--- title: Hamiltonian Expressibility in VQE url: https://www.emergentmind.com/topics/hamiltonian-expressibility type: topic --- # Hamiltonian Expressibility in VQE 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 [2507.22550]. 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 [2604.11688]. ## 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 $\varepsilon^{\mathcal H}=0$ $(\gamma^{\mathcal H}=1)$ iff the ansatz exactly reproduces the Haar-Hamiltonian frame potential [2507.22550]. 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 [2507.22550]. | Quantity | Definition | Role | |---|---|---| | $\varepsilon^{\mathcal H}(U,H)$ | $\|A_U(H^{\otimes2})\|_2$ | Hamiltonian expressibility | | $\gamma^{\mathcal H}(U,H)$ | $\mathcal F(U,H)/\mathcal F_{\rm Haar}(H)$ | Relative deviation from Haar | | $\mathcal F(U,H)$ | Ansatz-Hamiltonian frame potential | Circuit-dependent statistic | | $\mathcal F_{\rm Haar}(H)$ | 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 $\Delta E_0=E_0^{\rm VQE}-E_0^{\rm ED}$, the fidelity $f=|\langle\psi_{\rm VQE}|\psi_{\rm ED}\rangle|^2$, the gradient norm $\|\nabla_\theta E\|=\sqrt{\sum_i(\partial E/\partial\theta_i)^2}$ computed via the parameter-shift rule, bond-resolved correlator errors $C_{ij}=\langle Z_iZ_j\rangle_{\rm VQE}-\langle Z_iZ_j\rangle_{\rm ED}$, and excitation gaps $\Delta=E_1-E_0$ obtained via symmetry-resolved layers or the VQD cost \[ C(\theta)=\langle\psi(\theta)|H|\psi(\theta)\rangle+\beta\sum_{m0$, “order-by-disorder” lifts the degeneracy, but the competition between $-J\,Z_iZ_j$ and $-h\,X_i$ 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 [2604.11688]. The standard Hamiltonian Variational Ansatz with $p$ layers and $2p$ parameters is \[ |\psi(\gamma,\beta)\rangle =\prod_{k=1}^p e^{-i\beta_k H_X}\,e^{-i\gamma_k H_{ZZ}}\,|+\rangle^{\otimes N}, \] with \[ H_{ZZ}=-J\sum_{\langle i,j\rangle}Z_iZ_j,\qquad H_X=-\sum_i X_i. \] In the paramagnetic regime $h\gg|J|$, this HVA achieves $f>0.99$ at $p\le2$ for all $L$. In the frustrated regime $h\sim0.5$, however, the HVA infidelity at $L=3$ and $h=0.5$ saturates near $10^{-2}$ even at $p=8$, and to reach $f>0.99$ one needs $p_{\min}\approx24$, corresponding to CNOT count $\approx768$ at $p=24$. 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 $O(10^{-1})$ on frustrated bonds even at large $p$ [2604.11688]. 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 $ZZ$ angle cannot adapt to bond-dependent order. The diagnosis is expressibility failure induced by frustration and correlation inhomogeneity, not lack of gradient signal [2604.11688]. ## 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 $|+\rangle^{\otimes N}$ and, for each layer $k=1,\dots,p$, applies each $ZZ$ interaction with its own parameter $\gamma_{ij}^{(k)}$ followed by a global $X$ rotation. The state is \[ |\psi(\gamma,\beta)\rangle =\prod_{k=1}^p \Bigl[\prod_{\langle ij\rangle} e^{-i\gamma_{ij}^{(k)}Z_iZ_j}\Bigr] e^{-i\beta_k\sum_i X_i} |+\rangle^{\otimes N}. \] Its parameter count is $p(|E|+1)$ instead of $2p$ [2604.11688]. This modification restores expressibility. At $L=3$ and $p=8$, the bond-resolved HVA achieves energy error $\Delta E_0\approx10^{-3}$ versus approximately $10^{-1}$ for the standard HVA. At $h=0.5$, it reaches $f>0.99$ by $p=6$, corresponding to CNOT $\approx192$, compared with the standard HVA’s $p\approx24$ and CNOT $\approx768$. Remaining correlator errors drop below $10^{-3}$. The improvement is therefore not only in fidelity but in the ability to reproduce the inhomogeneous correlator pattern that frustration generates [2604.11688]. Low-energy excitations introduce a second expressibility challenge. Exact diagonalization shows very small gaps in the frustrated regime, and the gaps shrink with $L$. A symmetry-resolved VQE approach prepares even and odd parity slices under $\prod_i X_i$ and optimizes them separately with the bond-resolved HVA. The resulting variational gap reproduces the overall trend, but deviations grow in the $h=0.5$ regime and at larger $L$ because the low-lying states are densely packed. VQD matches exact diagonalization in the paramagnetic regime but becomes unstable for $h<1$, where $\beta$ must exceed the true gap and penalty terms mix near-degenerate states [2604.11688]. 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 $\prod_iX_i$ 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 [2604.11688]. ## 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 \[ A^T J + J A = 0,\qquad J=\begin{pmatrix}0&I_n\\-I_n&0\end{pmatrix}. \] For real $2n\times2n$ Hamiltonian matrices, the necessary spectral symmetry $\sigma=-\sigma$ 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 [1903.10313]. 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 \[ \dot x=A_kx^{k-1}+\cdots+A_2x \] admits a polynomial Hamiltonian \[ H(x)=\sum_{j=2}^k B_j x^j,\qquad \dot x=J\nabla H(x), \] if and only if each system tensor $A_j$ is a Hamiltonian cubical tensor. Equivalently, for a tensor $A\in\mathbb R^{[j,2n]}$, the following are equivalent: $A$ is Hamiltonian, $A=J R$ for a supersymmetric tensor $R$, and $J A$ is supersymmetric [2503.21487]. 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 \[ S[H](w) =\sum_{m=1}^\infty(-1)^{m-1}\lambda^{m-1}m!\;\partial_w^{\,m-1}\langle(H_*)^m\rangle(w), \] and in arbitrary dimension the $m$th normal-form term is written as a sum over full binary trees with Bernoulli-number weights [2605.02839]. A further algebraic-combinatorial usage appears in Hamiltonian-cycle counting. The Hamiltonian-cycle polynomial \[ P_{HC_n}(A)=\sum_{\sigma\in HC_n}\prod_{i=1}^n a_{i,\sigma(i)} \] satisfies the identity \[ P_{HC_n}(A)=\sum_{S\subseteq[n-1]}\det(-A_S)\,\mathrm{per}(A_{[n]\setminus S}), \] and the associated discussion describes expressibility as an exact determinant–permanent decomposition of Hamiltonian-cycle counts rather than as an ansatz metric [2510.02473]. 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. Source: https://www.emergentmind.com/topics/hamiltonian-expressibility