Quantum-assisted telescope arrays

Quantum networks provide a platform for astronomical interferometers capable of imaging faint stellar objects. In a recent work [E. T. Khabiboulline et al., Phys. Rev. Lett. 123, 070504 (2019)], we presented a protocol that circumvents transmission losses with efficient use of quantum resources and modest quantum memories. Here we analyze a number of extensions to that scheme. We show that it can be operated as a truly broadband interferometer and generalized to multiple sites in the array. We also analyze how imaging based on the quantum Fourier transform provides improved signal-to-noise ratio compared to classical processing. Finally, we discuss physical realizations including photon-detection-based quantum state transfer.

Figure 1

Memory-based interferometry scheme from Ref. [7]. The quantum state of an incoming photon and associated information is stored nonlocally between telescope sites in a binary qubit encoding that can be decoded using preshared entangled pairs. Encoding operations are performed in time bins set by the detector bandwidth, followed by decoding after one photon is expected to have arrived. Physical realization may involve qubits housed in cavities: (a) Reflecting the photon off cavities, interfering with a coherent state, and detecting photons performs the encoding, while (b) decoding is done with qubit-qubit interactions followed by measurement.

Figure 2

Efficient encoding of photon frequency and time information in ${log}_2(MR+1)$ memory qubits. (a) The incoming photon arrives in one of $M$ time bins (see Fig. 1) and one of $R$ frequency bands. The excitation is transferred to a frequency-matched receiving qubit (red) and later stored in memory qubits (green). (b) The time-frequency data have $MR+1$ possibilities ($+1$ for vacuum), which can be mapped to ${log}_2(MR+1)$ codewords in binary. An example of the encoding cnot gates is shown for the fifth time bin and second frequency band. The number of detectors scales linearly in $R$, whereas the coding operation consumes resources logarithmic in both $M$ and $R$.

Figure 3

Encoding of the time bin ($m=5$) for each frequency band in parallel. To keep track of frequency, the receiving atom is copied to another atom before being reinitialized. Before decoding, these ancillary atoms are compressed to ${log}_2(R+1)$ memory atoms (the nontrivial cnot for frequency band $r=2$ is shown). Once the frequency is determined via nonlocal parity checks, only the corresponding time memory is decoded, so total entanglement expenditure scales logarithmically in both $M$ and $R$. Memory qubits scale as $R{log}_{2}M$, to leading order.

Figure 4

Interferometry in a multiple-site array. (a) The photon arrival data are decoded using ${log}_2(M+1)$ GHZ states. A cz gate is performed between the memory register at each site and the qubit of the GHZ state. For those registers corresponding to a photon, the phase of the GHZ state is flipped from $+$ to $-$, which can be read by measuring the qubits of the GHZ state in the $X$ basis. (b) For pairwise readout, acting on a shared $W$ state with local cnot gates and measuring its qubits in the $Z$ basis projects the network state onto two sites, with probability $1-1/N$. Otherwise, the $W$ state is transformed into $|0,...,0\rangle$ and measurement can be retried with another $W$ state. Two-side readout proceeds as before, using Bell states for the other registers. After pairwise visibilities are collected, a classical Fourier transform is applied to acquire the desired intensity distribution.

Figure 5

After transferring the memory qubits to one site via quantum teleportation, a quantum Fourier transform is applied. The measurement probabilities $p_j$ of finding the excitation at site $j$ map directly to the source intensity distribution $\mathcal{I}\left(y\right)$.

Figure 6

The stellar photon's state is transferred to memory in three ways. (a) The photon is absorbed by an auxiliary atom. Interactions with code-specified memory atoms realize cnot gates. An $X$-basis measurement decouples the atom and imposes a phase correction on the memory to complete state transfer, as described in Ref. [7]. (b) The cnot gates are realized by reflecting the photon off cavities [21]. The photon is subsequently absorbed by an auxiliary atom to perform the $X$-basis measurement. (c) In photon-detection-based state transfer, measurement of the photonic qubit in the $X$ basis is approximately realized by interference at a beam splitter with an ancillary photonic state, followed by photon counting.

Figure 7

Extremal regimes for mixing at a beam splitter with the coherent state $|\alpha \rangle$ in the two-site case: accepting (a) all measurement outcomes (success probability $p=1$) and (b) only $|i-i^{\prime }|=|j-j^{\prime }|$ (fidelity $f=1$).

Figure 8

Fidelity $f$ and success probability $p$ of photonic state transfer when mixing with an ancillary $|+\rangle$ state and detecting photons with efficiency $\eta$. Measurement outcomes of zero or two total photons are postselected out. The performance of the operation drops with $\eta$.