Ancilla-Entangled VQE

• The paper demonstrates that ancilla-entangled VQE incorporates ancillary qubits into the variational ansatz to enforce state orthogonality and enable multi-eigenpair extraction.
• It employs methods like measurement-based schemes and symmetry filtering to optimize circuit performance and reduce readout errors.
• The approach offers practical advantages in eigenstate purification and circuit complexity while introducing overhead through additional ancilla-mediated measurements.

Ancilla-entangled variational quantum eigensolver refers to variational eigensolver constructions in which ancillary qubits or entangled auxiliary subsystems are part of the variational mechanism itself, rather than being absent by design. In the strongest form, the variational object is a joint system-ancilla state whose ancilla-sector decomposition represents several orthogonal system trial states simultaneously; in related forms, entangled resource states or temporary ancilla-system entanglement are used for measurement-based ansatz generation or symmetry filtering. The topic is therefore defined most clearly relative to the original ancilla-free variational quantum eigensolver, which evaluates $\langle \psi(\theta)|H|\psi(\theta)\rangle$ by direct Pauli-term estimation and explicitly “requires no control or auxiliary qubits” (Peruzzo et al., 2013), and relative to later constructions that either retain that ancilla-free philosophy for excited states (Nakanishi et al., 2018) or depart from it by using ancillas as purifying, screening, or resource-state degrees of freedom (Xu et al., 2022, Ferguson et al., 2020, Ahn et al., 30 Mar 2026).

1. Baseline formulation and the ancilla distinction

The foundational VQE formulation combines the Rayleigh-Ritz variational principle with a hybrid quantum-classical optimization loop. A parameterized trial state $\ket{\psi(\theta)}$ is prepared, its energy expectation value

$$E(\theta)=\langle \psi(\theta)|H|\psi(\theta)\rangle$$

is estimated on quantum hardware, and a classical optimizer updates $\theta$ to lower that energy. In the original formulation, the Hamiltonian is decomposed into Pauli products,

$$H = \sum_{i\alpha} h^i_\alpha \sigma_\alpha^i + \sum_{ij\alpha\beta}h^{ij}_{\alpha \beta} \sigma_\alpha^i \sigma_\beta^j + \ldots,$$

so that $\langle H\rangle$ is reconstructed termwise from directly measured Pauli-string expectations. The architecture is explicitly ancilla-free in both ansatz and demonstrated implementation: no Hadamard tests, no swap tests, no overlap tests with ancillas, no controlled-$U$ routines, and no ancilla-mediated nonlocal measurements are part of the baseline method (Peruzzo et al., 2013).

This baseline matters because later ancilla-entangled proposals modify one or more of three components that were ancilla-free in the original design: the ansatz, the measurement or cost-evaluation primitive, and the optimization target. The cited literature supports a technical distinction between at least three forms. A first form uses ancillas as a genuine purifying or orthogonality-enforcing resource inside the variational state itself; a second form uses ancillas as an operational add-on for tasks such as symmetry screening while keeping the core variational ansatz conventional; and a third form uses entanglement-rich auxiliary subsystems to realize a measurement-based or hardware-native variational family. By contrast, some influential excited-state extensions remain explicitly ancilla-free and should be read as counterexamples rather than instances of ancilla-entangled VQE (Nakanishi et al., 2018).

2. Purified joint-state ansatz for many eigenpairs

A clear example of an ancilla-entangled variational eigensolver is the multi-eigenpair construction in which ancillary qubits are not bookkeeping labels but an entangling and purifying resource. The setup uses $N_p$ physical qubits for the target Hamiltonian $\hat H$ and $N_a$ ancillary qubits, with

$\ket{\psi(\theta)}$0

so that the ancilla Hilbert space has dimension $\ket{\psi(\theta)}$1 and can represent at least $\ket{\psi(\theta)}$2 orthogonal labels. The initial state is a maximally entangled state between an $\ket{\psi(\theta)}$3-dimensional physical basis and an $\ket{\psi(\theta)}$4-dimensional ancilla basis,

$\ket{\psi(\theta)}$5

and the variational unitary $\ket{\psi(\theta)}$6 acts only on the physical register, giving

$\ket{\psi(\theta)}$7

Because $\ket{\psi(\theta)}$8 is unitary on the physical register,

$\ket{\psi(\theta)}$9

so orthogonality among the system trial states is preserved structurally throughout the optimization (Xu et al., 2022).

The objective is the sum of expectation values of the first $$E(\theta)=\langle \psi(\theta)|H|\psi(\theta)\rangle$$0 physical branches,

$$E(\theta)=\langle \psi(\theta)|H|\psi(\theta)\rangle$$1

with the generalized Rayleigh-Ritz bound

$$E(\theta)=\langle \psi(\theta)|H|\psi(\theta)\rangle$$2

Minimizing this single scalar loss drives the $$E(\theta)=\langle \psi(\theta)|H|\psi(\theta)\rangle$$3-dimensional orthonormal set toward the low-energy eigensubspace, but not necessarily toward a diagonal eigenbasis. The individual eigenpairs are therefore recovered by constructing the projected Hamiltonian

$$E(\theta)=\langle \psi(\theta)|H|\psi(\theta)\rangle$$4

which the paper rewrites as

$$E(\theta)=\langle \psi(\theta)|H|\psi(\theta)\rangle$$5

By expanding $$E(\theta)=\langle \psi(\theta)|H|\psi(\theta)\rangle$$6 on each ancilla qubit in the Pauli basis and measuring

$$E(\theta)=\langle \psi(\theta)|H|\psi(\theta)\rangle$$7

the full projected Hamiltonian matrix is reconstructed through

$$E(\theta)=\langle \psi(\theta)|H|\psi(\theta)\rangle$$8

A classical diagonalization then yields

$$E(\theta)=\langle \psi(\theta)|H|\psi(\theta)\rangle$$9

This formulation is ancilla-entangled in the strict sense: the variational object is a purified joint state, orthogonality is enforced by ancilla-sector decomposition rather than by penalties, and ancilla-dressed measurements provide both diagonal and off-diagonal projected matrix elements. The same construction also supports post-processing of mixtures and thermal observables through ancilla-side operators, for example

$\theta$0

with $\theta$1 encoding Boltzmann weights in the ancilla space (Xu et al., 2022).

3. Measurement-based and entangled-resource realizations

A second line of work replaces circuit-based ansatz preparation by a measurement-based variational quantum eigensolver. In this framework, the variational state is produced from a multipartite entangled resource state and local measurements on auxiliary qubits. The paper proposes two schemes: one constructs variational families by “decorating” a graph-state or stabilizer ansatz with additional auxiliary qubits, and the other translates a circuit-based VQE ansatz into a measurement-based scheme. The variational parameters are measurement-basis angles on those auxiliary qubits, with rotated basis

$\theta$2

A particularly explicit statement is that a single auxiliary qubit measured in $\theta$3 and connected to an arbitrary number of output qubits acts

$\theta$4

onto them. In the translated Schwinger-model example, a circuit VQE with $\theta$5 qubits, $\theta$6 single-qubit operations, and $\theta$7 entangling gates becomes an MB-VQE with $\theta$8 custom-state qubits and $\theta$9 measurements. The claimed benefit is a shift of difficulty from online entangling-gate execution to offline entangled-resource preparation, with problem-specific advantages in the required resources and coherence times (Ferguson et al., 2020).

A hardware-level analogue appears in silicon photonics, where the computation is not ancilla-based in the usual algorithmic sense but is built from an entangled bipartite resource state. Four identical spontaneous-four-wave-mixing sources prepare two path-entangled ququarts whose correlation provides the resource for generic trial-state preparation. The central state is

$$H = \sum_{i\alpha} h^i_\alpha \sigma_\alpha^i + \sum_{ij\alpha\beta}h^{ij}_{\alpha \beta} \sigma_\alpha^i \sigma_\beta^j + \ldots,$$0

or, in the four-qubit encoding,

$$H = \sum_{i\alpha} h^i_\alpha \sigma_\alpha^i + \sum_{ij\alpha\beta}h^{ij}_{\alpha \beta} \sigma_\alpha^i \sigma_\beta^j + \ldots,$$1

Here neither subsystem is designated as an ancilla register; both jointly encode the logical four-qubit state. Even so, the entanglement between idler and signal is operationally central because it supplies the variational trial-state family without explicit entangling gates such as probabilistic CNOTs. For $$H = \sum_{i\alpha} h^i_\alpha \sigma_\alpha^i + \sum_{ij\alpha\beta}h^{ij}_{\alpha \beta} \sigma_\alpha^i \sigma_\beta^j + \ldots,$$2 at $$H = \sum_{i\alpha} h^i_\alpha \sigma_\alpha^i + \sum_{ij\alpha\beta}h^{ij}_{\alpha \beta} \sigma_\alpha^i \sigma_\beta^j + \ldots,$$3, the reported VQE result is

$$H = \sum_{i\alpha} h^i_\alpha \sigma_\alpha^i + \sum_{ij\alpha\beta}h^{ij}_{\alpha \beta} \sigma_\alpha^i \sigma_\beta^j + \ldots,$$4

compared with

$$H = \sum_{i\alpha} h^i_\alpha \sigma_\alpha^i + \sum_{ij\alpha\beta}h^{ij}_{\alpha \beta} \sigma_\alpha^i \sigma_\beta^j + \ldots,$$5

which the paper presents as a proof of principle of an entangled-resource photonic VQE rather than a canonical ancilla-based VQE (Baldazzi et al., 2 Jan 2025).

4. Ancilla-assisted excited-state and symmetry-screening variants

Ancilla use in variational eigensolvers is not limited to purified ansätze. A distinct pattern is the ancilla-assisted screening add-on, exemplified by spin-filtering variational quantum deflation. The underlying excited-state optimizer remains VQD, with loss

$$H = \sum_{i\alpha} h^i_\alpha \sigma_\alpha^i + \sum_{ij\alpha\beta}h^{ij}_{\alpha \beta} \sigma_\alpha^i \sigma_\beta^j + \ldots,$$6

but a shallow QPE-like symmetry routine is inserted before energy estimation. The variational state is prepared by an $$H = \sum_{i\alpha} h^i_\alpha \sigma_\alpha^i + \sum_{ij\alpha\beta}h^{ij}_{\alpha \beta} \sigma_\alpha^i \sigma_\beta^j + \ldots,$$7-conserving symmetry-preserving ansatz, while a small ancilla register is coupled to the system through controlled rotations generated by

$$H = \sum_{i\alpha} h^i_\alpha \sigma_\alpha^i + \sum_{ij\alpha\beta}h^{ij}_{\alpha \beta} \sigma_\alpha^i \sigma_\beta^j + \ldots,$$8

Under the spin-free Hamiltonian assumption,

$$H = \sum_{i\alpha} h^i_\alpha \sigma_\alpha^i + \sum_{ij\alpha\beta}h^{ij}_{\alpha \beta} \sigma_\alpha^i \sigma_\beta^j + \ldots,$$9

and therefore

$\langle H\rangle$0

The ancilla routine estimates $\langle H\rangle$1, and shots with $\langle H\rangle$2 relative to the target sector are screened out. The key screening probability is

$\langle H\rangle$3

which equals $\langle H\rangle$4 for the minimal-spin sector $\langle H\rangle$5 and decreases for higher-spin contaminants. The joint evolution makes the ancilla role explicit; after inverse QFT the state is

$\langle H\rangle$6

so the ancilla stores the $\langle H\rangle$7 label while the system collapses into the corresponding $\langle H\rangle$8-eigencomponent upon measurement. The ancilla count is designed to be modest,

$\langle H\rangle$9

and the paper reports that in the singlet manifold of BeH$U$0, the method reduces the number of orthogonality checks by roughly a factor of four relative to VQD/SP and by about a factor of two relative to VQD/SSP (Ahn et al., 30 Mar 2026).

The contrast with ancilla-free excited-state VQE is instructive. Subspace-search variational quantum eigensolver prepares several orthogonal input states

$U$1

passes them through a common unitary ansatz $U$2, and uses unitarity to preserve orthogonality,

$U$3

Its basic cost is

$U$4

and the paper emphasizes that the algorithm “does not employ any ancilla qubits.” The stated advantage is precisely the disuse of ancilla qubits relative to swap-test-based overlap-penalty methods that “require us to double the number of qubits with additional gates” (Nakanishi et al., 2018). This makes SSVQE a direct reminder that excited-state variational methods do not inherently require ancillas; ancilla-entangled VQE is one design choice among several.

5. Adjacent ancilla-entangled eigensolvers outside standard VQE

The boundary of the topic is clarified further by ancilla-entangled eigensolvers that are not standard VQE. The quantum amplitude-amplification eigensolver is an ancilla-assisted, coherently amplified eigensolver with a non-variational core and a state-learning outer loop. One begins from a system ansatz

$U$5

and a single ancilla in

$U$6

so that the joint state is

$U$7

The ancilla controls short-time Hamiltonian evolutions through

$U$8

and the algorithm uses the Householder reflection

$U$9

to define one-round amplification

$N_p$0

After measuring the ancilla, the system collapses to

$N_p$1

and a state-learning step re-encodes this amplified target state into the ansatz for the next round (Baek et al., 15 Nov 2025).

This method is directly relevant to ancilla-entangled eigensolving because the ancilla is initialized in a coherent superposition, becomes entangled with the system under controlled evolution, and its measurement projects the system into an improved state. The paper is equally explicit that it is not a conventional VQE: it does not minimize $N_p$2, does not use energy gradients, and does not explore the energy landscape. Its closest relation to ancilla-entangled VQE is therefore contrastive. It shows that ancilla-system entanglement can be a genuine computational primitive inside a hybrid eigensolver without making the method an ancilla-entangled variational quantum eigensolver in the narrow technical sense (Baek et al., 15 Nov 2025).

6. Advantages, limitations, and recurrent misconceptions

The literature attributes several concrete advantages to ancilla-entangled or ancilla-assisted variants. In the purified multi-eigenpair formulation, ancillas keep the trial states orthogonal “throughout the whole optimization process,” the algorithm “allows these states to be efficiently computed in one quantum circuit,” and it “reduces significantly the complexity of circuits and the readout errors,” while enabling flexible post-processing on the optimized eigen-subspace (Xu et al., 2022). In the measurement-based formulation, entangled resource states and local measurements can offer “problem-specific advantages in terms of the required resources and coherence times” (Ferguson et al., 2020). In the spin-filtering VQD construction, the ancilla register carries spin information with only “modest circuit overhead,” avoids costly explicit evaluation of $N_p$3, and improves separation of singlet and triplet manifolds over conventional VQD without QPE-derived screening (Ahn et al., 30 Mar 2026).

The limitations are equally explicit. The purified multi-eigenpair method requires an ancilla overhead of at least $N_p$4 and a worst-case measurement overhead of $N_p$5 ancilla-dressed expectation values to reconstruct the effective Hamiltonian. It also relies on the initial physical basis states embedded in the purified state having nonzero overlap with the desired low-energy sector; otherwise some targeted states may be replaced by higher excited states (Xu et al., 2022). The measurement-based formulation increases qubit count, requires adaptive measurements and Pauli-frame handling, and shifts the hardware burden to preparation of a sufficiently large and suitable entangled resource state (Ferguson et al., 2020). The spin-filtering approach is probabilistic rather than an exact spin projector, its one-axis discrimination weakens for higher-spin contaminants, and its performance under realistic hardware noise and finite-shot conditions remains a stated future assessment (Ahn et al., 30 Mar 2026).

Several misconceptions are corrected by the cited work. Ancilla-entangled VQE is not synonymous with any use of entanglement in a variational algorithm: the original VQE already uses entanglement among system qubits yet remains ancilla-free (Peruzzo et al., 2013). Nor is every ancilla-assisted eigensolver an ancilla-entangled VQE: sfVQD is best understood as an ancilla-assisted screening add-on to VQD, and QAAE is an ancilla-assisted eigensolver with a non-variational amplification core rather than a standard VQE (Ahn et al., 30 Mar 2026, Baek et al., 15 Nov 2025). Conversely, ancillas are not required for excited-state variational algorithms as such, because SSVQE guarantees orthogonality structurally by unitary evolution of orthogonal inputs and uses no ancilla qubits (Nakanishi et al., 2018). The most precise use of the term therefore refers to constructions in which the ancilla is part of the variational representation or of the variational state-generation mechanism itself, as in purified subspace ansätze and measurement-based entangled-resource schemes (Xu et al., 2022, Ferguson et al., 2020).