Dimensional Expressivity Analysis, best-approximation errors, and automated design of parametric quantum circuits

The design of parametric quantum circuits (PQCs) for efficient use in variational quantum simulations (VQS) is subject to two competing factors. On one hand, the set of states that can be generated by the PQC has to be large enough to contain the solution state. Otherwise, one may at best find the best approximation of the solution restricted to the states generated by the chosen PQC. On the other hand, the PQC should contain as few parametric quantum gates as possible to minimize noise from the quantum device. Thus, when designing a PQC one needs to ensure that there are no redundant parameters. The dimensional expressivity analysis discussed in these proceedings is a means of addressing these counteracting effects. Its main objective is to identify independent and redundant parameters in the PQC. Using this information, superfluous parameters can be removed and the dimension of the space of states that are generated by the PQC can be computed. Knowing the dimension of the physical state space then allows us to deduce whether or not the PQC can reach all physical states. Furthermore, the dimensional expressivity analysis can be implemented efficiently using a hybrid quantum-classical algorithm. This implementation has relatively small overhead costs both for the classical and quantum part of the algorithm and could therefore be used in the future for on-the-fly circuit construction. This would allow for optimized circuits to be used in every loop of a VQS rather than the same PQC for the entire VQS. These proceedings review and extend work in [1, 2].

💡 Research Summary

The paper addresses the crucial problem of designing parametric quantum circuits (PQCs) for variational quantum simulations (VQS) on noisy intermediate‑scale quantum (NISQ) devices. Two opposing requirements must be balanced: the circuit must be expressive enough to span the target solution state, yet it should contain as few gates (and therefore parameters) as possible to limit noise. The authors introduce a systematic framework called Dimensional Expressivity Analysis (DEA) that simultaneously quantifies the expressive power of a PQC and identifies redundant parameters.

DEA treats a PQC as a map C : P → S from a real‑valued parameter space P to a quantum state space S (which may be the full Hilbert space or a physically relevant subspace). For each parameter θ_k the real Jacobian column J_k = ∂C/∂θ_k is constructed. By incrementally building the matrix J_k and its Gram matrix S_k = J_k† J_k, the rank increase (or, equivalently, the positivity of the smallest eigenvalue) determines whether θ_k adds an independent direction in state space. If the eigenvalue is zero, θ_k is linearly dependent on earlier parameters and can be fixed without loss of expressivity. This procedure scales as O(k³) for each step and, when the total number of parameters N is fixed, a single reduced‑row‑echelon transformation of S_N yields all independent parameters at once.

The authors also extend DEA to remove unwanted symmetries (e.g., global phase or gauge invariance). By augmenting the circuit with auxiliary parameters φ that generate the symmetry, DEA first tests φ; any original parameters that become dependent after φ are recognized as symmetry‑only and can be eliminated. This yields a minimal circuit that retains only physically relevant degrees of freedom.

A hybrid quantum‑classical implementation is proposed to evaluate the inner products ⟨∂_m C|∂_n C⟩ on quantum hardware. For rotation‑type gates, the derivative state ∂_m C can be expressed as the original circuit with an extra Pauli operator inserted after the m‑th rotation. The overlap ⟨∂_m C|∂_n C⟩ then reduces to measuring ⟨init|γ_m† γ_n|init⟩, which can be obtained using a single ancilla qubit and a controlled‑gate construction. The authors demonstrate this scheme experimentally on IBM Q devices (Ourense and Vigo) for a single‑qubit circuit C(θ)=R_Y(θ₄)R_Z(θ₃)R_X(θ₂)R_Z(θ₁)|0⟩. The measured smallest eigenvalues of S₂, S₃, and S₄ confirm that θ₁, θ₂, θ₃ are independent while θ₄ is redundant, matching theoretical predictions. Shot‑count scaling shows that statistical errors shrink as expected, validating the practical feasibility of DEA on current hardware.

Beyond full‑Hilbert‑space expressivity, the paper discusses physical state‑space restrictions. Many many‑body models possess symmetries or conserved quantities that confine the relevant dynamics to a subspace whose dimension grows only polynomially with system size. By targeting this subspace, one can design PQCs that are maximally expressive within the physically relevant manifold while using far fewer parameters than would be required for full‑space universality. The authors outline an automated construction scheme: compute the dimension d of the target subspace, generate a parametrized ansatz with ≥ d parameters, run DEA, and prune any excess parameters that are identified as dependent. This yields a circuit that is both minimal and sufficient for the problem at hand, enabling on‑the‑fly circuit redesign in each VQS iteration.

In summary, the work provides:

1. A rigorous, rank‑based method (DEA) to quantify circuit expressivity and detect redundant parameters.
2. A protocol for eliminating symmetry‑only degrees of freedom.
3. An efficient hybrid quantum‑classical algorithm to evaluate the necessary Jacobian overlaps on real quantum devices.
4. Experimental validation on IBM quantum processors.
5. A pathway to automated, problem‑specific PQC design that respects physical subspace constraints, promising reduced gate counts and lower noise for VQS and related quantum‑machine‑learning applications.