Quantum Enhanced Pauli Propagation

Abstract

Accurately estimating observables on noisy quantum devices remains a central challenge for near-term quantum algorithms. While quantum error mitigation techniques can reduce noise-induced bias, they often rely on unverifiable assumptions about the circuit noise, and cannot guarantee the magnitude of residual bias error. Here, rather than using classical resources to mitigate a noisy quantum circuit execution, we propose a hybrid algorithm that uses quantum resources to improve the accuracy of approximate classical Pauli-path simulation. Our protocol, Quantum Enhanced Pauli Propagation (QuEPP), uses Clifford perturbation theory (CPT) to construct a classically simulable ensemble of Clifford circuits from the low-order terms in CPT, which directly provide the approximate classical Pauli-path simulation of the target circuit. Noisy quantum expectation values of this ensemble are then used to infer a global rescaling factor that corrects quantum execution of the target circuit, providing higher-order contributions absent from the truncated low-order classical simulation. This approach requires no noise characterization, applies to arbitrary circuits, and provides a provable route to asymptotically unbiased estimates. Using IBM Heron hardware, we demonstrate QuEPP on 2D random mirror circuits of up to 49 qubits and circuit depth 80, as well as Trotterized Hamiltonian evolution, showing consistent improvements beyond classical CPT and unmitigated quantum results. QuEPP offers a simple, scalable, and model-free framework for enabling accurate quantum computation in the pre-fault-tolerant era.

Figures (5)

• Figure 1: Quantum Enhanced Pauli Path Simulation with QuEPP. a) The input to QuEPP is a quantum circuit and observable. The circuit consists of Clifford gates (blue), Pauli rotations (purple), and measurement. (b) Clifford Perturbation Theory (CPT) can be used to write the target expectation value as a sum over expectation values of many Clifford circuits. (c) This sum can be represented by Pauli paths describing how the observable is propagated through the circuit in a tree like graph. Here, nodes are Cliffords or Pauli rotations that compose the evolution of the observable, and edges indicate propagation of the observable along Pauli paths; the diagram shows path structure, not literal circuit edges. Clifford operations map a Pauli to a different Pauli, whereas Pauli rotations can split the Pauli into two different paths. (d) CPT truncates some of these paths based on the order to remain computationally feasible. Each path maps to a single Clifford circuit. (e) Running the target circuit directly gives us a noisy estimate of all the paths combined, and running the CPT ensemble Clifford circuits of the lower order terms gives the noisy estimate of the lower order paths. Different shades of red colors are used to denote the noisy paths. (f) Combining exact CPT estimate of lower order terms with rescaled estimate of the higher order terms, we get a better estimate of the expectation value than (d) or (e) alone.
• Figure 2: Mode of Operation for QuEPP This figure complements Fig. \ref{['fig1:Pauli Path']} by showing the operation workflow rather than the underlying Pauli-path structure. The user begins by specifying both a target quantum circuit and an observable on a classical computer. This circuit is sent to both a quantum computer for execution and an HPC cluster for further processing. The quantum computer collects shots for this target circuit to estimate the noisy expectation value of the observable. Meanwhile, the HPC cluster performs CPT up to a fixed order and collects Clifford circuits to form the CPT ensemble. Each of these Clifford circuits that has a non-zero overlap with the intended initial state is sent to be also run on the quantum computer. The noisy and exact expectation values of the Clifford circuits are used to estimate the impact of noise with QuEPP, and this correction is used to result in a mitigated expectation value of the observable with respect to the original (target) circuit.
• Figure 3: Unstructured mirror circuit a) The target circuit consists of preparing the all zero state on 49 qubits and 16 layers of a forward propagating random 2D circuit followed by the mirror inverse circuit. We are interested in measuring the expectation value of the observable $\langle Z_{0}Z_{11}Z_{18}Z_{41}Z_{48} \rangle$. b) Layers in the circuit consists of single qubit and two qubit Clifford gates with some non-Clifford Pauli RX rotations. c) Bias (as calculated by difference between ideal and computed expectation value) is plotted as a function of CPT order. We are comparing the purely classical method with CPT (in Purple) and QuEPP (in Pink). d) The circuit is mapped onto a heavy-hex topology while measured qubits are marked with yellow circles.
• Figure 4: Unstructured mirror circuit The target circuit consists of preparing all zero state on 40 qubits and 30 layers of a forward propagating random 1D circuit followed by the mirror inverse circuit. We are interested in measuring the expectation value of the observable $\langle Z_{1}Z_{4}Z_{12}Z_{13}Z_{15}Z_{21}Z_{26}Z_{27}Z_{28}Z_{32}Z_{33}Z_{34}Z_{37}Z_{39} \rangle$. a) Expectation value as a function of number of terms kept using CPT. We perform CPT until $10^{7}$ terms and then extrapolate to the ideal expectation values. We expect CPT will require $10^{12}$ terms to converge. b) Expectation value as a function of number of terms for QuEPP with Monte Carlo sampling method converges to the $0.99\pm 0.07$ within 293 terms. The inset shows the decay of the standard error of the mean of the estimator as a function of number of terms kept. We also plot the unmitigated expectation value as a red dotted line.
• Figure 5: Trotterized Hamiltonian Time Evolution a) One layer of the 10 qubit Trotterized circuit is shown. The circuit is parameterized by RX rotations between the entangling layers. The circuit being run repeats this layer 10 times. All qubits are initialized in $|+\rangle$ and finally we measure $\langle X^{\otimes10} \rangle$. b) Number of Clifford paths as a function of CPT order (plotted in purple) and number of these circuits being run on a quantum computer (in Pink). c) $\langle X^{\otimes10}$ plotted as a function of the RX rotation angle for ideal (simulated using state vector simulator - in green), CPT (truncated at order 3 - in dotted Purple) and QuEPP (with CPT truncated at order 3 - in pink). d) Estimated difference between exact expectation value and the ones predicted by CPT/QuEPP plotted as a function of the rotation angle. e) Zoomed in view of (c) focusing on the region where CPT alone struggles to predict the expectation value but QuEPP can recover the signal well.