Hamiltonian Expressibility in VQE

• Hamiltonian expressibility is a Hamiltonian-dependent metric that measures a circuit’s ability to uniformly explore the energy landscape in variational quantum eigenvalue problems.
• It leverages frame potentials and Monte Carlo sampling to compare the ansatz-generated unitary ensemble against the Haar measure, providing quantitative benchmarking for circuit design.
• The metric directly informs ansatz selection by correlating expressibility with VQE performance, noise resilience, and the need for bond-resolved parameters in frustrated systems.

Hamiltonian expressibility is a Hamiltonian-dependent measure for parametrized quantum circuits in variational quantum algorithms. In the formulation used for ansatz selection in VQE, it quantifies a circuit’s ability to uniformly explore the energy landscape associated with a Hamiltonian ground state search problem by comparing the unitary ensemble generated by the ansatz with the Haar measure, specialized to the problem Hamiltonian (Brozzi et al., 30 Jul 2025). Recent work further shows that, in frustrated many-body systems, poor variational performance can arise from insufficient expressibility of the ansatz rather than from optimization pathologies such as barren plateaus, with geometric frustration producing bond-dependent correlations and near-degenerate spectra that standard global-parameter Hamiltonian-inspired circuits fail to capture (Maiti, 13 Apr 2026).

1. Formal definition and core observables

For an $n$-qubit system with dimension $d=2^n$, the central object is the super-operator

$$A_U(X)=\int_{V\in\mathcal U(d)}d\mu_H(V)\;V^{\otimes2}X(V^\dagger)^{\otimes2} -\int_{\theta\in\Theta}d\theta\;U(\theta)^{\otimes2}X(U(\theta)^\dagger)^{\otimes2}.$$

Two specializations of $A_U$ give “state expressibility” and “Hamiltonian expressibility.” The Hamiltonian-specific quantity is

$$\varepsilon^{\mathcal H}(U,H)=\bigl\|A_U(H^{\otimes2})\bigr\|_2.$$

An equivalent formulation uses frame potentials. The ansatz-Hamiltonian frame potential is

$$\mathcal F(U,H) =\int\!\!\int_{\Theta\times\Theta} \Bigl[\mathrm{Tr}\bigl(H\,U(\theta')^\dagger U(\theta'')\,H\,U(\theta'')^\dagger U(\theta')\bigr)\Bigr]^2 \,d\theta'\,d\theta'',$$

and the Haar-Hamiltonian frame potential is

$$\mathcal F_{\rm Haar}(H) =\int_{V,W\in\mathcal U(d)} \Bigl[\mathrm{Tr}\bigl(H\,W^\dagger V\,H\,V^\dagger W\bigr)\Bigr]^2 \,d\mu_H(W)\,d\mu_H(V),$$

with closed form

$$\mathcal F_{\rm Haar}(H) =\frac{\mathrm{Tr}[H]^4+\mathrm{Tr}[H^2]^2}{2^{2n}-1} -\frac{2\,\mathrm{Tr}[H^2]\,\mathrm{Tr}[H]^2}{2^n(2^{2n}-1)}.$$

These definitions yield

$$\varepsilon^{\mathcal H}(U,H)=\sqrt{\mathcal F(U,H)-\mathcal F_{\rm Haar}(H)}, \qquad \gamma^{\mathcal H}(U,H)=\frac{\mathcal F(U,H)}{\mathcal F_{\rm Haar}(H)},$$

where $\gamma^{\mathcal H}\ge1$ is the “Hamiltonian-expressibility ratio” and $d=2^n$0 $d=2^n$1 iff the ansatz exactly reproduces the Haar-Hamiltonian frame potential (Brozzi et al., 30 Jul 2025).

The construction is explicitly problem-dependent: the same circuit family can have different Hamiltonian expressibility for different Hamiltonians. This distinguishes it from generic expressibility notions that ignore the target operator and instead assess coverage of the full unitary or state manifold. In the variational setting, the Hamiltonian dependence is the point: the metric is meant to probe the relevant search landscape rather than Haar closeness in the abstract (Brozzi et al., 30 Jul 2025).

Quantity	Definition	Role
$d=2^n$2	$d=2^n$3	Hamiltonian expressibility
$d=2^n$4	$d=2^n$5	Relative deviation from Haar
$d=2^n$6	Ansatz-Hamiltonian frame potential	Circuit-dependent statistic
$d=2^n$7	Haar-Hamiltonian frame potential	Reference value

This formalism is complemented, in frustrated-system studies, by observable-level diagnostics that expose whether a variational family captures the physically relevant sector of Hilbert space. Those diagnostics include the energy error $d=2^n$8, the fidelity $d=2^n$9, the gradient norm $$A_U(X)=\int_{V\in\mathcal U(d)}d\mu_H(V)\;V^{\otimes2}X(V^\dagger)^{\otimes2} -\int_{\theta\in\Theta}d\theta\;U(\theta)^{\otimes2}X(U(\theta)^\dagger)^{\otimes2}.$$0 computed via the parameter-shift rule, bond-resolved correlator errors $$A_U(X)=\int_{V\in\mathcal U(d)}d\mu_H(V)\;V^{\otimes2}X(V^\dagger)^{\otimes2} -\int_{\theta\in\Theta}d\theta\;U(\theta)^{\otimes2}X(U(\theta)^\dagger)^{\otimes2}.$$1, and excitation gaps $$A_U(X)=\int_{V\in\mathcal U(d)}d\mu_H(V)\;V^{\otimes2}X(V^\dagger)^{\otimes2} -\int_{\theta\in\Theta}d\theta\;U(\theta)^{\otimes2}X(U(\theta)^\dagger)^{\otimes2}.$$2 obtained via symmetry-resolved layers or the VQD cost

$$A_U(X)=\int_{V\in\mathcal U(d)}d\mu_H(V)\;V^{\otimes2}X(V^\dagger)^{\otimes2} -\int_{\theta\in\Theta}d\theta\;U(\theta)^{\otimes2}X(U(\theta)^\dagger)^{\otimes2}.$$3

(Maiti, 13 Apr 2026).

2. Estimation, sampling, and depth dependence

Direct integration over parameter space is intractable, so Hamiltonian expressibility is estimated by Monte Carlo sampling. Writing

$$A_U(X)=\int_{V\in\mathcal U(d)}d\mu_H(V)\;V^{\otimes2}X(V^\dagger)^{\otimes2} -\int_{\theta\in\Theta}d\theta\;U(\theta)^{\otimes2}X(U(\theta)^\dagger)^{\otimes2}.$$4

one samples $$A_U(X)=\int_{V\in\mathcal U(d)}d\mu_H(V)\;V^{\otimes2}X(V^\dagger)^{\otimes2} -\int_{\theta\in\Theta}d\theta\;U(\theta)^{\otimes2}X(U(\theta)^\dagger)^{\otimes2}.$$5 independent parameter pairs $$A_U(X)=\int_{V\in\mathcal U(d)}d\mu_H(V)\;V^{\otimes2}X(V^\dagger)^{\otimes2} -\int_{\theta\in\Theta}d\theta\;U(\theta)^{\otimes2}X(U(\theta)^\dagger)^{\otimes2}.$$6 and computes

$$A_U(X)=\int_{V\in\mathcal U(d)}d\mu_H(V)\;V^{\otimes2}X(V^\dagger)^{\otimes2} -\int_{\theta\in\Theta}d\theta\;U(\theta)^{\otimes2}X(U(\theta)^\dagger)^{\otimes2}.$$7

The Monte Carlo estimator is

$$A_U(X)=\int_{V\in\mathcal U(d)}d\mu_H(V)\;V^{\otimes2}X(V^\dagger)^{\otimes2} -\int_{\theta\in\Theta}d\theta\;U(\theta)^{\otimes2}X(U(\theta)^\dagger)^{\otimes2}.$$8

and then

$$A_U(X)=\int_{V\in\mathcal U(d)}d\mu_H(V)\;V^{\otimes2}X(V^\dagger)^{\otimes2} -\int_{\theta\in\Theta}d\theta\;U(\theta)^{\otimes2}X(U(\theta)^\dagger)^{\otimes2}.$$9

Confidence intervals are obtained by substituting $A_U$0. In practice $A_U$1 samples were used, which yields a relative error $A_U$2 on typical Hamiltonians (Brozzi et al., 30 Jul 2025).

With increasing repetition layers $A_U$3 in a hardware-efficient ansatz, $A_U$4 monotonically decreases until it saturates at the maximal-expressibility threshold $A_U$5 set by the Haar sampling bias. The required depth is Hamiltonian dependent. Diagonal Hamiltonians such as QUBO and MaxCut require fewer layers to reach saturation than non-diagonal Hamiltonians such as Heisenberg XXZ and the transverse-field Ising model, because their ground-state support is a computational basis state and the energy landscape is “narrower.” Across fixed depth, different base circuit patterns exhibit order-of-magnitude differences in $A_U$6; some templates become maximally expressive after $A_U$7, whereas others need $A_U$8–$A_U$9 (Brozzi et al., 30 Jul 2025).

These observations make Hamiltonian expressibility operational rather than merely descriptive. The metric is computed once per circuit–Hamiltonian pair and is then used to rank ansätze before training. This suggests a pre-optimization screening procedure in which expressibility is treated as a problem-conditioned architectural prior rather than as a post hoc diagnostic (Brozzi et al., 30 Jul 2025).

3. Relation to VQE performance and ansatz selection

The empirical value of Hamiltonian expressibility lies in its correlation with solution quality. In the reported experiments, all circuits were trained via VQE on $$\varepsilon^{\mathcal H}(U,H)=\bigl\|A_U(H^{\otimes2})\bigr\|_2.$$0-qubit and $$\varepsilon^{\mathcal H}(U,H)=\bigl\|A_U(H^{\otimes2})\bigr\|_2.$$1-qubit Hamiltonians under ideal and noisy conditions. Solution quality was measured by the normalized approximation ratio

$$\varepsilon^{\mathcal H}(U,H)=\bigl\|A_U(H^{\otimes2})\bigr\|_2.$$2

and correlations between $$\varepsilon^{\mathcal H}(U,H)=\bigl\|A_U(H^{\otimes2})\bigr\|_2.$$3 or $$\varepsilon^{\mathcal H}(U,H)=\bigl\|A_U(H^{\otimes2})\bigr\|_2.$$4 and A.R. were quantified by Pearson $$\varepsilon^{\mathcal H}(U,H)=\bigl\|A_U(H^{\otimes2})\bigr\|_2.$$5, Spearman $$\varepsilon^{\mathcal H}(U,H)=\bigl\|A_U(H^{\otimes2})\bigr\|_2.$$6, Kendall $$\varepsilon^{\mathcal H}(U,H)=\bigl\|A_U(H^{\otimes2})\bigr\|_2.$$7, and mutual information $$\varepsilon^{\mathcal H}(U,H)=\bigl\|A_U(H^{\otimes2})\bigr\|_2.$$8 (Brozzi et al., 30 Jul 2025).

For $$\varepsilon^{\mathcal H}(U,H)=\bigl\|A_U(H^{\otimes2})\bigr\|_2.$$9-qubit ideal results, diagonal Hamiltonians with basis-state ground states—MaxCut, MinVertex, MaxClique, and Random Diagonal—show Spearman and Kendall coefficients that are slightly positive or near zero, with $$\mathcal F(U,H) =\int\!\!\int_{\Theta\times\Theta} \Bigl[\mathrm{Tr}\bigl(H\,U(\theta')^\dagger U(\theta'')\,H\,U(\theta'')^\dagger U(\theta')\bigr)\Bigr]^2 \,d\theta'\,d\theta'',$$0 and $$\mathcal F(U,H) =\int\!\!\int_{\Theta\times\Theta} \Bigl[\mathrm{Tr}\bigl(H\,U(\theta')^\dagger U(\theta'')\,H\,U(\theta'')^\dagger U(\theta')\bigr)\Bigr]^2 \,d\theta'\,d\theta'',$$1, indicating that low-expressibility circuits perform better. Mutual information is small, $$\mathcal F(U,H) =\int\!\!\int_{\Theta\times\Theta} \Bigl[\mathrm{Tr}\bigl(H\,U(\theta')^\dagger U(\theta'')\,H\,U(\theta'')^\dagger U(\theta')\bigr)\Bigr]^2 \,d\theta'\,d\theta'',$$2. By contrast, non-diagonal, superposition-state problems—Heisenberg XXZ, Transverse-Field Ising, Adiabatic, and Random Non-Diagonal—show strong negative correlations: $$\mathcal F(U,H) =\int\!\!\int_{\Theta\times\Theta} \Bigl[\mathrm{Tr}\bigl(H\,U(\theta')^\dagger U(\theta'')\,H\,U(\theta'')^\dagger U(\theta')\bigr)\Bigr]^2 \,d\theta'\,d\theta'',$$3, $$\mathcal F(U,H) =\int\!\!\int_{\Theta\times\Theta} \Bigl[\mathrm{Tr}\bigl(H\,U(\theta')^\dagger U(\theta'')\,H\,U(\theta'')^\dagger U(\theta')\bigr)\Bigr]^2 \,d\theta'\,d\theta'',$$4, $$\mathcal F(U,H) =\int\!\!\int_{\Theta\times\Theta} \Bigl[\mathrm{Tr}\bigl(H\,U(\theta')^\dagger U(\theta'')\,H\,U(\theta'')^\dagger U(\theta')\bigr)\Bigr]^2 \,d\theta'\,d\theta'',$$5, and $$\mathcal F(U,H) =\int\!\!\int_{\Theta\times\Theta} \Bigl[\mathrm{Tr}\bigl(H\,U(\theta')^\dagger U(\theta'')\,H\,U(\theta'')^\dagger U(\theta')\bigr)\Bigr]^2 \,d\theta'\,d\theta'',$$6. In that regime, high expressibility, meaning small $$\mathcal F(U,H) =\int\!\!\int_{\Theta\times\Theta} \Bigl[\mathrm{Tr}\bigl(H\,U(\theta')^\dagger U(\theta'')\,H\,U(\theta'')^\dagger U(\theta')\bigr)\Bigr]^2 \,d\theta'\,d\theta'',$$7, correlates with better VQE performance (Brozzi et al., 30 Jul 2025).

For $$\mathcal F(U,H) =\int\!\!\int_{\Theta\times\Theta} \Bigl[\mathrm{Tr}\bigl(H\,U(\theta')^\dagger U(\theta'')\,H\,U(\theta'')^\dagger U(\theta')\bigr)\Bigr]^2 \,d\theta'\,d\theta'',$$8-qubit ideal results, the same tendencies persist but the correlations weaken for superposition problems, with $$\mathcal F(U,H) =\int\!\!\int_{\Theta\times\Theta} \Bigl[\mathrm{Tr}\bigl(H\,U(\theta')^\dagger U(\theta'')\,H\,U(\theta'')^\dagger U(\theta')\bigr)\Bigr]^2 \,d\theta'\,d\theta'',$$9, consistent with the onset of barren plateaus at larger $$\mathcal F_{\rm Haar}(H) =\int_{V,W\in\mathcal U(d)} \Bigl[\mathrm{Tr}\bigl(H\,W^\dagger V\,H\,V^\dagger W\bigr)\Bigr]^2 \,d\mu_H(W)\,d\mu_H(V),$$0. Under noisy $$\mathcal F_{\rm Haar}(H) =\int_{V,W\in\mathcal U(d)} \Bigl[\mathrm{Tr}\bigl(H\,W^\dagger V\,H\,V^\dagger W\bigr)\Bigr]^2 \,d\mu_H(W)\,d\mu_H(V),$$1-qubit conditions at realistic error rates $$\mathcal F_{\rm Haar}(H) =\int_{V,W\in\mathcal U(d)} \Bigl[\mathrm{Tr}\bigl(H\,W^\dagger V\,H\,V^\dagger W\bigr)\Bigr]^2 \,d\mu_H(W)\,d\mu_H(V),$$2, $$\mathcal F_{\rm Haar}(H) =\int_{V,W\in\mathcal U(d)} \Bigl[\mathrm{Tr}\bigl(H\,W^\dagger V\,H\,V^\dagger W\bigr)\Bigr]^2 \,d\mu_H(W)\,d\mu_H(V),$$3, $$\mathcal F_{\rm Haar}(H) =\int_{V,W\in\mathcal U(d)} \Bigl[\mathrm{Tr}\bigl(H\,W^\dagger V\,H\,V^\dagger W\bigr)\Bigr]^2 \,d\mu_H(W)\,d\mu_H(V),$$4, and overall $$\mathcal F_{\rm Haar}(H) =\int_{V,W\in\mathcal U(d)} \Bigl[\mathrm{Tr}\bigl(H\,W^\dagger V\,H\,V^\dagger W\bigr)\Bigr]^2 \,d\mu_H(W)\,d\mu_H(V),$$5, diagonal or basis-state problems show strongly positive $$\mathcal F_{\rm Haar}(H) =\int_{V,W\in\mathcal U(d)} \Bigl[\mathrm{Tr}\bigl(H\,W^\dagger V\,H\,V^\dagger W\bigr)\Bigr]^2 \,d\mu_H(W)\,d\mu_H(V),$$6, reaffirming that low-expressibility ansätze are most noise-resilient there. Non-diagonal problems split into two behaviors. For Transverse-Ising and Adiabatic problems, A.R. rises monotonically as $$\mathcal F_{\rm Haar}(H) =\int_{V,W\in\mathcal U(d)} \Bigl[\mathrm{Tr}\bigl(H\,W^\dagger V\,H\,V^\dagger W\bigr)\Bigr]^2 \,d\mu_H(W)\,d\mu_H(V),$$7 increases, so low expressibility is preferred under noise. For Heisenberg XXZ and Random Non-Diagonal, a bell-shaped dependence $$\mathcal F_{\rm Haar}(H) =\int_{V,W\in\mathcal U(d)} \Bigl[\mathrm{Tr}\bigl(H\,W^\dagger V\,H\,V^\dagger W\bigr)\Bigr]^2 \,d\mu_H(W)\,d\mu_H(V),$$8 appears, with best performance at intermediate expressibility and poor performance at both extremes; in that case Pearson, Spearman, and Kendall are approximately $$\mathcal F_{\rm Haar}(H) =\int_{V,W\in\mathcal U(d)} \Bigl[\mathrm{Tr}\bigl(H\,W^\dagger V\,H\,V^\dagger W\bigr)\Bigr]^2 \,d\mu_H(W)\,d\mu_H(V),$$9, while mutual information is $$\mathcal F_{\rm Haar}(H) =\frac{\mathrm{Tr}[H]^4+\mathrm{Tr}[H^2]^2}{2^{2n}-1} -\frac{2\,\mathrm{Tr}[H^2]\,\mathrm{Tr}[H]^2}{2^n(2^{2n}-1)}.$$0 (Brozzi et al., 30 Jul 2025).

The practical guidelines are therefore conditional. In small, noiseless or very low-noise VQE, low-expressibility ansätze are favored when the ground state is expected to be a basis state, while high-expressibility ansätze are favored when the ground state is superposed or entangled. As system size grows, the benefit of high expressibility decreases because expressibility must be balanced against trainability. Under realistic noise, low expressibility remains best for basis-state problems, whereas for some superposition-state problems an intermediate level of expressibility often maximizes performance by avoiding both under-exploration and over-complexity (Brozzi et al., 30 Jul 2025).

4. Frustration as a physical source of expressibility failure

A complementary perspective arises in frustrated quantum many-body systems, where expressibility failure can be traced to the physics of the target Hamiltonian rather than to generic optimization barriers. The model analyzed is the transverse-field Ising model on a square lattice with diagonal couplings,

$$\mathcal F_{\rm Haar}(H) =\frac{\mathrm{Tr}[H]^4+\mathrm{Tr}[H^2]^2}{2^{2n}-1} -\frac{2\,\mathrm{Tr}[H^2]\,\mathrm{Tr}[H]^2}{2^n(2^{2n}-1)}.$$1

where $$\mathcal F_{\rm Haar}(H) =\frac{\mathrm{Tr}[H]^4+\mathrm{Tr}[H^2]^2}{2^{2n}-1} -\frac{2\,\mathrm{Tr}[H^2]\,\mathrm{Tr}[H]^2}{2^n(2^{2n}-1)}.$$2 are nearest neighbors, $$\mathcal F_{\rm Haar}(H) =\frac{\mathrm{Tr}[H]^4+\mathrm{Tr}[H^2]^2}{2^{2n}-1} -\frac{2\,\mathrm{Tr}[H^2]\,\mathrm{Tr}[H]^2}{2^n(2^{2n}-1)}.$$3 are diagonal pairs, $$\mathcal F_{\rm Haar}(H) =\frac{\mathrm{Tr}[H]^4+\mathrm{Tr}[H^2]^2}{2^{2n}-1} -\frac{2\,\mathrm{Tr}[H^2]\,\mathrm{Tr}[H]^2}{2^n(2^{2n}-1)}.$$4 is antiferromagnetic, and $$\mathcal F_{\rm Haar}(H) =\frac{\mathrm{Tr}[H]^4+\mathrm{Tr}[H^2]^2}{2^{2n}-1} -\frac{2\,\mathrm{Tr}[H^2]\,\mathrm{Tr}[H]^2}{2^n(2^{2n}-1)}.$$5 is the transverse field (Maiti, 13 Apr 2026).

At $$\mathcal F_{\rm Haar}(H) =\frac{\mathrm{Tr}[H]^4+\mathrm{Tr}[H^2]^2}{2^{2n}-1} -\frac{2\,\mathrm{Tr}[H^2]\,\mathrm{Tr}[H]^2}{2^n(2^{2n}-1)}.$$6, each square plaquette is subdivided into two triangles, and on each triangle one bond must be “unsatisfied” with two antiparallel and one parallel configuration. The ground-state manifold is therefore macroscopically degenerate. For $$\mathcal F_{\rm Haar}(H) =\frac{\mathrm{Tr}[H]^4+\mathrm{Tr}[H^2]^2}{2^{2n}-1} -\frac{2\,\mathrm{Tr}[H^2]\,\mathrm{Tr}[H]^2}{2^n(2^{2n}-1)}.$$7, “order-by-disorder” lifts the degeneracy, but the competition between $$\mathcal F_{\rm Haar}(H) =\frac{\mathrm{Tr}[H]^4+\mathrm{Tr}[H^2]^2}{2^{2n}-1} -\frac{2\,\mathrm{Tr}[H^2]\,\mathrm{Tr}[H]^2}{2^n(2^{2n}-1)}.$$8 and $$\mathcal F_{\rm Haar}(H) =\frac{\mathrm{Tr}[H]^4+\mathrm{Tr}[H^2]^2}{2^{2n}-1} -\frac{2\,\mathrm{Tr}[H^2]\,\mathrm{Tr}[H]^2}{2^n(2^{2n}-1)}.$$9 generates strongly inhomogeneous, bond-dependent correlations even in a translationally invariant Hamiltonian. This is the physical background for the observed variational limitations: a uniform circuit can be matched to a uniform Hamiltonian while still failing to represent its nonuniform correlational structure (Maiti, 13 Apr 2026).

The standard Hamiltonian Variational Ansatz with $$\varepsilon^{\mathcal H}(U,H)=\sqrt{\mathcal F(U,H)-\mathcal F_{\rm Haar}(H)}, \qquad \gamma^{\mathcal H}(U,H)=\frac{\mathcal F(U,H)}{\mathcal F_{\rm Haar}(H)},$$0 layers and $$\varepsilon^{\mathcal H}(U,H)=\sqrt{\mathcal F(U,H)-\mathcal F_{\rm Haar}(H)}, \qquad \gamma^{\mathcal H}(U,H)=\frac{\mathcal F(U,H)}{\mathcal F_{\rm Haar}(H)},$$1 parameters is

$$\varepsilon^{\mathcal H}(U,H)=\sqrt{\mathcal F(U,H)-\mathcal F_{\rm Haar}(H)}, \qquad \gamma^{\mathcal H}(U,H)=\frac{\mathcal F(U,H)}{\mathcal F_{\rm Haar}(H)},$$2

with

$$\varepsilon^{\mathcal H}(U,H)=\sqrt{\mathcal F(U,H)-\mathcal F_{\rm Haar}(H)}, \qquad \gamma^{\mathcal H}(U,H)=\frac{\mathcal F(U,H)}{\mathcal F_{\rm Haar}(H)},$$3

In the paramagnetic regime $$\varepsilon^{\mathcal H}(U,H)=\sqrt{\mathcal F(U,H)-\mathcal F_{\rm Haar}(H)}, \qquad \gamma^{\mathcal H}(U,H)=\frac{\mathcal F(U,H)}{\mathcal F_{\rm Haar}(H)},$$4, this HVA achieves $$\varepsilon^{\mathcal H}(U,H)=\sqrt{\mathcal F(U,H)-\mathcal F_{\rm Haar}(H)}, \qquad \gamma^{\mathcal H}(U,H)=\frac{\mathcal F(U,H)}{\mathcal F_{\rm Haar}(H)},$$5 at $$\varepsilon^{\mathcal H}(U,H)=\sqrt{\mathcal F(U,H)-\mathcal F_{\rm Haar}(H)}, \qquad \gamma^{\mathcal H}(U,H)=\frac{\mathcal F(U,H)}{\mathcal F_{\rm Haar}(H)},$$6 for all $$\varepsilon^{\mathcal H}(U,H)=\sqrt{\mathcal F(U,H)-\mathcal F_{\rm Haar}(H)}, \qquad \gamma^{\mathcal H}(U,H)=\frac{\mathcal F(U,H)}{\mathcal F_{\rm Haar}(H)},$$7. In the frustrated regime $$\varepsilon^{\mathcal H}(U,H)=\sqrt{\mathcal F(U,H)-\mathcal F_{\rm Haar}(H)}, \qquad \gamma^{\mathcal H}(U,H)=\frac{\mathcal F(U,H)}{\mathcal F_{\rm Haar}(H)},$$8, however, the HVA infidelity at $$\varepsilon^{\mathcal H}(U,H)=\sqrt{\mathcal F(U,H)-\mathcal F_{\rm Haar}(H)}, \qquad \gamma^{\mathcal H}(U,H)=\frac{\mathcal F(U,H)}{\mathcal F_{\rm Haar}(H)},$$9 and $\gamma^{\mathcal H}\ge1$0 saturates near $\gamma^{\mathcal H}\ge1$1 even at $\gamma^{\mathcal H}\ge1$2, and to reach $\gamma^{\mathcal H}\ge1$3 one needs $\gamma^{\mathcal H}\ge1$4, corresponding to CNOT count $\gamma^{\mathcal H}\ge1$5 at $\gamma^{\mathcal H}\ge1$6. At the same time, gradients remain finite. The failure is therefore not diagnosed as a barren plateau but as an insufficient variational manifold. Bond correlation errors remain $\gamma^{\mathcal H}\ge1$7 on frustrated bonds even at large $\gamma^{\mathcal H}\ge1$8 (Maiti, 13 Apr 2026).

This reframes a common misconception. Poor VQE performance in hard many-body regimes need not indicate optimization collapse. In this case, the limiting factor is that a single global $\gamma^{\mathcal H}\ge1$9 angle cannot adapt to bond-dependent order. The diagnosis is expressibility failure induced by frustration and correlation inhomogeneity, not lack of gradient signal (Maiti, 13 Apr 2026).

5. Bond-resolved ansätze, excitations, and design rules

To address the failure of global-parameter HVAs, the frustrated-system study introduces a bond-resolved Hamiltonian Variational Ansatz. Its circuit initializes $d=2^n$00 and, for each layer $d=2^n$01, applies each $d=2^n$02 interaction with its own parameter $d=2^n$03 followed by a global $d=2^n$04 rotation. The state is

$d=2^n$05

Its parameter count is $d=2^n$06 instead of $d=2^n$07 (Maiti, 13 Apr 2026).

This modification restores expressibility. At $d=2^n$08 and $d=2^n$09, the bond-resolved HVA achieves energy error $d=2^n$10 versus approximately $d=2^n$11 for the standard HVA. At $d=2^n$12, it reaches $d=2^n$13 by $d=2^n$14, corresponding to CNOT $d=2^n$15, compared with the standard HVA’s $d=2^n$16 and CNOT $d=2^n$17. Remaining correlator errors drop below $d=2^n$18. The improvement is therefore not only in fidelity but in the ability to reproduce the inhomogeneous correlator pattern that frustration generates (Maiti, 13 Apr 2026).

Low-energy excitations introduce a second expressibility challenge. Exact diagonalization shows very small gaps in the frustrated regime, and the gaps shrink with $d=2^n$19. A symmetry-resolved VQE approach prepares even and odd parity slices under $d=2^n$20 and optimizes them separately with the bond-resolved HVA. The resulting variational gap reproduces the overall trend, but deviations grow in the $d=2^n$21 regime and at larger $d=2^n$22 because the low-lying states are densely packed. VQD matches exact diagonalization in the paramagnetic regime but becomes unstable for $d=2^n$23, where $d=2^n$24 must exceed the true gap and penalty terms mix near-degenerate states (Maiti, 13 Apr 2026).

The design guidelines derived from these results are specific and restrictive. Geometry and correlation inhomogeneity must inform ansatz structure; local or bond-resolved parameters can restore expressibility at moderate depth and reduce circuit resources; symmetry resolution, such as $d=2^n$25 parity, aids excitation extraction; and poor performance should not be attributed to barren plateaus alone, because healthy gradient norms are compatible with an overly restrictive variational manifold (Maiti, 13 Apr 2026).

6. Broader meanings of “Hamiltonian expressibility”

Outside variational quantum algorithms, the phrase “Hamiltonian expressibility” is used in several distinct senses. In the spectral theory of real Hamiltonian matrices, it refers to realizability of prescribed spectra under the structural constraint

$d=2^n$26

For real $d=2^n$27 Hamiltonian matrices, the necessary spectral symmetry $d=2^n$28 together with closure under complex conjugation is also sufficient, and constructive block realizations are given for real pairs, pure imaginary pairs, and general complex quadruples (Manzaneda et al., 2019).

In tensor-based polynomial systems, the analogous question is whether a polynomial ODE can be expressed as a Hamiltonian system with a polynomial Hamiltonian. There, a tensor-based polynomial system

$d=2^n$29

admits a polynomial Hamiltonian

$d=2^n$30

if and only if each system tensor $d=2^n$31 is a Hamiltonian cubical tensor. Equivalently, for a tensor $d=2^n$32, the following are equivalent: $d=2^n$33 is Hamiltonian, $d=2^n$34 for a supersymmetric tensor $d=2^n$35, and $d=2^n$36 is supersymmetric (Cui et al., 27 Mar 2025).

In Hamiltonian normal-form theory, expressibility denotes direct closed-form dependence of the normal form on the original Hamiltonian, bypassing step-by-step normalization. For one degree of freedom, the normal form is recovered from an explicit nonlinear functional

$d=2^n$37

and in arbitrary dimension the $d=2^n$38th normal-form term is written as a sum over full binary trees with Bernoulli-number weights (Treschev, 4 May 2026).

A further algebraic-combinatorial usage appears in Hamiltonian-cycle counting. The Hamiltonian-cycle polynomial

$d=2^n$39

satisfies the identity

$d=2^n$40

and the associated discussion describes expressibility as an exact determinant–permanent decomposition of Hamiltonian-cycle counts rather than as an ansatz metric (Sawczuk et al., 2 Oct 2025).

This suggests that the term is polysemous across subfields. In VQAs it is a quantitative, Hamiltonian-conditioned measure of circuit coverage of an energy landscape. In matrix, tensor, normal-form, and combinatorial settings, it denotes exact structural realizability or explicit algebraic representation. The common thread is not a shared metric but a shared concern with how faithfully a constrained formalism can represent the Hamiltonian object of interest.