Recurrent Gaussian quantum networks for online processing of quantum time series -

While in the previous section we saw that using RGQN instead of RC-inspired strategies can improve performance on benchmark tasks, in this section we show that the greater flexibility of the model makes it suitable to tackle quantum communication tasks. For these two tasks, RGQN removes some of the hardware limitations of currently available methods.

Superadditivity

First, we show that RGQNs can enhance the transmission speed of quantum channels that exhibit memory effects. When a state is transmitted through such a “memory channel”, it interacts with the channel's environment. Subsequent input states also interact with the environment, resulting in correlations across different channel uses. In contrast to memoryless channels, it is known that the transmission speed of a memory channel can be scaled up by providing entangled input states across subsequent channel uses.^{twenty four}This is a phenomenon known as “superadditivity”, and here we aim to create such entangled input states using RGQN.

Note the similarity to the definition of the entanglement task, except that here the goal is not to create maximal entanglement between different states, but to create a specific type of entanglement that relies on memory effects in the channel and improves the transmission speed.

The setup of the “superadditivity task” is shown in Figure 5. $m_\text {io} = 1$ We transform a vacuum state into a quantum time series. Information is encoded by moving each state of the series over successive distances in phase space. These distances are provided by classical complex-valued information streams. Their probabilities follow a Gaussian distribution with mean zero and covariance matrix. $\varvec{\gamma }_{\textrm{mod}}$.

The resulting time series is sent through a memory channel. hair Consecutive use of a channel is modeled as a single parallel hairMode Channel. The memory effects considered here are modeled by correlated noise arising from a Gauss-Markov process.¹⁶The environment has a classical noise covariance matrix: $\varvec{\gamma }_{\textrm{env}}$:

$$\begin{aligned}{} & {} \varvec{\gamma }_{\textrm{env}} = \left( \begin{array}{cc}{\textbf{M}}(\phi ) &{} 0 \\ 0 &{} {\textbf{M}}(-\phi )\end{array}\right) ~, \end{aligned}$$

(1)

$$\begin{aligned}{} & {} M_{ij}(\phi ) = N \phi ^{|ij|} ~. \end{aligned}$$

(2)

here, $\phi \in [0, 1)$ denotes the strength of the nearest-neighbor correlations and $N \in {\mathbb {R}}$ is the variance of the noise. In Eq. (1), ${\textbf{M}}(\phi )$ correlates the q quadratures, while ${\textbf{M}}(-\phi )$ anti-correlates the p quadratures.

The transmission rate of the channel is calculated from the von Neumann entropy of the states that pass through the channel (i.e. from the Holevo information). Here we adopt the approach and the parameter values outlined in Ref.¹⁶.

Note that the average photon number that is transmitted per channel use (${\bar{n}}$) has a contribution from both the RGQN (i.e. from its squeezers) and from the displacer. Given a value for ${\bar{n}}$, the transmission rate is maximized by training both the circuit ${\textbf{S}}$ and $\varvec{\gamma }_{\textrm{mod}}$ under the energy constraint imposed by ${\bar{n}}$. Nonzero squeezing values are obtained, leading to an information carrier. This highlights the counter-intuitive quantum nature of the superadditivity phenomenon: by spending a part of the available energy on the carrier generation rather than on classical modulation, one can reach higher transmission rates, something that has no classical analog.

We now define a quality measure for the superadditivity task. The gain G is the ratio of the achieved transmission rate to the optimal transmission rate for separable input states. For 30 channel uses, Fig. 6 shows G as a function of the average photon number ${\bar{n}}$ per use of the channel and for different values of the correlation parameter $\phi$. We take the signal-to-noise ratio ${\text{SNR}} = {\bar{n}}/N = 3$, where N is defined in Eq. (2). We observe that superadditivity is achieved, as the gain is higher than 1 and can reach as high as 1.10. These results agree with the optimal gain values that were derived in prior numerical studies of this memory channel (cfr. Fig. 7 of Ref.²⁵).

While a scheme already exists to generate carriers sequentially¹⁷ (i.e., generating carriers without creating all channel entries simultaneously), our model provides a simpler and more versatile alternative. Unlike the existing scheme, our model eliminates the need for Bell measurements, while achieving the same near-optimal gains. Additionally, it is known that reconfigurable circuits can learn to compensate for fabrication imperfections as these imperfections are taken into account during training^18,19.

Quantum channel equalization

In this section, we use the RGQN as a model for a quantum memory channel. This time, we assume its memory effects to be unwanted (unlike the previous section) and compensate for them by sending the channel’s output through a second RGQN instance.

Figure 7 shows the setup for the quantum channel equalization (QCE) task in more detail. An ‘encoder’ RGQN acts as a model for a memory channel. Because such channels normally do not increase the average photon number of transmitted states, we restrict the encoder’s symplectic circuit to be orthogonal and denote it as ${\textbf{O}}_\text {enc}$. This circuit is initialized randomly and will not be trained later. A second ‘decoder’ RGQN is trained to invert the transformation caused by the encoder. Similar to the STQM task, we will show that an orthogonal symplectic circuit ${\textbf{O}}_\text {dec}$ is enough to lead to the desired performance, without requiring optical squeezers, which is beneficial for experimental realizations. We will further denote the number of memory modes of the encoder and decoder as $m_\text {mem,enc}$ and $m_\text {mem,dec}$ respectively. Finally, we introduce a delay of $D$ iterations between the input and output time series, similar to the definition of the STQM task (see Fig. 2a).

Assume for a moment that the input time series of the encoder only consists of a single state, i.e. we are looking at an impulse response of the system. We send this state to the encoder at iteration 0, and expect the decoder to reconstruct it at iteration $D$. However, a part of the input state may be initially stored in the encoder’s memory modes. By choosing $D>0$, multiple states are sent from the encoder to the decoder, thereby depleting the encoder’s memory modes. When we increase $D$, more information about the original input state reaches the decoder by the time it needs to start the reconstruction process. A similar reasoning applies when the input time series consists of multiple states. This approach effectively addresses the challenge posed by the no-cloning principle, which prevents the decoder from accessing information stored in the encoder’s memory or in the correlations between the encoder’s memory and output.

For the RC-inspired model of Ref.¹¹, only the case where $D=0$ was considered. The no-cloning problem was addressed by redundantly encoding the input signals of the encoder. I.e., multiple copies of the same state were generated based on classical input information and subsequently fed to the model through different modes (‘spatial multiplexing’) or at subsequent iterations (‘temporal multiplexing’). Here, we show that taking $D>0$ allows us to solve the QCE task without such redundancy, ultimately using each input state only once. This not only simplifies the operation procedure but also enables us to perform the QCE task without prior knowledge of the input states, which is often missing in real-world scenarios such as quantum key distribution. As these input states cannot be cloned, our approach significantly increases the practical use of the QCE task. It is important to note that such an approach where $D>0$ was also attempted for the RC-inspired model²⁶, but this was unsuccessful, which can be attributed to its limited number of trainable interactions.

Additionally, we will show that the QCE performance of the RGQN is influenced by two key factors: the memory capacity of the decoder (as determined by the value of $m_\text {mem,dec}$), and the response of the encoder to a single input state (observed at the encoder’s output modes).

More formally, we measure the impulse response of the encoder by sending in a single squeezed vacuum state (with an average photon number of ${\bar{n}}_\text {impulse}$) and subsequently tracking the average photon number $h_{\text {enc}}$ in its output modes over time. We denote the impulse response at iteration k by $h_{\text {enc}}^k$.

We now define:

$$\begin{aligned} I_{\text {enc}} = \frac{1}{{\bar{n}}_\text {impulse}} \sum _{k=0}^{D} h_{\text {enc}}^k ~. \end{aligned}$$

(3)

$I_{\text {enc}}$ is a re-normalized cumulative sum that represents the fraction of ${\bar{n}}_\text {impulse}$ that leaves the encoder before the decoder has to reconstruct the original input state.

We now consider 20 randomly initialized encoders with $m_\text {mem,enc}=2$. The input states are randomly sampled from a set of squeezed thermal states (more details in Methods). Figure 8a shows the average fidelity²³ between an input state of the encoder at iteration k and an output state of the decoder at iteration $k+D$ as a function of $I_{\text {enc}}$ and for different values of $D- m_\text {mem,dec}$. We see that if $D\le m_\text {mem,dec}$ (blueish dots), the decoder potentially has enough memory, and the quality of constructing the input states increases as the decoder receives more information from the encoder (i.e. as $I_{\text {enc}}$ increases). If $D> m_\text {mem,dec}$ (reddish dots), we ask the decoder to wait for a longer time before starting to reconstruct the input. This explains why the dots are clustered on the right side of the diagram because more information about the input will be received and $I_{\text {enc}}$ will be higher. On the other hand, if the delay is too long, it will exceed the memory of the decoder, and the input will start to be forgotten. This explains that the darkest dots with the longest delay have the worst performance.

Note that $D$ is a hyperparameter that can be chosen freely. Also note that the optimal choice for the value of $D$ is not necessarily $D= m_\text {mem,dec}$ (light grey dots) and the actual optimum depends on the exact configuration of the encoder. Figure 8b shows a subset of the results in Fig. 8a, where the optimal value of $D$ is chosen for every encoder initialization and every value of $m_\text {mem,dec}$. We conclude that the task can be tackled without redundantly encoded input signals. As discussed earlier, such redundancy was required in Ref.¹¹, but often cannot be provided in real-world scenarios. We further observe that the performance increases with both $m_\text {mem,dec}$ and $I_{\text {enc}}$. For $m_\text {mem,dec}=3$, all 20 encoders are equalized better than is done in Ref.¹¹.

Experimental demonstration of quantum channel equalization

As the RGQN does not contain any non-Gaussian components, it can be constructed using optical components that are readily available in the laboratory^13,14,15. In this section, we perform the QCE task on the recently introduced quantum processor Borealis¹⁵. Because of hardware restrictions, we only consider the case where $m_\text {mem,enc}= m_\text {mem,dec}= 1$. The setup for this task is visualized in Fig. 9. The input time series consists of squeezed vacuum states (whereas squeezed thermal states were previously used to simulate the QCE task). Both the encoder and decoder are decomposed using beam splitters and phase shifters. Here we use the following definitions for those respective components:

$$\begin{aligned} BS(\theta )= & {} e^{\theta (a_i a_j^\dagger – a_i^\dagger a_j)}~, \end{aligned}$$

(4)

$$\begin{aligned} PS(\phi )= & {} e^{i \phi a_i^\dagger a_i}~, \end{aligned}$$

(5)

where $a_i^\dagger$ ($a_i$) is the creation (annihilation) operator on mode i. Note that the transmission amplitude of the beamsplitter is $\cos (\theta )$.

Whereas Fig. 8 presented the results of training the decoder, here we will visualize a larger part of the cost function landscape (including sub-optimal decoder configurations). By doing so, we are agnostic to the exact training procedure, such as the parameter-shift rule^27,28. Note that while evaluating a certain point of the cost function landscape, i.e. while processing a single time series, the parameters of the beam splitters and phase shifters are kept fixed. Hence, in Fig. 9, the measurement results are not influenced by the phase shifters outside of the loops (i.e. outside of the memory modes). These components can be disregarded. Consequently, we can parameterize the encoder (decoder) using only a beam splitter angle $\theta _{\text {enc}}$ ($\theta _{\text {dec}}$) and a phase shift angle $\phi _{\text {enc}}$ ($\phi _{\text {dec}}$).

We use a photon-number-resolving (PNR) detector to detect output states. This provides us with the average photon numbers of the output states, but not with their phases. Unlike in Fig. 8, we cannot use fidelity as a quality measure. Therefore, we will assess the performance of the RGQN using the following cost function:

$$\begin{aligned} \text {cost} = \sum _{k=0}^{K} |{\bar{n}}_\text {out}^k-{\bar{n}}_\text {target}^k| ~, \end{aligned}$$

(6)

where ${\bar{n}}_\text {out}^k$ and ${\bar{n}}_\text {target}^k$ are the average photon numbers of the actual output state and the target output state at iteration k respectively. K is the total number of states in the input time series. As the average photon number of a squeezed vacuum state increases monotonously with the squeezing parameter r²⁹, i.e. ${\bar{n}} = \sinh ^2(r) > 0$, the cost function quantifies how much the squeezing levels of the output states deviate from the target values. As is explained in more detail in Methods, the experimentally obtained PNR results are re-scaled prior to evaluating Eq. (6) in order to compensate for hardware imperfections.

The small-scale version of the QCE task (depicted in Fig. 9) is run on the photonic processor Borealis¹⁵ (depicted in Fig. 10). It consists of a single optical squeezer that generates squeezed vacuum states. These states are sent to a sequence of three dynamically programmable loop-based interferometers of which the delay lines have different lengths, corresponding with propagation times of T, 6T, and 36T ($T = 36 \mu s$). For our experiment, we only use the two leftmost loop-based interferometers. More formally, we choose $\theta =0$ for the rightmost BS and $\phi =0$ for the rightmost PS.

As is explained in more detail in Methods, we can virtually make the lengths of Borealis’ delay lines equal. We do so by lowering the frequency at which we send input states and by putting the $\theta =0$ for the leftmost beam splitter in between input times.

We first consider the case where $\phi _{\text {enc}}= \phi _{\text {dec}}= 0$, such that all phase shifters can be disregarded. Figure 11 compares the experimental and numerical performance of the QCE task for $D=1$. We observe that the task is solved perfectly when either the encoder or the decoder delays states by $D=1$ and the other RGQN transmits states without delay. The performance is worst when both the encoder and decoder delay states with an equal number of iterations (either $D=0$ or $D=1$). Indeed, the cost of Eq. (6) is then evaluated between randomly squeezed vacuum states. For beam splitter angles between 0 and $\pi /2$, we find that the cost follows a hyperbolic surface. We observe good agreement between simulation and experiment.

We now consider the case where $\phi _{\text {enc}}$ and $\phi _{\text {dec}}$ are not restricted to 0. Note that the Borealis setup (Fig. 10) does not have phase shifters inside the loops. However, as is explained in more detail in Methods, we can virtually apply the phase shifts $\phi _{\text {enc}}$ and $\phi _{\text {dec}}$ inside Borealis’ loops by dynamically adjusting the phase shifters positioned before those loops over time. Unfortunately, this procedure is hampered by hardware restrictions. The range of Borealis’ phase shifters is restricted to $[-\pi /2,\pi /2]$This is a problem because applying a single virtual phase shift value within a particular loop requires that the phase shifters preceding that loop be dynamically adjusted throughout the loop. $[-\pi ,\pi ]$ As a result, approximately half of the phase shifter parameter values are not applied correctly. When such values fall outside the acceptable range, $\pi$ It is artificially added to ensure proper operation of the hardware.

Figure 12 shows the QCE ($D=1$) for three different encoders (corresponding to the three columns). We compare classical simulation results (rows 1 and 2) with experimental results (row 3). The limited range of the Borealis phase shifter is considered in rows 2 and 3, but not in row 1. Figure 11 ( $\phi _{\text {enc}}= \phi _{\text {dec}}= 0$) is the decoder, $\theta _{\text {dec}}= 0$ and $\theta _{\text {dec}}= \pi /2$,This optimization is, $\phi _{\text {enc}}\ne 0$ and $\phi _{\text {dec}}\ne 0$Although row 3 is affected by experimental imperfections, we can see that the general trend of the cost function landscape is consistent between rows 2 and 3.