Analytical Model for Unitary Magnon-Mediated Quantum Gates in Hybrid Magnon-Photon Systems

Abstract

Modern quantum computing architectures are built to provide low-error environments within which to perform the unitary operations that form the bases of quantum computations. These low-error environments necessary for successful quantum computation require use of complex structures that make large-scale implementation difficult, leading to the investigation of coupled magnon-photon systems [1-6] that can achieve strong coupling rates (> 100 MHz) as a potential platform for hybrid quantum computing architectures. Such coupled magnon-photon systems benefit from the magnetic field tunability of the resonant magnon frequency, where dynamic tuning of resonant magnon frequencies with a pulsed magnetic field at rates comparable to or less than the order of the magnon-photon energy exchange time can realize unique operations not possible with static tuning [7-8]. Previously, we analyzed this problem numerically and found that a parabolically shaped pulsed magnetic field profile can realize the coherent exchange of quantum data and the generation of entanglement by changing the curvature λ and maximum frequency Δ of the parabolic profile [8]. That is, in the same physical structure, depending on the shape of the pulsed magnetic field, different unitary magnon-mediated quantum gates can be realized [8].Here, we consider an analytical approach to study this system. Namely, we consider a hybrid magnon-photon system, consisting of two photonic resonators having degenerate frequencies ωp and one magnonic resonator with a dynamically-tuned resonance frequency ωm(t). The photonic resonators are not directly coupled with each other but are both coupled with the magnonic mode with the coupling rate κ.We consider a simple analytically-solvable interaction protocol in this hybrid system, which can be experimentally realized with rectangular magnetic field pulses of sufficiently large magnitude. We assume that initially (t<0) the frequency difference │ωm–ωp│>>κ and interaction between the subsystems can be ignored. At t=0 the magnonic mode frequency ωm(t) is brought within the efficient interaction interval Δ=ωm–ωp∼κ and is held constant for the duration T. After t=T, the magnonic frequency is once again moved out of the photonic resonance region and any interaction between the subsystems stops. Depending on the control pulse parameters Δ and T, different behaviors of the hybrid system can be realized.Of particular interest is the situation, in which the magnonic and photonic subsystems stay completely decoupled after the action of the control pulse. Such a case corresponds to a unitary operation on the photonic modes, realized through the indirect magnon-mediated coupling, and is of great importance for practical applications of magnonic resonators in quantum computing systems.Using our model, we derived a simple analytical condition of realizing unitary magnon-mediated operations, namely, for each detuning Δ there exist an infinite series of pulse durationsTn(Δ)=2nπ/(8κ2+Δ2)½providing unitary conditions (here, n is a positive integer). The resulting effect of the magnon interaction on the photonic subsystem, in this case, can be described by 2×2 unitary matrix Un=Uei(nπ/2–Tn(Δ)Δ/4) relating photonic states before and after the pulse, which has components:U11=ei2πcos(θn), U12=eiπ/2sin(θn), U21=–e–iπ/2sin(θn), U22=e–i2πcos(θn)Here θn=nπ/2–Tn(Δ)Δ/4 has the meaning of inter-mode energy exchange angle. For example, for θn=π/2+mπ (m is an integer number) a complete exchange of two photonic states is achieved, while θn=π/4+mπ describes 50% mixing behavior that allows one to entangle the two photonic modes.The obtained results allow one to select pulse parameters realizing the desired unitary operation and, also, to analyze the limitations on the performance of the proposed magnon-mediated gates. Thus, it is clear that the simple rectangular pulse cannot realize all possible unitary gates since the matrix Un depends only on one continuous parameter (θn), while the general 2×2 unitary matrix has 4 independent parameters. On the other hand, this pulse profile is sufficient to achieve an arbitrary exchange angle θn. The shortest pulse duration for which a complete exchange (θn=π/2) is possible is Tmin=π/(κ√2)and is realized at exact resonance Δ=0.Our results, also, show a possibility to realize magnon-mediated unitary gates working in off-resonance conditions │Δ│≥κ. For example, θn=π/2 gate can be realized with Δ=18(2/19)½κ and T10=(19/2)½π/κ. While such off-resonance operations require longer interaction times, their advantage in comparison to resonance-type operations is that the magnonic and photonic subsystems remain only weakly coupled throughout the whole process. This may lead to weaker decoherence of the quantum photonic subsystem caused by spin-lattice relaxation processes in the magnonic resonator. In general, n-th unitary branch Tn(Δ) has n different detuning values Δ, for which a given exchange angle θn (up to multiplies of π) is achieved, which leaves a lot of space for optimization of practical magnon-mediated unitary gates.In summary, we proposed a simple model of magnon-mediated interaction in dynamically-tuned hybrid magnon-photon quantum systems. We derived analytically the conditions under which such interaction results in a unitary operation on the photonic subsystem and obtained expressions for the parameters of the control pulse necessary for achieving the desired operation. Our results can be used for estimation and optimization of performance parameters of practical hybrid magnon-photon systems. **