Abstract
Excitonpolariton condensates have attractive features for quantum computation, e.g., room temperature operation, high dynamical speed, ease of probe, and existing fabrication techniques. Here, we present a complete theoretical scheme of quantum computing with excitonpolariton condensates formed in semiconductor micropillars. Quantum fluctuations on top of the condensates are shown to realize qubits, which are externally controllable by applied laser pulses. Quantum tunneling and nonlinear interactions between the condensates allow SWAP, squarerootSWAP and controlledNOT gate operations between the qubits.
Introduction
Excitonpolaritons are hybrid lightmatter particles, where the small dephasing of a photonic system is blended with the particle–particle interactions from a condensed matter system. The overwhelming majority of research into excitonpolaritons has been focused on semiclassical physics, that is, physics where the quantization of the particle number according to quantum theory is not itself apparent: BoseEinstein condensation and polariton lasing;^{1} quantum fluids and solitons;^{2} and topological polaritons^{3} are all welldescribed within semiclassical theory. Nevertheless, it has always been expected that excitonpolaritons are ultimately quantum particles,^{4} theoretically capable of entanglement,^{5,6} and it has been speculated that polaritons could perform as quantum computers^{7,8} or quantum simulators.^{9,10}
As the quality factor of microcavities has steadily progressed, there has been a recent resurgence of interest in quantum excitonpolariton physics. Through exciting a polariton with an entangled photon it was proven that excitonpolaritons are indeed quantum particles, capable of carrying correlations at the quantum level.^{11} The onset of the polariton blockade^{12} has also shown that the repulsive interaction energy between just two polaritons is considerable,^{13,14} and that suggestions of strongly interacting polaritons can be reachable in the future.^{15,16}
However, serious challenges must be addressed before claiming that excitonpolaritons can perform as a quantum computer. First, a suitable basis for encoding information must be chosen and it must be shown that excitonpolaritons are compatible with such a basis. Typically, architectures for quantum computation are based on qubits, which require a two level quantum system. It is not immediately obvious that excitonpolaritons can serve in this regard, since they typically form multiparticle states. Even though binary degrees of freedom (e.g., spin or vorticity^{17,18}) exist, this does not automatically mean that polaritons can operate as qubits as the different degrees of freedom can themselves be populated by a range of particle number (Fock) states.
Even if a quantum computational basis can be established, a significant task remains in designing quantum logic gates (assuming that one aims for emulation of the most wellknown quantum circuit model of quantum computation). If a set of universal quantum gates can be established, then in principle any unitary operation can be reproduced on a given input state. For example, a set of single qubit logic gates combined with squarerootSWAP gate (or controlledNOT gate) is universal. However, it is essential that the gates can be arbitrarily connected and formed into scalable circuits, which is already a challenging task for classical polaritonic logic gates.^{19}
Finally, it is important that the logic gates should be operable on a timescale faster than the dissipation and dephasing rates of excitonpolaritons. While classical information carried in the form of a polariton wavepacket can be reamplified with a nonresonant external laser,^{20} which replaces lost particles with those of the same classical properties (e.g., phase and polarization), applying the same to a quantum polariton state actually causes quantum information to be lost faster.^{21} Essentially, if a polariton carrying quantum correlations is lost from the system it can not be replaced with a fully identical particle. Attempting to do so necessarily worsens the situation as the replacement particle can not possibly have the same quantum correlations with the rest of the system as the particle that was lost. Thus all gates need to act before polaritons escape the system. Fortunately, stateoftheart microcavities support polariton lifetimes exceeding hundreds of picoseconds,^{22} while applied laser pulses can be operated on sub picosecond scale or faster.
Here we address the above challenges from the theoretical point of view, considering excitonpolariton condensates in micropillar arrays^{9} as a platform. Even though a polariton condensate is composed of many particles, we show that number fluctuations on top of the meanfield value themselves correspond to an anharmonic oscillator in the presence of moderate polariton–polariton interactions. Transitions between the lowest two energy levels are nonresonant with other levels, effectively allowing a qubit basis, which can be measured with homodyne detection techniques. As outlined in Fig. 1, single qubit gates are controllable by laser parameters and the availability of quantum tunneling between neighboring micropillars, it is possible to realize SWAP and squarerootSWAP (sSWAP) gates, which allow for moving qubits between different pillars and universal quantum operations to establish a complete scalable quantum circuit architecture. Additionally, the availability of cross–Kerr interactions between polaritons, which are allowed via the presence of cross–spin interactions, are shown to realize a quantum controlledNOT (cNOT) gate.
The excitonpolariton condensate
We consider a semiconducting micropillar that supports a nonlinear polariton mode \({\hat{\mathrm{a}}}\). The mode is excited with a coherent optical field \({\mathcal{P}}(t)\). The Hamiltonian is thus
where Δ is a detuning between the optical field and the polariton mode and α is the Kerr nonlinear interaction strength. We define a condensate as a coherent state, which is readily generated by the application of a constant coherent optical field \({\mathcal{P}}(t)={P}_{0}\) with the occupation number, \({N}_{{\mathrm{c}}}=\langle {\hat{{\rm{a}}}}^{\dagger }\hat{{\rm{a}}}\rangle\). We aim to engineer the quantum operations by switching the optical pump from \({\mathcal{P}}(t)={P}_{0}\) to \({\mathcal{P}}(t)=P{e}^{i\varphi }\) for a very short time compared to the polariton lifetime. Since the condensate formed by P_{0} does not have enough time to decay, the short excitation Pe^{iφ} will only induce fluctuations on top of the preformed condensate. We now consider the following change of variable: \({\hat{{\rm{a}}}}^{\dagger }=\sqrt{\hat{{\rm{N}}}}\exp (i\hat{\theta })\), where \(\hat{{\rm{N}}}\) and \(\hat{\theta }\) are the number and phase operators of the condensate satisfying the commutation relation \([\hat{\theta },\hat{{\rm{N}}}]=i\). In terms of these operators the Hamiltonian reads:
The polariton number operator can be expressed as \({\hat{\mathrm{N}}} = {\hat{\mathrm{a}}}^{\dagger}{\hat{\mathrm{a}}} = N_{c} + {\hat{\mathrm{n}}}\), where \({N}_{{\mathrm{c}}}=\langle {\hat{{\rm{a}}}}^{\dagger }\hat{{\rm{a}}}\rangle\) is the meanfield part of the condensate and \(\hat{{\rm{n}}}\) is the quantum part. Note that the mean field N_{c} cannot carry quantum information itself. Instead, the quantum part \(\hat{{\rm{n}}}\) will be used for encoding quantum information. We express the Hamiltonian in terms of \(\hat{{\rm{n}}}\):
where C = ΔN_{c} + αN_{c}(N_{c} − 1) is a classical number and Ω = Δ + α(2N_{c} − 1) is defined as the effective detuning. Here we now consider that the magnitude of N_{c} is much larger than the quantum fluctuations \(\hat{{\rm{n}}}\) (which is the standard case for polariton condensates), such that the effective Hamiltonian is given by,
Note that inside the squareroot, we have neglected the operator \(\hat{{\rm{n}}}\) which is justified when \(\sqrt{\langle {\hat{{\rm{n}}}}^{2}\rangle }/{N}_{{\mathrm{c}}}\ll 1\). In Fock space, the number operator satisfies \(\hat{{\rm{n}}}\leftn\right\rangle =n\leftn\right\rangle\) and the corresponding phase operator satisfies \(\exp (\!\pm i\hat{\theta })\leftn\right\rangle =\leftn\pm 1\right\rangle\) (see the Supplementary Information). The effective Hamiltonian in the Fock space then reads,
This effective system is formed with an excitonpolariton condensate in a micropillar interacting with a laser field. Note that a coherent condensate can be achieved even in the strongly interacting regime (see Supplementary Information). The system has similarity with a superconducting qubit formed in a Josephson junction between two superconducting islands. While one among the two superconducting islands is analogous to the condensate in the micropillar, the second one is replaced by the classical laser field which can be manipulated (through controlling its phase, detuning and amplitude) with much more ease and accuracy than a cryogenically cooled superconducting island (controlling phase, energy and amplitude of the superconducting wavefunction are much more challenging). Moreover, the condensate in a semiconductor micropillar itself can operate at much higher temperatures (even at room temperature) than the operating temperatures of superconducting qubits.
Anharmonic spectrum and the qubit
In Fig. 2, we find that the spectrum of the Hamiltonian is periodic in the effective detuning Ω with a period 2α. Let us replace Ω with an effective detuning parameter ω that varies between ±α such that the spectrum is restricted within one period. Moreover, in Fig. 2 we see that the gaps between the lowest two energy levels and the next ones are different. According to our need, we can operate in some regime where the difference in the gaps can be obtained by diagonalizing the Hamiltonian given by Eq. (5). These unequal gaps are originating from the anharmonic behavior of the Hamiltonian, which can be explicitly seen in Eq. (4) in the form of the cosine term (if instead of the cosine term one had a \({\hat{\theta }}^{2}\) term, one would have a harmonic oscillator). Let us consider that δE_{1} is the gap between the first two energy levels and δE_{2} is the same between the second and third energy levels. In the regime, where ∣δE_{1} − δE_{2}∣ is larger than the linewidth γ, we can consider only the lowest two energy levels as our qubit basis \(\left0\right\rangle\) and \(\left1\right\rangle\). In this low energy qubit subspace, the effective qubit Hamiltonian reads \({\hat{H}}_{{\mathrm{q}}}={E}_{{\mathrm{x}}}{\hat{\sigma }}_{{\mathrm{x}}}+{E}_{{\mathrm{y}}}{\hat{\sigma }}_{{\mathrm{y}}}+{E}_{{\mathrm{z}}}{\hat{\sigma }}_{{\mathrm{z}}}\), where \({\hat{\sigma }}_{i}\) are the Pauli matrices, \({E}_{{\mathrm{x}}}\,=\,P\sqrt{{N}_{{\mathrm{c}}}}\cos \varphi\), \({E}_{{\mathrm{y}}}\,=\,P\sqrt{{N}_{{\mathrm{c}}}}\sin \varphi\), and E_{z} = (α − ω)/2. Note that here we have excluded the overall energy shift \({\mathbb{1}}(\alpha  \omega)/2\) from the Hamiltonian. Formally, this Hamiltonian can be rewritten in a compact form:
where the vectors E = (E_{x}, E_{y}, E_{z}) and \(\hat{{\boldsymbol{\sigma }}}=({\hat{\sigma }}_{{\mathrm{x}}},{\hat{\sigma }}_{{\mathrm{y}}},{\hat{\sigma }}_{{\mathrm{z}}})\). This qubit Hamiltonian can be externally controlled by tuning the vector E through controlling the laser detuning ω, strength P, and the phase φ.
Singlequbit gates
For universal quantum computing, singlequbit gates are required. A singlequbit gate operation is equivalent to a unitary rotation of a qubit around a given axis. Here, we show that the evolution of an initially prepared state under the effective Hamiltonian \({\hat{H}}_{\rm{q}}\) can induce an arbitrary singlequbit quantum gate operation. For specific operation, the parameters involved in the Hamiltonian must be chosen appropriately. If the duration of the application of the pump Pe^{iφ} is τ, then the evolution operator is given by \(\exp (i\tau {\hat{H}_{\mathrm{q}}}/\hbar )\), which can be rewritten as,
where \(\beta = \tau E / \hbar\), and ϵ = E/E is a unit vector controlled by the system parameters, and \(E=\sqrt{{P}^{2}{N}_{{\mathrm{c}}}+{(\alpha \omega )}^{2}/4}\). This is a general form of a unitary matrix in singlequbit space. More explicitly the operator can be written in matrix form:
where we parameterized the unit vector in angular coordinates \({\boldsymbol{\epsilon }}=(\sin \xi \cos \varphi ,\ \sin \xi \sin \varphi ,\ \cos \xi )\). By choosing appropriate β, ξ, and φ we can obtain different quantum operations, e.g., a rotation \({\hat{R}}_{{\mathrm{x}}}(\beta )\) around \({\hat{\sigma }}_{x}\) with (ξ = π/2, φ = 0), and a rotation \({\hat{R}}_{\rm{y}}(\beta )\) around \({\hat{\sigma }}_{\rm{y}}\) with (ξ = π/2, φ = π/2). Since any single qubit unitary operation can be composed as \({\hat{R}}_{{\mathrm{x}}}({\beta }_{1}){\hat{R}}_{{\mathrm{y}}}({\beta }_{2}){\hat{R}}_{{\mathrm{x}}}({\beta }_{3})\) (where β_{1,2,3} correspond to three time durations),^{23} arbitrary singlequbit gates can be obtained with a fixed ξ = π/2 and controlling only β and φ. This shows that arbitrary singlequbit operations are allowed by a pulse with a fixed detuning ω = α (equivalently E_{z} = 0) but a time dependent phase φ. However, additional control on the laser field amplitude P provides control on ξ, which allows more flexibility on achieving singlequbit gates, e.g., a frequently used singlequbit gate, known as the Hadamard gate, can be readily realized with (β = π/2, ξ = π/4, φ = 0).
SWAP and sSWAP gates
Let us now consider that two physically separated polariton condensates with associated field operators \({\hat{{\rm{a}}}}_{1}\) and \({\hat{{\rm{a}}}}_{2}\) are coupled via a coherent (Josephson) tunneling term. The corresponding Hamiltonian is given by,
where \({H}_{j}={\Delta }_{j}{\hat{{\rm{a}}}}_{j}^{\dagger }{\hat{{\rm{a}}}}_{j}+{\alpha }_{j}{\hat{{\rm{a}}}}_{j}^{\dagger }{\hat{{\rm{a}}}}_{j}^{\dagger }{\hat{{\rm{a}}}}_{j}{\hat{{\rm{a}}}}_{j}+{P}_{j}({\hat{{\rm{a}}}}_{j}{e}^{i{\varphi }_{j}}+{\hat{{\rm{a}}}}_{j}^{\dagger }{e}^{i{\varphi }_{j}})\) corresponds to a single condensate. Following the same procedure as we have shown for a single condensate, we obtain a low energy effective Hamiltonian:
where E_{T} is the tunneling amplitude. This Hamiltonian allows both efficient SWAP and sSWAP operations. While a SWAP operation swaps the quantum states between the two quibts, sSWAP operations establish universality (any quantum operation can be achieved with sSWAP and singlequbit gates).
Our considered system is a driven dissipative system. Excitonpolaritons have a finite lifetime \({\hbar}\)/γ. An accurate description of this driven dissipative system is given by the quantum master equation:
where \(\hat{\rho }\) is the density matrix describing the qubits, \({{\mathcal{L}}}^{j}(\hat{\rho })=2{\hat{\sigma }}_{}^{j}\hat{\rho }{\hat{\sigma }}_{+}^{j}{\hat{\sigma }}_{+}^{j}{\hat{\sigma }}_{}^{j}\hat{\rho }\hat{\rho }{\hat{\sigma }}_{+}^{j}{\hat{\sigma }}_{}^{j}\), and \({\hat{\sigma }}_{\pm }^{j}\) are the raising and lowering operators. In Fig. 3b, we show the gate fidelities of sSWAP and SWAP gates as the function of the ratio of the polariton–polariton interaction strength to the dissipation rate.
For efficient gate operations we need to externally control the tunneling amplitude E_{T}, such that the quantum tunneling can be activated at will. In principle, Josephson coupling can be modulated by applying stress^{24} or a magnetic field,^{25} however, here we opt for a scheme where two qubits in two physical semiconductor micropillars are connected by a third micropillar placed between them (see the inset of Fig. 3b). This three site system is considered with a fixed nearestneighbor tunneling amplitude, g (see Fig. 4). The onsite energy of the middle site could be controlled via an external laser field. In Fig. 4, we show the probability of the transition \(\left10\right\rangle \to \left01\right\rangle\) as a function of the ratio between the relative detuning Δ_{s} and the tunneling amplitude g between the neighboring micropillars. We find that the transition probability can be tuned from a high value to a low one by tuning the parameter Δ_{s}/g.
ControlledNOT gate
Although a cNOT operation can be obtained as a combination of singlequbit and sSWAP gates, here we show that a direct realization of cNOT gate is also possible in semiconductor micropillars. Let us consider that two polariton condensates with associated field operators \({\hat{{\rm{a}}}}_{1}\) and \({\hat{{\rm{a}}}}_{2}\) are interacting with a crossKerr type nonlinearity. The corresponding Hamiltonian is given by,
where the Hamiltonian \({H}_{j}={\Delta }_{j}{\hat{{\rm{a}}}}_{j}^{\dagger }{\hat{{\rm{a}}}}_{j}+{\alpha }_{j}{\hat{{\rm{a}}}}_{j}^{\dagger }{\hat{{\rm{a}}}}_{j}^{\dagger }{\hat{{\rm{a}}}}_{j}{\hat{{\rm{a}}}}_{j}+{P}_{j}({\hat{{\rm{a}}}}_{j}{e}^{i{\varphi }_{j}}+{\hat{{\rm{a}}}}_{j}^{\dagger }{e}^{i{\varphi }_{j}})\) describes a single condensate, α_{j} and α_{12} are the Kerr and crossKerr nonlinearity strengths, and P_{j}, φ_{j}, and Δ_{j} are the amplitudes, phases and detuning values of the applied laser fields. The crossKerr nonlinearity can be physically realized with excitonpolaritons by considering the spin degree of freedom. Excitonpolaritons can have one of two spin projections on the structure growth axis,^{26} corresponding to two independent quantum modes. Noise in the spin is drastically suppressed in the regime of polariton condensates.^{27} While the quantum gates in our schemes are aimed to operate in a fraction of the polariton lifetime, timescales associated to spin noise and the number fluctuations are much larger for large condensate occupation.^{27} It has also been well established that crossKerr type nonlinearity is present between the opposite spin states, and a variety of interaction strengths have been achieved experimentally.^{28} Alternatively, crossphase modulation between neighboring optical cavity modes has been discussed in different works,^{29,30} although not yet realized experimentally for excitonpolariton systems.
Following the same procedure as we have shown for a single condensate, we obtain a low energy effective Hamiltonian:
where \({{\boldsymbol{E}}}_{j}=[{P}_{j}\sqrt{{N}_{{\mathrm{c}}}}\cos {\varphi }_{j},\ \ {P}_{j}\sqrt{{N}_{{\mathrm{c}}}}\sin {\varphi }_{j},\ \ ({\alpha }_{j}{\omega }_{j})/2]\), \({E}_{z}^{12}={\alpha }_{12}/2\), and ω_{j} varies between ±α_{j}. For demonstrating cNOT operations, we consider the twoqubit Hamiltonian \({H}_{{q}_{1}{q}_{2}}\) with φ^{j} = 0. It has been shown earlier that this Hamiltonian can be used for inducing cNOT operation.^{31} For instance, under a pulse of duration τ, the evolution operator is given by,
We find that this evolution operator becomes a cNOT operation for \({E}_{z}^{1}=0\), \({E}_{y}^{2}/{E}_{z}^{2}\ll 1\), and a suitable τ. In Fig. 3b, we show the fidelity of an excitonpolariton cNOT gate as a function of the ratio of the polaritonpolariton interaction strength to the dissipation rate.
Discussion
We have arrived at a theoretical scheme for implementation of the quantum circuit model of quantum computation using excitonpolariton condensates. Qubits are encoded in an anharmonic oscillator formed by the quantized fluctuations of the particle number about the meanfield value. A universal set of quantum gates (singlequbit and sSWAP gates) can be engineered using individual micropillars and quantum tunneling among them. SWAP gates are introduced to allow the coherent transport of quantum information between qubits. We further showed that cNOT gates can be directly induced by exploiting the available crossKerr interactions between modes with different polarizations. Other quantum gates, e.g., iSWAP and squarerootiSWAP can also be implemented straightforwardly within our scheme (see Supplementary Information). These gates would be controllable via the use of external laser pulses.
Our scheme is best suited for condensates with large number of polaritons. While a condensate with 50–100 polaritons can induce high quality quantum gates, smaller polariton occupancies introduce losses in the gate fidelity (see Supplementary Information). Technological imperfections can also introduce effects like small polarization splitting between the spin modes in a micropillar. However, the gate fidelity remains almost unaffected for splittings smaller than the polariton–polariton interaction strength (see Supplementary Information).
Aside exciton–exciton interaction another source of nonlinearity comes from the saturation of the exciton–photon coupling, which can be mapped to a renormalized Kerr nonlinear term in our considered Hamiltonian in Eq. (1).^{2} Similarly, the biexciton nonlinearity can also be accounted for by renormalizing the interaction constant between polaritons with opposite spins.^{32}
While excitonpolariton condensates allow high fidelity quantum gates for lifetimes much larger than the gate operation time, limited lifetime can introduce errors in the quantum circuits. However, faulttolerant quantum computers can be achieved by encoding a logical qubit in multiple exciton–polariton qubits (allowing quantum error correction).^{23,33,34,35} In principle, dynamical decoupling schemes appropriate for open quantum systems could also extend the effective loss time.^{36} Finally, it is notable that typical polariton micropillar lattices can be fabricated with micron scale precision,^{3} while the coherence of polariton condensates has been reported extending over a fraction of a millimeter.^{37} The stateoftheart in quantum computing has been developing rapidly in recent years, with companies developing systems with around 50 qubits,^{38} while superconducting qubit systems^{39} and ion trap^{40} systems have already achieved 8 and 20 qubits, respectively. We hope that polariton lattices, which can potentially have a size of around 100 × 100 = 10^{4} qubits (or double accounting for spin), will also be seen as relevant candidates.
We note that typically a fraction of a milliwatt of laser power is needed to coherently excite a micron sized condensate. A typical 1 mm^{2} sized sample, which could contain 10^{6} individual condensates, would require almost kilowatt power to achieve cryogenic liquid Helium temperature. As superconducting qubits occupy a much larger area, their cryogenic power required per qubit is much higher. In addition, excitonpolaritons can operate faster and at higher temperature.
Data availability
The numerical data presented in this study is available from the authors upon reasonable request.
Code availability
The codes for solving the quantum master equation for coupled mode systems are available from the authors upon reasonable request.
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Acknowledgements
This work was supported by the Singapore Ministry of Education, grant MOE2017T21001.
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S.G. conceived the project through discussions with T.L. S.G. performed the theoretical calculations. Both authors interpreted the results and wrote the manuscript. T.L. supervised the project.
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Ghosh, S., Liew, T.C.H. Quantum computing with excitonpolariton condensates. npj Quantum Inf 6, 16 (2020). https://doi.org/10.1038/s415340200244x
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DOI: https://doi.org/10.1038/s415340200244x
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