Abstract
Quantum photonic integrated circuits, composed of linear-optical elements, offer an efficient way for encoding and processing quantum information on-chip. At their core, these circuits rely on reconfigurable phase shifters, typically constructed from classical components such as thermo- or electro-optical materials, while quantum solid-state emitters such as quantum dots are limited to acting as single-photon sources. Here, we demonstrate the potential of quantum dots as reconfigurable phase shifters. We use numerical models based on established literature parameters to show that circuits utilizing these emitters enable high-fidelity operation and are scalable. Despite the inherent imperfections associated with quantum dots, such as imperfect coupling, dephasing, or spectral diffusion, we show that circuits based on these emitters may be optimized such that these do not significantly impact the unitary infidelity. Specifically, they do not increase the infidelity by more than 0.001 in circuits with up to 10 modes, compared to those affected only by standard nanophotonic losses and routing errors. For example, we achieve fidelities of 0.9998 in quantum-dot-based circuits enacting controlled-phase and – not gates without any redundancies. These findings demonstrate the feasibility of quantum emitter-driven quantum information processing and pave the way for cryogenically-compatible, fast, and low-loss reconfigurable quantum photonic circuits.
1 Introduction
Reconfigurable quantum photonic integrated circuits (qPICs) are versatile tools capable of simulating molecular dynamics [1], executing quantum logic [2], and generating multidimensional entanglement [3]–[7]. They utilize quantum properties such as entanglement and indistinguishability for information processing, which is unachievable through classical means. This capability is crucial for developing emerging quantum communication and computation technologies.
To date, qPICs have predominantly operated at room temperature, harnessing the advancements of foundry photonics to create increasingly complex devices [8], [9]. As depicted in Figure 1, the core of these circuits is a mesh of Mach–Zehnder interferometers (MZIs) [10], where each MZI is comprised of two directional couplers and two phase shifters (see Figure 1(b)), typically thermo-optical in nature [11], [12]. That is, this mesh is fully classical, based on low-loss and highly accurate (albeit slow and hot) phase shifters. More recently, electro-optic [13], [14] and opto-mechanical [15], [16] phase shifters have become popular alternatives, however, they are limited to large footprints (
The quantum nature of qPICs stems from the individual photons that propagate through the circuits. Single photons are often generated by optical nonlinearities such as spontaneous parametric down-conversion, which is compatible with room-temperature chips but inherently probabilistic [23]–[25]. In contrast, single-photon sources based on solid-state quantum emitters, such as the quantum dots (QDs) we consider, can operate on-demand but require cryogenic temperatures [26]–[29]. Similarly, high-efficiency integrated single-photon detectors also necessitate a cryogenic environment [30]–[32]. As a result, both state-of-the-art sources and detectors cannot be heterogeneously integrated with qPICs and instead currently rely on lossy interconnects [33].
Here, we propose that QDs can be used not only as single-photon sources, but also as reconfigurable phase shifters for creating fast, cryogenically-compatible meshes. As photons scatter from QDs, and indeed all solid-state emitters, the imparted phase shift depends on the detuning between the photon and emitter transition frequency. For the case of QDs, the detuning can be modulated electrically [29], [34], optically [35], [36], or by using strain [37]. Our model of QD-based qPICs builds from this operating principle, yet additionally incorporates standard nanophotonic imperfections, such as losses and routing errors, along with QD-specific non-idealities, like imperfect interactions and both fast and slow noise processes. We use this model to evaluate the fidelity of both the resultant unitary operations and the desired output states. Our findings reveal that these QD-based meshes can be optimized to achieve remarkable scalability, with a unitary infidelity less than 0.001 for circuits up to 10 × 10 in dimension, using state-of-the-art QD parameters from the literature. We further consider QD-based controlled-phase and – not gates as examples, where we find that state-of-the-art circuits process logical states with fidelities of 0.9998. In sum, our results offer a roadmap to cryogenically-compatible, reconfigurable qPICs based on solid-state quantum emitters.
2 Quantum-emitter phase shifters
The scattering of photons from a quantum emitter embedded in a nanophotonic waveguide is a complex process [38] that may modulate the photons’ amplitude or phase [39], [40] or, when more than a single photon is present, induce complex correlations [35], [41]. The exact response depends on the properties of the emitter and the efficiency with which it couples to the various available modes, as sketched in Figure 2(a), yet in the most general case, an input single-photon state with phase φ0,
where, in the presence of losses the sum of the probabilities to scatter coherently and incoherently, |αco|2 and |αinc|2, do not sum to unity.
The description of Eq. (1) only holds if the single-photon pulse is not reshaped during scattering, requiring that the pulse linewidth σp be much shorter than that of the emitter. More formally, we require that σp ≤ Γ/1000, ensuring the phase shifter response is linear. As shown in the Supplementary Information, this condition holds for any two-level system, including QDs.
In this regime, the single-photon transmission coefficient t and total transmission T are the same as that of a weak coherent beam (see Supplementary Information for derivation),
where βR is the coupling efficiency for right-traveling photons (with a total coupling efficiency β = βR + βL), Δp is the detuning between the photon and emitter-transition frequencies, Γ is the decay rate of the emitter, and Γ2 = Γ/2 + Γdp where Γdp is the pure dephasing rate (i.e. fast noise). We note that in the presence of dephasing,
Eqs. (2) and (3) allow us to quantify the results of the scattering. The coefficients, αco and αinc, are related to the transmission, as shown in Figure 2(b). In the ideal case, where βR = 1 and Γdp = 0, the transmission is always unity (orange curve) meaning that αco = 1 and αinc = 0. Conversely, in the presence of losses and/or dephasing, the situation is more complex with
The phase of the coherently-scattered photons is likewise calculated from
3 QD-based qPICs
Having seen that quantum emitters such as QDs can serve as reconfigurable phase shifters, we quantify the performance of qPICs based on this technology. To do so, we first compare how well we can reproduce any unitary (i.e. operation) with our emitter-based qPICs relative to the ideal, summarizing the results in Figure 3. Here, we show the dependence of the mean circuit infidelity
As an example, consider the β-dependence of
which we then average. In the figure, we show the cases for N = 2, 6, 10 mode circuits, where in all cases, we observe a monotonic increase from a baseline
Encouragingly, we can optimize the performance of the emitter-based qPICs, following a fast and efficient routine that finds the optimal set
Overall, we summarize the circuit scaling in Figure 3(e), where we plot the raw and optimized infidelity as a function of circuit size, both using typical and state-of-the-art parameters (see Table 1). For typical values (blue curve), we see that the infidelity quickly approaches unity, yet by adding imperfections sequentially (purple and orange curves), we see that this is almost entirely caused by the residual dephasing. This is consistent with the optimized state-of-the-art qPIC performance (green curve), where
4 Examples: CZ gate
To demonstrate the possibilities of emitter-based qPICs, we consider the controlled-phase (CZ) gate, which can be used to generate entanglement [52], and, in the Supplementary Information, a similar realization of a controlled-not (CNOT) gate that enables universal quantum computation [53]. A linear-optical unheralded CZ gate can be realized on a 6 × 6 mesh [52], and in Figure 4(a) we plot the unitary infidelity
More significantly, we consider the fidelity with which logical states are processed by the CZ gate
5 Conclusions
Our research demonstrates that qPICs built with reconfigurable QD-phase shifters can perform comparable to those using classical phase shifters. Notably, we show that with state-of-the-art QD parameters, circuits can be scaled up to 10 modes without significant increases in unitary infidelity. This advancement allows QD-based qPICs to efficiently perform operations such as multi-qubit gates, as demonstrated in our study, and to simulate molecular dynamics [54] at cryogenic temperatures. In this respect, QD-based phase shifters join a select class that includes electro-optical [13], [14] and opto-mechanical [15], [16] shifters, but with a much smaller footprint, low operational energy and fast response times. As with current quantum photonic circuits, the performance of emitter-based systems could be further enhanced by incorporating redundancies, where extra MZIs and phase shifters provide better compensation for imperfections [17], [19]–[21].
We recognize that while QDs are the only quantum emitters currently integrated into photonic circuits, alternative technologies based on single organic molecules [55], [56] and defects in diamond [57], [58], silicon [59], or silicon carbide [60], [61] are rapidly maturing. In fact, even with QDs, not all state-of-the-art parameters have been demonstrated using a single chiral quantum photonic platform (c.f. Table 1). To date, however, high quality chiral quantum interfaces have been demonstrated with nanobeams [48], glide-plane waveguides [44] and topological photonics [62], [63]. Recent calculations suggest that, were the emitter to be placed at exactly the correct location within either photonic resonators [64] or waveguides [65], these could act as near-ideal chiral interfaces. This is particularly promising in light of recent developments demonstrating the ability to pre-select specific QDs and integrate them deterministically within a circuit [66]–[71].
Finally, we note that fabricating and managing circuits with numerous emitters remains a topic of inquiry. Recent experiments with QDs have successfully demonstrated the integration of deterministic QD-based single-photon sources with qPICs [72], [73], and independent control of multiple emitters within a circuit [74]–[76], respectively. These advancements open the door to larger-scale implementations where emitters would function both as sources and processing elements. Such circuits would be entirely cryogenic, and thus compatible with deterministic sources and detectors, with phase shifters whose operational speeds are determined by emitter lifetimes, potentially enabling GHz rate operation with mild enhancement [50].
Funding source: Vector Institute
Funding source: Natural Sciences and Engineering Research Council of Canada
Funding source: Canada Foundation for Innovation
Acknowledgment
The authors thank J. Carolan for generating stimulating discussions on quantum photonic integrated circuits, and gratefully acknowledge support by the Natural Sciences and Engineering Research Council of Canada (NSERC), the Canadian Foundation for Innovation (CFI), the Vector Institute, and Queen’s University.
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Research funding: Natural Sciences and Engineering Research Council of Canada (https://doi.org/10.13039/501100000038); Canadian Foundation for Innovation (https://doi.org/10.13039/501100000196); Vector Institute (https://doi.org/10.13039/501100019117).
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Author contributions: The project was conceived and planned by N.R. with support from B.J.S. A.M. programmed the model of the emitter-based-network, with help from J.E., and performed all simulations. Analysis and interpretation was led by A.M. and N.R. with support from all authors. All authors contributed to the writing and editing of the manuscript. All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
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Conflict of interest: Authors state no conflict of interest.
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Data availability: The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.
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