Protocol for nonlinear state discrimination in rotating condensate

A variety of atomtronic architectures have been proposed for quantum computing and quantum technology applications Amico et al. (2021, 2022). Two main Bose-Einstein condensate (BEC) types have been considered for realizing qubits: multi-component condensates and multi-mode condensates. Multi-component and spinor condensate approaches Cirac et al. (1998); Tian and Zoller (2003); Byrnes et al. (2012, 2015); Luiz et al. (2015); Großardt encode a single qubit in two (or more) metastable atomic states, such as spin or hyperfine levels, with all atoms in the same translational mode (for example the motional ground state). Multi-mode approaches Solenov and Mozyrsky (2011); Amico et al. (2014); Aghamalyan et al. (2016) encode a single qubit using two (or more) translational modes in a scalar condensate, such as a BEC in a double-well trapping potential. In the limit where there are a large number of condensed bosons in each well, the system becomes equivalent to two (or more) BECs, each with a well defined phase, connected by tunneling barriers that act as Josephson junctions Solenov and Mozyrsky (2011); Amico et al. (2014); Aghamalyan et al. (2016). Arrays of such BECs can be produced in optical lattices and are described by the Bose-Hubbard model Amico et al. (2014); Aghamalyan et al. (2016). Another multi-mode approach, which we adopt here, uses circulating states in a ring geometry Ryu et al. (2020); Kapale and Dowling (2005); Ryu et al. (2007); Ramanathan et al. (2011); Eckel et al. (2014); Kim et al. (2018); Pezzè et al. for the translational modes.

Given the demonstrated high performance and scalability of trapped ion qubits Moses et al. (2023), superconducting qubits Krinner et al. (2022); Google Quantum AI (2023); McKay et al. , and of neutral atom arrays Bluvstein et al. (2022); Graham et al. (2022), what does a BEC qubit offer? We argue that it offers a platform for an alternative approach to quantum information processing that leverages the special properties of condensates. In this approach, a BEC is used to prepare a register of qubits in a product state $|\psi\rangle^{\!\otimes n}$ and control its subsequent evolution. From a quantum computing perspective, having multiple identical copies of an unknown input is already a useful resource (whereas classical information is freely cloned). In standard circuit-model quantum computation, illustrated in Fig. 1a, initialized qubits are subsequently entangled using two-qubit gates. Here we do the opposite and try to suppress entanglement, Fig. 1b. This is achieved by making $n$ large, interactions weak, and by preserving permutation symmetry. In this limit entanglement monogamy Coffman et al. (2000); Zong et al. bounds the pairwise concurrence to zero, and the BEC is exactly described by a nonlinear Schrödinger equation (the Gross–Pitaevskii equation Gross (1961); Pitaevskii (1961)), enabling novel dynamics Abrams and Lloyd (1998); Wu and Niu (2000); Riedel et al. (2010); Meyer and Wong (2013); Amico et al. (2014); Childs and Young (2016); Aghamalyan et al. (2016); Ryu et al. (2020); Xu et al. (2022); Deffner (2022); Geller (2023a). The theory is developed in a large $n$ limit with a rigorous bound on the error resulting from the mean field approximation. The nonlinear approach trades exponential time complexity for space complexity, requiring $n$ to be large.

Does this mean that $n$ has to be exponentially large? Actually the requirements on $n$ are not that bad. This is because the BEC is assumed to be initialized in a product state, and it takes time $t_{\rm ent}$ for the atomic collisions to produce entanglement. Ideally, the whole experiment is performed in a short-time regime. We measure the accuracy of mean field theory by $\epsilon:=\|\rho_{\rm eff}(t)-\rho_{1}(t)\|_{1}$, and call this the model error. Here $\rho_{\rm eff}$ is the mean field state, $\rho_{1}$ is the exact state traced over all atoms but one, $t$ is the gate duration, and $\|\cdot\|_{1}$ is the trace norm. In a large family of condensate models Erdős and Schlein (2009); Geller (2023b)

where $c$ and $t_{\rm ent}$ are positive constants (model-dependent quantities independent of $t$ and $n$). Although the error might grow exponentially in time, there is always a short-time window $t<t_{\rm ent}$ where the required number of condensed atoms $n\approx ct/t_{\rm ent}\epsilon$ is sub-exponential in $t$.

We propose a demonstration of quantum information processing using this nonlinearity. In the remainder of this section we discuss the qubit encoding and state discrimination subroutine. The protocol is explained in Sec. II. Conclusions are given in Sec. III, and additional information about the BEC model and large $n$ limit are provided in the Appendix.

The process is illustrated in Fig. 2. A single state $|\psi\rangle\in\{|a\rangle,|b\rangle\}$ is input to the discriminator, which ideally returns output $|0\rangle$ if $|\psi\rangle\!=\!|a\rangle$, or returns $|1\rangle$ if $|\psi\rangle\!=\!|b\rangle$. The single-input discriminator regarded as a channel must be nonunitary, because the overlap $\langle a|b\rangle$ is not preserved. Equivalently, the distance $\|\rho_{a}-\rho_{b}\|_{1}$ between their density matrices in trace norm is not preserved in time (here $\|X\|_{1}:={\rm tr}\sqrt{X^{\dagger}X}$). For pure states, $\|\rho_{a}-\rho_{b}\|_{1}=2|\sin(\theta_{ab}/2)|$, where $\theta_{ab}$ is the angle between their Block vectors. Linear completely-positive trace preserving (CPTP) channels satisfy $\frac{d}{dt}\|\rho_{a}-\rho_{b}\|_{1}\leq 0$; they are either distance preserving or strictly contractive on the inputs Ruskai (1994). Because the discriminator orthogonalizes the potential inputs, it is expansive on those inputs: $\frac{d}{dt}\|\rho_{a}-\rho_{b}\|_{1}>0$. Thus, the discriminator is described by a nonlinear PTP channel Abrams and Lloyd (1998); Mielnik (1980); Geller (2023a).

The implementation proposed here does not discriminate an unknown input (produced by a previous computation), but instead uses a black box state preparation step to randomly prepare $|a\rangle$ or $|b\rangle$, with a small Bloch vector angle $\theta_{ab}\geq 0$ between them. Then $|\langle a|b\rangle|^{2}=\cos^{2}(\theta_{ab}/2)\approx 1-(\theta_{ab}/2)^{2}$. This can be accomplished by initializing in the $|\Phi^{n}_{0}\rangle$ state and using $V_{01}$ and $B_{z}$ in (7) to apply $x$ and $z$ rotations. (Ideally, this step is hidden from the remainder of the experiment.) The discrimination gate itself follows Refs. Abrams and Lloyd (1998); Childs and Young (2016) and uses the $z$-axis torsion to increase the angle between $|a\rangle$ and $|b\rangle$. It’s clear that $|a\rangle$ and $|b\rangle$ should begin with equal and opposite $z$ components $z_{a}=-z_{b}$ [here $r^{\mu}_{a,b}={\rm tr}(\rho_{a,b}\sigma^{\mu}),\ \mu\in\{1,2,3\}$]. Consider a simple option with $y_{a,b}=0$, namely

which has

After switching on $g$, the two input options evolve as $R_{z}(\pm gt\theta_{ab})$ and orthogonalize after a time $t\approx\pi/g\theta_{ab}$. However this implementation does not have a favorable scaling with $\theta_{ab}$. The optimal protocol for nonlinear discrimination was derived by Childs and Young (CY) in Childs and Young (2016). Instead of (11), the CY gate begins with

and applies $x$ rotations to hold $y_{a,b}=z_{a,b}$ during the subsequent evolution in order to reach antipodal points on the Bloch sphere. The options orthogonalize in a time $t=O(\log\frac{1}{\theta_{ab}})$, after which a readout gate $U_{\rm read}$ (defined with respect to time-evolved $|a\rangle,|b\rangle$) transforms them to circulation states $|\Phi_{0}^{n}\rangle$ or $|\Phi_{1}^{n}\rangle$, which are then measured via time-of-flight Amico et al. (2005); Moulder et al. (2012).

In an idealized context where (7) is regarded as exact, and where there are no control errors, readout errors, decoherence errors, or noise, the nonlinear discriminator works perfectly every time. We refer to this idealization as a single-input discriminator to distinguish it from the more familiar minimum error and unambiguous discriminators based on linear CPTP channels Barnet and Croke ; Bae and Hwang (2013); Bae and Kwek (2015); Rouhbakhsh and Ghoreishi . Of course any actual atomtronic realization is likely to suffer from all such errors, and may fail to give the correct answer or return an answer at all. Although the theoretically achievable performance depends sensitively on the system and device details, and is beyond the scope of this work, we note that combining torsion with non-CP dissipation is predicted to implement an autonomous discriminator Geller (2023a), whose control sequence and operation is (mostly) independent of $|a\rangle$ and $|b\rangle$. In this implementation, the nonlinearity and dissipation create two basins of attraction with a shared boundary in the Bloch ball, one with an attracting fixed point near $|0\rangle$ and the other with an attracting fixed point near $|1\rangle$, giving the discriminator a degree of intrinsic fault-tolerance.

The presence of channel nonlinearity indicates a breakdown of the superposition principle. Figure 3 illustrates a nice example of this effect: In Fig. 3a, the evolution of a superposition $\psi_{0}|0\rangle+\psi_{1}|1\rangle$ is given by a superposition of evolved basis states $e^{-it}|0\rangle$ and $e^{it}|1\rangle$, shown as a velocity field. However in Fig. 3b, the evolved states $e^{-it}|0\rangle$ and $e^{-it}|1\rangle$ are now static (phase factors are a global phase), whereas the actual dynamics is not, except on the equatorial plane.

We have discussed an approach to quantum information processing that leverages the special properties of condensates, including their nonlinearity, and proposed an atomtronic implementation of a “nonlinear” qubit. An experimental demonstration of nonlinear state discrimination, while striking, would not by itself constitute a computation, because the qubit isn’t coupled to anything. To implement a useful computation, the BEC qubit must be entangled with other qubits (for example trapped ions) in a scalable circuit-model quantum computer, which is not addressed here.

The standard models of quantum computation assume gates and errors based on linear CPTP channels. Physical hardware, however, might admit initial correlation and be better described by more general maps Dominy et al. (2016); Dominy and Lidar (2016). It is therefore interesting to investigate any additional computational power enabled by quantum channels beyond the linear CPTP paradigm, as we did here. Another example was investigated by Chen et al. Chen et al. , who experimentally demonstrated unambiguous state discrimination in a linear but non-Hermitian optical system. After completing this work, Großardt posted a preprint Großardt proposing the use of a two-component BEC coupled to a neutral atom computer to simulate a large family of nonlinear Schrödinger equations. Given their potential for fast quantum state discrimination and simulation of the nonlinear Schrödinger equation, the non-Hermitian and nonlinear approaches to quantum information processing deserve further exploration.