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arXiv:2608.23389

Exact Fock-State Preparation with $n^{1/4}$ Circuit Depth

Tanay Roy

quant-phSubmitted Aug 24, 2026• Announced Aug 25, 2026• 0 citations
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Abstract

Efficient, deterministic, and high-fidelity preparation of large Fock states is essential for scaling bosonic quantum technologies and exploring quantum phenomena at large excitation energies. We introduce a deterministic one-parameter (D1p) protocol that maps Fock-state preparation in an infinite-dimensional Hilbert space onto two-dimensional amplitude amplification. Starting from a coherent state with $|α|\simeq\sqrt{n}$, the initial target-state population scales as $n^{-1/2}$, yielding an iteration count and circuit depth of $\mathcal{O}(n^{1/4})$. Phase matching guarantees unit fidelity in the ideal model; remarkably, preparing $|{10^6}\rangle$ requires only 39 iterations. The protocol uses only displacements and number-selective phase operations, requires no numerical optimization, and further extends to state transfer, general superpositions, finite-dimensional systems, and multipartite entangled states. In the large-amplitude regime, its multi-target form prepares $L$-legged cat states with an iteration count determined only by $L$; cats with up to ten legs require only two iterations, independent of the coherent-state amplitude. This framework provides a broadly applicable route to highly excited bosonic states on platforms supporting these elementary controls.

Comments
8 pages, 3 figures

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Exact Fock-State Preparation with n1/4n^{1/4}n1/4 Circuit Depth
Efficient, deterministic, and high-fidelity preparation of large Fock states is essential for scaling bosonic quantum technologies and exploring quantum phenomena at large excitation energies. We introduce a deterministic one-parameter (D1p) protocol that maps Fock-state preparation in an infinite-dimensional Hilbert space onto two-dimensional amplitude amplification. Starting from a coherent state with ∣α∣≃n|α|\simeq\sqrt{n}∣α∣≃n​, the initial target-state population scales as n−1/2n^{-1/2}n−1/2, yielding an iteration count and circuit depth of O(n1/4)\mathcal{O}(n^{1/4})O(n1/4). Phase matching guarantees unit fidelity in the ideal model; remarkably, preparing ∣106⟩|{10^6}\rangle∣106⟩ requires only 39 iterations. The protocol uses only displacements and number-selective phase operations, requires no numerical optimization, and further extends to state transfer, general superpositions, finite-dimensional systems, and multipartite entangled states. In the large-amplitude regime, its multi-target form prepares LLL-legged cat states with an iteration count determined only by LLL; cats with up to ten legs require only two iterations, independent of the coherent-state amplitude. This framework provides a broadly applicable route to highly excited bosonic states on platforms supporting these elementary controls.
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Large Fock states are important resources for bosonic quantum information and quantum-enhanced metrology, but preparing them with high probability at large excitation numbers remains challenging, as deterministic methods become increasingly control-intensive, while measurement-based approaches typically suffer from low heralding probabilities. Here we propose a protocol that combines quantum nondemolition photon-number encoding with quantum amplitude amplification to enable high-probability heralded generation of large Fock states. Starting from a cavity mode prepared in a coherent state, Quantum Phase Estimation encodes photon-number information into a multi-qubit register, while Quantum Amplitude Amplification boosts the probability of a desired target outcome before measurement. The scheme has an immediate implementation in dispersive circuit-QED, but can be analogously adapted to other bosonic platforms with QND photon-number readout, such as cavity-QED. With a register of up to eight qubits, near-deterministic preparation of Fock states with hundreds of excitations is possible. We also show that the protocol can serve as the first stage of an extension toward generating a two-mode NOON state via a conditional beam-splitter operation.
In this work, we develop a coherent control technique for cavity QED based on amplitude amplification. We consider two physical platforms. In the first setting, we study a three-level quantum emitter coupled to a single mode of an optical cavity and introduce a protocol for producing traveling single photons based on oblivious amplitude amplification. As the key ingredient of our protocol, we propose an extension of oblivious amplitude amplification which uses reflection unitaries solely on the signal qubit, along with UUU and U†U^\daggerU†, where the unitary UUU prepares the initial state. Our approach improves the scaling of the single-photon-generation protocol length from N∼1/pN\sim 1/pN∼1/p to N∼1/pN\sim 1/\sqrt{p}N∼1/p​, with ppp denoting the success probability of obtaining a short single photon from a single application of the weak control pulse. Furthermore, our protocol also reduces the error from intrinsic cavity loss compared to protocols using a single strong control pulse in various experimentally relevant regimes, suggesting the application of our methods for error reduction. In the second setting, we consider a superconducting qubit coupled to a single bosonic mode of a microwave cavity in a circuit QED architecture in the dispersive regime for preparing Fock states. Using fixed-point amplitude amplification, we obtain a protocol for preparing Fock states whose length scales as O(n1/4)O(n^{1/4})O(n1/4), where nnn is the number of photons. Additionally, as an application of our methods for state preparation, we describe a protocol for preparing NOON states.
The scalable preparation of bosonic quantum states with macroscopic excitations poses a fundamental challenge in quantum technologies, limited by control complexity and photon-loss rates that severely constrain prior theoretical and experimental efforts to merely dozens of excitations per mode. Here, based on the duality of the quantum state evolution in Fock state space and the optical wave-function propagation in a waveguide array, we introduce a Kerr-engineered multi-lens protocol in a single bosonic mode to deterministically generate Fock states exceeding 10,00010,00010,000 photons. By optimizing phase and displacement operations across lens groups, our approach compensates for non-paraxial aberrations, achieving fidelities above 73%73\%73% in numerical simulations for photon numbers up to N=100,000N=100,000N=100,000. Counterintuitively, the protocol's execution time scales as N−1/2N^{-1/2}N−1/2 with the target photon number NNN, exhibiting robustness against the photon loss. Our framework enables exploration of quantum-to-classical transitions of giant Fock states, paving the way for advanced quantum metrology with significant quantum gains, and error-corrected quantum information processing in high-dimensional Hilbert spaces.
The scalable preparation of large photon-number (Fock) states is a long-standing frontier in quantum science, with direct implications for quantum metrology and bosonic quantum information processing. Despite substantial progress at small photon numbers, extending state generation to large photon numbers while maintaining high fidelity and operating deterministically remains a significant challenge. Here we demonstrate a scalable and experimentally accessible control protocol for generating large photon-number states using only native spin--oscillator operations. The protocol alternates Jaynes--Cummings interactions with phase-space displacements to imprint photon-number--dependent phases and convert them into selective interference in photon-number space. It already achieves high preparation fidelity unconditionally, while an optional final qubit projection removes residual qubit--field correlations and further enhances the fidelity. Conditioned on this final projection, photon-number state preparation with fidelities exceeding 0.950.950.95 is achieved for photon numbers in the few-hundred regime, with a success probability exceeding 0.900.900.90, placing the protocol in a near-deterministic operating regime. The resulting control sequences remain shallow and are robust against detuning, control noise, and experimentally relevant dissipation. Our results establish a practical route to scalable, high-fidelity photon-number state preparation at large photon numbers and provide a versatile interference-engineering toolbox for nonclassical bosonic state synthesis.
Preparing matrix product states (MPS) on a quantum device is a key subroutine in many quantum algorithms. The most competitive methods, based on the renormalisation group, prepare translationally invariant MPS of size LLL and bond dimension χχχ, up to an error ε\varepsilonε, in circuit depth O~(χ4log⁡(L/ε))\tilde O(χ^{4}\log(L/\varepsilon))O~(χ4log(L/ε)) or O~(χ6log⁡log⁡(L/ε))\tilde O(χ^{6}\log\log(L/\varepsilon))O~(χ6loglog(L/ε)). We improve multiple aspects of these methods. First, using block-encoded correction maps, whose post-selection succeeds with constant probability, we render the preparation exact without sacrificing the scaling in LLL. Second, through a generalisation of oblivious amplitude amplification to isometries, we reduce the bond-dimension dependence, improving the depth to O~(χ2log⁡L+χ4)\tilde O(χ^{2}\log L + χ^{4})O~(χ2logL+χ4) or O~(χ2log⁡log⁡L+χ4)\tilde O(χ^{2}\log\log L + χ^{4})O~(χ2loglogL+χ4), and even to O~(χ3log⁡L)\tilde O(χ^{3}\log L)O~(χ3logL) for incoherent preparations. Finally, we extend the framework to non-translationally invariant MPS and prove logarithmic-depth exact preparation for independent and identically distributed random tensor sequences. Confirmed by numerical studies, these results constitute, to the best of our knowledge, the most efficient exact MPS preparation protocols in the relevant parameter regimes.