Gaussian tomography for cold-atom simulators

arXiv:2510.23591 · quant-ph, cond-mat.quant-gas · Submitted 2025-10-27 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: Today's paper: "Gaussian tomography for cold-atom simulators".

Mira: The gist The authors propose experiment-friendly schemes to measure chargeoff-diagonal correlations in cold-atom simulators by using non-interacting dynamics for random times followed by standard quantum gas microscope measurements to effectively…

Kai: First, who's behind it and why it matters.

Paper summary: Kai: So we're looking at this paper, "Gaussian tomography for cold-atom simulators," which is about figuring out how to measure things that aren't just simple densities in these cold-atom setups.

Mira: Exactly. The main point here is that analog quantum simulators, like using atoms in optical lattices, are usually limited because you can only easily measure observables that are diagonal in the charge basis, meaning you get densities and density correlation functions.

Kai: But what the authors propose is an experiment-friendly way to measure those off-diagonal correlations, which would be things like currents. They claim they have a scheme using non-interacting dynamics for random times followed by standard quantum gas microscope measurements to measure in random bases.

Lev: From an error correction standpoint, I’m interested in the requirement for turning off interactions because that's what allows them to use these non-interacting Hamiltonians, which is key when you're trying to simulate things on real hardware.

Kai: Right, and they outline two main schemes: a local scheme and a global scheme. The global scheme estimates the whole correlation matrix by expanding a one-dimensional chain into two dimensions.

Mira: That sounds like they're trying to get around the limitation of only measuring diagonal observables by making the measurement basis random in a structured way across many copies of the system.

Kai: The procedure involves evolving many copies of the system with randomly chosen non-interacting Hamiltonians, which results in a final state described by a unitary transformation of an initial covariance matrix C s = U* s (C zero Canc) U T where s is the random choice <ref:2510.23591#pg1>.

Lev: If you're talking about running this on actual hardware, that random evolution part sounds like it’s going to demand a lot of time or many repetitions just to get a good sample of the resulting state, which is something I think we need to watch carefully.

Kai: The measurement step then involves measuring particle occupation on each site, yielding random variables n, which they then collate into z = n e s. The goal is to estimate the initial correlation matrix C zero by taking the expectation value of this collated data <ref:2510.23591#pg1>.

Mira: They show that if the map F has full row rank, you can find a left inverse G, and then the desired correlation matrix C zero is estimated by looking at = G(z - danc), which they say is an unbiased estimator of C zero <ref:2510.23591#pg1>.

Lev: So, if we look at the numbers, the paper says that for local observables in one-dimensional and two-dimensional systems, the worst-case sample complexity for nearest-neighbor observables plateaus around three thousand samples.

Paper summary: Kai: And they also mention that for grids larger than four by four, a randomly chosen or average case observable would only require about ten five samples to reach an accuracy of less than zero point zero five, even for chains of one hundred sites.

Mira: That's a pretty solid number if you want to estimate those local correlation functions accurately with this specific method described in "Gaussian tomography for cold-atom simulators."

Kai: The paper also discusses robustness, showing that the scheme is robust against errors due to bias in the evolution Hamiltonian, with the maximal deviation dictated by the largest eigenvalue of G opt F* - I.

Lev: That sounds like a manageable error bound if you can control those parameters, but I wonder how sensitive it really is to tuning errors in those random potentials they introduce.

Kai: The local scheme shows mild growth in error with system size because the reconstruction is essentially local, whereas the global scheme displays a stronger sensitivity to tuning errors because of that long time evolution.

Mira: That distinction between the two schemes is important because it tells us how much effort we need to put into controlling things if we want to get better results for different types of correlations.

Kai: The method generalizes to higher-order observables, like k-point functions, by showing that non-interacting evolution connects those correlators only to other k-point correlators.

Lev: That’s interesting because it suggests you don't need a whole new measurement protocol for every kind of correlation if you stick to this framework of random non-interacting evolution.

Kai: Overall, the paper presents a practical and flexible method for efficiently estimating correlation matrices using between ten cubed and ten five occupation measurements, featuring easily implemented quench Hamiltonians and classically efficient post-processing.

Mira: So, to sum up the main points of "Gaussian tomography for cold-atom simulators," it's a protocol that lets us measure things like currents in cold atoms by using random non-interacting dynamics and specific measurement techniques.

Kai: And the conclusion is that this framework offers a way to efficiently estimate correlation matrices using between ten cubed and ten five occupation measurements, making it competitive for local observables with a few thousand shots and low evolution times of at most five hopping times.

Lev: For someone listening who doesn't do quantum computing, what does this actually change? It means we might be able to get some information about how particles are moving around in these simulators that we couldn't before without needing much more complex control hardware.

Kai: The paper lays a foundation for many promising areas of future development, suggesting this approach can be adapted to continuous-space systems and even time-dependent Hamiltonians for noise robustness.

Mira: That sounds like the big picture here—taking a technique that works on these specific cold atom setups and seeing if it can be applied more broadly to other physical systems.

Conclusion: Kai: The authors are using non-interacting dynamics for random times and then using standard quantum gas microscope measurements to read out the results in random bases.

Mira: It’s essentially building a technique—tomography—to reconstruct the full correlation matrix from those limited occupation number measurements.

Lev: I see how that works mathematically, but it makes me wonder how robust this whole process is when you try to run it on actual hardware with real noise.

Kai: The authors show that for local observables, we can get a reasonable estimate with a few thousand shots and low evolution times.

Mira: It’s not just about getting an estimate; they are giving us the sample complexity numbers, showing you exactly how many measurements you need for different systems.

Lev: Those ten five samples for larger grids sound like they might be feasible if the post-processing is fast enough, which is a big deal when we think about real experimental time.

Kai: And it shows that this method can handle higher-order correlations too, connecting them in a predictable way through the non-interacting evolution.

Mira: So what this really means is that we might be able to get much richer information about how these atoms are interacting and moving around in complex systems than just density measurements alone.

Lev: It’s a foundation for seeing what kind of observables are even measurable before we try to build the next generation of simulators.

Technical University of Munich School of Natural Sciences · IQM Quantum Computers

quant-ph, cond-mat.quant-gas

Submitted: 2025-10-27

Updated: 2025-10-27

Comments: 6 + 2 pages, 2 figures, 1 table

Journal ref: Phys. Rev. A 114, 013311 (2026)

DOI: 10.1103/nkwy-23tw

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 89/100

The gist: The gist The authors propose experiment-friendly schemes to measure chargeoff-diagonal correlations in cold-atom simulators by using non-interacting dynamics for random times followed by standard

Key concepts

Chargeoff-diagonal correlations
These are complex relationships between particles on different sites in the cold atom system. Standard measurements often only capture simple density information. This paper focuses on measuring these more intricate connections to fully characterize the quantum state of the simulator.
Quantum Gas Microscope Measurements
This is a technique used to measure where every single particle is located in an optical lattice at the end of an experiment. However, it typically only allows measurements in the particle number basis, which limits what types of correlations can be directly measured.
Global Scheme
This method estimates the entire correlation matrix by treating a 1D chain as a 2D system. It involves evolving many copies of the system under random non-interacting Hamiltonians and then using classical post-processing to estimate the desired correlation matrix from particle occupation measurements.

Terminology

Summary

The gist The authors propose experiment-friendly schemes to measure chargeoff-diagonal correlations in cold-atom simulators by using non-interacting dynamics for random times followed by standard quantum gas microscope measurements to effectively measure in random bases

Limitations and Motivation

Analog quantum simulators based on cold atoms in optical lattices are limited because readout is typically restricted to observables diagonal in the charge basis, such as densities and density correlation functions The quantum gas microscope allows single site-resolved measurement of each particle at the end of the experiment but only permits measurements in the particle number (or computational) basis, which excludes important observables such as currents Efforts to overcome this limitation involve using time evolution under an interacting or non-interacting Hamiltonian for random times or in the presence of ancillas before measuring Schemes using interacting dynamics require either local control or the ability to simulate long-time evolution during classical post-processing, which is assumed to be prohibitive

Proposed Schemes

The proposed scheme utilizes three main ingredients: (i) the ability to turn off interactions leading to non-interacting dynamics, (ii) an additional laser used to impart a quasiperiodic potential that breaks all lattice symmetries, and (iii) an additional laser used to impart a quasiperiodic potential that breaks all lattice symmetries The global scheme estimates the whole correlation matrix by expanding a 1d chain into two dimensions, which requires evolution times of order N

  1. Evolve many copies of the system by randomly chosen non-interacting Hamiltonians of the form in Eq. (2), resulting in a final state described by a unitary transformation of the initial covariance matrix Cs = U∗s(C0 ⊕ Canc)U Ts

  2. Measure the particle occupation on each site, yielding random variables nˆ, which are collated into zˆ = nˆ ⊗ es

The expectation value of this collated data is E[zˆ] = (p1d1 · · · pSdS)T = FC0) + FancCanc) If the map F has full row rank, a left inverse G can be found such that GF = I The desired correlation matrix C0) is then estimated by the expectation value of Yˆ):= G(z−danc), which is an unbiased estimator of C0)

Sample Complexity and Performance

The sample complexity required to achieve a desired level of statistical uncertainty can be determined by calculating the variance of the estimator Eq. (7) The optimal sample complexity for estimating a specific observable O is given by Var[ˆθO]optimal = 1/R(oL−1o) For local observables in 1d and 2d systems, the worst-case sample complexity for nearest-neighbor observables plateaus at approximately R = 3000 samples, independent of the system size N The global scheme shows that for grids greater than 4x4, a randomly chosen or average-case observable would require only ∼ 10 5 samples to reach an accuracy of ε < 0.05 even for chains of 100 sites

Robustness and Generalization

The scheme is robust against errors due to bias in the evolution Hamiltonian, with the maximal deviation dictated by the largest eigenvalue of GoptFerr − I The local scheme exhibits mild growth in error with system size as reconstruction is essentially local, whereas the global scheme displays a stronger sensitivity to tuning errors due to long time evolution The method generalizes to higher-order observables (k-point functions) by showing that noninteracting evolution connects k-point correlators only to other k-point correlators The approach can be adapted to continuous-space systems and the exploration of time-dependent Hamiltonians may offer advantages for noise robustness or sample efficiency

Conclusion

The framework provides a practical and flexible method for efficiently estimating correlation matrices using 10 cubed − 10 5 occupation measurements, featuring easily implemented quench Hamiltonians and classically efficient post-processing The scheme is competitive for local observables with few thousand shots and low evolution times of at most five hopping times The work lays a foundation for many promising areas of future development

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Gaussian tomography for cold-atom simulators

Matthew Kiser,

1, 2, ∗ Max McGinley,

3 and Daniel Malz 4

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Local scheme Global scheme Atoms hopping in 1d or 2d Expansion from 1d→2d → No ancillas → Ω(N 2) ancillas Random times ti ≤ const Fixed t ∝ N Applied quasiperiod pot. (orange) Same Random pot. strength hi ≤ hmax Same Random pot. phase φi No phase

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In the following, we present a general protocol, but will select two concrete variants for numerical simulation, illustrated in Table I.

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III. METHODS We now give a precise mathematical description of our protocol and show how to compute an estimator for the correlation matrix from the measurement data.

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Postprocessing.—Running the experiment R times, with a different randomly chosen s in each repetition, we generate multiple pairs of data R r=1 ≡ zˆr> R r=1 each of which can be used to construct an independent estimator of the full correlation matrix Yˆ) as described above

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VI. RESULTS We now present numerical results to estimate the sample complexities of our methods in the two proposed schemes shown in Table I: the “local scheme” to estimate local observables in 1d chains and 2d lattices and the “global scheme” to estimating the arbitrary correlations in 1d chains

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C. Robustness to Errors We evaluate the error due to bias in the evolution Hamiltonian by adding a random potential to the Hamiltonian, sampled from a Gaussian distribution of variance ν

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Appendix B: Variance and optimality of the estimator Suppose we use the scheme described in Section III to estimate a particular quadratic observable ⟨A⟩ = (aC0), where A = P ij aijc†i cj†i. For a given run of the experiment, we obtain measurements of the occupation numbers nˆ ∈ 0, 1 Ntot, and using the method described in the main text we construct an estimator of the correlation matrix Yˆ) = G(zˆ − danc), where G is the left inverse of F The quantity ˆθA:= (aŶ) is then our desired estimator, E[ˆθA] = ⟨A⟩. As explained in the main text, depending on the dimensions of the linear map F, there might not be a unique choice of left inverse G, and thus, there will be different possible estimators

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Appendix C: Approximate local inverse The measurement map F defined in Eq. (6) has SN3 entries, which can require an excessive amount of memory to store For example, for S = 1000 and N = 121 = 112, storing F requires approximately 26 GB of memory In the local schemes, in which we aim to only recover local correlation functions, we only use constant time evolution, and thus we can instead define an approximate local version of F that becomes independent of system size

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[1] R. P. Feynman, Simulating physics with computers, Int. J. Theor. Phys.

Improvements for AI systems

  1. Improved ability to measure chargeoff-diagonal correlations: The protocol allows for experiment-friendly schemes to measure chargeoff-diagonal correlations (such as currents) by using non-interacting dynamics for random times followed by standard quantum gas microscope measurements to effectively measure in random bases. This enables the measurement of observables beyond the particle number (or computational) basis, specifically currents.

  2. Improved scalability and efficiency: The scheme demonstrates efficient estimation of bilinear correlation functions, requiring less than 4000 samples to measure local currents to 5% error (system-size independent) and 10 4 samples to simultaneously measure all non-local correlations in 70-site systems. This suggests the method is scalable and requires only modest requirements in terms of total evolution time and number of repetitions.

  3. Improved estimation of arbitrary observables: The protocol provides a global scheme that can estimate the whole correlation matrix, allowing for the reconstruction of arbitrary non-local correlation functions by expanding a 1D chain into 2D, with hopping times of order N being sufficient to measure all correlation functions.

  4. Improved resource management in local schemes: For estimating local observables without ancillas, the worst-case sample complexity plateaus at approximately R = 3000 samples, independent of the system size N, indicating that the observable is recovered locally. This allows for practical implementation with only a few thousand shots.

  5. Improved noise robustness: The analysis shows that in the local scheme employing constant time evolution, the errors grow only very mildly with system size as the reconstruction is essentially local, suggesting better performance than schemes relying on long-time evolution for global measurements when dealing with Hamiltonian errors.

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