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arXiv:2610.06113

Collective Quantum State Preparation Is Nonadditive

Lin Zhu, Ranyiliu Chen, Xin Wang, Shenggen Zheng

quant-phSubmitted Oct 5, 2026• Announced Oct 6, 2026• 0 citations
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Abstract

Quantum state preparation is a basic primitive of quantum computation and control, but its resource cost is usually analyzed for one target state or under an independent-copy construction. Here we study the many-copy problem under weighted Pauli control, where a Pauli-product rotation $e^{iφP}$ is charged by its rotation angle $|φ|$, independently of the weight of $P$. We derive the exact single-qubit preparation cost and show that it is strictly nonadditive: a four-pulse collective protocol prepares two $T$ states at lower cost than two optimal one-copy protocols. We further identify finite regions of the Bloch sphere with collective saving, including trajectories that remain product throughout, showing that transient entanglement is not the only mechanism behind the advantage. Complementarily, a dimension-independent Rényi-entropy speed limit gives an extensive lower bound for every coherent target and an exponential fidelity penalty below a finite preparation rate. These results establish a many-copy geometry of state synthesis in which uncorrelated outputs can nevertheless benefit from collective control.

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Quantum state preparation is a basic primitive of quantum computation and control, but its resource cost is usually analyzed for one target state or under an independent-copy construction. Here we study the many-copy problem under weighted Pauli control, where a Pauli-product rotation eiφPe^{iφP}eiφP is charged by its rotation angle ∣φ∣|φ|∣φ∣, independently of the weight of PPP. We derive the exact single-qubit preparation cost and show that it is strictly nonadditive: a four-pulse collective protocol prepares two TTT states at lower cost than two optimal one-copy protocols. We further identify finite regions of the Bloch sphere with collective saving, including trajectories that remain product throughout, showing that transient entanglement is not the only mechanism behind the advantage. Complementarily, a dimension-independent Rényi-entropy speed limit gives an extensive lower bound for every coherent target and an exponential fidelity penalty below a finite preparation rate. These results establish a many-copy geometry of state synthesis in which uncorrelated outputs can nevertheless benefit from collective control.
Quantum state preparation is a fundamental primitive in quantum algorithms for encoding classical data into quantum amplitudes. We compare the cost of preparing general nnn-qubit states with real amplitudes using two common paradigms: rotation-based methods, based on controlled rotations, and sampling-based methods, based on a structured representation of the target state. Although these approaches are often theoretically compared using CNOT count and TTT-count, their relative performance in total gate count remains less well understood practically. We compare representative rotation-based and sampling-based methods using TTT-count and total gate count, and analyze how compilation overhead affects their relative performance. We also develop a software package for compiling state preparation circuits, designed as a practical subroutine for more general quantum computations. Numerical experiments on resource states and quantum states related to quantum chemistry, condensed matter physics, and simulation via Magnus expansion over a range of target accuracies εεε support the analysis. Our results show that sampling-based methods achieve asymptotically lower TTT-count and retain an overall advantage after accounting for total gate count and compilation overhead.
Can two independently prepared quantum systems require a joint measurement to read a fluctuation with the least possible variance? We prepare two copies of either candidate state. A measured Pearson quantity compares the outcome distributions of the two candidate preparations. Fully-PPT detectors include measurements made separately on each copy. The class maximum is the product of the one-copy optima, attained by measuring an optimal score on each copy. Noncommuting states allow an unrestricted joint measurement to do strictly better. We then fix a state-calibrated collective score as a theoretical witness. Its spectral readout adds no variance, but every unbiased fully-PPT readout must add some. A moment inequality strengthens this necessary bound; the exact restricted minimum remains unknown. Driven and equilibrium models illustrate the bounds without a device demonstration. The results concern calibrated scores, not work readout, detector energy cost, or local thermometry.
Probing multiple properties of ground states is central to understanding quantum matter, yet conventional readout can be prohibitively costly because it consumes many independently prepared copies. Here, in copy-scarce regimes, we establish optimal trade-offs between the number of initial ground-state copies and elapsed Hamiltonian-evolution time required for estimating multiple observables or reconstructing a full classical description of a unique gapped ground state. We derive a unifying inequality that quantifies how well ground states can be distinguished using initial copies and controlled dynamics, even when all copies may be consumed. It yields worst-case lower bounds on elapsed Hamiltonian time that any protocol must incur to perform either task. We construct parallel catalytic readouts that return all initial copies nearly unchanged while attaining these bounds up to poly-logarithmic factors. Our protocols require no coherent preparation circuit or its inverse, and their elapsed Hamiltonian time depends only logarithmically on the inverse of the desired return error. Thus, optimal copy--dynamics trade-offs can be attained together with catalytic return. Numerical benchmarks further show substantial reductions in end-to-end elapsed Hamiltonian time relative to a task-specialized copy-consuming protocol, even under an optimistic preparation-cost model. Our results reveal a copy--dynamics resource landscape connecting copy-scarce learning to copy-only endpoints realized by known protocols for shadow tomography and pure-state tomography.
Abstract The W state is a key resource for quantum communication and distributed quantum computing. Its entanglement survives the loss of any single qubit. Preparing the W state exactly is nevertheless hard in fault-tolerant architectures. The amplitude 1/√3 lies outside the ring Z[1/√2, i]. So no Clifford+T unitary can prepare the three-qubit W state exactly, and deterministic circuits must approximate the required rotation at a T-count that grows as log(1/ϵ). We present a heralded protocol that prepares the three-qubit W state exactly using Clifford gates and a single non-Clifford operation. A shallow Clifford circuit with two CNOT gates first prepares an equal superposition of |001⟩, |010⟩, |100⟩ and |111⟩. A logical-AND measurement on one ancilla then removes the |111⟩ component without disturbing the remaining coherent superposition. The protocol succeeds with probability 3/4, a failed attempt leaves the known product state |111⟩, and each attempt consumes four T gates or one |CCZ⟩ magic state. This gives an expected cost of about 5.33 T gates independent of target precision. We prove that some non-Clifford step is necessary, and by exhaustively enumerating all 315,057,600 six-qubit stabilizer tensors we show that no post-selected Clifford circuit with three or fewer T gates outputs the W state. Every heralded Clifford+T preparation of the W state therefore uses at least four T gates per successful run, and our protocol attains this bound. The protocol uses 6 to 19 times fewer T gates than gate synthesis at precisions between 10−3 and 10−10. We compare it with a deterministic circuit using an exact repeat-untilsuccess rotation, which achieves a slightly lower expected cost of about 4.67 T gates, and discuss implementation in modular architectures and extensions to N-qubit W states. Note on this version. This version replaces and corrects the first version of this preprint (“A Tcount Zero Protocol for Heralded W-state Preparation via Clifford-only Gates”). The claim there that the W state can be prepared with Clifford gates alone is incorrect: as shown here, some non-Clifford step is necessary, and at least four T gates are needed per successful run.
Entanglement is a key quantum resource in various quantum protocols, with a rich set of laws governing its manipulation. In this context, catalysis refers to the possibility of an auxiliary state that enables a previously forbidden manipulation, while being completely returned at the end. While the catalytic setting has been thoroughly examined in the single-copy regime, much less is known in the asymptotically many copy regime. In this work, we focus on the entanglement cost of preparing asymptotically many copies of a given state exactly. We show that catalysis can significantly lower the exact entanglement cost by constructing an explicit catalytic protocol. Additionally, these findings generalize readily to other resource theories, showing a general catalytic advantage in the resource dilution task.
We study collective state preparation and channel encoding in fermionic systems under parity superselection and composition governed by the canonical anticommutation relations. For a two-mode Gaussian state τττ, we obtain the exact entanglement of formation, EFphys(τ⊗m)=⌈m/2⌉log⁡2E_{\mathrm F}^{\mathrm{phys}}(τ^{\otimes m})=\lceil m/2\rceil\log2EFphys​(τ⊗m)=⌈m/2⌉log2 for every m≥1m\geq1m≥1. Two copies have the same formation value as one, violating additivity and strong superadditivity, while regularization halves the single-copy value. For an associated Gaussian channel, we determine the minimum output entropy and Holevo information at every blocklength and prove nonadditivity of both. Together, these results give exact answers to the fermionic counterparts of the four additivity questions associated with Shor and Hastings. One physical use carries no classical information, whereas two uses transmit one bit perfectly. The unassisted classical and zero-error classical capacities both equal one half bit per mode. A sharp finite-block bound gives a strong converse: the decoding success probability vanishes at every rate above capacity. These optimizations cover all physical pure-state decompositions and channel inputs, with Gaussian ensembles and classical codes attaining the optima. The formation and minimum output entropy formulas hold for every Rényi order α∈(0,∞]α\in(0,\infty]α∈(0,∞]. We also determine the unassisted quantum and zero-error quantum capacities, both equal to one half logical qubit per mode, and prove a strong converse: entanglement fidelity vanishes at every rate above capacity. For a noisy Gaussian family, we determine the one- and two-copy formation values and prove strict nonadditivity at full rank. For τττ, Bell violation under local parity-preserving measurements first appears at three copies.