⚛️ quantum physics

Zero noise extrapolation of transport coefficients

This paper demonstrates that standard zero-noise extrapolation fails for transport coefficients in noisy quantum systems due to universal non-linear hydrodynamic corrections that induce a non-analytic $\gamma^{1/2}$ dependence, and proposes a modified extrapolation method to accurately recover noiseless transport properties in one-dimensional spin chains.

Original author: Hansveer Singh, Pieter Claeys, Sarang Gopalakrishnan, Anushya Chandran

Published 2026-11-05

📖 5 min read🧠 Deep dive

CC BY 4.0

Original author: Hansveer Singh, Pieter Claeys, Sarang Gopalakrishnan, Anushya Chandran

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the author. For technical accuracy, refer to the original paper. Read full disclaimer

In the quantum world, scientists are learning to build machines that can solve problems far beyond the reach of ordinary computers. However, these new devices are fragile. They are easily disturbed by their environment, a phenomenon known as noise, which scrambles the delicate information they hold. To get useful answers, researchers have developed a clever trick called zero-noise extrapolation. The idea is simple: run the same experiment multiple times, each time making the noise slightly stronger. By watching how the results change as the noise grows, they try to draw a line back to where the noise would be zero, hoping to reveal what the perfect, noiseless machine would have shown. This method has become a standard tool for fixing errors in quantum experiments, working well for many types of measurements.

But there is a catch. This trick relies on the assumption that the results change in a smooth, predictable way as the noise increases. When scientists try to use it to measure how particles move or flow through a material—a property known as transport—the method often breaks down. In many quantum systems, the movement of particles is governed by strict rules of conservation, much like how water in a closed pipe must stay within the pipe. When noise is introduced, it can break these rules, causing the particles to behave in ways that do not follow a smooth path back to zero noise. If researchers ignore this, they might calculate a value for how well a material conducts electricity or spin that is completely wrong, even if they use the most sophisticated equipment.

A team of physicists has now uncovered exactly why this happens and how to fix it. They studied one-dimensional chains of tiny magnetic particles, known as spins, which are a common model for understanding how matter behaves at the quantum level. In a perfect, noiseless world, these chains can transport spin in two distinct ways: either like a standard fluid, where the particles spread out at a steady rate, or in a slower, more restricted manner called subdiffusion, which happens when the particles are constrained by additional conservation laws. The researchers found that when noise is added, the system develops a new, effective way of transporting spin that depends on the strength of the noise.

The key discovery is that the relationship between the noise and the transport is not smooth. Instead of changing gradually, the transport coefficient—the number that tells us how fast the particles move—changes in a way that is mathematically sharp and unpredictable near zero noise. The researchers showed that this happens because of universal, hidden interactions within the fluid of particles. Even in a noisy environment, the particles still interact with each other in complex ways that create long-lasting ripples in their movement. These ripples, which are a fundamental feature of how fluids behave, cause the transport coefficient to approach its true, noiseless value at a specific, slow rate. The team proved that for certain systems, this approach follows a square-root pattern, meaning the value changes in proportion to the square root of the noise strength, rather than the noise strength itself.

To verify this, the researchers ran detailed computer simulations of these quantum chains. They tested two specific scenarios. In the first, they looked at a chain where the particles could move freely in all directions, a system with multiple conservation laws. In the second, they looked at a chain where the particles were more restricted, conserving a property called the dipole moment, which leads to the slower, subdiffusive movement. In both cases, they measured how the spin density waves decayed over time under different levels of noise. The data confirmed their theory: the effective transport coefficient did not change smoothly. Instead, it followed the predicted square-root curve, deviating significantly from the straight-line assumption that standard zero-noise extrapolation relies on.

The team also addressed a second problem: what happens if the noise changes the very state of the system? In some cases, the noise forces the particles into a new arrangement that is different from the one they would have in a perfect world. If researchers simply try to extrapolate back to zero noise without accounting for this shift, they will end up with the wrong answer. The researchers showed that by carefully controlling the timing of the measurements and the strength of the noise, it is possible to isolate the true transport properties of the system at a specific state. They demonstrated that by holding the product of the noise strength and the time constant, they could effectively "freeze" the system in a desired configuration and extract the correct, noiseless transport value.

This work provides a crucial guide for the future of quantum computing. It tells experimentalists that when they are trying to measure how things flow or conduct in a quantum system, they cannot simply assume a straight line back to perfection. They must look for these specific, non-smooth curves that arise from the fundamental nature of the particles. By using the correct mathematical shape to guide their extrapolation, they can finally recover the true, noiseless values that these fragile machines are trying to reveal. The findings suggest that the errors in current quantum measurements are not just random glitches, but are deeply rooted in the physics of how conserved quantities interact with their environment. Understanding this allows scientists to correct their data with much higher precision, bringing them closer to the reliable quantum computers of tomorrow.

Technical Summary: Zero Noise Extrapolation of Transport Coefficients

Problem Statement
Zero-noise extrapolation (ZNE) is a standard protocol in noisy intermediate-scale quantum (NISQ) computing where observables are measured at varying noise strengths ( $\gamma$ ) and extrapolated to the zero-noise limit ( $\gamma \to 0$ ) to estimate properties of the ideal system. While effective for observables that depend analytically on noise strength, ZNE is known to fail when estimating transport coefficients of conserved quantities. This failure occurs because noise generically violates conservation laws, causing transport to vanish at any non-zero $\gamma$ . Furthermore, even when noise preserves conservation laws on average, it can alter the stationary state or induce non-analytic dependencies on $\gamma$ due to universal hydrodynamic corrections. Consequently, standard ZNE schemes often yield incorrect estimates for diffusion and subdiffusion coefficients.

Methodology
The authors investigate ZNE in one-dimensional, non-integrable spin chains subject to noise modeled by the Gorini-Kossakowski-Lindblad-Sudarshan (GKLS) master equation with strength $\gamma$ . The study employs a combination of analytical hydrodynamic arguments and numerical simulations using tensor network methods (ITensor).

Two-Rate Ansatz: The authors propose that for $\gamma > 0$ , the decay rate $R(k, \gamma)$ of Fourier modes of a conserved quantity (spin density) follows the form:
$R(k, \gamma) = D_{\text{eff}}(\gamma)k^z + R(0, \gamma)$
where $z$ is the dynamical exponent and $R(0, \gamma)$ is the decay rate of the total charge. This allows for the definition of an effective, noisy diffusion constant $D_{\text{eff}}(\gamma)$ even when true transport is suppressed.
Handling Stationary State Evolution: When noise causes the system's stationary state to evolve (changing magnetization density), the authors argue that the correct zero-noise limit is recovered by taking the limits $\gamma \to 0$ and $t \to \infty$ simultaneously while holding the product $\gamma t$ fixed. This ensures the extrapolation targets a specific magnetization density $M^*$ rather than the noise-induced steady state.
Hydrodynamic Corrections: The paper derives the noise dependence of $D_{\text{eff}}(\gamma)$ using fluctuating hydrodynamics. They posit that $D_{\text{eff}}(\gamma)$ consists of an extrapolated value $D_{\text{exp}}$ , an analytic correction $f_a(\gamma)$ , and a non-analytic correction $f_h(\gamma)$ arising from universal long-time tails in current-current correlators:
$D_{\text{eff}}(\gamma) = D_{\text{exp}} + f_a(\gamma) + f_h(\gamma)$
The non-analytic term scales as $\gamma^{\alpha-1}$ , where $\alpha$ is determined by the power-law decay of current correlations ( $\sim \tau^{-\alpha}$ ) in the noiseless limit.

Key Results
The authors benchmark their analytical predictions against numerical simulations in two specific models:

Dissipative Heisenberg Chain (Diffusive Transport):
- Multiple Conservation Laws (SU(2) symmetric): In the absence of a staggered field, the system has multiple conserved quantities. The current-current correlator decays as $\tau^{-3/2}$ ( $\alpha = 3/2$ ), leading to a non-analytic correction scaling as $\sqrt{\gamma}$ . Numerical data and perturbative calculations confirm that $D_{\text{eff}}(\gamma) \approx D_{\text{exp}} - c\sqrt{\gamma}$ .
- Single Conservation Law (Staggered field): When a staggered field breaks SU(2) symmetry, leaving only one conserved quantity, the leading non-analytic correction vanishes (or becomes subleading), and the dependence becomes analytic (linear in $\gamma$ ).
Dipole-Conserving Fracton Chain (Subdiffusive Transport):
- The authors analyze a chain conserving total dipole moment, which exhibits subdiffusive transport ( $z_{\text{true}} = 4$ ) in the noiseless limit.
- They demonstrate that the effective subdiffusion coefficient approaches its noiseless value as $\gamma^{1/2}$ . This confirms that universal hydrodynamic corrections produce non-analytic behavior even in subdiffusive regimes.
Extrapolation Protocol: The study validates that by fixing $\gamma t$ (and thus the magnetization density) during the extrapolation, one can successfully recover the true noiseless diffusion constant $D_{\text{true}}(M^*)$ even when the noise drives the system to a different stationary state.

Significance and Claims
The paper claims that standard ZNE fails for transport coefficients not merely because noise destroys conservation laws, but because of universal non-linear corrections to hydrodynamics that induce non-analytic dependencies on noise strength. Specifically:

Standard ZNE assumes analyticity in $\gamma$ , but for systems with multiple conserved quantities (or specific symmetries in subdiffusive systems), the leading correction is non-analytic (e.g., $\gamma^{1/2}$ ).
The effective transport coefficient in the noisy system approaches the noiseless value via a power law $\gamma^{\alpha-1}$ , where $\alpha < 2$ in the presence of multiple conservation laws.
The authors provide a modified extrapolation protocol (fixing $\gamma t$ ) to correctly identify transport coefficients corresponding to specific magnetization densities when the stationary state is noise-dependent.

The work serves as a theoretical and numerical benchmark, demonstrating that without accounting for these non-analytic hydrodynamic tails, ZNE will systematically misestimate transport coefficients in quantum simulators.

More like this