Exploring superadditivity of coherent information of noisy quantum channels through genetic algorithms

Machine learning techniques are increasingly being used in fundamental research to solve various challenging problems. Here we explore one such technique to address an important problem in the quantum communication scenario. While transferring quantum information through a noisy quantum channel, the utility of the channel is characterized by its quantum capacity. Quantum channels, however, display an intriguing property called superadditivity of coherent information. This makes the calculation of quantum capacity a hard computational problem involving optimization over an exponentially increasing search space. In this work, we first utilize a neural network Ansatz to represent quantum states, and then we apply an evolutionary optimization scheme to address this problem. We find regions in the three-parameter space of qubit Pauli channels where coherent information exhibits this superadditivity feature. We characterized the quantum codes that achieve high coherent information, finding several nontrivial quantum codes that outperform the repetition codes for some Pauli channels. For some Pauli channels, these codes display very high superadditivity of the order of 0.01, much higher than the observed values in other well-studied quantum channels. We further compared the learning performance of the neural network Ansatz with the raw Ansatz to find that in the three-shot case, the neural network Ansatz outperforms the raw representation in finding quantum codes of high coherent information. We also compared the learning performance of the evolutionary algorithm with a simple particle swarm optimization scheme, and we show empirical results indicating comparable performance, suggesting that the neural network Ansatz coupled with the evolutionary scheme is indeed a promising approach to finding nontrivial quantum codes of high coherent information.

Figure 1

Evolution of neural networks: An illustration showing how neural network states evolve to create new generations.

Figure 2

The neural network Ansatz: A simple illustration of the neural network Ansatz used in this study. The given NN represents a 4-qubit quantum state. The inputs given to the NN are binary strings of length four and represent the corresponding computational basis states. The two outputs from the NN represent the real and imaginary parts of the amplitude corresponding to the input basis state. The first layer has a cosine activation, while all subsequent layers are given a tanh activation function.

Figure 3

Result of GA optimization: The single-shot capacity of the class of Pauli channels is shown in the density plot, color-coded as detailed on the color bar. Each point in the $p_1{p}_2{p}_3$ space represents a Pauli channel. We studied the superadditivity of points in a 3D grid of width 0.01. The black spots in the graph represent Pauli channels that display superadditivity of coherent information among these points.

Figure 4

Regions of two-shot superadditivity obtained from the three quantum codes: The blue region (marked with “I”) shows the highest two-shot coherent information when code $|\mathrm{\Psi }\left(\text{I}\right)\rangle$ is used. Similarly, the red region (marked “II”) corresponds to code $|\mathrm{\Psi }\left(\text{II}\right)\rangle$ and the green region (marked “III”) corresponds to code $|\mathrm{\Psi }\left(\text{III}\right)\rangle$. $p_1,p_2,p_3$ are, respectively, the probabilities of three Pauli errors.

Figure 5

2D density plot of superadditivity for fixed $p_3$: max$[Q^{\left(2\right)}(\mathrm{\Psi },{\mathrm{\Lambda }}_{\vec{p}})-Q^{\left(1\right)}\left({\mathrm{\Lambda }}_{\vec{p}}\right)$,0] is plotted for fixed values of $p_3\in \{0,0.02,0.05,0.15,0.2\}$, where $Q^{\left(2\right)}(\mathrm{\Psi },{\mathrm{\Lambda }}_{\vec{p}})\equiv max[Q^{\left(2\right)}(|\mathrm{\Psi }\left(\text{I}\right)\rangle ,{\mathrm{\Lambda }}_{\vec{p}}),Q^{\left(2\right)}(|\mathrm{\Psi }\left(\text{II}\right)\rangle ,{\mathrm{\Lambda }}_{\vec{p}}),Q^{\left(2\right)}(|\mathrm{\Psi }\left(\text{III}\right)\rangle ,{\mathrm{\Lambda }}_{\vec{p}})]$. All colored regions correspond to regions of superadditivity, with the amount of superadditivity shown by the “color bar” to the right of the plot.

Figure 6

Pauli channels with $Q^{\left(1\right)}\left({\mathrm{\Lambda }}_{\vec{p}}\right)=0$ but $Q^{\left(2\right)}({\mathrm{\Lambda }}_{\vec{p}},\rho )>0$: max$[Q^{\left(2\right)}({\lambda }_{\vec{p}},\rho )-Q^{\left(1\right)}\left({\mathrm{\Lambda }}_{\vec{p}}\right)$,0] is plotted for fixed values of $p_3\in \{0,0.02,0.05,0.15,0.2\}$, whenever $Q^{\left(1\right)}\left({\mathrm{\Lambda }}_{\vec{p}}\right)=0$. All colored regions correspond to regions of superadditivity, with the amount of superadditivity shown by the “color bar” to the right of the plot.

Figure 7

Regions of three-shot superadditivity obtained from the three quantum codes: The blue region (marked with “I”) shows the highest three-shot coherent information when code $|\mathrm{\Phi }\left(\text{I}\right)\rangle$ is used. Similarly, the red region (marked “II”) corresponds to code $|\mathrm{\Phi }\left(\text{II}\right)\rangle$ and the green region (marked “III”) corresponds to code $|\mathrm{\Phi }\left(\text{III}\right)\rangle$. $p_1,p_2,p_3$ are, respectively, the probabilities of three Pauli errors.

Figure 8

Regions of three-shot superadditivity obtained from the quantum codes in Eq. (15): The blue region (marked with “I”) shows the highest three-shot coherent information when code $|\chi \left(\text{I}\right)\rangle$ is used. Similarly, the red region (marked “II”) corresponds to code $|\chi \left(\text{II}\right)\rangle$ and the green region (marked “III”) corresponds to code $|\chi \left(\text{III}\right)\rangle$. $p_1,p_2,p_3$ are, respectively, the probabilities of three Pauli errors.

Figure 9

Regions of three-shot superadditivity obtained from the quantum codes in Eqs. (15) and (13): The red region (marked “I”) corresponds to points where Eq. (13) shows maximum superadditivity, while the blue region (marked “II”) corresponds to points where Eq. (15) outperforms Eq. (13).

Figure 10

Learning curves for GA/PSO with NNs of increasing depth. A population of 100 individuals is evolved in both cases. RAW represent the case when the NN Ansatz is not used and the coefficients of the wave function are treated as the optimization variables. Each learning curve shows the average fitness of the population averaged over 100 learning trajectories. (a) Going down the $y$-axis near the asymptote, the lines are in the order “2 layers,” “3 layers,” “5 layers,” “4 layers,” and “RAW.” (b) Top curve corresponds to “5 layers” followed by “4 layers,” “3 layers,” “2 layers,” and finally “RAW.”

Figure 11

Learning curves for GA/PSO with NNs of increasing depth. A total of 100 individuals are evolved in each case. RAW represents the case when the NN Ansatz is not used and the coefficients of the wave function are treated as the optimization variables. Each learning curve shows the best solution of a generation averaged over 100 learning trajectories. (a) Going down the $y$-axis near the asymptote, the lines are in the order “2 layers,” “RAW,” “3 layers,” “4 layers,” and “5 layers.” (b) Top curve corresponds to “5 layers” followed by “4 layers,” “3 layers,” “2 layers,” and finally “RAW.”

Figure 12

Learning curves for GA with NNs of increasing depth. Each learning curve shows the average fitness of the population averaged over 100 learning trajectories. (a) The initial population size is set to 50 individuals. The top curve corresponds to “2 layers,” followed by “3 layers,” “4 layers,” “5 layers,” and finally “RAW” at the bottom. (b) The initial population size is set to 300 individuals. The high population size resulted in an insensitivity to NN depth and hence the learning curves are overlapped and indistinguishable.

Figure 13

Learning curves for GA and PSO with increasing population size. A NN with a depth of 4 layers is used as the optimization Ansatz. Each learning curve shows the average fitness of the population averaged over 100 learning trajectories. (a) A population size of 300 (bottom curve) shows optimal performance, followed by a population size of 100 (middle) and then by a size of 50 (top). (b) Going down the $y$-axis near the asymptote, the lines are in the order “pop. size = 50,” “pop. size = 100,” “pop. size = 300.”

Figure 14

Learning curves for GA/PSO with NNs of increasing depth. RAW represent the case when the NN Ansatz is not used and the coefficients of the wave function are treated as the optimization variables. Each learning curve shows the average fitness of the population averaged over 100 learning trajectories. (a) The top curve corresponds to RAW Ansatz. Learning curves corresponding to 6 layers, 5 layers, and 3 layers are largely indistinguishable. (b) The top curve corresponds to “6 layers,” followed by “5 layers,” “3 layers,” and finally the RAW Ansatz.

Figure 15

Learning curves for GA and PSO with increasing population size. A NN with a depth of 6 layers is used as the optimization Ansatz. Each learning curve shows the average fitness of the population averaged over 100 learning trajectories. (a) The top curve corresponds to a population size of 50, followed by a population size of 100 and finally a population size of 300 individuals. (b) The top curves, corresponding to a population size of 50 and 100, are largely indistinguishable near the asymptote; these are followed by a green curve corresponding to a population size of 300.

Figure 16

Learning curves for PSO with different max. velocities. A 6-layered NN is used as the optimization Ansatz. Each learning curve shows the average fitness of the population, averaged over 100 learning trajectories. The top curve corresponds to smax = 1, followed by the curve with a maximum speed of 2. The curves corresponding to smax = 3 and smax = 1/2 are largely indistinguishable from each other and show maximum performance.