Logical operations with single x-ray photons via dynamically-controlled nuclear resonances

Photonic qubits lie at the heart of quantum information technology, often encoding information in their polarization state. So far, only low-frequency optical and infrared photons have been employed as flying qubits, as the resources that are at present easiest to control. With their essentially different way of interacting with matter, x-ray qubits would bear however relevant advantages: they are extremely robust, penetrate deep through materials, and can be focused down to few-nm waveguides, allowing unprecedented miniaturization. Also, x-rays are resonant to nuclear transitions, which are very well isolated from the environment and present long coherence times. Here, we show theoretically that x-ray polarization qubits can be dynamically controlled by nuclear Mössbauer resonances. The control knob is played by nuclear hyperfine magnetic fields, that allow via fast rotations precise processing of single x-ray quanta polarization. With such rotations, single-qubit and binary logical operations such as a destructive C-NOT gate can be implemented.


A scalable physical system with well characterized qubits
In our approach, the polarization of single photons is used in order to encode information.
The following qubit notation is adopted: |0 = π, |1 = σ. The scalability of x-ray photonic qubits is similar to the one of optical qubits, with the difference that potentially, x-ray photons can be much tighter focused than their longer-wavelength counterparts. For instance, a 7 nm focus has been recently achieved [7]. In conjunction with 10-nm x-ray waveguides [8], focusing might bring a significant improvement in the spatial scalability of x-ray qubits.
2. The ability to initialize the state of the qubits to a simple fiducial state, such as |000 . . .
For photonic qubits the initialization goes back to a suitable single-photon source. In the keV energy regime, x-ray parametric down-conversion [31, S2-S4] is a way to construct a heralded single x-ray photon source. In this process a non-linear crystal (typically diamond) is used to split up an incoming pump x-ray into an idler and a signal photon. Detection of the idler photon heralds the emission of the signal single photon. As an alternative, in this work we envisage a proof-of-principle experiment that uses the low spectral density of synchrotron radiation (SR) pulses to construct a single-photon x-ray source. The SR pulse drives a narrow-width nuclear transition in 57 Fe. Typically, at most one single photon in the SR pulse is resonant to the nuclear transition. The resonant photon will be filtered out by the nuclear transition and re-emitted as a single quantum long after the rest of the SR pulse has left the setup.

Long relevant decoherence times, much longer than the gate operation time
In general, photonic qubits have very long coherence times. Moreover, the loss rates are low when moving in free space. The magnetic field rotations envisaged in this work take place on a nanosecond scale, while the coherence time is on the order of 100 ns.

A universal set of quantum gates
For a universal set of quantum gates, single-and two-qubit operations are required. Since photons do not interact with each other in free space, two-qubit operations are difficult to realize. Here, we have shown that logical operations on single x-ray photons can be simulated by using nuclear interfaces. Moreover, an effective photon-photon interaction has been artificially introduced by correlating the gate operation with the detection time of the control photon in order to realize a destructive version of the two-qubit CNOT gate. The basic requirement for the realization of the two-qubit CNOT gate is that the single-qubit gates can be operated at a single nuclear target solely by varying the magnetic field rotation instant.

A qubit-specific measurement capability
The polarization of x-rays can be measured by instruments based on the principle of Comp- After separation by the polarizer, the single x-ray photons can be detected, time-resolved, with fast avalanche photo-diodes, which have approx. 1 ns time resolution [27].

APPLICATION TO SUPERPOSITION STATES
By applying timed rotations of an external magnetic field, single-photon nuclear interfaces can be used in order to simulate logical operations. Initially either π-or σ-polarized x-rays have been considered. According to the introduced qubit notation and by using Eq. (1), the 2 following gate transformations are realized: Here, the time-dependent amplitudes ψ describe the resonant scattering process after the magnetic field rotation (t > t 0 ). They are determined via Eq. (1). The small perturbations from the "wrong" polarization state (see Fig. 2) have been neglected in Eqs. (S1). For an initial single x-ray photon in a superposition state |Ψ = α |0 + β |1 with α, β ∈ C, the gate transformation properties can be straightforwardly generalized to Note that the scattering amplitudes ψ occurring in Eqs. (S2) may imprint an additional time-dependent phase between |0 and |1 because different hyperfine transitions are involved depending on the initial polarization state and/or on the nature of the induced gate transformation. The time dependence of this phase difference is well-known but may be difficult to control in the same operation.
After the gate transformations, high-precision polarizers can be used to filter out and detect the pure polarization states |0 and |1 .