Noise mitigation of quantum observables via learning from Hamiltonian symmetry decays

Abstract

We present a new quantum error mitigation technique (QEM), called GUiding Extrapolations from Symmetry decayS (GUESS), which exploits Hamiltonian symmetries to improve accuracy of noisy quantum computations. This method is explicitly designed for quantum algorithms that estimate expectation values of observables and consists in learning the extrapolation coefficients from a symmetry observable of the system to then estimate the value of a target observable. Furthermore, we propose a Hamiltonian impurity technique to enforce symmetries allowing the mitigation of local observables of interest. We employ the IBM Heron r2 quantum processing unit '\texttt{ibm\_basquecountry}' to simulate the time evolution of average magnetization and nearest-neighbor correlator observables for transverse field Ising and $XZ$ Heisenberg models in 1D with open boundary conditions. We benchmark the accuracy of our method against baseline Zero Noise Extrapolation (ZNE) and tensor network simulations for systems of $100$ qubits. Remarkably, GUESS achieves a relative error around $10\%$ for circuits containing up to $8000$ CZ gates, while showcasing lower variance than ZNE on average across $20$ observables and requiring only twice the number of shots per observable compared to baseline ZNE. Furthermore, we demonstrate that GUESS enables statistical post-selection based on the outcomes of the symmetry observable, which provides critical information about the quality of the target qubits by means of its mean and variance. These results indicate that GUESS is a powerful QEM technique capable of mitigating utility-scale circuit outcomes, delivering high accuracy and reduced variance for large-scale circuits with minimal quantum overhead.

Figures (8)

• Figure 1: Schematic representation of the GUESS method. The top left panel shows a representative Trotter circuit used to estimate the time evolved expectation value of the target observable $Z_{85}$ at qubit $85$ for different noise factors $(1, 1.2, 1.5)$. The bottom left panel illustrates the same circuit with a local impurity added, which generates a symmetry observable at the same qubit, $S_{85}$. In STEP 1, we measure expectation values of the symmetry observable at each noise factor and we use them together with the known ideal value $\langle S_{85} \rangle_{ideal} = 1$ to learn the GUESS coefficients $\{x_i\}_{i=1}^{n}$. In STEP 2, these coefficients are applied to the measured expectation values of $\langle Z_{85} \rangle$ at the same noise factors to estimate its noise‑free value. The GUESS estimate is compared to the ideal (target) result, which here is obtained with tensor networks, and to a standard ZNE extrapolation. We used the IBM heron r2 'ibm_basquecountry' for simulating the dynamics of an transverse field Ising model at Trotter step $20$.
• Figure 2: Simulation of the transverse field Ising chain with open boundaries, up to $44$ trotter steps and measuring each $4$ steps. The maximum two-qubit depth of the reported circuits is $176$, with a total number of $8712$ CZ gates for the baseline case without noise amplification $G=1$. We show the result for the best $20$ observables based on our statistical postprocessing method (see Appendix \ref{['Sec: Postpro']}). In panels (a) and (b) we show the dynamics of the average magnetization (weight-$1$ observables) and in panels (c) and (d) of the average nearest-neighbor correlators (weight-$2$ observables); using the exponential and the linear model , respectively. The blue points (cross) refer to the ideal values, obtained from the MPS simulation using a time evolving block decimation (TEBD) algorithm with bond dimension of $150$ and $dt=t/\#\textrm{Trotter}$; the red (square) points to the extrapolated values using GUESS; and the dark (triangle) and light (circle) to ZNE and the raw machine measurements for $100$k shots, respectively.
• Figure 3: Simulation of transverse field $XZ$ Heisenberg chain with open boundaries, up to $24$ trotter steps measuring each $4$ steps. The maximum two-qubit depth of the reported circuits is $192$, with a total number of $9504$ CZ gates for the baseline case without noise amplification $G=1$. We show the result for the best $20$ observables based on our statistical postprocessing method (see Appendix \ref{['Sec: Postpro']}). We show the averaged magnetization for (a) the exponential and (b) the linear models, where the blue points (cross) refer to the ideal values, obtained from the MPS simulation using a TEBD algorithm with bond dimension of $150$; the red (square) points to the extrapolated values using GUESS; and the dark (triangle) and light (circle) to ZNE and the raw machine measurements for $100$k shots, respectively.
• Figure 4: Selected qubit layouts and harwdware status of the 'ibm_basquecountry' at the times of circuit runs. (a) Transverse field Ising model experiments for magnetization observables, (b) Transverse field Ising model experiments for correlator observables, (c) Transverse field $XZ$ Heisenberg model experiments for magnetization observables.
• Figure 5: Relative error of (a) the average magnetization and the (b) average nearest-neighbor correlators over 20 observables for the Ising model; and of (c) the average magnetization of the $XZ$ Heisenberg model over 20 observables.