Probabilistic Representation of Commutative Quantum Circuit Models

In commuting parametric quantum circuits, the Fourier series of the pairwise fidelity can be expressed as the characteristic function of random variables. Furthermore, expressiveness can be cast as the recurrence probability of a random walk on a lattice. This construction has been successfully applied to the group composed only of Pauli-Z rotations, and we generalize this probabilistic strategy to any commuting set of Pauli operators. We utilize an efficient algorithm by van den Berg and Temme (2020) using the tableau representation of Pauli strings to yield a unitary from the Clifford group that, under conjugation, simultaneously diagonalizes our commuting set of Pauli rotations. Furthermore, we fully characterize the underlying distribution of the random walk using stabilizer states and their basis state representations. This would allow us to tractably compute the lattice volume and variance matrix used to express the frame potential. Together, this demonstrates a scalable strategy to calculate the expressiveness of parametric quantum models.

💡 Research Summary

The paper develops a probabilistic framework for quantifying the expressiveness of commuting parametric quantum circuits (PQCs). A PQC is defined as U(θ)=∏_{j=1}^{N}e^{iθ_j H_j}, where each H_j is a Pauli string and all H_j commute. Because of commutation, the pairwise fidelity F_U(θ,θ′)=|⟨0^{⊗n}|U(θ)†U(θ′)|0^{⊗n}⟩|^2 reduces to |f_U(θ−θ′)|^2 with f_U(θ)=⟨0^{⊗n}|U(θ)|0^{⊗n}⟩. By Bochner’s theorem, f_U is the characteristic function of a discrete random variable K.

The authors first show that there exists a Clifford unitary W that simultaneously diagonalizes all H_j, i.e., Λ_j = W H_j W† are diagonal and consist only of Z and identity operators. After conjugation, the circuit becomes U(θ)=e^{i∑θ_j Λ_j} and the reference state becomes |ψ_0⟩ = W|0^{⊗n} , a stabilizer state. The diagonal entries of each Λ_j are ±1 and can be encoded in a binary matrix A∈{0,1}^{N×n} where A_{jm}=1 iff the j‑th Pauli string has a Z on qubit m.

Using the stabilizer tableau formalism, the authors express |ψ_0⟩ as |ψ_0⟩ = 2^{-r/2}∑_{z∈{0,1}^r} i^{c·f(z)+f(z)^T Q f(z)} |f(z)⟩, where f(z)=Rz+t, R∈{0,1}^{n×r} has rank r, and t∈{0,1}^n. Consequently the measurement probabilities P(X=x)=|⟨ψ_0|u_x⟩|^2 are either 0 or 2^{-r}, meaning X is uniformly distributed over a subset of size 2^r of the computational basis. The random variable K can be written as K = (−1)^{A u_X}. Its probability mass function becomes

P(K = (−1)^b) = 2^{-rank(AR)} if b ∈ range(A·f) else 0,

where AR∈{0,1}^{N×r}. This generalizes the earlier result for pure Z‑only circuits (where r=n and AR=A).

The next step connects K to a random walk W_t on ℤ^n with increments Δ_t = K_t − K’_t, where K_t and K’_t are independent copies of K. The frame potential F_U(t) = ∫∫|f_U(θ−θ′)|^{2t} dθ dθ′/(2π)^{2N} equals the probability that the walk returns to the origin after t steps: F_U(t)=P(W_t=0). By the central limit theorem, for large t

F_U(t) ≈ V_U (4π t)^{-n/2} det(Cov(K))^{-1/2},

where V_U is the volume of the lattice generated by the walk and Cov(K) is the covariance matrix computed from the stabilizer expectation values ⟨ψ_0|Λ_iΛ_j|ψ_0⟩−⟨ψ_0|Λ_i|ψ_0⟩⟨ψ_0|Λ_j|ψ_0⟩.

The authors provide an algorithmic pipeline: (1) use the van den Berg‑Temme tableau method to find W; (2) extract A, R, and t from the stabilizer tableau of |ψ_0⟩; (3) compute rank(AR) and the covariance matrix; (4) evaluate the asymptotic frame potential via the CLT formula. All steps run in polynomial time in n, making the approach scalable to many qubits.

A concrete three‑qubit example demonstrates the method, showing excellent agreement between the exact frame potential (computed by brute force) and the asymptotic approximation. The paper also discusses two possible extensions: handling non‑commuting Pauli sets by grouping into commuting subsets, and applying the technique to higher‑order design measures beyond the second moment.

In summary, the work extends the probabilistic representation of pairwise fidelity from Z‑only commuting circuits to arbitrary commuting Pauli sets, leverages stabilizer tableau techniques to characterize the underlying random variable, and translates expressiveness (frame potential) into a tractable lattice‑walk problem. This provides a scalable, theoretically grounded tool for assessing the capacity of variational quantum circuits, with immediate relevance to quantum machine learning architecture design.