label style=label position=below,anchor=north,yshift=-0.3cm ]layer $1$        R_y(ϕ^0_1)         …         1[3,steps=3,style=dashed,
rounded corners,fill=blue!20, inner xsep=1pt, inner ysep = 1pt,background, label style=label position=below,anchor=north,yshift=-0.3cm ]layer $k$        R_y(ϕ^0_k)
R_y(ϕ^1_0)              1     R_y(ϕ^1_1)         …              1      R_y(ϕ^1_k)

R_y(ϕ^2_0)                  R_y(ϕ^2_1)         …                 R_y(ϕ^2_k)
Quantum variational form using Pauli-$Y$ rotations and $CZ$ gates with depth $k$.

Our  preliminary experiments involved training a qGAN, as conceptualised in <cit.>, for the calorimeter problem – however this immediately revealed a problem: as the image itself is encoded in the amplitudes of the computational basis states, this meant that only the average of all the training samples could be learned, and the GAN did not, therefore, sample typical images. To put this in more precise terms, in order to achieve the exponential compression of representing a $N = 2^n$ pixel image using $n$ qubits, it follows that the qGAN prepares a quantum state of the form:

\begin{equation}
\ket{\psi} =  {\textstyle\sum}_{j=0}^{2^{n}-1}\sqrt{I_{j}}\ket{j}
\end{equation}

where $I_j$ is the intensity of the $j^{th}$ pixel. So we can see
that, in a sense, the qGAN encodes a single image as
a probability distribution, and so there is no room left to also
encode a probability distribution, representing the full dataset, to
sample images from. This problem does not arise in classical GANs,
where a single sample from a $\mathcal{O}(N)$ sized neural network
generates a single image in one go.

In this section we describe our solution, a new type of quantum GAN,
the dual Parameterized Quantum Circuit (PQC) GAN model
(dual-PQC GAN model). It aims to reproduce a set of image samples from
real training data while preserving the exponential compression
achieved by amplitude encoding.  The work in
<cit.> follows from similar motivation.

The dual-PQC GAN is a hybrid qGAN architecture which has one classical
discriminator and two parameterized quantum circuits, PQC1 and PQC2,
sharing the role of the generator.  PQC1, with $n_1$ qubits, learns
the probability distribution over image samples and PQC2, with $n_2$
qubits, learns the amplitude distribution over pixels of each
image.  The classical discriminator takes the training set and the
images generated by PQC2, and it classifies them into real and
fake. The predicted labels are used to tune alternatively $\phi_1$,
$\phi_2$ and $\theta$, the parameters for PQC1, PQC2, and the
discriminator, respectively.

In this study, both PQCs consist of alternating layers of
single-qubit Pauli-rotations and a set of two-qubit entanglement
gates, as shown in fig:vf It is widely used in quantum
machine learning thanks to its strong expressive power, offering an
effective way of reconstructing an expected behaviour
<cit.>. We use RY rotation gates and CZ
entanglement gates, but other choices are possible.

Generator2 = [rectangle, draw, fill=green!20,
text width=4.6em, text centered, rounded corners, minimum height=1.3cm]
Discriminator2 = [rectangle, draw, fill=blue!20,
text width=6.8em, text centered, rounded corners, minimum height=1cm]
input2 = [rectangle, draw, fill= white,
text width=0.3em, minimum height = 2.3em, align= center,
copy shadow=draw, shadow xshift=1mm, shadow yshift=-1mm ]
meter/.append style=draw, inner sep=4, rectangle, font=, minimum width=16, line width=.36,
path picture=[black] ([shift=(.04,.12)]path picture bounding box.south west) to[bend left=50] ([shift=(-.04,.12)]path picture bounding box.south east);[black,-latex] ([shift=(0,.04)]path picture bounding box.south) – ([shift=(.12,-.04)]path picture bounding box.north);

line = [draw, -latex']

[align=center,node distance = 2cm, auto]

(6.5,-0.5) node[input2, label = below:
Fake Data
by sampling] (FakeData) input;
[Discriminator2, right of = FakeData, node distance=2.3cm] (Disc) Classical
[input2, above of = Disc, node distance=1.3cm, label = right:Real Data] (RealData) input;
[text width = 3.5em, right of = Disc, node distance = 2.5cm, minimum height = 1.6cm](PredictedLabel) Predicted Labels;
[text width = 3.8em](Discard) at (7,0.5) Discard;
[rectangle, text width=6em, minimum height = 1em, align= center] (Basis) at (0,-0.5) ${\ket{0},...,\ket{2^{n_1}-1}}$;
[rectangle, text width=4em, minimum height = 1em, align= center] (Zeros) at (0,0.5) $\ket{0}^{\otimes n_2 - n_1}$;
[Generator2, below of = Basis, node distance = 1.9cm] (PQC1)   PQC1
$n_1$ qubits
depth $d_{g,1}$;
[Generator2] (PQC2) at (3,0)  PQC2
$n_2$ qubits
depth $d_{g,2}$;

[thick] (4.4, 0.35) – (4.7, 0.65);
[thick] (4.4, -0.65) – (4.7, -0.35);

[thick] (1.4, 0.35) – (1.7, 0.65);
[thick] (1.4, -0.65) – (1.7, -0.35);

[text width = 5em] at (4.55,0.72) $n_2 - n$;
[text width = 1em] at (4.55,-0.28) $n$;

[text width = 5em] at (1.55,0.72) $n_2 - n_1$;
[text width = 1em] at (1.55,-0.28) $n_1$;

[line] (PQC1) – (Basis);
(Basis)  |- (PQC2.west |-Basis.east);
(Zeros)  |- (PQC2.west |-Zeros.east);
[line] (PQC2.east|-FakeData.west) |- (FakeData);
[line] (PQC2.east|-Discard.west) |- (Discard);
[line] (FakeData.east) – (Disc);
[line] (Disc) – (PredictedLabel);
[line] (RealData) – (Disc);
[line, style = dashed] (PredictedLabel.south) – ++(0,-1.1cm)  -| (Disc.south);
[line, style = dashed] (PredictedLabel.south) – ++(0,-1.1cm)  -|node [pos = 0.25, below] (TextNode) Classical Optimization  (PQC2.south);
[line, style = dashed] (PredictedLabel.south) – ++(0,-1.1cm)  – (PQC1.east);

[meter, above of = PQC1, node distance = 1.1cm](meter);
[meter, left of = Discard, node distance = 1.45cm](meter);
[meter, left of = FakeData, node distance = 0.95cm](meter);

Schematic Diagram of dual-PQC GAN to reproduce images of $2^n$ pixels.

Consider a training set $X \subset \mathbb{R}^{2^n}$ of $N = 2^n$
pixel images. To begin, the output state of PQC1 is measured
producing $n_1$ bits.  Then, via a set of Pauli-$X$ gates, this bit string
is used to initialise PQC2 with the corresponding computational basis
state in the $2^{n_1}$ dimensional Hilbert space, $\ket{i} \in
\{\ket{0},..., \ket{2^{n_1} - 1}\}$.  Then, by repeatedly measuring
$n$ output qubits of PQC2, the probability distribution over the
computational basis in the $2^{n}$ dimensional Hilbert space, $\ket{i}
\in \{\ket{0},..., \ket{2^n - 1}\}$, is constructed and translated as
an image of size $2^n$ for each input state.

Note that since PQC2 performs a unitary operation, and since its
inputs are always computational basis states, the output quantum
states are necessarily orthogonal. This puts an unwanted restriction
on the possible images.  Relying on the Stinespring dilation theorem,
we can remove this restriction by choosing $n_2 > n_1, n$ and using
some ancilla qubits which are discarded at the end.  We will return to
this point at the  end of the  section.

Let $p_g^i$ be the probability that state $\ket{i}$ is
measured by PQC1; let $\mathcal{I}_i$ denote the normalized image
produced by PQC2 when given the input state $\ket{i}$; and  let $I_{ij}$ be
the amplitude of $j^{th}$ pixel of $\mathcal{I}_i$ with $i\in \{0,..., 2^{n_1}-1\}$ and $j \in \{0,..., 2^n-1\}$. Then the output states generated by PQC1, $G_{1,\phi_1}$, and PQC2, $G_{2,\phi_2}$, are explicitly given as :

\begin{equation}
G_{1,\phi_1}\ket{\psi_{initial}} = \ket{g_{1,\phi_1}} = \sum_{j=0}^{2^{n_1}-1}\sqrt{p_g^j}\ket{j},
\end{equation}

\begin{equation}
%G_{2,\phi_2}\ket{i} = \ket{g_{2,\phi_2}} = %\sum_{j=0}^{2^{n}-1}\sqrt{I_{ij}}\ket{j}\otimes\sum_{j=0}^{2^{n-n_2}-1}c_j\ket{j},
%SJH alternative -- original still kept in comments above.
G_{2,\phi_2}\ket{\mathbf0}\ket{i} = \ket{g_{2,\phi_2}} = \sum_{j=0}^{2^{n}-1}\sqrt{I_{ij}}\ket{\Psi_j} \ket{j}
\end{equation}

where $\ket{\psi_{initial}}$ is the input state of PQC1 fixed during the whole training and $\ket{\Psi_j}$ are some $n-n_2$ qubit states that we discard.

During the training, PQC1 learns the distribution $p_g(i)$ over $\mathcal{I}_i$, so that it approaches to the real distribution, $p_{real}$ over $X$. On the other hand, PQC2 learns the  amplitude over $2^n$ pixels for $2^{n_1}$ images, to make $\mathcal{I}_i$ as close as possible to real images. At the end of the training, the true/fake probabilty  predicted by the discriminator for $\mathcal{I}_i$, $D(\mathcal{I}_i)$ should converge to $1/2$.

For the following simulations, we use a modified min-max loss, given as:

\begin{equation}
L_G =  -\frac{1}{m}\sum_{i = 1}^m \log D(G( \mathbf{z}_i)) = -\sum_{i=0}^{2^{n_1} - 1} p_g^i\log(D(\mathcal{I}_i))
\label{eq:lg}
\end{equation}

\begin{equation}
L_D = \frac{1}{m} \sum_{i = 1}^{m} \big[\log D(\mathbf{x}_i) + \log ( 1 - D(G( \mathbf{z}_i)))\big] = \frac{1}{m} \sum_{i = 1}^{m} \log D(\mathbf{x}_i) + \sum_{i=0}^{2^{n_1} - 1} p_g^i\ \log ( 1 - D(\mathcal{I}_i)) ,
\label{eq:ld}
\end{equation}

where $m$ is the batch size, $x_i$ the real data and $z_i$ random input. The first equality gives the definition of the loss in the classical GAN and the second equality the practical formula used in dual-PQC GAN simulations.

Based on the calculated loss, the parameters in the quantum generator are tuned by computing the analytic quantum gradient descent <cit.>, while the discriminator is optimized in the exactly same way as in classical GAN. Further details on the method used for analytic quantum gradient are explained in Ref. <cit.>.

Ultimately, the dual-PQC GAN model can generate $2^{n_1}$ images of
size $2^n$. Increasing the number of qubits used in PQC1 and PQC2
allows to increase both the number and size of produced images. This
model shows an advantage in terms of computational resources
by using only $\mathcal{O}(\log(N))$ qubits,     compared to the
classical neural networks with $\mathcal{O}(N)$ neurons to reproduce
an image of size $N$. Specifically, the potential advantages are
threefold: firstly, there is an exponential reduction in space
(memory) requirement; secondly, the resultant exponential reduction in
number of tunable parameters (i.e. the number of tunable parameters is
proportional to the number of gates in the PQC or neurons in the
classical neural network) suggests that training could be performed
more efficiently; finally, it is potentially advantageous to have the
images encoded in quantum states if further processing is to be
performed (for example if that image processing can itself be
performed more efficiently using quantum computing). It should,
however, be noted that if all one wants to do is to generate
an image, then the number of samples required from PQC2 is exponential
in the number of qubits.

How many ancillas are needed?

Since we want to be able to reproduce any collection of images, we
have to use some ancillary qubits.  Stinespring's theorem gives an
upper bound on the number of ancillas but it is not tight.  We will
now show that $n_2 = 2n$ will suffice when we assume for simplicity that $n_1 = n$.

Let $U$ denote the unitary
matrix corresponding to PQC2.  When $U$ is applied to the input state
$\ket0^{\otimes n}\otimes \ket i$, where $\ket i$ is an $n$-qubit basis
state, the resulting state is always one of the columns of $U$; in
particular it always one of the first $2^n$ columns.  We'll construct
an example of $U$ which realises $2^n$ arbitrary images.  Let

\[
\ket{\mathcal{I}_i}
= (\sqrt{I_{i,0}}e^{i\phi_{i,0}}, \ldots, \sqrt{I_{i,0}}e^{i\phi_{i,0}})^T
\]

be a state whose amplitudes encode the pixels of image $i$.  Consider
the state $\ket{U(i)} = \ket i \otimes \ket{\mathcal{I}_i}$, and
observe that if its first $n$ qubits are measured in the computational
basis and discarded, then the remaining quantum state is precisely
$\ket{\mathcal{I}_i}$.  Since $\bra{U(i)}\ket{U(j)} = \delta_{ij}$, ie, they are orthonormal, we
can construct the required unitary matrix, $U$, by setting the first $2^n$
columns to be $\ket{U(i)}$ for $i\in\{0,\ldots 2^n-1\}$ and choosing
the remainder arbitrarily.  Since these columns correspond to states
which will not be selected by any input to PQC2, they don't matter.
It should be noted that it is unlikely that the trained dual-PQC GAN
would actually converge on unitaries of such a form, and thus this
construction is given merely to demonstrate that $2n$ qubits suffice;
further improvements are surely possible.

§ TRAINING DUAL-PQC GAN

Classification of 20,000 normalized real image samples (only 100 samples displayed) into $4$ classes via K-means clustering <cit.> (a) and their average (b). The mean image of Set $i$ is assigned to a computational basis state $\ket{i}$.

This section tests  dual-PQC GAN described in sec:dual_PQC and shows its potential to generate a set of image samples from a training set with a certain degree of fidelity.  We emphasize that, in order to work with a manageable number of qubits,  this study simplifies the original 3DGAN problem by reducing it to a 1D problem: reproducing the energy pattern along the calorimeter depth. In other words, the training dataset is composed by 1D energy profiles along the calorimeter $z$ dimension, averaged over $N=4$ pixels. It should be noted that such drastic reductions in problem size are common-place in quantum machine learning, in order to obtain useful proof-of-principle results.

In order to evaluate the performance of the model, the original data set of 20,000 sample images is classified into $2^{n}$ classes of via K-means clustering <cit.> as shown on fig:classification for the case $n = 2$. The average image for each class, displayed on fig:mean_images, gives an insight on the shape of images which should be produced by PQC2. Note that this clustering is purely for evaluation of results - raw data are used for the training.

Although, theoretically,  it is sufficient  to take $n_1 = n = 2$, several preliminary simulations have shown that $n_1 = 2n = 4$ gives better stability in the results. Therefore, for the following simulations, PQC1 takes $n_1 = 4$ but still builds a probability distribution over $2^2 = 4$ images, by only measuring $n = 2$ qubits among four. PQC1 is initialized with an equiprobable superposition over the computational basis, $\{\ket{0},..., \ket{2^{n_1}-1}\}$ with $n_1 = 4$. Furthermore, the initial parameters for both PQC1 and PQC2 are sampled from a uniform distribution over $[-\delta, \delta]$, with $\delta = 10^{-1}$. The discriminator is implemented in PyTorch, using an input layer with 4 nodes, two hidden layer with 256, 128 nodes, respectively, and a single node output layer. After the first two layers follows a Leaky ReLU function <cit.> with $\alpha = 0.2$ and a sigmoid function <cit.> is applied after the output layer. In addition, a gradient penalty <cit.> for real images is added to help stability and convergence of the model, with the parameters $\lambda = 7$, $k = 0.01$ and $c = 1$.
The dual-PQC GAN is trained using the AMSGRAD optimizer with initial learning rate of $10^{-4}$ for PQC1 and discriminator and $10^{-3}$ for PQC2.

Results of dual-PQC GAN simulations with $n = 2$, $n_1 = n_2 = 4$, $d_{g,1} = 2$ and $d_{g,2} = 6$ (a, b, c) / $d_{g,2} = 16$ (d, e, f). The mean image amplitudes calculated by eq:mean_image (a,d), the generator and the discriminator losses (b, e), and the progress in relative entropy for the mean image (c,f) are illustrated.

fig:mean_loss_dual displays progress in the loss functions and relative entropy, as well as the average of generated images at the end of the training, weighted with their probability, given by:

\begin{equation}
\mathcal{I}_{mean} = {\textstyle\sum}_{i=0}^{2^n - 1}p_g^i\cdot \mathcal{I}_i.
\label{eq:mean_image}
\end{equation}

Both cases of $d_{g,2} = 6$ and  $d_{g,2} = 16$ exhibit convergence in mean energy distribution towards the target, as well as convergence of generator and discriminator losses. Furthermore, the relative entropy between real and generated mean images reaches below $10^{-4}$ as shown on fig:rel_entr_N4_D2D6 and fig:rel_entr_N4_D2D16. Note that after the convergence in loss function, the relative entropy does not cease decreasing in case of $d_{g,2} = 16$, while it starts to oscillate with large amplitude in case of $d_{g,2} = 6$, reflecting certain degree of instability in the simulation.

As an analysis on the mean images is not enough to validate the GAN result, it is necessary to evaluate the individual images produced by PQC2 as well as the probability distribution generated by PQC1. The images generated by PQC2 with $d_{g,2} = 16$, displayed on fig:images_N4_D2D16_opt3, are similar to the mean real images shown in fig:mean_images with the peaks at $x = 2$ or $x = 3$. On the other hand, those generated by PQC2 with $d_{g,2} = 6$ on fig:images_N4_D2D6_opt3 contain two images, $\mathcal{I}_0$ and $\mathcal{I}_2$, which are far from the real image profile.

The probability distribution generated by PQC1 can explain this discrepancy. As shown on fig:prob_N4_D2D6_opt3, the weights for $\mathcal{I}_0$ and $\mathcal{I}_2$ are negligible compared to those for $\mathcal{I}_1$ and $\mathcal{I}_3$, meaning that their labels, produced by the classical discriminator, are suppressed in the loss function given by eq:lg. Therefore, the mean images and the  generator loss could converge to the correct value, despite considerable errors in the produced images themselves.
Contrarily, in case of $d_{g,2} = 16$, the probability on fig:prob_N4_D2D16_opt3 implies that all 4 images have non-negligible weights, thus leading to consistency between the quality for the mean and individual images. This result certainly highlights the importance of choosing a correct structure of dual-PQC GAN model in order to prevent any bias during training.

Images generated by PQC2 (a,c) and its distribution obtained from PQC1 (b,d) with $n = 2$, $n_1 = n_2 = 4$ and  $d_{g,1} = 2$, but with different $d_{g,2}$ :  1) $d_{g,2} = 6$ (a,b), 2) $d_{g,2} = 16$ (c,d). Although the mean images shown on fig:mean_loss_dual are close to the real value in both cases, PQC2 with $d_{g,2} = 6$ cannot imitate correctly the real data (c.f. fig:mean_images), while the one with $d_{g,2} = 16$ achieves to reproduce very similar images.

Relative entropy of generated images $\mathcal{I}_0,..., \mathcal{I}_3$ with respect to average of real image classes, shown on fig:mean_images. The images are generated via dual-PQC GAN with $n = 2$, $n_1 = n_2 = 4$, $d_{g,1} = 2$ and $d_{g,2} = 16$. According to minimum relative entropy at the end of the training, one-to-one correspondence can be established between real and generated images.

Finally, the quality of individual images is evaluated by calculating the relative energy between the real images Set $i$, shown on fig:mean_images, and the generated images $\mathcal{I}_i$ for $i = 0,1,2,3$, as displayed on fig:individual_rel_entr. Considering the lowest relative entropy values across the last 4 epochs (200 epochs are run in total), a bijection between real images and generated images can be constructed : $\mathcal{I}_0 \to$ Set0, $\mathcal{I}_1 \to$ Set3, $\mathcal{I}_2 \to$ Set1 and $\mathcal{I}_3 \to$ Set2. This result gives a quantitative proof that PQC2 can reproduce four different sets of images in the real training data. Despite this optimistic affirmation, the relative entropy large instability is the main point that should be improved in future studies.

Unfortunately, the number of qubits in the quantum generator scales not only with the number of pixels in one image but also with the number of images that can be produced, while in original GAN, the system size mainly scales with image size. This fact represents the largest limitation of the model and it requires further improvement. We are currently investigating a solution feeding  extra noise to the remaining PQC2 $n_2 - n_1$ qubits, while PQC1 keeps generating a probability distribution over the zero-noise image.

§ CONCLUSION

This work presents a dual-PQC GAN model, a prototype of quantum GAN with two quantum generators, sharing the role of a single generator. One of the generators is responsible for reproducing the distribution over images, while the other for building amplitude distributions over pixels on a single image. If we supplement the input of PQC2 with a sample from PQC1 and some (continuous) noise, then we can see that PQC1 really is a distribution over means of different classes of images, and the noise allows generation of typical samples from the class. The results obtained prove that this model can generate individual image samples and their probability distribution, similar to the training set.

It is also worth noting that, as the number of possible images (or classes of images if noise is added) grows exponentially with $n_1$, the fact that we sample from a finite set of images (or classes of images if noise is added) rather than a continuum of typical images (as in the corresponding classical case) is unlikely to seriously compromise performance.

An interesting question for future research would be how to reproduce an arbitrary number of outputs with the dual-PQC GAN model.  One possible way is to introduce a complex-valued noise to the remaining qubits in the PQC2. Using a fixed sampler instead of a trained quantum circuit for PQC1 to pass an entangled quantum state to PQC2 is another possible approach. Throughout future studies, we look forward to building a more advanced quantum GAN model to imitate the performance of classical GAN.