Probabilistic Representation of Commutative Quantum Circuit Models

1. Problem Statement

The paper addresses the challenge of quantifying the expressiveness (or capacity) of Parametric Quantum Circuits (PQCs), which are central to Quantum Machine Learning (QML). Expressiveness refers to the range of quantum states or unitaries a circuit can generate. A key metric for this is the Frame Potential ($F_U(t)$), which measures the average similarity between pairs of states produced by the circuit.

• The Bottleneck: Calculating the frame potential is generally intractable for large systems due to the exponential scaling of the Hilbert space ($2^n$).
• The Gap: Previous work (specifically reference [8]) established a probabilistic representation for the frame potential, but only for circuits where the Hamiltonian generators ($H_j$) were restricted to the Cartan subalgebra of Pauli-Z operators (diagonal matrices). This paper aims to generalize this strategy to any commuting set of Pauli operators, which may include $X$ and $Y$ terms, not just $Z$.

2. Methodology

The authors propose a scalable, algorithmic strategy to compute the asymptotic behavior of the frame potential for any commuting set of Pauli operators. The methodology proceeds in three main stages:

A. Simultaneous Diagonalization via Clifford Unitaries

Since the set of Pauli operators $\{H_j\}$ commutes, they are simultaneously diagonalizable.

• The authors utilize a unitary operator $W$ from the Clifford group such that $W H_j W^\dagger = \Lambda_j$, where $\Lambda_j$ are diagonal matrices containing only $I$ and $Z$ operators.
• This transformation maps the original circuit $U(\theta)$ to a diagonal form, allowing the fidelity function $f_U(\theta)$ to be expressed as an expectation over a transformed initial state $|\psi_0\rangle = W|0\rangle^{\otimes n}$.

B. Probabilistic Representation as a Random Walk

The core theoretical insight is interpreting the quantum expectation $f_U(\theta)$ as the characteristic function of a discrete random variable $K$.

• Random Variable Construction: The diagonal entries of the transformed Hamiltonians define a set of vectors $k_x$. The random variable $K$ takes values from these vectors based on the probability distribution defined by the amplitudes of the initial state $|\psi_0\rangle$.
• Frame Potential as Recurrence Probability: The frame potential $F_U(t)$ is shown to be equivalent to the probability that a random walk on a lattice (with increments drawn from the distribution of $K$) returns to the origin at step $t$.
• Asymptotic Approximation: Using the Central Limit Theorem, the frame potential is approximated as:
  $$\tilde{F}_U(t) \approx \frac{V_U}{\sqrt{(4\pi t)^N \det(\text{Cov}(K))}}$$
  where $V_U$ is the volume of the lattice generated by the random walk, and $\text{Cov}(K)$ is the covariance matrix of the random variable.

C. Characterizing the Distribution via Stabilizer States

To compute the distribution of $K$ efficiently, the authors leverage the fact that $|\psi_0\rangle$ is a stabilizer state (since $W$ is a Clifford operator).

• Tableau Formalism: They use the tableau representation of stabilizer states to characterize the support of the state $|\psi_0\rangle$.
• Algorithmic Derivation:
  1. They define a binary matrix $A$ encoding the Pauli strings of the diagonalized Hamiltonians.
  2. They derive that the random variable $K$ takes values in a subset of $\{-1, 1\}^N$ determined by the linear map $A$ acting on the support of the stabilizer state.
  3. Using the rank of the tableau's $X$-matrix, they determine the dimension $r$ of the subspace where the state has non-zero amplitude.
  4. The probability mass function is shown to be uniform over a specific set of size $2^{\text{rank}(AR)}$, where $R$ is derived from the stabilizer tableau.

3. Key Contributions

1. Generalization of Probabilistic Representation: The paper extends the probabilistic framework from [8] from Pauli-Z only circuits to any commuting set of Pauli operators. This is achieved by utilizing the Clifford group to diagonalize arbitrary commuting Pauli strings.
2. Stabilizer State Characterization: The authors fully characterize the underlying distribution of the random walk using stabilizer state theory. They provide an explicit formula (Proposition 1) for the probability mass function of the random variable $K$ based on the rank of the matrix product $AR$ (where $A$ encodes the Hamiltonians and $R$ encodes the stabilizer state support).
3. Scalable Algorithm: The paper presents a polynomial-time algorithm (using tableau simulation) to compute the necessary parameters (lattice volume $V_U$ and covariance matrix) for the frame potential approximation, bypassing the exponential cost of direct integration.
4. Extension to Non-Commutative Cases (Future Work): The authors discuss a potential pathway to extend these methods to non-commuting sets by decomposing the fidelity function into even and odd components and utilizing higher-order Suzuki-Trotter decompositions.

4. Results

• Theoretical Validation: The paper proves that for any commuting set of Pauli operators, the frame potential can be approximated by the recurrence probability of a random walk on a lattice defined by the stabilizer structure of the circuit.
• Example Application: The authors demonstrate the method on a 5-qubit circuit with a specific set of commuting Pauli operators (including $X$ and $Y$ terms).
  • They successfully computed the diagonalizing unitary $W$, the matrix $A$, and the stabilizer parameters ($R$ and $t$).
  • They calculated the lattice volume ($V_U = 64$) and the covariance matrix (Identity), yielding an approximate frame potential of $\tilde{F}_U(t) = 2(\pi t)^{-5/2}$.
  • Comparative Analysis: By comparing this to a different circuit configuration with full-rank $R$ and minimum volume ($V_U=32$), they demonstrated that the frame potential is halved, indicating twice the expressiveness. This confirms the utility of the metric in distinguishing circuit architectures.

5. Significance

• Efficient Architecture Search: This work provides a tractable tool for evaluating the expressiveness of PQC architectures without simulating the full quantum state space. This is crucial for designing better quantum machine learning models.
• Bridging Quantum and Probability: It deepens the theoretical link between quantum circuit properties (expressiveness) and classical probabilistic concepts (random walks, characteristic functions, and lattice volumes).
• Scalability: By reducing the problem to linear algebra operations on binary matrices (tableaus) rather than exponential state vectors, the method scales efficiently with the number of qubits, provided the circuit consists of commuting Pauli rotations.
• Foundation for Future Research: The paper lays the groundwork for analyzing non-commuting circuits and exploring the Fourier series expansions of quantum circuits, which could reveal periodic behaviors relevant to QML training dynamics.