Encoded Quantum Signal Processing for Heisenberg-Limited Metrology

Abstract

Entangled quantum probes can achieve Heisenberg-limited measurement precision, but this advantage is typically destroyed by noise. We address this issue by introducing a framework that we call encoded quantum signal processing, which unifies quantum error detection and quantum signal processing into an effective single-qubit framework, and provides a paradigm for constructing logical sensors that are robust to noise while remaining sensitive to the signal of interest. We show that encoding sensor qubits into a repetition code and using syndrome measurements as a signal-processing primitive restores Heisenberg scaling under realistic noise, without applying recovery operations. We prove that product-state sensing with syndrome post-processing is fundamentally limited to standard quantum limit (SQL) scaling, and develop four protocols that overcome this barrier through entanglement or sequential signal amplification, achieving Heisenberg-limited precision with exponential error suppression in code distance. For spatially inhomogeneous fields, Bayesian marginalization preserves Heisenberg scaling provided noise decreases sufficiently with system size. The underlying mechanism, which we formalize as encoded quantum signal processing, reduces multi-qubit metrology to an effective single-qubit problem where syndrome measurement implements nonlinear signal transformations. Numerical simulations validate the theoretical predictions: syndrome-based inference achieves near-Heisenberg scaling at noise levels where bare probes approach the SQL, and a concatenated protocol maintains this scaling under joint transverse noise and longitudinal inhomogeneities.

Figures (5)

• Figure 1: Plot of $\arctan\left( \frac{x^{2L+1}}{(1-x^2)^{(2L+1)/2}}\right)$ for $L=1$ ($N=3$, left) and $L=2$ ($N=5$, right). The resultant phase angle approaches a sigmoid function very rapidly and exactly upon success.
• Figure 2: Schematic of Encoded QSP with the repetition code. $N$ physical qubits, each subject to the signal unitary $U_s$, are encoded in a repetition code. Syndrome measurement extracts the error count $j$ and projects the multi-qubit state onto an effective single-logical-qubit rotation $\Theta_j = \arctan(\tan^{N-2j}(\phi))$ (Theorem \ref{['thm:syndrome-rotation']}). The syndrome outcome conditions the classical post-processing, replacing the explicit processing rotations of standard QSP.
• Figure 3: Error-detected protocol achieves near-Heisenberg scaling where bare GHZ degrades toward the SQL. Resource scaling $T$ (total qubits consumed) vs. target precision $\epsilon$ for bare GHZ (solid) at depolarizing noise rates $\gamma = 1\%$ and $10\%$, alongside error-detected post-selection (dashed, $L{=}1$, $N{=}3$ qubits) at $\gamma = 10\%$. The scaling exponent $\alpha$ is defined by $T \propto \epsilon^{-\alpha}$, where $\alpha = 1$ is the Heisenberg limit and $\alpha = 2$ is the SQL. At $\gamma = 10\%$, bare GHZ scaling approaches the SQL ($\alpha = 1.95$), while post-selection on the trivial syndrome reduces the exponent to $\alpha = 1.06$, recovering near-Heisenberg scaling. At $\gamma = 1\%$, bare GHZ achieves $\alpha = 1.66$; all noise levels tested show similar improvement under error detection (Table \ref{['tab:error-detected-results']}). Lines show mean power-law fits; shaded bands indicate 95% confidence intervals on the scaling exponent $\alpha$ (slope uncertainty only) over 40 independent seeds.
• Figure 4: (a) Scaling exponent $\alpha$ (where $T \propto \epsilon^{-\alpha}$) and (b) convergence rate (fraction of precision targets reaching convergence within the experiment budget) across protocols and inference modes. Bar color encodes noise rate ($\gamma = 1\%, 5\%, 10\%$). Solid bars denote post-selection (retaining only error-free rounds); hatched bars denote full-likelihood (using all rounds with syndrome-adjusted likelihoods). Each group shows bare GHZ (no error detection, single bar per noise level) or the error-detected protocol at code distances $L = 1, 2, 3$. Both inference modes achieve near-Heisenberg scaling at all tested noise levels. Full-likelihood consistently improves both scaling exponents and convergence rates (e.g., $\alpha = 1.04$ at 100% convergence vs. $\alpha = 1.06$ at 90% for post-selection at $L{=}1$, $\gamma{=}10\%$) by utilizing all measurement data including error-flagged rounds. Fitted exponents slightly below the Heisenberg limit ($\alpha < 1$) for some full-likelihood configurations are finite-range OLS fitting artifacts, not violations of the theoretical bound. Error bars represent standard error of the mean over 40 seeds.
• Figure 5: Combined protocol under joint transverse noise ($\gamma$) and longitudinal inhomogeneities ($\sigma_\epsilon = 0.01$). Resource scaling $T$ (total qubits consumed) vs. target precision $\epsilon$ for the concatenated code, which distributes a logical GHZ state across three $[[2L{+}1, 1]]$ repetition code blocks to simultaneously suppress both noise types. Results shown for $L = 1$ ($N_{\mathrm{total}} = 9$ qubits, solid) at $\gamma = 1\%$ and $10\%$, and $L = 2$ ($N_{\mathrm{total}} = 15$, dashed) at $\gamma = 10\%$. Post-selection retains only rounds where no error is detected in any of the three blocks. All configurations maintain near-Heisenberg scaling ($\alpha \leq 1.19$, Table \ref{['tab:combined-results']}) even under joint noise, with the $L = 2$ code adding modest resource overhead at the same scaling exponent. Lines show mean power-law fits; shaded bands indicate 95% confidence intervals on the scaling exponent $\alpha$ (slope uncertainty only) over 40 independent seeds.

Theorems & Definitions (88)

• Definition 1: Encoded Quantum Signal Processing
• Proposition 2
• proof
• Proposition 3
• proof
• Corollary 4
• proof
• Lemma 5: Hamiltonian to Signal Rotation Mapping
• proof
• Theorem 6: Syndrome-Dependent Logical Rotation