Quantum Anchor States

Updated 4 July 2026

Quantum anchor states are structured quantum objects that serve as stable reference points, pinning quantum information in both cryptography and quantum chemistry.
In cryptographic applications, these states use techniques like random linear scrambling and decoy qubits to securely anchor quantum data to a spacetime location.
In quantum chemistry, engineered anchor states guide quantum phase estimation by ensuring high overlap with the true ground state, facilitating efficient active-space selection.

Searching arXiv for the cited works and nearby usage of the term. Quantum anchor states are structured quantum objects that serve as a stable reference for verification tasks in settings where naïve representations become fragile or ambiguous. In recent arXiv literature, the term appears in two distinct senses. In position-based cryptography, a quantum anchor state is a bipartite anchor/vessel state used to pin quantum information to a spacetime location and to support quantum localization, trajectory verification, state localization, and functionality localization (Bartusek et al., 29 May 2026). In quantum chemistry, an anchor state is a guiding state for quantum phase estimation (QPE) whose defining property is large overlap with the true ground state, often engineered through quantum embedding so that the overlap remains sufficiently high even for large molecular systems (Erakovic et al., 2024). A related but distinct phrase, “quantum anchor,” appears in the quantization of higher Koszul brackets, where the classical $L_\infty$ -morphism called the anchor is quantized into a single linear operator, namely a formal Fourier integral operator (Shemyakova et al., 2024).

1. Terminological scope

The expression “quantum anchor state” does not denote a single universal construction across the literature. It names different technical objects whose common role is to preserve a verifiable relation between a manageable quantum object and a target structure that would otherwise be difficult to certify directly.

Domain	Object	Function
Position-based cryptography	bipartite anchor/vessel state	pin quantum information to a spacetime location
Electronic structure	guiding or anchor state	provide significant overlap with the true ground state for QPE
Higher Koszul brackets	quantum anchor operator	quantize the classical anchor into a single linear operator

In the cryptographic usage, the anchor register remains with the verifier and the vessel register is sent into the prover’s control. In the electronic-structure usage, the anchor state is the QPE input state, and the central issue is whether it has non-negligible overlap $\eta = |\langle \psi | \psi_0 \rangle|$ with the true ground state $|\psi_0\rangle$ . In the higher-Koszul setting, the term refers not to a state but to an operator-level quantization of the cotangent $L_\infty$ -algebroid anchor. This suggests a shared functional motif: the “anchor” is the object that keeps the relevant quantum or homotopical information tied to a tractable representation, even though the concrete mathematics differs substantially across fields.

2. Cryptographic quantum anchor states as bipartite localization resources

In "How To Track Qubits Through Space and Time (Or: Sailing in a Quantum Boat)" (Bartusek et al., 29 May 2026), the quantum anchor state is the core bipartite object used to localize quantum information. The state is built from a vessel register, which is sent into the prover’s control, and an anchor register, which stays with the verifier. The paper’s informal description states that the anchor state is formed by taking several EPR pairs, adding decoy qubits, and applying a random linear change of basis $U_{\sf shift}$ to scramble where the special subspaces live. The stated effect is that the prover receives a large quantum register in which the meaningful entanglement is hidden among decoys.

Formally, the construction depends on polynomial parameters $n_e(\lambda), n_s(\lambda), n_h(\lambda)$ , a uniformly random invertible matrix $U_{\sf shift}$ of size $(n_e+n_h+n_s)\times(n_e+n_h+n_s)$ , and subspaces $S$ and $T$ determined by $\eta = |\langle \psi | \psi_0 \rangle|$ 0, where $\eta = |\langle \psi | \psi_0 \rangle|$ 1 is spanned by the first $\eta = |\langle \psi | \psi_0 \rangle|$ 2 columns and $\eta = |\langle \psi | \psi_0 \rangle|$ 3 is spanned by the first $\eta = |\langle \psi | \psi_0 \rangle|$ 4 columns. The resulting anchor state is a bipartite state on an $\eta = |\langle \psi | \psi_0 \rangle|$ 5-qubit anchor register and a $\eta = |\langle \psi | \psi_0 \rangle|$ 6-qubit vessel register, parameterized by a secret key $\eta = |\langle \psi | \psi_0 \rangle|$ 7 that includes $\eta = |\langle \psi | \psi_0 \rangle|$ 8 and shifts $\eta = |\langle \psi | \psi_0 \rangle|$ 9.

The generation procedure is specified as follows. One prepares a maximally entangled state between three auxiliary blocks $|\psi_0\rangle$ 0, $|\psi_0\rangle$ 1, and $|\psi_0\rangle$ 2 and the vessel register; the block sizes are $|\psi_0\rangle$ 3, $|\psi_0\rangle$ 4, and $|\psi_0\rangle$ 5, respectively. One then measures $|\psi_0\rangle$ 6 in the Hadamard basis to get $|\psi_0\rangle$ 7 and $|\psi_0\rangle$ 8 in the standard basis to get $|\psi_0\rangle$ 9, computes

$L_\infty$ 0

and finally applies the linear map $L_\infty$ 1 to the vessel register. The resulting state is described as

$L_\infty$ 2

up to notational variants in the text. The anchor register is entangled with an index $L_\infty$ 3, the vessel register encodes a coset-like superposition around that index, and the whole state is maximally entangled across the anchor/vessel cut.

A central conceptual point is that this state generalizes ordinary coset states. The paper describes the anchor state as a coherent superposition over many coset states,

$L_\infty$ 4

where each $L_\infty$ 5 is a coset state. The paper further notes that when the entangled-half size $L_\infty$ 6 is set to $L_\infty$ 7, the anchor state degenerates to an ordinary coset state. In this precise sense, the anchor-state framework extends unclonable cryptography from isolated coset states to entangled, coherently indexed families of such states.

3. Localization, trajectory verification, and proof architecture

The primary application of the cryptographic anchor state is entanglement localization. The goal is to certify that the prover holds a quantum state that is genuinely entangled with the verifier’s state at a specific spacetime point $L_\infty$ 8. The paper defines this in extractor-based terms: if a prover strategy is accepted with probability $L_\infty$ 9, then there exists an extractor acting only on the prover’s state near $U_{\sf shift}$ 0 that recovers a register $U_{\sf shift}$ 1 such that the verifier’s $U_{\sf shift}$ 2 together with the extracted $U_{\sf shift}$ 3 is close to the intended entangled state (Bartusek et al., 29 May 2026). The paper explicitly contrasts this with existing security definitions for quantum position verification, which only guarantee that part of the successful adversarial party is in the claimed location.

The protocol structure uses the anchor-state geometry directly. In the setup, the verifiers sample $U_{\sf shift}$ 4, prepare the anchor state, keep the anchor register, send the vessel register to the prover, and publish a public oracle

$U_{\sf shift}$ 5

In the online phase, the verifiers send additive shares $U_{\sf shift}$ 6 and a basis choice $U_{\sf shift}$ 7. The honest prover must coherently query the oracle on the vessel register in the standard basis when $U_{\sf shift}$ 8 and in the Hadamard basis when $U_{\sf shift}$ 9. The construction is designed so that standard-basis measurement yields something in $n_e(\lambda), n_s(\lambda), n_h(\lambda)$ 0, while Hadamard-basis measurement yields something in $n_e(\lambda), n_s(\lambda), n_h(\lambda)$ 1.

The same anchor-state mechanism underlies trajectory verification, state localization, and functionality localization. Trajectory verification is obtained by repeatedly applying entanglement-localization tests along a claimed path $n_e(\lambda), n_s(\lambda), n_h(\lambda)$ 2, so that passing the protocol at successive checkpoints is evidence that the same entangled object is being transported through spacetime. State localization removes the entanglement-tracking objective and instead localizes an unclonable state family $n_e(\lambda), n_s(\lambda), n_h(\lambda)$ 3, using the encoding

$n_e(\lambda), n_s(\lambda), n_h(\lambda)$ 4

Functionality localization localizes the ability to compute a function $n_e(\lambda), n_s(\lambda), n_h(\lambda)$ 5 at a spacetime point by combining a quantum state obfuscator for a signed functionality, a coset-state or anchor-state encoding, a MAC or signature system, and the same localization machinery.

The proof architecture relies on five named ingredients. First, random linear scrambling via $n_e(\lambda), n_s(\lambda), n_h(\lambda)$ 6 hides which qubits correspond to the important cosets and which are decoys. Second, standard-vs-Hadamard duality provides complementary tests. Third, a collapsing-vs-non-collapsing oracle argument allows coherent oracle access to be related to a collapsing test. Fourth, monogamy-of-entanglement reductions show that if a prover could answer both basis tests, then the state at the midpoint must be close to many EPR pairs. Fifth, sequential repetition is used to amplify soundness. All of these results are proven in the classical oracle model, identified also as the ideal obfuscation model. The paper states that constructions in that model may be heuristically instantiated in the plain model using post-quantum indistinguishability obfuscation, and that the oracle model itself can be heuristically realized via iO plus a pseudorandom oracle or hash function assumption.

4. Anchor states in quantum embedding and phase estimation

In "High ground state overlap via quantum embedding methods" (Erakovic et al., 2024), the anchor-state idea appears in a different form. Here the relevant task is ground-state energy estimation by QPE, which requires an initial guiding state with significant overlap with the true ground state. The paper states that QPE only needs an initial state $n_e(\lambda), n_s(\lambda), n_h(\lambda)$ 7 with non-negligible overlap $n_e(\lambda), n_s(\lambda), n_h(\lambda)$ 8 with the true ground state $n_e(\lambda), n_s(\lambda), n_h(\lambda)$ 9, because the success probability scales with that overlap. It further notes that if the overlap is close to $U_{\sf shift}$ 0, circuit depth can be reduced by parallelization.

The obstacle is the orthogonality catastrophe: as the number of orbitals grows, even small local imperfections in an approximate state can cause the many-body overlap with the exact ground state to decay rapidly, often exponentially or at least very unfavorably. The paper argues that quantum embedding is a natural way to address this for large molecules and materials. An embedding method allows a reduction to one or multiple smaller quantum cores embedded in a larger quantum region. The basic decomposition is a small region where electron correlation is important and must be treated accurately, together with an environment treated approximately, typically at the mean-field or DFT level.

For a single core, the guiding state is written in antisymmetrized form as

$U_{\sf shift}$ 1

where $U_{\sf shift}$ 2 is an arbitrary state on the active/core subspace $U_{\sf shift}$ 3, and $U_{\sf shift}$ 4 is a Slater determinant on the environment $U_{\sf shift}$ 5. The paper examines two embedding families. In density-matrix embedding, illustrated through bootstrap embedding, the full system is represented by a global mean-field determinant whose Schmidt decomposition defines fragment and bath orbitals; the fragment plus bath are then treated with a higher-level solver, while the rest is frozen into a determinant. In projection-based embedding, specifically Huzinaga embedding, one starts from a DFT calculation and constructs an effective fragment Hamiltonian in which the environment enters through an embedding potential, with projection terms preventing unwanted mixing of fragment and environment orbitals.

A key claim is that embedding is not only a Hamiltonian-reduction technique. The paper emphasizes that the embedding choice changes the fragment orbital basis and therefore the entanglement structure of the quantum core, which matters directly for state preparation quality. This is the sense in which embedding is used to engineer a ground-state anchor state rather than merely to reduce system size.

5. Overlap-driven active-space selection and empirical performance

The paper’s formal contribution is to connect impurity-style structure theorems to active-space selection for high-overlap state preparation (Erakovic et al., 2024). Bravyi and Gosset had proved favorable structure for quantum impurity problems via bounds on the one-particle reduced density matrix (1-RDM) and derived quasipolynomial classical algorithms under certain conditions. The later work extends that analysis from Gaussian states to the Slater-determinant or fixed-particle-number setting relevant to chemistry and reinterprets the result as a statement about selecting orbital spaces for overlap rather than directly for energy minimization.

The central theorem states that for impurity-like Hamiltonians with localized correlation,

$U_{\sf shift}$ 6

there exists a Slater determinant $U_{\sf shift}$ 7 on $U_{\sf shift}$ 8 modes and an arbitrary state $U_{\sf shift}$ 9 on $(n_e+n_h+n_s)\times(n_e+n_h+n_s)$ 0 modes such that

$(n_e+n_h+n_s)\times(n_e+n_h+n_s)$ 1

has overlap

$(n_e+n_h+n_s)\times(n_e+n_h+n_s)$ 2

with a ground state $(n_e+n_h+n_s)\times(n_e+n_h+n_s)$ 3 of $(n_e+n_h+n_s)\times(n_e+n_h+n_s)$ 4. The paper treats this as the formal basis for overlap-driven active-space selection. The relevant criterion is state overlap or fidelity, not immediate energy accuracy of the approximate wavefunction.

The 1-RDM $(n_e+n_h+n_s)\times(n_e+n_h+n_s)$ 5 supplies the practical orbital-selection principle. Its eigenvectors are the natural orbitals, and its eigenvalues $(n_e+n_h+n_s)\times(n_e+n_h+n_s)$ 6 indicate occupation. Orbitals with $(n_e+n_h+n_s)\times(n_e+n_h+n_s)$ 7 or $(n_e+n_h+n_s)\times(n_e+n_h+n_s)$ 8 can be frozen as unoccupied or occupied, while only orbitals with intermediate occupations need to remain active. The appendices state that if the 1-RDM eigenvalues decay appropriately, one can choose a small $(n_e+n_h+n_s)\times(n_e+n_h+n_s)$ 9 such that the active-space approximation has overlap at least $S$ 0. A common misconception addressed directly by the paper is to conflate active-space quality for chemistry with correlation-energy recovery alone; for QPE, the relevant metric is explicitly overlap with the true ground state.

The numerical studies examine a five-atom fragment of tryptophan’s aromatic side chain under bootstrap embedding and under Huzinaga embedding, and also a ruthenium-containing anticancer drug with a coordination sphere around Ru in two charge/spin states. The paper reports qualitatively different orbital entanglement in the two embedding strategies. Bootstrap embedding uses HF orbitals and a Schmidt decomposition of the mean-field state, often producing relatively modest single-orbital entropies. Huzinaga embedding uses DFT orbitals and localized orbitals (IBOs), retains the full virtual space, and yields significantly larger orbital entropies for some modes. In the tryptophan calculations, the entanglement is concentrated in roughly a dozen orbitals associated with the aromatic $S$ 1-system and a C–H $S$ 2-interaction.

Across all examples, the mean-field Hartree–Fock state already has large overlap with the exact or high-quality reference ground state of the embedded Hamiltonian, with overlaps larger than $S$ 3 in almost all cases. For the tryptophan fragment, even a very small-bond-dimension MPS has overlap around $S$ 4 with the reference MPS, and sum-of-Slater expansions also achieve substantial overlaps. As the active space grows, the overlap of simple Slater-based ansätze decreases, but if the most entangled orbitals are included, smaller active spaces can still produce essentially the same overlap as larger ones. The paper therefore concludes that embedding can postpone or mitigate the orthogonality catastrophe by restricting the quantum part to the chemically relevant region while also shrinking Hamiltonian-simulation cost.

"A quantum anchor for higher Koszul brackets" (Shemyakova et al., 2024) uses the anchor terminology in a third, explicitly non-state-based sense. The paper studies a $S$ 5-structure in which an ordinary Poisson bivector is replaced by an even multivector

$S$ 6

with $S$ 7 the canonical Schouten bracket. The higher Koszul brackets are defined by Voronov’s higher derived bracket construction,

$S$ 8

In this setting, the classical anchor of the cotangent $S$ 9-algebroid is a nonlinear map of graded manifolds

$T$ 0

whose pullback obeys

$T$ 1

The paper emphasizes that in the $T$ 2 case the correct classical structure is not a single linear bracket-preserving map but an $T$ 3-morphism, encoded using Voronov’s thick morphisms. The dual anchor is described by a generating function

$T$ 4

and its nonlinear pullback yields the $T$ 5-morphism from higher Koszul brackets to the Schouten bracket.

The quantization problem asks whether this entire $T$ 6-morphism can be represented by a single operator. The paper’s main answer is yes: the quantum anchor is a quantum pullback associated with a quantum thick morphism, and it is a formal Fourier integral operator. The higher Koszul brackets are generated by

$T$ 7

and the quantum Mackenzie–Xu transform satisfies

$T$ 8

$\eta = |\langle \psi | \psi_0 \rangle|$ 05

This usage is terminologically adjacent to quantum anchor states but mathematically distinct. The commonality is structural rather than ontological: in both the cryptographic and geometric settings, the “anchor” is the object that mediates between two descriptions while preserving the relevant invariant, whether that invariant is localization of quantum information, overlap with a target ground state, or compatibility between higher Koszul and Schouten structures.