Classically estimating observables of noiseless quantum circuits

TL;DR

The paper presents a classical algorithm based on Pauli propagation to estimate expectation values for observables on random, unstructured quantum circuits, applicable across architectures and depths. By truncating to low-weight Pauli terms and leveraging the orthogonality and Pauli-mixing properties of locally scrambling layers, it proves mean-squared-error bounds with polynomial-time complexity in the number of qubits and circuit depth for fixed accuracy, and quasi-polynomial time for inverse-polynomial accuracy. The results imply that observables of circuits with chaotic and locally scrambling dynamics are classically tractable on average, while numerical experiments extend the reach beyond the proven assumptions and hint at broader applicability to real-time dynamics and ground-state search. The work also provides certified error estimates and discusses practical implications for variational quantum algorithms and potential quantum-classical hybrid strategies.

Abstract

We present a classical algorithm based on Pauli propagation for estimating expectation values of arbitrary observables on random unstructured quantum circuits across all circuit architectures and depths, including those with all-to-all connectivity. We prove that for any architecture where each circuit layer is randomly sampled from a distribution invariant under single-qubit rotations, our algorithm achieves a small error $\varepsilon$ on all circuits except for a small fraction $δ$. The computational time is polynomial in qubit count and circuit depth for any small constant $\varepsilon, δ$, and quasi-polynomial for inverse-polynomially small $\varepsilon, δ$. Our results show that estimating observables of quantum circuits exhibiting chaotic and locally scrambling behavior is classically tractable across all geometries. We further conduct numerical experiments beyond our average-case assumptions, demonstrating the potential utility of Pauli propagation methods for simulating real-time dynamics and finding low-energy states of physical Hamiltonians.

Figures (5)

• Figure 1: Schematic depiction of Pauli propagation equipped with weight truncation. Pauli operators generally split into a weighted sum of several Pauli operators when acted upon by non-Clifford operations and to higher weight. We sketch the truncation of Pauli operators above a threshold of $k=2$.
• Figure 2: Classical simulation of a local Pauli expectation value with 64 qubits on a $8 \times 8$ grid. The quantum circuit ansatz consists of randomly sampled SU(4) gates in a 2D staircase topology zhang2023absence. After one circuit repetition, the back-propagated observable contains fully global Pauli operators, which is pathological for approaches relying solely on small entanglement light cones. a) Average simulation error as a function of quantum circuit depth for different operator weight truncations. This error is numerically estimated using the Monte Carlo sampling approach. We compare against the general bound in Theorem \ref{['thm:errorbound']}. As the parameterised expectation value exponentially concentrates with more circuit repetitions, the variance of the expectation value $\mathrm{Var}[f]$ decays exponentially. This variance (gray line) is exactly the MSE achieved by the trivial estimator (Supplemental Section \ref{['app:XQUATH']}), and therefore constitutes a baseline for quantifying the performance of our algorithm. The average simulation errors (blue lines) also drop exponentially and becomes more accurate as $k$ increases. These errors are always better than the trivial estimator but the relative improvement reduces as circuit depth increases. b) Simulation time of one expectation value using low-weight Pauli propagation. For example, three circuit repetitions can be simulated on a single CPU thread of an i7-12850HX processor on a laptop to below $10^{-4}$ MSE in approximately $10$ seconds. c) Weight distribution of Pauli operators for up to four circuit repetitions. The inset shows the expected contribution of all operators per weight over the landscape. We observe an exponentially decaying contribution of high-weight Pauli operators.
• Figure 3: Supplemental Figure 3: Numerical verification of the effectiveness of weight truncation for a 127-qubit quantum circuit that does not strictly comply with our assumptions. We employ an ansatz consisting of repeated RX and RZZ rotations equivalent to a Trotter time evolution circuit of the transverse field Ising Hamiltonian. Thus, one layer of this circuit is not locally scrambling. The entangling topology is chosen to be the heavy-hex lattice, and the measurement is $\sigma^z_{63}$ in the middle of the lattice.
• Figure 4: Supplemental Figure 4: Numerical verification of the effectiveness of weight truncation for correlated angles on 16 qubits. The circuit ansatz consists of repeated RX and RZ rotations on each qubit followed by RZZ gates in a staircase ordering. The observable is a local Pauli Z operator on the first qubit. We either draw a) random parameters for all gates or b) one random parameter for all gates. We also report the variance of the un-truncated loss function $\mathrm{Var}[f]$, which indicates the presence and absence of exponential concentration in the case of uncorrelated and correlated parameters respectively.
• Figure 5: Supplemental Figure 5: Variational ground state optimization with low-weight Pauli propagation. We consider a Heisenberg Hamiltonian on a $4 \times 4$ grid, with all Hamiltonian coefficients equal to 1. As an ansatz, we choose three layers of the 2D staircase SU(4) unitary circuit discussed in the main text. a) Exact statevector optimization with Pauli propagation evaluation of the circuits throughout optimization with varying Pauli weight truncation. b) Low-weight Pauli propagation optimization with exact statevector evaluation of the circuits throughout optimization.

Theorems & Definitions (69)

• Theorem 1: Mean squared error
• Theorem 2: Time complexity
• Theorem 3: Certified error estimate
• Definition 4: Locally scrambling distribution
• Definition 5: Locally scrambling circuit
• Lemma 6
• proof
• Lemma 7: Orthogonality and Pauli-mixing
• proof
• Lemma 8: Vanishing cross-terms