Accurate ground state energy estimation with noise and imperfect state preparation

Abstract

We introduce a classical estimator for the post-processing of quantum phase estimation data generated either by quantum-Fourier-transform-based or quantum-signal-processing-based methods. We focus on the estimation of a single target phase promised to be within an interval where no other phases are present, which is typical of e.g. ground state energy estimation of gapped quantum systems. This allows us to perform phase estimation by filtering the signal within the promise region and recovering the phase through a moment-projection estimator. We show that our methods are robust in the presence of both additional phases outside the promise region and global depolarizing noise. In the noiseless case our estimator can achieve an exponential suppression of bias with respect to a naive mean estimator. In the presence of global depolarizing noise our estimator achieves a bias exponentially small in the circuit depth $t$ at fixed circuit fidelity $F$, and a variance proportional to $t^{-2}$, improving by a factor of $t^2$ over the naive shifted-and-rescaled-mean approach. To mitigate realistic circuit-level noise, we combine our method with the explicit unbiasing scheme described in [Dutkiewicz et al., 2025]. As an illustrative example, we implement these estimators on a small-scale simulation of the Ising model, validating our theoretical results and finding better-than-expected performance for a global depolarizing noise approximation. The robustness of the moment-projection estimator in the presence of both multiple eigenvalues and realistic noise makes phase estimation with limited depth practical for early fault tolerant quantum experiments.

Figures (6)

• Figure 1: Schematic plot illustrating the construction of the signal distribution of a filtered QPE experiment. (Top left) A unitary operator $U$ and initial state $|\psi\rangle$ define the spectral distribution $a(x)$ in Def. \ref{['def:spectral_distribution']} -- a sum of Dirac deltas centered at the eigenphases $\phi_j$ of $U$ with amplitudes $a_j$. (Top right) We define a kernel function $f(x)$ (Def. \ref{['def:kernel_function']}), with a shape and width $\sigma$ depending on the details of the circuit. (Bottom left) The QPE circuits can sample from a distribution obtaining by convolution of $a$ and $f$ -- the distribution in Def. \ref{['def:kernel_function']}, shown by the dashed black line. (The colored lines indicate the contributions of each eigenphase $\phi_j$ to the total distribution.) The filtering procedure discards all samples outside of a filtering region $\mathcal{D}$, which is chosen assuming that the phase of interest $\phi_0$ is within the interval (farther than an inner buffer distance $c$ from the interval edges) and all other phases are outside the interval (farther than an outer buffer distance $d$ from the interval edges) -- see Def. \ref{['def:promise_interval']}. The unnormalized distribution of filtered outcomes in the absence of noise is highlighted in blue. (Bottom right) Adding noise to the distribution $f*a$ yields the distribution $p(x)$ (Def. \ref{['def:noisy-distribution']}). Normalizing this distribution within the filtering interval $\mathcal{D}$ we obtain the filtered noisy distribution $P(x)$ (Def. \ref{['def:filtered_distribution']}).
• Figure 2: Schematic representation of the fMPE (Def. \ref{['def:m-projection-phase-estimator-with-gdn']}): among a family of models $Q(x|\phi)$ [Eq. \ref{['eq:qphi_with_noise']}, colored as per the colorbar based on the value of the parameter $\phi$] supported on $\mathcal{D}$, the one that minimizes the inverse KL divergence with $P(x)$ (Def. \ref{['def:filtered_distribution']}) is $Q(x|\phi^*)$. The target distribution $P(x)$ is the final result of the filtered-QPE scheme represented in Fig. \ref{['fig:scheme']}. We also highlight the model distribution for the ideal value of $\phi=\phi_0$: $Q(x|\phi_0)$ -- the difference between $\phi_0$ and the optimal parameter $\phi_*$ is the bias of the fMPE.
• Figure 3: Comparison of the bias of the moment projection and mean estimators for the distribution $P(x)$ in Eq. \ref{['eq:gaussian-prop-dist-no-noise']}, with phases $\phi_0$, $\phi_1$, amplitudes $a_0 = 1-a_1 = 0.7$, filtering region $\mathcal{D} = [-1,1]$, and kernel width $\sigma = 0.3$. (Left) The first order bias $b^{(\text{M-proj})}$ of the moment projection estimator [first term in Eq. \ref{['eq:m-proj-1st-order-bias']}]. (Center) The bias of the mean estimator $b^{(\text{mean})} = \left| \int_{\mathcal{D}}P(x)x\dd x - \phi_0 \right|$. (Right) The ratio of the two biases, $b^{(\text{M-proj})}/b^{(\text{mean})}$. The dashed lines represent $d = |\phi_0 - \phi_{guess}|$ and mark the regime in which the dominant source of the mean estimator's bias is the distance of $\phi_0$ from the center of the interval [Eq. \ref{['eq:estimator-vs-mean-gaussian-no-noise']}].
• Figure 4: Circuits used in this work for example numerical QPE implementation. Top row: (Left) Noiseless circuit (Right) Decomposition of the system unitary $U = \prod_j e^{-i t Z_j Z_{j+1}}\prod_j e^{-i h Z_j}$. Bottom row: (Left) First layers of the noiseless circuit for $n=4$ control qubits; (Centre) insertion of stochastic gates $G_{i,j}$ after each moment; (Right) one explicit noisy circuit realization obtained by sampling Pauli errors.
• Figure 5: Spectral distribution and QPE results on a 4-qubit Ising model, with $n=4$ QPE control qubits. The arrows represent the eigenphases and corresponding amplitudes for the fast-forwarded unit-time evolution unitary of the considered model, i.e. the spectral function $a(x)$. The blue line shows the smoothed spectral function $f*a(x)$, with the kernel function corresponding to the continuous Fejér kernel expected from textbook QPE: $f(x) = f^{\text{(Fejer)}}K(x)$ from Eq. \ref{['eq:continuous-fejer-kernel']} with $K=2^{n}$. The ochre histogram shows the samples obtained from simulating $100k$ shots of textbook QPE with circuit local depolarizing noise, with error probabilities tuned such that the total circuit fidelity is $F=e^{-1}$ [i.e. samples from the noisy QPE distribution $p(x)$, Eq. \ref{['eq:numerics_pec_decomposition']}]. The maroon histogram represents the subset of samples coming from simulations where at least one noise event happened [i.e. samples from the pure noise distribution, $p_1(x)$, Eq. \ref{['eq:numerics_pec_decomposition']}], which we use to simulate a simplified version of PEC/Explicit Unbiasing. The un-shaded area represents the filtering interval $\mathcal{D}$.

Theorems & Definitions (33)

• Definition 1: Def. \ref{['def:classical-subroutine']}, informal
• Lemma 2: Lemma \ref{['lem:m-projection-expansion']}, informal
• Theorem 3: Theorem \ref{['thm:nme-expansion']}, informal
• Theorem 4: Theorem \ref{['thm:fmpe-cost-nonoise']}
• Theorem 5: Theorem \ref{['thm:fmpe-cost-gdn']}, informal
• Definition 6: Spectral distribution
• Definition 7: Kernel function
• Definition 8: Gaussian kernel function
• Definition 9: Noisy distribution
• Definition 10: Promise interval