Quantum computing · Theory
Logarithmic Bias of Zero-Noise Extrapolation under 1/f Noise
arXiv:2611.02444v1 ↗PDF ↗Submitted 2 November 2026
- Question
- Zero-noise extrapolation amplifies noise, for example by folding a circuit into \(U(U^\dagger U)^n\), and fits the measured values to extrapolate to zero noise. The fit assumes a functional form, usually a polynomial or an exponential. Low-frequency noise with a \(1/f\) spectrum is common in solid-state qubits, and folding samples it at longer and longer delays. What response does that produce, and what does it do to the fit?
- Result
- At second order in the coloured noise, the folded response contains a term \(\theta r^2\ln r\), with \(\theta\) set by the noise amplitude and the circuit's forward and inverse response. Any minimal polynomial Richardson rule leaves a bias \(\theta M\) with \(M>0\), equal to \(2.73\theta\) for the nodes 1, 3, 5. Adding \(r^2\ln r\) to the fitting basis removes this bias at that order, without knowing \(\theta\).
The paper in detail
The circuit is folded into \(r=2n+1\) blocks \(F,I,F,\dots,F\), each of duration \(T\), and the coloured noise continues across block boundaries. The noise enters as \(\lambda\sum_\alpha\xi_\alpha(t)B_\alpha(t)\) with stationary covariance \(\mathsf A_{\alpha\beta}C(t-s)\). At second order, the change in the observable is an ordered double integral of a circuit kernel against \(C\). Markovian noise contributes a term affine in \(r\), because there are \((r+1)/2\) forward and \((r-1)/2\) inverse blocks.
Counting block pairs exactly, the second difference \(E(r+2)-2E(r)+E(r-2)\) removes the affine Markovian part. What remains is the covariance integrated over delays near \(rT\), weighted by a fixed kernel. So curvature in \(r\) measures memory: white noise gives none, frozen noise a constant, exponential memory a decaying contribution, and logarithmic memory a logarithmic one.
For a two-sided spectrum \(A/|\omega|\) between \(\omega_{\mathrm{IR}}\) and \(\omega_{\mathrm{UV}}\), the covariance is \(C(\tau)=[\mathrm{Ci}(\omega_{\mathrm{UV}}|\tau|)-\mathrm{Ci}(\omega_{\mathrm{IR}}|\tau|)]/\pi\), logarithmic at intermediate delays. Inserting it gives \[E(r)=E^*+\delta+a_1r+a_2r^2+\theta r^2\ln r+R(r),\] where \(\theta\) is \(\lambda^2/2\pi\) times the observable's double-commutator response to the average of the integrated forward and inverse couplings. Here \(\delta\) is a native offset and \(R\) has explicitly bounded parts from control shape, band edges and higher orders.
The response-matched estimator uses intercept weights on the design \((1,r,r^2,r^2\ln r)\), on at least four odd nodes. The log-domain version, which the paper calls PE_log, applies them to \(\ln y_k\). The linear version carries an exact certificate: its bias is \(\delta+\sum_kw_kR(r_k)\). In a ten-qubit Clifford kicked-Ising model, the exponential bias across 106 stabiliser predictions was almost proportional to the total \(\theta\) (\(R^2=0.997\)), and PE_log suppressed the trend. The paper presents these as numerical illustrations, not as a guarantee for every circuit.
Transferable methods
Second differences isolate memory
Time-local noise adds a little error per block, which is affine in the number of blocks. Taking the second difference removes it, leaving only correlations between blocks at a distance, which is exactly what memory provides.
If each block contributes independently, \(E(r)\) is affine in \(r\) and \(\Delta_2^2E=0\). Cross-block terms involve \(C(dT+u-v)\) for blocks separated by \(d\). The second difference keeps only the pairs gained or lost at the largest separations, which gives Eq. (2): an integral of a fixed kernel against \(C(rT+u-v)\) for \(u,v\in[0,2T]\).
With \(C(\tau)\approx[\ell_T-\ln(\tau/T)]/\pi\), write \(\ln(rT+u-v)=\ln(rT)+\ln(1+(u-v)/rT)\). For \(r\ge3\) the ratio is at most \(2/3\), so the expansion converges, and the second difference takes the form \(d_0+d_L\ln r+\sum_pd_p/r^p\). Summing it twice gives \(r^2\ln r\) as the leading new term, since \[\Delta_2^2(r^2\ln r)=8\ln r+12+O(r^{-2}),\] so its coefficient is \(d_L/8\).
Editorial check: the second difference of \(r^2\ln r\) is \(24.77\) at \(r=5\) and \(54.4264\) at \(r=201\), against \(24.88\) and \(54.4264\) from \(8\ln r+12\), so the error falls as \(r^{-2}\). Numerical integration of the band-limited \(1/|\omega|\) spectrum, with \(\omega_{\mathrm{IR}}=10^{-3}\) and \(\omega_{\mathrm{UV}}=10^3\), matched \((-\gamma_E-\ln\omega_{\mathrm{IR}}\tau)/\pi\) to \(2\times10^{-6}\) at \(\tau=10\) and to \(6\times10^{-4}\) at \(\tau=0.1\), where the ultraviolet edge begins to matter.
How to reuse it
For any amplification protocol that repeats blocks, the second difference in the repetition count is a model-free probe of noise memory. Measured curvature that does not fit white or frozen noise indicates which extra coordinate a fit needs. The probe needs repeatable block implementations and enough precision to resolve a curvature signal.
Moment conditions for unbiased extrapolation
A linear extrapolation rule removes every response term whose node moments vanish. Richardson cancels powers of \(r\) but not \(r^2\ln r\), and the leftover moment can be computed exactly.
Degree-\(p\) Richardson weights \(g_k\) satisfy \(\sum_kg_k=1\) and \(\sum_kg_kr_k^m=0\) for \(1\le m\le p\). Inserting Eq. (3) leaves the bias \(\delta+\theta M_{2\log}+\sum_kg_kR(r_k)\), with \[M_{2\log}=\sum_kg_kr_k^2\ln r_k=\int_0^\infty\frac{t\prod_jr_j}{\prod_j(t+r_j)}\,dt>0.\] The integral form shows that the sign is fixed for every minimal rule with \(p\ge2\), so more shots at the same nodes cannot remove this term.
On nodes 1, 3, 5 the weights are \((15/8,-5/4,3/8)\), and \(M=\tfrac{75}8\ln5-\tfrac{45}4\ln3\approx2.729\). The response-matched rule instead takes \(w^{\mathsf T}=e_1^{\mathsf T}X^+\) for rows \((1,r_k,r_k^2,r_k^2\ln r_k)\), so all four moments are cancelled by construction and \(\theta\) does not need to be known.
Editorial check: we confirmed the weights and \(M=2.72909\), and the integral form gave \(2.72908\). On a toy response \(1+0.01r-0.002r^2+0.004r^2\ln r\), Richardson at 1, 3, 5 is off by \(0.010916=\theta M\) exactly, while the four-node rule at 1, 3, 5, 7 recovers the intercept to \(10^{-15}\). Its weights are \((2.509,-3.152,2.277,-0.634)\), with \(\|w\|_1=8.57\), about 2.4 times the Richardson value. That ratio is roughly the price in statistical amplification. For the quadratic least-squares rule on 1, 3, 5, 7 used in the paper's bias-improvement condition, \(\beta=4.18\) and \(\|q\|_1=2.75\).
How to reuse it
Before choosing an extrapolation form, derive the response's coordinates from the noise model and check which moments a candidate rule cancels. The leftover moment times the coefficient is the bias. Adding coordinates costs variance, so compare \(\|w\|\) against the bias removed for the expected noise level.
Context & limitations
Why it matters. It identifies a specific missing extrapolation coordinate and derives it from the physics, rather than from goodness of fit, and it gives a forward prediction that can be tested by varying the noise. In the paper's simulations the corrected estimator cuts single-qubit bias from 5.55% (exponential) and 1.71% (Richardson) to −0.26%.
Limitations. All results are model predictions; the authors say hardware experiments are under way. The response is second order in the coloured noise and assumes Gaussian, stationary noise with a logarithmic covariance over the sampled delays, so strong noise, non-Gaussian fluctuators or short memory windows can break it. Fitting an extra coordinate amplifies statistical error: on nodes 1, 3, 5, 7 the weights have \(\|w\|_1=8.57\), against 3.5 for Richardson on 1, 3, 5. Native offsets that are not amplified are not removed.