Quantum reservoir computing using arrays of Rydberg atoms

Let us now extend the classical RNN in (II) to the quantum setting. We replace each of the $N$ neurons with a spin-1/2 particle for which a spin measurement along the $z$-axis yields the values $\{-1,1\}$. Thus, each neuron $n$ is in a normalized quantum state in the Hilbert space $\mathcal{H}_{n}$ with basis vectors $\{|\text{-}1\rangle_{n},|1\rangle_{n}\}$ which are eigenstates of the Pauli-Z operator $\sigma_{n}^{z}=|1\rangle\langle 1|_{n}-|\text{-}1\rangle\langle\text{-}1|_{n}$. The state of the composite system lives in the product Hilbert space $\mathcal{H}=\bigotimes_{n=1}^{N}\mathcal{H}_{n}$.

We choose spins interacting via the time-dependent Hamiltonian

	$\displaystyle H(t)$	$\displaystyle=-\sum_{n=1}^{N}\Delta_{n}(t)\sigma_{n}^{z}+\sum_{nm}J_{nm}\sigma_{n}^{z}\sigma_{m}^{z}$
		$\displaystyle+\frac{\Omega(t)}{2}\sum_{n=1}^{N}\sigma_{n}^{x},$		(5)

where $\sigma_{n}^{x}=|1\rangle\langle\text{-}1|_{n}+|\text{-}1\rangle\langle 1|_{n}$ is the Pauli-X operator. Indeed, the evolution under (III.1) encompasses the update rule in (II). To see this, note that in the classical case of (II), the RNN evolves under the rules

	$\displaystyle\text{If }h_{n}>0,\text{ }s_{n}\text{ doesn't change.}$		(1C)
	$\displaystyle\text{If }h_{n}<0,\text{ }s_{n}\text{ flips.}$		(2C)

Here “$C$” stands for “classical”. Now, consider a qRNN starting in the configuration $|s_{1},s_{2},...,s_{N}\rangle$ and evolving for a time $t=2\pi\Omega^{-1}$. In the limit where $\Delta_{n}\gg\Omega$ or $J_{nm}\gg\Omega$, each spin experiences the Ham7iltonian $H_{n}=h_{n}\sigma_{n}^{z}+\frac{\Omega}{2}\sigma_{n}^{x}$ where $h_{n}=-\Delta_{n}+\sum_{m}J_{nm}s_{m}$ is the effective field generated by the rest of the spins where $s_{m}$ stands for the measurement result of $\sigma_{m}^{z}$ on the initial configuration. We then obtain the quantum update rules

	$\displaystyle\text{If }{\color[rgb]{0,0,0}{|h_{n}|}}\gg\Omega,\text{ }{\color[rgb]{0,0,0}{|s_{n}\rangle}}\text{ doesn't change.}$		(1Q)
	$\displaystyle\text{If }{\color[rgb]{0,0,0}{|h_{n}|}}\ll\Omega,\text{ }{\color[rgb]{0,0,0}{|s_{n}\rangle}}\text{ flips.}$		(2Q)

Here, “Q” stands for “quantum”. Therefore, (III.1) can implement (1C)-(2C) but without the use of the nonlinear activation function in (II). Nonetheless, (III.1) allows for more general dynamics beyond the perturbative limit for which (1Q)-(2Q) holds. We now highlight three features arising from the quantum evolution of the qRNN: (i) the ability to compute complex functions on the input by using quantum interference, (ii) exploiting the choice of measurement basis, and (iii) efficiently achieving stochastic processes inaccessible to classical RNNs with no hidden neurons.

Having seen how (III.1) recovers the discrete update rule (II), we now show that a qRNN under dissipation naturally evolves under continuous-time dynamics analogous to (3). This establishes only mathematical similarities between the evolution of NISQ devices and neural circuits, allowing us to use available quantum hardware for cognitive tasks, an idea that we explore further in Sec. V.

Consider the qRNN in $(\ref{eq:HGeneral})$ under spontaneous emission where a spin relaxes from $|1\rangle$ to $|\text{-}1\rangle$ at a rate $\gamma$. To extract the dynamics of continuous variables, we focus on the dynamics of the expectation values of local Pauli operators.

The expectation value of an observable $A$ is $\left\langle A\right\rangle=\text{Tr}(A\rho)$ where $\rho$ is the density matrix describing the system. In particular, we focus on the expectations of the operators $\sigma^{x}_{n}$, and $\sigma^{y}_{n}=i|\text{-1}\rangle\langle 1|_{n}-i|1\rangle\langle\text{-1}|_{n}$. If we start the qRNN at a state for which $\langle\sigma_{n}^{z}(0)\rangle=-1$ then (see Appendix B)

	$\displaystyle\dot{\left\langle\sigma^{y}_{n}\right\rangle}=$	$\displaystyle-\frac{1}{\tau}\left\langle\sigma^{y}_{n}(t)\right\rangle-\frac{\Omega}{2\gamma}\sum_{m}J_{nm}\left\langle\sigma_{n}^{x}(t)\sigma_{m}^{y}(t)\right\rangle$
		$\displaystyle+\Delta_{n}(t)\left\langle\sigma_{n}^{x}(t)\right\rangle,$		(11)

where we have defined the neural time-scale $\tau^{-1}=\gamma/2+\Omega^{2}/4\gamma$ which is different than that in (3) but bears the analogous significance of the time-scale in which $\left\langle\sigma_{n}^{y}\right\rangle$ decays.

Differently, that (3), notice that the dynamics of $\left\langle\sigma_{n}^{y}\right\rangle$ are influenced by the spin’s value along the $x$-axis, a consequence of the nontrivial commutation relation of spin variables. The commutation relations also make (III.2) quadratic, and therefore nonlinear. The quadratic term in (III.2) is analogous to the nonlinear term that gives RNNs their computational power.

In Appendix B, we explore the dynamics of $\left\langle\sigma_{n}^{x}\right\rangle$ as well and show that together with $\left\langle\sigma_{n}^{y}\right\rangle$, we recover dynamics analogous to integrate-and-fire RNN model [54], a more realistic model of neural networks in the brain than the one in $(\ref{eq:EOMClassical})$.

A hallmark of classical RNNs is their ability to multitask. Multitasking consists of simultaneously learning several output functions. Dale’s principle defines an inhibitory neuron, indexed by $n$, as one with a negative sign in its interactions with all other neurons [57]

$$J_{nm}\leq 0\quad\forall m.$$

(14)

Two Rydberg atoms with different principal quantum numbers $n_{Q}$, and $n_{Q}^{\prime}$ and angular momentum quantum numbers the same can interact with a $1/r^{6}$ attractive potential $V_{n_{Q},n_{Q}^{\prime}}$ [45]. Using the Python package PairInteraction [58], we note that if $n_{Q}$ represents the state $|r\rangle=|70S_{1/2},m_{j}=-1/2,m_{I}=-3/2\rangle$, and $n_{Q}^{\prime}$ represents $|r^{\prime}\rangle=|73S_{1/2},m_{j}=-1/2,m_{I}=-3/2\rangle$, then the interaction $V_{n_{Q},n_{Q}}=V\approx-V_{n_{Q},n_{Q}^{\prime}}$ where $V$ is the strength between atoms with principal quantum numbers $n_{Q}$ (see Appendix D). We can use this fact to encode inhibitory neurons. We restrict the concentration of $n_{Q}^{\prime}$ Rydberg atoms to be sparse such that pairs of $n_{Q}^{\prime}$ atoms are placed as far as possible at a distance $d_{max}$ in a 1D chain arrangement. We choose the field strength $V$ so that $V/d_{max}^{6}=10^{-2}$, and as a result we can neglect the interactions between pairs of $n_{Q}^{\prime}$ atoms, but not the interactions between pairs $(n_{Q})(n_{Q}^{\prime})$ and $(n_{Q})(n_{Q})$. This amounts to saying that if atom $n$ is driven to $n_{Q}^{\prime}$ then for all $m$ $J_{nm}\lesssim 0$ as in (14). By implementing this in on our reservoir we can learn XOR, AND, and OR simultaneously for different concentrations of inhibitory neurons as illustrated in Fig. 5(A).

Fig. 5(B) shows the errors of simultaneously learning XOR, OR, and AND as a function of the system size $N$ for a different number of inhibitory neurons in the array. The network is initialized in the state $|g\rangle^{\otimes N}$, and the network receives two binary inputs $x,y\in\{0,1\}$ (in units of MHz) for a time $\Delta t$ (in units of $\mu$s) with input noise $\sigma_{in}=0.1$. Afterwards, the network is interrogated to give XOR$(x,y)$, OR$(x,y)$, and AND$(x,y)$. $W^{\text{out}}$ is trained using the loss in (4). The errors shown in Fig. 5(B) are the minimum achieved over a wide range of choices of interaction time $\Delta t\in[0,5]$ $\mu$s. This shows that in some cases our reservoir can benefit from having a connectivity matrix $J_{nm}$ with both positive (excitatory) and negative (inhibitory) values, analogously to the mammalian brain. For small system sizes, it seems that a ratio of 1:4 inhibitory neurons betters the learning performance, similar to the results in [56]. This is supported by the performance at 4 and 8 neurons in Fig. 5(B). Particularly, $N=8$ neurons, two of which are inhibitory, result in a 40% decrease in the loss. Nonetheless, we observe that having no inhibitory neurons is best when dealing with $N=$6 and 10 neurons. No inhibitory neurons are ever the worse choice. Fig. 5(C)-(E) shows the results of learning XOR, OR, and AND simultaneously using $N=8$ and two inhibitory neuron. Note that the network is fully capable of classification errors well below the input noise threshold $\sigma_{in}$.

Lastly, while this task shows the success of the Rydberg reservoirs at approximating Boolean functions of the input, we note that one may also want to calculate different nonlinear functions of the input. We remark that our Rydberg reservoir can approximate biologically relevant nonlinear functions such as ReLu and sigmoid.

One of the great successes of classical RNNs is their ability to integrate sensory stimuli to choose between two actions. Here, we present the Rydberg reservoir with a variant of the dot motion decision-making task initially studied in monkeys in which several inputs are analyzed to produce a scalar nonlinear function [59]. This function represents a decision. This task shows the Rydberg reservoir’s ability to produce nonlinear functions of the input and perform simple cognitive tasks, a feature of most reservoirs proposed thus far [60].

In this task, a reservoir is presented with two inputs $\Delta_{1}^{in}$ and $\Delta_{2}^{in}$, and the goal is to train the network to choose which input is the largest. That is,

$$y^{\text{targ}}=\text{sign}\left(\Delta_{1}^{in}-\Delta_{2}^{in}\right).$$

(15)

The stimuli, which in the case of a qRNN are local detunings on a pair of atoms, are turned on for a normally distributed time $\Delta t$ with variance also $\sigma_{in}=0.1$ and mean $\langle\Delta t\rangle=0.1$ $\mu s$ (see Fig. 6(A)). The stimuli are then turned off, and the network chooses a relaxation time $t_{out}$ after which it “makes a decision” by approximating (15). This is known as the fixed-duration protocol since the experimentalist fixes the stimulation period, and the subject, the reservoir in this case, learns to choose a response time $t_{out}$.

In the brain, we expect the performance of a decision-making task to follow a sigmoidal psychometric response [44, 59]. A psychometric response maps out the accuracy of a decision-making task as a function of stimuli distinguishability. As an example of a psychometric response, the reader could think about paying a routine visit to the eye doctor and having to discern the letters “b” and “p” written on the wall. If the letters are large enough, they become distinguishable, and if the letters are too small one often fails to make out the right letter.

Classically, a decision-making task benefits from connectivity between all neurons. Since our connectivity is limited by physical constraints, a 2D square lattice structure was chosen to prevent neurons from being isolated from the rest. Moreover, a 2D square lattice is experimentally friendly. We set up a Rydberg reservoir of $3\times 2$ atoms with two input atoms and two different output atoms (for details see Appendix D). The reservoir is then trained by optimizing over $t_{out}$, and $W^{\text{out}}$ such that the reservoir’s output approximates (15) while keeping the network parameters $J_{nm}$, $\Omega$ and $\Delta^{in}_{n}$ fixed. We observe that $t_{out}\approx 1$ $\mu$s is regularly obtained as this is the time scale in which the information about $\Delta_{1,2}^{in}$ propagates through the network. In our case, $c_{1}=\Delta^{in}_{1}-\Delta^{in}_{2}$ is a natural choice for a measure of stimuli distinguishability. Fig. 6(B) shows the psychometric response of the task which is qualitatively similar to the ones obtained in classical RNNs [44]. Moreover, we see in Fig. 6(B) that if $|c_{1}|\geq\sigma_{in}$ such that it is above the input noise level our network success more than $80\%$ of the time. The success of this task shows the Rydberg reservoir’s ability to emulate simple cognitive tasks.

Our next neurological task is that of parametric working memory. Working memory, which is one of the most important cognitive functions, deals with the brain’s ability to retain and manipulate information for the later execution of a task. Here, we train a network to perform a task based on the decision-making task in Sec. V.2 but with two temporally separate stimuli (see Fig. 7(A)). We use the fixed-time protocol where the separation between stimuli, denoted by $t_{delay}$ if fixed by us. The stimuli are both turned on for a time $\Delta t$, and after the second input the network is left to relax for a time $t_{out}$ before two output neurons are used to approximate (15. To avoid overfitting, we add Gaussian noise to the times $\Delta t$, $t_{out}$, and $t_{delay}$ with zero mean and standard deviation $\sigma_{in}=0.1$. The network optimizes over $W^{\text{out}}$. Thus, the network has to retain information about $\Delta_{1}^{in}$ for a few “seconds” to then compare against $\Delta_{2}^{in}$ and make a decision.

We set a Rydberg reservoir of $3\times 2$ atoms with two input atoms and two different output atoms (for details see Appendix D). Fig. 7(B) shows the loss of the network as a function of the total time the inputs are injected into the network ($\tau=2\Delta t+t_{delay}$). We note that the loss function is high for small $\tau$ since it takes the input neurons to correlate with the rest of the reservoir. Accordingly, in Fig. 7(B) we show that growth of the entanglement entropy of the input qubits accompanies a decrease in the loss function. For Fig. 7(C) we fixed $t_{out}=0.1$, a choice which has little effect on the reservoir’s performance.

In Fig. 7(C) we show the accuracy of the reservoir at reproducing (15) as a function of the time the inputs are turned on ($\Delta t$) and for different choices of $t_{out}$. For these plots $t_{delay}=0.1$ is fixed. We notice that the accuracy is largely invariant to our sampled choices of $t_{out}$.

Lastly, in Fig. 7(D), we probe the reservoir’s accuracy as a function of $t_{delay}$. For these experiments, we fix $t_{out}=0.5$ and $\Delta t=0.15$. Importantly we set $V=2\pi\times 10$ MHz and $\Omega=2\pi\times 4.2$ MHz such that $V>\Omega$ and neighboring Rydberg excitations are off-resonance putting our reservoir in the so-called blockaded regime [61, 62]. While one initially might expected the accuracy to decrease for increasing $t_{delay}$, we found that this is not the case and instead the accuracy oscillates persistently reaching high accuracies as shown in Fig. 7(D) blue curve. Interestingly, this behavior disappears when the coupling $V=2\pi\times 0.1$ MHz such that $V<\Omega$ as shown in the red curve in Fig. 7(D), although the performance is statistically significant even for long $t_{d}elay$ with an accuracy greater than $50\%$. We can conclude that, in the blockaded regime, the reservoir can hold information for longer periods. We can understand this dependence on $V/\Omega$ as follows. In the disordered regime, atoms are mostly uncorrelated and the atoms are allowed to freely oscillate with the dynamics being dominated by the drive $\Omega$. Thus, after a short time, the inputs coming through a $z$-field are largely irrelevant and the network is unable to hold the information about the first input. On the other hand, when $V>\Omega$ the atoms are largely correlated since neighboring excitations of Rydberg atoms are blockaded and the dynamics are slowed down. These slow dynamics in the system allow for longer memory times. In Sec. V.4, we will explore the longer-term memory in the blockaded regime and show that long-term memory in a reservoir can be stabilized due to the presence of quantum many-body scars.

Finally, we turn to examine a reservoirs ability to encode long-term memory. The task consists of encoding a classical bit $m$ in the initial state of a reservoir $|\psi_{m}(0)\rangle$ so that after the system is left to evolve under its inherent dynamics for a time $T$, local measurements of the state $|\psi_{m}(T)\rangle$ are used to recover $m$. However, $m$ cannot be recovered from local measurements if the dynamics obey the eigenstate-thermalization hypothesis (ETH) [63]. Instead, local measurements of $|\psi_{m}(T)\rangle$ obey thermal statistics described by the energy spectrum of the Hamiltonian and bear no information on the initial condition $|\psi_{m}(0)\rangle$. Thus, reservoirs that violate the ETH are naturally suited for memory tasks, since they can locally retain information about their initial state. Indeed this notion has begun to be studied in quantum reservoirs [25, 27]. Recent experiments using quench dynamics in arrays of Rydberg atoms have revealed quantum many-body scaring behavior [35], which can be stabilized [46, 47] to delay the thermalization of the system. Here, we use these results to enlarge the memory lifetime of a reservoir. Simulation details are found in Appendix D.

In the case of a kicked ring of Rydberg atoms experiencing nearest-neighbor blockade, the dynamics are captured in the so-called PXP-model [35, 47, 64, 65]

	$\displaystyle H(t)$	$\displaystyle=H_{PXP}+\hat{N}\sum_{k\in\mathbb{Z}}\theta_{k}\delta(t-k\tau)$		(16)
	$\displaystyle H_{PXP}$	$\displaystyle=\Omega\sum_{n=1}^{N}P_{n-1}\sigma_{n}^{x}P_{n+1}\qquad\hat{N}=\sum_{n}\hat{n}_{n}$		(17)

where $P_{n}=|g\rangle\langle g|_{n}$ projects the atom at the $n^{th}$ site onto the ground state and we choose periodic boundary conditions to mitigate edge effects. In (16) we let $\theta_{k}=\pi+\epsilon_{k}$ where $\epsilon_{k}$ is a Gaussian random variable with mean $\epsilon$ and variance $\sigma_{in}^{2}$. That is $\epsilon_{k}$ plays the role of added noise in the reservoir. For this discussion we let $\gamma=0$ since we know from experiments that the quantum scaring behavior is robust to the atom’s decoherence, and the choice to work with the Hamiltonian evolution helps speed up the acquisition of numerical data.

We denote $\chi_{\tau}=\exp{(-i\pi\hat{N})}\exp{(-i\tau H_{PXP})}$. It has been empirically observed that $\chi_{\tau}$ approximately exchanges the Neel states $|AF\rangle=|1010...\rangle$ and $|AF^{\prime}\rangle=|0101...\rangle$ for $\tau\approx 1.51\pi\Omega^{-1}$ [35]. Note that $\chi_{\tau}\chi_{\tau}=\mathds{1}$, and so under no noise, any state $|\psi\rangle$ is recovered after a cycle of evolution of $2\tau$. However, the noise $\epsilon_{k}$ destroys the revival of all initial states except for $|AF\rangle$ and $|AF^{\prime}\rangle$ (see Appendix C). This leads to many-body quantum scars stabilized by the operator $\exp{(-i\pi\hat{N})}$ [46, 47].

Given the dynamics in (16), we propose the following scheme for encoding a binary memory $m\in\{0,1\}$. We choose a reference state $|\psi\rangle$, and let $|\psi_{0}(0)\rangle=|\psi\rangle$ and $|\psi_{1}(0)\rangle=\chi_{\tau}|\psi\rangle$. Subsequently, the state $|\psi_{m}(0)\rangle$ is left to evolve for $n$ cycles of duration $2\tau=2(1.51\pi)$ after which the populations $\bm{r}_{m}(n)=(P_{g}(2n\tau|m),P_{r}(2n\tau|m))$ of the single-atom reduced density matrix are used to retrieve $m$. The retrieval is done by training a vector $W^{\text{out}}_{n}$ on $M$ instances of $\bm{r}_{m}(n)$ in order to minimize (4) with $\bm{y}^{\text{targ}}=\bm{m}$ the binary vector of memories and $\bm{y}^{\text{out}}(n)=W_{n}^{\text{out}}\bm{r}(n)$ our networks’ output after $n$ cycles.

To quantify the quality of the memory retrieval $R(n)$, we use the squared Pearson’s $r$-factor

$$R(n)=\frac{\text{cov}^{2}(\bm{m},\bm{y}^{\text{out}}(n))}{\sigma^{2}(\bm{m})\sigma^{2}(\bm{y}^{\text{out}}(n))}.$$

(18)

Fig. 9(a) shows the memory retrieval error as a function of the number of cycles for three different choices of reference states. Fig. 9(b) shows the average entanglement entropy $(\bar{S}_{E})$ of the left-most atom in the ring. Saturation of $\bar{S}_{E}$ signals growth in the memory retrieval error as the state “forgets” the initial condition. From other studies, we see that memory is retrieved at longer times due to the slow thermalization of the Neel states due to quantum many-body scars [35, 47, 64, 65, 46, 47]. The time-crystalline nature of the reservoir using $|\psi\rangle=|AF\rangle$ signals long-time correlations, and thus the reservoir can be used to encode and predict series with long-time correlations [17].

The Neel states exhibit long-term memory due to the evolution’s scaring behavior. This can be understood by analyzing the average evolution produced by a single cycle. Up to second order in $\epsilon_{k}$, the state at time $2\tau n$, $\rho(n)$, evolves to the state at time $2\tau(n+1)$, $\rho(n+1)$, where (see Appendix C)

	$\displaystyle\rho(n+1)$	$\displaystyle=\rho(n)-i\epsilon[H^{+},\rho(n)]$
		$\displaystyle+\sigma^{2}_{in}\left(H^{+}\rho(n)H^{+}-\frac{1}{2}\{H^{+}H^{+},\rho(n)\}\right)$
		$\displaystyle+\sigma^{2}_{in}\left(H^{-}\rho(n)H^{-}-\frac{1}{2}\{H^{-}H^{-},\rho(n)\}\right).$		(19)

Here, $H^{\pm}=\hat{N}\pm\chi_{\tau}\hat{N}\chi_{\tau}$ are Hermitian operators. We can rewrite (V.4) as $\rho(n+1)=\rho(n)+\mathcal{L}_{\epsilon,\sigma}(\rho(n))$. Since $[H^{+},\chi_{\tau}]=0$, the operator $H^{+}$ has an emergent $\mathbb{Z}_{2}$ symmetry which means that the ground states of $H^{+}$ are well approximated by the states $|\pm\rangle=\frac{1}{\sqrt{2}}\left(|AF\rangle\pm|AF^{\prime}\rangle\right)$ [47]. Note that

	$\displaystyle H^{+}|+\rangle$	$\displaystyle\approx N|+\rangle,\quad$	$\displaystyle H^{-}|+\rangle\approx 0,$		(20)
	$\displaystyle H^{+}|-\rangle$	$\displaystyle\approx N|+\rangle,\quad$	$\displaystyle H^{-}|-\rangle\approx 0,$		(21)

were $N$ is the system size We conclude that if $\rho(n)=|AF\rangle\langle AF|$ then $\rho(n+1)\approx\rho(n)$ as this state is (approximately) in the kernel of $\mathcal{L}_{\epsilon,\sigma}$. Therefore, the Neel states are suitable memory states.

Equation (V.4) also tells us that any density matrix in the kernel of $\mathcal{L}_{\epsilon,\sigma}$ may also serve as a memory state since it is a steady state of the evolution. This would allow us to enlarge the number of memories accessible in a qRNN. In Appendix C we show the existence of a large number of steady states, and we present a scheme to prepare a number of them. It’s worth noting, however, that these memories may have to be distinguished from one another via global measurements. The questions of how to efficiently prepare and distinguish these memory states remain importantly both open and key in telling us if a memory quantum advantage can be claimed in qRNNs. As it stands, using quantum scars signals that Rydberg-inspired RNNs may present enhanced memory since quantum scars are classically simulatable due to their low entanglement entropy. However, it’s unclear whether the system can be classically simulated at late times due to the onset of the thermalization. These questions are left for future studies.

Quite recently, another proposal to enlarge the number of memories accessible in a quantum reservoir has been introduced using the emergent scale-free network dynamics of a melting discrete time-crystal in an Ising chain [25]. The proposal in [25] can be seen as a generalization of the quantum reservoir presented in (16) by dropping the constraint of the Rydberg blockade. Our results, as well as those in [25], pose the possibility of having an RNN with a memory capacity that outpaces that of classical RNNs such as the Hopfield network [66].