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arXiv:2501.18884 · Physical Review A

Deterministic carving of quantum states with Grover's algorithm

Omar Nagib, M. Saffman, Klaus Mølmer

Published Jun 30, 2025• 2 citations
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Abstract

We show that iteration of a few ( $\sim N^{1/4}$) unitary steps of Grover's algorithm suffices to perfectly prepare a Dicke state of $N$ atoms in a cavity. We also show that a few subsequent Grover steps can be employed to generate GHZ and Cat states. The Grover iteration is physically realized by global qubit rotations and by the phase shift of single photons reflected on the cavity. Our protocols are deterministic and require no individual addressing of the atoms. A detailed error analysis accounting for spatial mode matching of the photon to the cavity, spontaneous emission, mirror scattering, and the finite bandwidth of the photon mode is used to predict the fidelity of the prepared states as a function of system parameters and atom-cavity cooperativity. The fidelity can be increased by heralding on detection of the reflected photon.

DOI
10.1103/s3vs-xz7w

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We show that iteration of a few ( ∼N1/4\sim N^{1/4}∼N1/4) unitary steps of Grover's algorithm suffices to perfectly prepare a Dicke state of NNN atoms in a cavity. We also show that a few subsequent Grover steps can be employed to generate GHZ and Cat states. The Grover iteration is physically realized by global qubit rotations and by the phase shift of single photons reflected on the cavity. Our protocols are deterministic and require no individual addressing of the atoms. A detailed error analysis accounting for spatial mode matching of the photon to the cavity, spontaneous emission, mirror scattering, and the finite bandwidth of the photon mode is used to predict the fidelity of the prepared states as a function of system parameters and atom-cavity cooperativity. The fidelity can be increased by heralding on detection of the reflected photon.
We propose to employ the amplification mechanism of Grover's search algorithm to efficiently prepare entangled states of an ensemble of qubits. The conditional change of sign employed in the algorithm can be implemented by the phase shift of photons scattered on an optical cavity hosting an atomic ensemble. We show that collective Dicke states, Greenberger-Horne-Zeilinger states, and Schrödinger cat superpositions of N atoms may be prepared deterministically by few (∼N^{1/4}) photon scattering events without individual addressing of the atoms.
We demonstrate entanglement generation of two neutral atoms trapped inside an optical cavity. Entanglement is created from initially separable two-atom states through carving with weak photon pulses reflected from the cavity. A polarization rotation of the photons heralds the entanglement. We show the successful implementation of two different protocols and the generation of all four Bell states with a maximum fidelity of (90±2)%. The protocol works for any distance between cavity-coupled atoms, and no individual addressing is required. Our result constitutes an important step towards applications in quantum networks, e.g., for entanglement swapping in a quantum repeater.
The so-called state-carving protocol generates high-fidelity entangled states at an atom-cavity interface without requiring high cavity cooperativity. However, this protocol is limited to 50% efficiency, which restricts its applicability. We propose a simple modification to the state-carving protocol to achieve efficient entanglement generation, with unit probability in principle. Unlike previous two-photon schemes, ours employs only one photon which interacts with the atoms twice-avoiding separate photon detections which causes irrecoverable probability loss. We present a detailed description and performance evaluation of our protocol under nonideal conditions. High fidelity of 0.999 can be achieved with cavity cooperativity of only 34. Efficient state carving paves the way for large-scale entanglement generation at cavity interfaces for modular quantum computing, quantum repeaters, and creating arbitrary shaped atomic graph states, essential for one-way quantum computing.
We present a method for realizing efficiently Grover's search algorithm in an array of coupled cavities doped with three-level atoms. We show that by encoding information in the lowest two ground states of the dopants and through the application of appropriately tuned global laser fields, the reflection operator needed for the quantum search algorithm can be realized in a single physical operation. Thus, the time steps in which Grover's search can be implemented become equal to the mathematical steps , where N is the size of the register. We study the robustness of the implementation against errors due to photon loss and fluctuations in the cavity frequencies and atom-photon coupling constants.
We present a deterministic protocol for the preparation of entangled states in the symmetric Dicke subspace of N spins coupled to a common cavity mode that prepares entangled states useful for quantum sensing, achieving a precision significantly better than the standard quantum limit in the presence of photon cavity loss, spontaneous emission and dephasing. The protocol combines a new geometric phase gate which can be utilized for exact unitary synthesis on the Dicke subspace, an analytic solution of the noisy quantum channel dynamics and optimal control methods. This Letter opens the way to entanglement-enhanced sensing with cold trapped atoms in cavities and is extendable to other spin systems coupled to a bosonic mode.
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