# Learning quantum systems via out-of-time-order correlators

Thomas Schuster<EMAIL_ADDRESS>Google Quantum AI, Venice, CA 90291,
USA  Department of Physics, University of California, Berkeley, CA 94720 USA
Murphy Niu  Google Quantum AI, Venice, CA 90291, USA  Jordan Cotler  Society
of Fellows, Harvard University, Cambridge, MA, USA  Black Hole Initiative,
Harvard University, Cambridge, MA, USA  Thomas O’Brien  Google Quantum AI,
Venice, CA 90291, USA  Jarrod R. McClean  Google Quantum AI, Venice, CA 90291,
USA  Masoud Mohseni<EMAIL_ADDRESS>Google Quantum AI, Venice, CA 90291,
USA

###### Abstract

Learning the properties of dynamical quantum systems underlies applications
ranging from nuclear magnetic resonance spectroscopy to quantum device
characterization. A central challenge in this pursuit is the learning of
strongly-interacting systems, where conventional observables decay quickly in
time and space, limiting the information that can be learned from their
measurement. In this work, we introduce a new class of observables into the
context of quantum learning—the out-of-time-order correlator—which we show can
substantially improve the learnability of strongly-interacting systems by
virtue of displaying informative physics at large times and distances. We
identify two general scenarios in which out-of-time-order correlators provide
a significant advantage for learning tasks in locally-interacting systems:
($i$) when experimental access to the system is spatially-restricted, for
example via a single “probe” degree of freedom, and ($ii$) when one desires to
characterize weak interactions whose strength is much less than the typical
interaction strength. We numerically characterize these advantages across a
variety of learning problems, and find that they are robust to both read-out
error and decoherence. Finally, we introduce a binary classification task that
can be accomplished in constant time with out-of-time-order measurements. In a
companion paper Cotler  _et al._ (2022), we prove that this task is
exponentially hard with any adaptive learning protocol that only involves
time-ordered operations.

##  I Introduction

Learning properties of quantum systems can pose challenges not present in
their classical counterparts Mohseni  _et al._ (2008); Huang  _et al._ (2022).
These differences often stem fundamentally from the existence of
entanglement—measurements of a quantum system that is highly entangled with
another system or the environment reveal little information from which to
learn. In practical settings, these difficulties are most commonly encountered
in _strongly-interacting_ quantum systems. Strong interactions can introduce
non-local entanglement throughout the system at short time scales, and are
found to thereby inhibit the learning of system properties (e.g. the
Hamiltonian) from physical observables Wiebe  _et al._ (2014); Aharonov  _et
al._ (2021); Huang  _et al._ (2021); Zhou  _et al._ (2020); O’Brien  _et al._
(2021).

The ubiquity of strong interactions in experimental applications of quantum
learning has spurred a variety of solutions to this problem. For instance, in
nuclear magnetic resonance (NMR) spectroscopy, a suite of technologies have
been developed to controllably dampen undesired strong interactions between
solid-state nuclear spins, which has enabled the identification of hitherto
inaccessible molecular structures Laws  _et al._ (2002). In a similar spirit,
in quantum device characterization Wang  _et al._ (2015) and quantum sensing
Zhou  _et al._ (2020), dynamical decoupling control sequences Ezzell  _et al._
(2022) can effectively eliminate unwanted interactions and improve learning of
the residual interactions. Other approaches include learning by transducing
quantum data from the system onto a quantum simulator Wiebe  _et al._ (2014);
Aharonov  _et al._ (2021); Huang  _et al._ (2021); Wang  _et al._ (2017);
Huang  _et al._ (2022), or learning from high-precision local measurements at
early times, before entanglement has formed Bairey  _et al._ (2019); Haah  _et
al._ (2021). Nonetheless, owing to incomplete control or limited experimental
precision, many physical systems remain unlearnable with existing approaches.

Figure 1:  Schematic of time-ordered correlators (TOCs) and out-of-time-order
correlators (OTOCs) in strongly-interacting systems. TOCs typically decay in
$O(1)$ times and distances (top, red), making it hard to learn features
(yellow bond) that manifest only at late times. OTOCs utilize backwards time-
evolution to “refocus” many-body correlations (bottom, blue), enabling
learning of such features.

In this paper, we introduce a different paradigm for learning in strongly-
interacting quantum systems—learning via out-of-time-order correlators
(OTOCs). First studied in early works on semi-classical methods Larkin and
Ovchinnikov (1969) and NMR Baum  _et al._ (1985), the OTOC has more recently
initiated a renaissance of work at the intersection of quantum information
theory, many-body dynamics, and quantum gravity (e.g. Shenker and Stanford
(2014); Stanford (2016); Maldacena  _et al._ (2016); Maldacena and Stanford
(2016); Hosur  _et al._ (2016); Cotler  _et al._ (2017, 2018); Nahum  _et al._
(2018); Von Keyserlingk  _et al._ (2018)). Physically, OTOCs quantify the
spread of local quantum information into highly non-local, entangled
correlations Xu and Swingle (2022). Experimental measurements of the OTOC
typically employ _reversed_ time-evolution to refocus these correlations, and
have been performed on dozens of qubits in superconducting quantum processors
and trapped ion quantum simulators, and hundreds of nuclear spins in NMR
spectroscopy Gärttner  _et al._ (2017); Mi  _et al._ (2021); Sánchez  _et al._
(2021); Domínguez  _et al._ (2021).

In this work, we utilize the OTOC as a _tool_ for learning properties of
strongly-interacting quantum systems. Our application is motivated by a simple
intuition: while time-ordered observables decay quickly as a system becomes
entangled, out-of-time-order observables continue to fluctuate up to long
times (Fig. 1). Guided by this intuition, we demonstrate the power of learning
via OTOCs across a range of physical systems, supported by numerical studies,
phenomenological estimates, and rigorous information-theoretic proofs. We
begin in locally-interacting systems, where we identify two general scenarios
in which OTOCs provide a strong learning advantage: ($i$) when experimental
access to the system is spatially-restricted, for example via a single “probe”
qubit Burgarth  _et al._ (2009); Lovchinsky  _et al._ (2016); Gentile  _et
al._ (2021), and ($ii$) for detecting weak interactions in an otherwise
strongly-interacting system Wang  _et al._ (2015); Zhou  _et al._ (2020). We
characterize these advantages using both information-theoretic measures (the
Fisher information) and performance metrics for concrete learning tasks.
Moreover, we find that the advantages are robust to experimental read-out
error and time-reversal imperfections arising from strong coupling with an
environment or decoherence. Finally, motivated by recent advances in
_provable_ learning advantages Aharonov  _et al._ (2021); Chen  _et al._
(2021a); Huang  _et al._ (2021), we introduce a learning task involving
distinguishing two classes of unitary operations if given oracle access. In a
companion work Cotler  _et al._ (2022), we prove that OTOCs provide an
exponential advantage in performing this task over any time-ordered learning
protocol.

##  II Behavior of time-ordered vs. out-of-time-order correlators

Figure 2:  Learning with state preparation and read-out restricted to a probe
qubit, and local unitary control over the remaining system. (a) Results from
SVM regression for learning the distance, $d$, in the spin geometry shown,
with access to TOCs (red) or both TOCs and OTOCs (blue). Color bars (black
ticks) denote $75\%$ ($100\%$) percentiles of predictions on $200$ disorder
realizations, and grey step function represents the actual $d$. (b) Fisher
information, $\text{FI}(J_{d}|C)$, of an interaction, $J_{d}$ (top; red line),
a distance $d$ away from the probe (top; purple circle), maximized over all
correlators, $C$, in an $L$-qubit 1D chain. The FI decays exponentially in $d$
when $C$ is time-ordered (red), and algebraically, $\sim 1/d$, when $C$ is
out-of-time-order (blue).

We begin by reviewing the phenomenology of time-ordered and out-of-time-order
correlators in ergodic locally-interacting systems (Fig. 1). A time-ordered
correlator (TOC) is defined as any correlation function that takes the
following general form:

$C_{\text{TOC}}=\tr(A_{k}(t_{k})\ldots
A_{1}(t_{1})\,\rho\,B_{1}(t^{\prime}_{1})\ldots B_{\ell}(t^{\prime}_{\ell})).$
(1)

where the operators $A,B$ increase in time away from the initial density
matrix $\rho$, i.e. $t_{k}>\cdots>t_{1}$ and
$t^{\prime}_{\ell}>\cdots>t^{\prime}_{1}$. Time-ordered correlators can be
measured by evolving the state $\rho$ forward in time (e.g. via Hamiltonian
evolution $O(t)=e^{iHt}Oe^{-iHt}$) while applying intermediary quantum
operations at each time $t_{i},t^{\prime}_{j}$ fn (4). Any correlation
function that does not obey this form is called an _out-of-time-order_
correlator.

A common example of a time-ordered correlator is the two-point function,

$C_{\text{TOC}}=\expectationvalue{V_{x}(t)W_{x^{\prime}}(0)},$ (2)

where $\langle\cdot\rangle\equiv\tr(\cdot)/2^{L}$ denotes the infinite
temperature trace for $L$ qubits, and $V_{x},W_{x^{\prime}}$ are local
operators at sites $x,x^{\prime}$. Such correlators measure the spread of
local quantities in space and time; for instance, how much spin prepared at
site $x^{\prime}$ at time zero has transferred to site $x$ at time $t$. A wide
range of literature on thermalization in strongly-interacting systems has
found that local TOCs typically decay quickly, i.e. in $O(1)$ times, to their
thermal values D’Alessio  _et al._ (2016). This quick decay can inhibit
learning tasks, since no additional information can be acquired from the TOC
at times after the decay has occurred O’Brien  _et al._ (2021).

Meanwhile, the prototypical out-of-time-order correlator is the four-point
function Xu and Swingle (2022),

$C_{\text{OTOC}}=\expectationvalue{V_{x}(t)W_{x^{\prime}}(0)V^{\dagger}_{x}(t)W_{x^{\prime}}(0)},$
(3)

with local operators $V_{x},W_{x^{\prime}}$. Unlike time-ordered measurements,
OTOCs typically require both forwards _and_ backwards time-evolution to
measure Xu and Swingle (2022); Cotler  _et al._ (2022). (Importantly for our
application, nearly all experimental techniques for time-reversal rely only on
the type of interaction being reversed and require no knowledge of the
specific Hamiltonian, which one might wish to learn. For example, the same
pulse sequence reverses an arbitrary dipole-dipole coupling Hamiltonian in an
NMR experiment Sánchez  _et al._ (2021).) Physically, the OTOC probes whether
information encoded at site $x^{\prime}$ at time zero is contained in
correlations involving site $x$ at time $t$. This is quantified by the squared
commutator of a time-evolved operator at $x$ with a local operator at
$x^{\prime}$,
$\expectationvalue{\left|\left[V_{x}(t),W_{x^{\prime}}(0)\right]\right|^{2}}=1-C_{\text{OTOC}}$.
In local strongly-interacting systems, operators are expected to spread
ballistically according to the connectivity of the system Lieb and Robinson
(1972); Aleiner  _et al._ (2016); Nahum  _et al._ (2018). Crucially, this
spread continues for a duration proportional to the system’s spatial extent
$\sim L$ by which time the information has been delocalized across the entire
system.

This phenomenology leads to two central intuitions for learning from OTOCs.
First, the dynamics of the OTOC contain information primarily about the
connectivity of the system under study. Second, the OTOC continues to reveal
such information up to $O(L)$ times, long after TOCs have decayed. Notice that
this timescale increases as the system size increases. In what follows, we
apply these intuitions to identify two broad regimes where access to OTOCs
provides a significant learning advantage.

##  III Learning with restricted access

The first regime we consider is learning in systems with _restricted access_.
Specifically, motivated by recent advances in solid-state defects Boss  _et
al._ (2016); Lovchinsky  _et al._ (2016); Gentile  _et al._ (2021) and NMR
Theis  _et al._ (2016); Blanchard  _et al._ (2021); Burgarth and Ajoy (2017),
we focus on the scenario where an experimenter has state preparation and read-
out capabilities over only a single “probe” qubit interacting with a larger
system that one wishes to learn. We note that high-fidelity OTOC measurements
have already been achieved in similar setups by using rapid global pulse
sequences to reverse time-evolution Wei  _et al._ (2018); Sánchez  _et al._
(2019, 2021). Previous theoretical approaches to learning in this scenario
have been limited to non-interacting dynamics Burgarth  _et al._ (2009);
Burgarth and Maruyama (2009); Di Franco  _et al._ (2009); Zhang and Sarovar
(2014); Burgarth and Ajoy (2017); Sone and Cappellaro (2017). Meanwhile,
experiments have found that it is in general difficult to learn features of a
system that are distant from the probe qubit Lovchinsky  _et al._ (2016);
Gentile  _et al._ (2021). In strongly-interacting systems, this difficulty can
be understood from the quick decay of correlation functions in space and time.
Here, we provide evidence via phenomenological estimates (Appendix B) and
numerical simulations (Fig. 2) that access to OTOCs can exponentially improve
the learnability of distant features.

To be concrete, we will assume for now that the experimenter has local unitary
control over the qubits of the larger system fn (1). We will also assume that
the larger system begins in an infinite temperature (i.e. maximally mixed)
state, which is the natural scenario in NMR and solid-state defect setups Boss
_et al._ (2016); Theis  _et al._ (2016). Within these assumptions, a simple
class of measurement protocols proceeds as follows:

  1. 1.

Prepare the probe qubit $p$ in an eigenstate of an operator $V_{p}$, such that
the density matrix of the entire system is
$\rho=\frac{1}{2}(\mathbbm{1}_{p}+V_{p})\otimes\frac{1}{2^{L-1}}\,\mathbbm{1}_{\text{sys}}$.

  2. 2.

Time-evolve by time $\tau$.

  3. 3.

Perturb the system by a unitary operation $W_{x}$ on a qubit $x$.

  4. 4.

Time-evolve by a time $\tau^{\prime}$.

  5. 5.

Read out the expectation value of $V_{p}$ on the probe qubit.

Taking $\tau,\tau^{\prime}$ to be positive (e.g. $\tau=\tau^{\prime}=t/2$),
this allows measurement of time-ordered correlation functions of the form
$\langle V_{p}(t)\,W_{x}(t/2)\,V_{p}(0)\,W^{\dagger}_{x}(t/2)\rangle$. With
access to reversible time-evolution (e.g. $\tau=-\tau^{\prime}=t$), the above
protocol also allows measurement of out-of-time-order correlation functions
$\langle V_{p}(0)\,W_{x}(t)\,V_{p}(0)\,W^{\dagger}_{x}(t)\rangle$. In Appendix
B and C, we discuss how learning is modified when $W$ is instead a _global_
spin rotation over the larger system.

Figure 3:  (a) Learning a weak “link” interaction (red line) in a 1D spin
chain with otherwise strong interactions (black lines). (b) Accuracy of binary
SVM classification of whether the link is present or absent, as a function of
the link strength $J_{\ell}$ and at fixed read-out error $\delta=3\%$.
Learning via OTOCs can detect smaller $J_{\ell}$ as $L$ increases, while the
TOC can only detect relatively large $J_{\ell}$, independent of $L$. (c) The
maximum Fisher information $\text{FI}(\log(J_{\ell})|C)$ of $J_{\ell}$ decays
$\sim\\!J_{\ell}^{4}$ for small $J_{\ell}$, and is enhanced in OTOCs (blue)
compared to TOCs (red) by a factor that increases with $L$.

We begin our exploration of learning via OTOCs by introducing a concrete
learning task. We consider the following scenario: one is given access to a
quantum system consisting of two spin chains intersecting at a distance $d$
from a probe qubit [Fig. 2(b)]. The value of $d$ as well as the specific
Hamiltonian parameters of the system are unknown (see below for the specific
distribution that the Hamiltonian is drawn from). The goal is to learn the
value of $d$, i.e. the geometry of the system, from measurements of the
system’s correlation functions.

To solve this task, we assume that the experimenter is capable of simulating
quantum dynamics on either a classical or quantum computer. Since the task
involves high-dimensional input data (i.e. the correlators for every $x,t$),
we will approach it using machine learning techniques. Specifically, we
envision using the quantum simulator to compute the correlation functions of
an ensemble of Hamiltonians for each value of $d$. These ensembles can then be
used to train a classical learning model to predict an unknown Hamiltonian’s
value of $d$ given its correlation functions.

Let us briefly summarize our numerical simulations in more detail (see
Appendix A for a complete description). Throughout this work, we consider spin
systems with disordered on-site fields,
$H_{f}=\sum_{i,\alpha}h_{i}^{\alpha}\sigma^{\alpha}_{i}$ with
$h_{i}^{\alpha}\in[-1,1]$ and $\alpha=x,y,z$, and dipolar interactions between
neighboring spins, $H_{c}=\sum_{\langle
ij\rangle}J_{ij}(\sigma^{x}_{i}\sigma^{x}_{j}+\sigma^{y}_{i}\sigma^{y}_{j}-2\sigma^{z}_{i}\sigma^{z}_{j})$
with $J_{ij}\in[0.6,1.4]$. We specify to Floquet dynamics consisting of
alternating applications of $H_{f}$ and $H_{c}$ for time $T=\pi/2$, and
simulate time-evolution via Krylov-subspace methods dyn . We expect that
learning Floquet dynamics will be qualitatively similar to learning time-
independent Hamiltonian dynamics at moderate times and distances, which we are
restricted to in our numerics (see Appendix C for numerical support of this
statement). At larger distances we expect Hamiltonian dynamics to be dominated
by hydrodynamics of the conserved energy (Appendix B) and the two will differ.

Returning to the learning task at hand, we train a support vector machine
(SVM) on 3000 randomly drawn Hamiltonians (300 for each value of
$d=0,\ldots,9$), and test its performance on 2000 additional Hamiltonians. To
ensure that learning is not sensitive to fine-tuned features of the
correlation functions, we add a Gaussian distributed “read-out error” to all
correlation functions, with mean zero and standard deviation $\delta=3\%$. The
model’s predictions as a function of the actual value of $d$ are displayed in
Fig. 2(b), for learning either via TOCs (red) or both TOCs and OTOCs (blue).
We find that learning via OTOCs allows accurate predictions of $d$ within $\pm
1$ of its actual value for all distances probed (up to $d=9$). In contrast,
with access to only TOCs, the model performs significantly worse for all $d$
and resorts to nearly random guessing for $d\gtrsim 3$.

To evaluate the learning advantage of OTOCs independent of a specific learning
task, we turn to the Fisher information (FI). The FI quantifies the amount of
information that a random variable (e.g. a correlation function $C$, measured
within some read-out error $\delta$) carries about an unknown parameter (e.g.
a coupling strength, $J$), and thereby bounds the ultimate learnability of the
parameter Kay (1993). If one assumes that read-out errors are normally
distributed, the FI is simply a squared derivative,
$\text{FI}(J|C)\equiv\delta^{2}\text{FI}(J|C;\delta)=\left|\partial C/\partial
J\right|^{2}$, where we remove the $\delta$-dependence by introducing a factor
$\delta^{2}$.

We numerically compute the FI in ergodic 1D spin chains, where one seeks to
learn a coupling $J_{d}$ lying a distance $d$ away from a probe qubit [Fig.
2(b) inset] Burgarth  _et al._ (2009); Burgarth and Maruyama (2009); Di Franco
_et al._ (2009); Zhang and Sarovar (2014); Burgarth and Ajoy (2017); Sone and
Cappellaro (2017). We consider the same set of correlation functions as
specified for the learning task in Fig. 2(a). In Fig. 2(b), we plot the
maximum Fisher information $\max_{C}\text{FI}(J|C)$ over all correlation
functions (i.e. over all $x,t$), averaged over 200 and 1000 disorder
realizations for TOCs and OTOCs respectively. We find that the maximum FI of
TOCs (red) decays exponentially in the distance $d$ from the probe qubit. In
contrast, the maximum FI of OTOCs (blue) follows a slow algebraic decay, $\sim
1/d$, thereby achieving a multiple-order-of-magnitude advantage over TOCs even
at modest distances, $d\gtrsim 3$. This algebraic decay arises from the
$\sim\\!\sqrt{t}$ broadening of the OTOC wavefront in time Nahum  _et al._
(2018), see Appendix B for a full phenomenological derivation.

##  IV Learning weak interactions

We now turn to our second learning scenario: characterizing weak interactions
in an otherwise strongly-interacting system. Such characterization is
notoriously difficult because weak interactions take long times to manifest
(of order the inverse interaction strength), at which point TOCs have decayed
due to the strong interactions. Previous approaches require either dynamical
decoupling of the strong interactions Wang  _et al._ (2015); Zhou  _et al._
(2020) or high-precision measurements at early times Bairey  _et al._ (2019);
Haah  _et al._ (2021). We will now show that access to OTOCs allows one to
side-step these requirements when characterizing weak interactions that
_change the connectivity_ of a strongly-interacting system. Notably, in
contrast to the previous learning scenario, this advantage holds when the
experimenter is capable of measuring all local correlation functions of the
system of interest.

For concreteness, we specialize to 1D spin chains with a single “weak link”
interaction, of strength $J_{\ell}$ much less than the typical interaction
strength $J$ [see Fig. 3(b) inset]. We consider TOCs and OTOCs of the form Eq.
(2) and Eq. (3), where $x,x^{\prime}$ run over all qubits in the system. We
anticipate that access to more general correlators within a given time-
ordering, e.g. via shadow tomography or related techniques Aaronson (2019);
Cotler and Wilczek (2020); Huang  _et al._ (2020), will not qualitatively
change the observed physics (see Appendix A).

Figure 4:  Learning as a function of experimental error, in the “weak
interaction” learning task of Fig. 3(a). (a) Accuracy of binary SVM
classification as in Fig. 3(a), now with a coupling $g$, to an extrinsic
cavity mode that is not time-reversed (cavity frequency $\omega=1.7$). Despite
imperfect time-reversal, learning via OTOCs continues to provide an advantage
up to large spin-cavity couplings $g\sim 0.5$. (b) The minimum link strength
$J_{\ell}^{*}$ classifiable with $>90\%$ accuracy as a function of read-out
error $\delta$, obtained by repeating Fig. 3(a) for each $\delta$. The minimum
link strength in general decreases with decreasing $\delta$; for learning via
TOCs, this decrease plateaus for $\delta\lesssim 0.1\%$, indicating that
learning below this value is not limited by read-out error.

We begin as before with a concrete learning task. Specifically, we suppose
that one is given access to a spin chain with unknown Hamiltonian parameters
and either no link interaction ($J_{\ell}\rightarrow 0$) or a fixed non-zero
weak link interaction strength $J_{\ell}$. For each fixed value of $J_{\ell}$,
we train a binary SVM classifier on the correlation functions [Eqs. (2), (3)]
of 300 disorder samples, again including a read-out error $\delta=3\%$ in each
correlator value. We test model performance on 200 additional samples; the
resulting classification accuracies are shown in Fig. 3(b). We observe the
following general trends: ($i$) the accuracy decreases as $J_{\ell}$
decreases; ($ii$) learning via both OTOCs and TOCs (blue) allows detection of
$\sim\\!10$ times smaller $J_{\ell}$ than learning via only TOCs (red); and
($iii$) OTOCs allow detection of increasingly small $J_{\ell}$ as the size $L$
of the chain increases.

To understand this behavior analytically, we first note that the optimal
correlation functions for detecting the link will typically involve operators
lying immediately adjacent to that link, on both of its sides. These
correlators measure either the transfer of spin polarization (for TOCs) or
operator support (for OTOCs) across the link, and will be non-trivial only if
the link interaction strength is nonzero. For TOCs, one expects spin
polarization to cross the link incoherently, at a rate
$\sim\\!J_{\ell}^{2}/J$, where $J$ is the typical strong interaction strength.
Combined with an overall exponential decay of spin in time (if the system has
no conserved quantities), we expect
$C_{\text{TOC}}\sim(J_{\ell}^{2}/J)\,t\,e^{-Jt}$. For OTOCs, one expects an
operator’s support to cross the link at a similar rate,
$1-C_{\text{OTOC}}\sim(J_{\ell}^{2}/J)t$. Crucially however, this growth
persists until much _later_ times, $t\sim L/J$, at which information traveling
“around” the chain will abruptly cause the OTOC to decay to zero. The optimal
time for detecting the link occurs when these correlators are maximized, since
each is zero in the absence of the link. The TOC is maximized at an order one
time $t\sim 1/J$, at which the correlator magnitude $C_{\text{TOC}}\sim
J_{\ell}^{2}/J^{2}$ is suppressed by the square of the weak link interaction
strength. In contrast, the OTOC is maximized at a much later time $t\sim L/J$,
and thereby features a magnitude $1-C_{\text{OTOC}}\sim
L(J_{\ell}^{2}/J^{2})$. In both cases we see that detection of the link
becomes more difficult as the link strength decreases. Detection via the OTOC
is enhanced by a factor of $L$, which captures the connectivity change
associated with the link.

We confirm these estimates quantitatively by computing the Fisher information
of the link interaction strength. In Fig. 3(b), we plot the maximum Fisher
information $\max_{C}\text{FI}(\log(J_{\ell})|C)$ over all local correlation
functions, averaged over 100 disorder realizations. Here, we consider the
logarithm of the link interaction strength in order to appropriately compare
the Fisher information over multiple orders of magnitude of the interaction.
The Fisher information of $\log(J_{\ell})$ bounds the learnability of the
interaction strength as a _percentage_ of its actual value. Applying our
phenomenological estimates, we predict that $\text{FI}\sim J_{\ell}^{4}/J^{4}$
for TOCs, and $\text{FI}\sim L^{2}J_{\ell}^{4}/J^{4}$ for OTOCs. Observing
Fig. 3(b), we indeed find that the FI is suppressed by $\sim\\!J_{\ell}^{4}$
(dashed lines) for small $J_{\ell}$, and displays a multiplicative advantage
for OTOCs (blue) compared to TOCs (red), which grows as $L$ increases.

##  V Effect of experimental errors

Let us now address the impact of experimental errors on learning. We begin
with errors that accumulate throughout time-evolution. These may occur from
extrinsic decoherence or imperfect time-reversal dynamics, each of which
disrupt the non-local correlations probed by the OTOC Swingle and Halpern
(2018); Mi  _et al._ (2021); Sánchez  _et al._ (2021); Domínguez  _et al._
(2021); Schuster and Yao (2022). While this disruption can be mitigated via
independent error estimates Swingle and Halpern (2018); Mi  _et al._ (2021),
for sufficiently large errors these estimates involve measuring quantities of
small magnitude (comparable to the TOC), squandering the OTOC’s learning
advantage. In Appendix B we estimate that our previous results are modified in
the presence of a small local error rate $\varepsilon\ll J$ as follows: in the
first learning regime, the OTOC maintains its advantage up to distances
$d\lesssim J/\varepsilon$; in the second regime, the $L$-fold advantage is
replaced by a $(\min\\{L,\sqrt{J/\varepsilon}\\})$-fold advantage.

Figure 5:  Solution to the disjoint unitary problem with out-of-time-order
measurements. The state $\ket{0}^{\otimes n}$ is prepared and the unknown
unitary (either $U$ or $U_{1}\otimes U_{2}$) is applied. Next $\sigma_{x}$ is
applied to the first qubit, followed by the inverse of the unknown unitary.
Finally, it is checked if the second block of $n/2$ qubits ends up in the all
zero state. If so, then the hidden unknown unitary is $U_{1}\otimes U_{2}$ as
per case (ii); if not, then the unknown unitary is $U$ as per case (i).

In practice, we find that learning via OTOCs remains robust even to relatively
large amounts of imperfect time-reversal [Fig. 3(d)]. We study this
numerically in the “weak interaction” learning problem of Fig. 3(b). As a
concrete instance of imperfect time-reversal, we take the spins to be coupled
to an extrinsic cavity mode and assume that the spin dynamics are perfectly
reversed but the cavity dynamics and spin-cavity coupling $g$ are unreversed.
We find that access to OTOCs substantially improves the classification
accuracy even for quite large spin-cavity couplings $g\sim 0.5$, up to half
the spin-spin interaction strength.

We can also examine the dependence of learning on read-out errors, namely
where one measures a correlator $C$ up to additive error. Indeed, we have
already incorporated a realistic read-out error $\delta=3\%$ in our previous
numerical studies [Figs. 2(a), 3(a), 4(a)]. Intuitively, we expect larger
read-out errors to make learning more difficult; however, we have little
reason to expect read-out error to change the _relative_ advantage of OTOCs
compared to TOCs. We test this numerically by repeating the analysis of Fig.
3(b) for various read-out errors, $\delta$. For each $\delta$, we compute the
minimum link strength $J_{\ell}^{*}$ that can be learned with $>90\%$ accuracy
[Fig. 3(c)]. For errors $\delta\gtrsim 10^{-3}$, our results agree well with
analytic estimates, which predict $(J_{\ell}^{*}/J)^{2}\sim\delta$ for TOCs
and $(J_{\ell}^{*}/J)^{2}\sim\delta/L$ for OTOCs. Intriguingly, for
sufficiently small errors $\delta\lesssim 10^{-3}$, the minimum link strength
detectable with TOCs _saturates_ to a finite value $J_{\ell}^{*}\sim 0.2$.
Below this value, sample-to-sample fluctuations of the TOC cause the learning
task to be difficult regardless of the read-out error.

##  VI Provable learning advantage

We have so far demonstrated the learning power of OTOCs using phenomenological
arguments and numerical simulations, owing to the difficulty of obtaining
analytic results for ergodic Hamiltonian systems. Complementary to these
results, we now introduce a binary classification task in which the OTOC is
provably efficient. The task is as follows:

Disjoint unitary problem: One is given oracle access to either: ($i$) a fixed,
$n$-qubit Haar-random unitary $U$, or ($ii$) a tensor product of two fixed,
$n/2$-qubit Haar-random unitaries, $U_{1}\otimes U_{2}$. The task is to
determine which of (i) or (ii) is realized.

Qualitatively, this problem resembles the Hamiltonian learning scenarios
identified previously. First, the feature we seek to learn—the connectivity of
the unitary—directly determines how information spreads through the system, as
measured by the OTOC. Second, a Haar-random unitary is inherently “strongly-
interacting”, which causes time-ordered measurements to decay and thus provide
little information.

In Fig. 5 we show that the disjoint unitary problem can be solved with a
constant number (with respect to $n$) of queries to the oracle _and its time-
reverse_ $U^{\dagger}$, by measuring an out-of-time-order observable. Letting
$V$ denote the unknown unitary (either $U$ or $U_{1}\otimes U_{2}$), the OTOC
is

$\textsf{OTOC}(V)=\text{tr}\\!\left(\mathds{1}_{\frac{n}{2}}\\!\otimes|0\rangle\langle
0|^{\otimes\frac{n}{2}}\\!\left\\{V^{\dagger}\sigma_{x}^{1}V|0\rangle\langle
0|^{\otimes n}V^{\dagger}\sigma_{x}^{1}V\right\\}\right).$

In case ($i$), the OTOC is near zero with probability exponentially close to
one Cotler  _et al._ (2022). In case ($ii$), the OTOC is one, since the two
subsystems are not coupled by $U_{1}\otimes U_{2}$. Thus, with probability
exponentially close to one, the two cases may be distinguished with a single
query to the unknown unitary and its time-reverse. In contrast, in a companion
work Cotler  _et al._ (2022), we prove that any time-ordered learning protocol
requires an exponential number $\Omega(2^{n/4})$ of queries of the unknown
unitary to solve the disjoint unitary problem. Our proof applies even to
adaptive measurement strategies, and leverages novel contemporary techniques
from quantum learning theory Aharonov  _et al._ (2021); Chen  _et al._
(2021a); Huang  _et al._ (2021).

##  VII Discussion

In this work, we have shown that out-of-time-order measurements can provide
powerful advantages for learning the dynamics of quantum systems. Our results
thus highlight the potential gains that can be achieved by quantum experiments
if they have sufficient control and coherence to apply time-reversed dynamics.
Extraordinary experimental progress has led to an ever-increasing number of
such platforms Li  _et al._ (2017); Gärttner  _et al._ (2017); Landsman  _et
al._ (2019); Blok  _et al._ (2020); Mi  _et al._ (2021); Sánchez  _et al._
(2021); Domínguez  _et al._ (2021), and we envision that learning via OTOCs
might find applications across these diverse physical contexts. Specific
future directions include learning long-range cross-talk in quantum processors
Sarovar  _et al._ (2020), and strongly-interacting problems in NMR O’Brien
_et al._ (2021).

On the theoretical front, our results follow in the footsteps of recent works
in quantum learning theory Aharonov  _et al._ (2021); Huang  _et al._ (2021);
Chen  _et al._ (2021a, b); Cotler  _et al._ (2021) to provide new avenues for
quantum advantage. Our applications pertain to genuine questions of
experimental interest, providing a new bridge between the theoretical tools of
quantum learning theory and problems of practical importance in experiments.

_Acknowledgements_ —We are grateful to Ryan Babbush, Soonwon Choi, Kostyantyn
Kechedzhi, Bryce Kobrin, Lev Ioffe, Vadim Smelyanskiy and Norman Y. Yao for
insightful discussions. The numerical simulations performed in this work used
the dynamite Python frontend dyn , which supports a matrix-free implementation
of Krylov subspace methods based on the PETSc and SLEPc packages Hernandez
_et al._ (2005). T.S. acknowledges support from the National Science
Foundation Graduate Research Fellowship Program under Grant No. DGE 1752814.
J.C. is supported by a Junior Fellowship from the Harvard Society of Fellows,
the Black Hole Initiative, as well as in part by the Department of Energy
under grant DE-SC0007870.

## References

  * Cotler  _et al._ (2022) J. Cotler, T. Schuster, and M. Mohseni, forthcoming (2022).
  * Mohseni  _et al._ (2008) M. Mohseni, A. T. Rezakhani, and D. A. Lidar, Physical Review A 77, 032322 (2008).
  * Huang  _et al._ (2022) H.-Y. Huang, M. Broughton, J. Cotler, S. Chen, J. Li, M. Mohseni, H. Neven, R. Babbush, R. Kueng, J. Preskill, and J. R. McClean, Science 376, 1182 (2022), https://www.science.org/doi/pdf/10.1126/science.abn7293 .
  * Wiebe  _et al._ (2014) N. Wiebe, C. Granade, C. Ferrie, and D. Cory, Physical Review A 89, 042314 (2014).
  * Aharonov  _et al._ (2021) D. Aharonov, J. Cotler, and X.-L. Qi, arXiv:2101.04634 (2021).
  * Huang  _et al._ (2021) H.-Y. Huang, M. Broughton, J. Cotler, S. Chen, J. Li, M. Mohseni, H. Neven, R. Babbush, R. Kueng, J. Preskill, _et al._ , arXiv:2112.00778 (2021).
  * Zhou  _et al._ (2020) H. Zhou, J. Choi, S. Choi, R. Landig, A. M. Douglas, J. Isoya, F. Jelezko, S. Onoda, H. Sumiya, P. Cappellaro, _et al._ , Physical Review X 10, 031003 (2020).
  * O’Brien  _et al._ (2021) T. E. O’Brien, L. B. Ioffe, Y. Su, D. Fushman, H. Neven, R. Babbush, and V. Smelyanskiy, arXiv:2109.02163 (2021).
  * Laws  _et al._ (2002) D. D. Laws, H.-M. L. Bitter, and A. Jerschow, Angewandte Chemie International Edition 41, 3096 (2002).
  * Wang  _et al._ (2015) S.-T. Wang, D.-L. Deng, and L.-M. Duan, New Journal of Physics 17, 093017 (2015).
  * Ezzell  _et al._ (2022) N. Ezzell, B. Pokharel, L. Tewala, G. Quiroz, and D. A. Lidar, arXiv preprint arXiv:2207.03670 (2022).
  * Wang  _et al._ (2017) J. Wang, S. Paesani, R. Santagati, S. Knauer, A. A. Gentile, N. Wiebe, M. Petruzzella, J. L. O?Brien, J. G. Rarity, A. Laing, _et al._ , Nature Physics 13, 551 (2017).
  * Bairey  _et al._ (2019) E. Bairey, I. Arad, and N. H. Lindner, Physical Review Letters 122, 020504 (2019).
  * Haah  _et al._ (2021) J. Haah, R. Kothari, and E. Tang, arXiv:2108.04842 (2021).
  * Larkin and Ovchinnikov (1969) A. Larkin and Y. N. Ovchinnikov, Sov Phys JETP 28, 1200 (1969).
  * Baum  _et al._ (1985) J. Baum, M. Munowitz, A. N. Garroway, and A. Pines, The Journal of Chemical Physics 83, 2015 (1985).
  * Shenker and Stanford (2014) S. H. Shenker and D. Stanford, Journal of High Energy Physics 2014, 67 (2014).
  * Stanford (2016) D. Stanford, Journal of High Energy Physics 2016, 1 (2016).
  * Maldacena  _et al._ (2016) J. Maldacena, S. H. Shenker, and D. Stanford, Journal of High Energy Physics 2016, 1 (2016).
  * Maldacena and Stanford (2016) J. Maldacena and D. Stanford, Physical Review D 94, 106002 (2016).
  * Hosur  _et al._ (2016) P. Hosur, X.-L. Qi, D. A. Roberts, and B. Yoshida, Journal of High Energy Physics 2016, 1 (2016).
  * Cotler  _et al._ (2017) J. Cotler, N. Hunter-Jones, J. Liu, and B. Yoshida, Journal of High Energy Physics 2017, 1 (2017).
  * Cotler  _et al._ (2018) J. S. Cotler, D. Ding, and G. R. Penington, Annals of Physics 396, 318 (2018).
  * Nahum  _et al._ (2018) A. Nahum, S. Vijay, and J. Haah, Physical Review X 8, 021014 (2018).
  * Von Keyserlingk  _et al._ (2018) C. Von Keyserlingk, T. Rakovszky, F. Pollmann, and S. L. Sondhi, Physical Review X 8, 021013 (2018).
  * Xu and Swingle (2022) S. Xu and B. Swingle, arXiv:2202.07060 (2022).
  * Gärttner  _et al._ (2017) M. Gärttner, J. G. Bohnet, A. Safavi-Naini, M. L. Wall, J. J. Bollinger, and A. M. Rey, Nature Physics 13, 781 (2017).
  * Mi  _et al._ (2021) X. Mi, P. Roushan, C. Quintana, S. Mandra, J. Marshall, C. Neill, F. Arute, K. Arya, J. Atalaya, R. Babbush, _et al._ , arXiv:2101.08870 (2021).
  * Sánchez  _et al._ (2021) C. M. Sánchez, A. K. Chattah, and H. M. Pastawski, arXiv:2112.00607 (2021).
  * Domínguez  _et al._ (2021) F. D. Domínguez, M. C. Rodríguez, R. Kaiser, D. Suter, and G. A. Álvarez, Physical Review A 104, 012402 (2021).
  * Burgarth  _et al._ (2009) D. Burgarth, K. Maruyama, and F. Nori, Physical Review A 79, 020305 (2009).
  * Lovchinsky  _et al._ (2016) I. Lovchinsky, A. Sushkov, E. Urbach, N. P. de Leon, S. Choi, K. De Greve, R. Evans, R. Gertner, E. Bersin, C. Müller, _et al._ , Science 351, 836 (2016).
  * Gentile  _et al._ (2021) A. A. Gentile, B. Flynn, S. Knauer, N. Wiebe, S. Paesani, C. E. Granade, J. G. Rarity, R. Santagati, and A. Laing, Nature Physics 17, 837 (2021).
  * Chen  _et al._ (2021a) S. Chen, J. Cotler, H.-Y. Huang, and J. Li, arXiv:2111.05881 (2021a).
  * fn (4) To see this explicitly, we specify to the case that $A_{k},B_{l}$ are Pauli operators. Consider four possible operations on the density matrix $\rho$: the two unitary rotations $\rho\rightarrow\frac{\mathbbm{1}\pm iA_{k}}{\sqrt{2}}\rho\frac{\mathbbm{1}\mp iA_{k}}{\sqrt{2}}$ (where the signs are chosen together) and the two projections $\rho\rightarrow\frac{\mathbbm{1}\pm A_{k}}{2}\rho\frac{\mathbbm{1}\pm A_{k}}{2}$. By taking linear combinations of these four possibilities in classical post-processing, we can in effect implement the matrix multiplications $\rho\rightarrow A_{k}\rho$ (and similarly $\rho\rightarrow\rho B_{l}$). Applying the appropriate operation at each time $t_{k}$ (and $t^{\prime}_{l}$) gives Eq. (1) .
  * D’Alessio  _et al._ (2016) L. D’Alessio, Y. Kafri, A. Polkovnikov, and M. Rigol, Advances in Physics 65, 239 (2016).
  * Lieb and Robinson (1972) E. H. Lieb and D. W. Robinson, in  _Statistical mechanics_ (Springer, 1972) pp. 425–431.
  * Aleiner  _et al._ (2016) I. L. Aleiner, L. Faoro, and L. B. Ioffe, Annals of Physics 375, 378 (2016).
  * Boss  _et al._ (2016) J. M. Boss, K. Chang, J. Armijo, K. Cujia, T. Rosskopf, J. R. Maze, and C. L. Degen, Physical Review Letters 116, 197601 (2016).
  * Theis  _et al._ (2016) T. Theis, G. X. Ortiz Jr, A. W. Logan, K. E. Claytor, Y. Feng, W. P. Huhn, V. Blum, S. J. Malcolmson, E. Y. Chekmenev, Q. Wang, _et al._ , Science Advances 2, e1501438 (2016).
  * Blanchard  _et al._ (2021) J. W. Blanchard, D. Budker, and A. Trabesinger, Journal of Magnetic Resonance 323, 106886 (2021).
  * Burgarth and Ajoy (2017) D. Burgarth and A. Ajoy, Physical Review Letters 119, 030402 (2017).
  * Wei  _et al._ (2018) K. X. Wei, C. Ramanathan, and P. Cappellaro, Physical review letters 120, 070501 (2018).
  * Sánchez  _et al._ (2019) C. M. Sánchez, A. K. Chattah, K. X. Wei, L. Buljubasich, P. Cappellaro, and H. M. Pastawski, arXiv:1902.06628 (2019).
  * Burgarth and Maruyama (2009) D. Burgarth and K. Maruyama, New Journal of Physics 11, 103019 (2009).
  * Di Franco  _et al._ (2009) C. Di Franco, M. Paternostro, and M. Kim, Physical Review Letters 102, 187203 (2009).
  * Zhang and Sarovar (2014) J. Zhang and M. Sarovar, Physical Review Letters 113, 080401 (2014).
  * Sone and Cappellaro (2017) A. Sone and P. Cappellaro, Physical Review A 95, 022335 (2017).
  * fn (1) To be precise, our numerical simulations are relevant to the scenario where the experimenter can perform $O(1)$ local spin rotations within the time-scale set by the inverse spin interaction strength. If substantially faster local rotations are possible, then the Hamiltonian interactions could be individually tuned via local dynamical decoupling, which would allow substantially faster learning Wang  _et al._ (2015) .
  * (50) For more information visit: https://dynamite.readthedocs.io/en/latest/ .
  * Kay (1993) S. Kay, Technometrics 37, 465 (1993).
  * Aaronson (2019) S. Aaronson, SIAM Journal on Computing 49, STOC18 (2019).
  * Cotler and Wilczek (2020) J. Cotler and F. Wilczek, Physical Review Letters 124, 100401 (2020).
  * Huang  _et al._ (2020) H.-Y. Huang, R. Kueng, and J. Preskill, Nature Physics 16, 1050 (2020).
  * Swingle and Halpern (2018) B. Swingle and N. Y. Halpern, Physical Review A 97, 062113 (2018).
  * Schuster and Yao (2022) T. Schuster and N. Yao, forthcoming (2022).
  * Li  _et al._ (2017) J. Li, R. Fan, H. Wang, B. Ye, B. Zeng, H. Zhai, X. Peng, and J. Du, Physical Review X 7, 031011 (2017).
  * Landsman  _et al._ (2019) K. A. Landsman, C. Figgatt, T. Schuster, N. M. Linke, B. Yoshida, N. Y. Yao, and C. Monroe, Nature 567, 61 (2019).
  * Blok  _et al._ (2020) M. Blok, V. Ramasesh, T. Schuster, K. O’Brien, J. Kreikebaum, D. Dahlen, A. Morvan, B. Yoshida, N. Yao, and I. Siddiqi, arXiv:2003.03307 (2020).
  * Sarovar  _et al._ (2020) M. Sarovar, T. Proctor, K. Rudinger, K. Young, E. Nielsen, and R. Blume-Kohout, Quantum 4, 321 (2020).
  * Chen  _et al._ (2021b) S. Chen, J. Cotler, H.-Y. Huang, and J. Li, arXiv:2111.05874 (2021b).
  * Cotler  _et al._ (2021) J. Cotler, H.-Y. Huang, and J. R. McClean, arXiv:2112.00811 (2021).
  * Hernandez  _et al._ (2005) V. Hernandez, J. E. Roman, and V. Vidal, ACM Transactions on Mathematical Software (TOMS) 31, 351 (2005).
  * Buitinck  _et al._ (2013) L. Buitinck, G. Louppe, M. Blondel, F. Pedregosa, A. Mueller, O. Grisel, V. Niculae, P. Prettenhofer, A. Gramfort, J. Grobler, R. Layton, J. VanderPlas, A. Joly, B. Holt, and G. Varoquaux, in  _ECML PKDD Workshop: Languages for Data Mining and Machine Learning_ (2013) pp. 108–122.
  * Lin (1991) J. Lin, IEEE Transactions on Information theory 37, 145 (1991).
  * Nahum  _et al._ (2017) A. Nahum, J. Ruhman, S. Vijay, and J. Haah, Physical Review X 7, 031016 (2017).
  * Qi and Streicher (2019) X.-L. Qi and A. Streicher, Journal of High Energy Physics 2019, 1 (2019).
  * Revathi and Balakrishnan (1993) S. Revathi and V. Balakrishnan, Journal of Physics A: Mathematical and General 26, 5661 (1993).

##  Appendix A Details of numerical simulations

Here we provide further details on the numerical simulations displayed in
Figs. 2, 3, 4 of the main text.

###  A.1 Correlation functions

We begin by explicitly writing down the correlation functions used in Figs. 2,
3, 4. Throughout, we denote time-evolved operators as $V(t)\equiv
U(t,0)VU(t,0)^{\dagger}$, where the time-evolution unitary is
$U(t_{2},t_{1})=\mathcal{T}\left\\{e^{-i\int_{t_{1}}^{t_{2}}dt\,H(t)}\right\\}$
and $H(t)$ is the time-dependent stroboscopic Floquet Hamiltonian specified in
the main text (unless otherwise stated, in Fig. 7).

In the restricted access scenario considered in Figs. 2, 7, we use the
following correlation functions:

$\begin{split}C_{\text{TOC}}(t)&=\left\langle
V_{p}(t)\,V_{p}(0)\right\rangle\\\ C_{\text{TOC}}(x,t)&=\left\langle
V_{p}(t)\,W_{x}(t/2)\,V_{p}(0)\,W_{x}(t/2)\right\rangle\\\
C_{\text{OTOC}}(x,t)&=\left\langle
V_{p}(t)\,W_{x}(0)\,V_{p}(t)\,W_{x}(0)\right\rangle\end{split}$ (4)

where $p$ denotes the probe qubit, and $\langle\cdot\rangle\equiv
2^{-L}\tr(\cdot)$ is an infinite temperature average. Each of these
correlation functions can be measured using state preparation and read-out on
the probe qubit, combined with time-evolution and a single local unitary
operation on the larger system. (In the case of the auto-correlation function
$C_{\text{TOC}}(t)$ no local unitary operation is needed,
$W_{x}=\mathbbm{1}$.)

In principle, we envision allowing $V,W$ to run over all local operators in
the system. For instance, they could run over all $4^{w}\binom{N}{w}$ Pauli
operators of weight $\leq w$, where $w\sim O(1)$. This is naturally achieved
by randomized measurement strategies such as shadow tomography with local
Clifford unitaries and $O(3^{w})$ measurements Cotler and Wilczek (2020);
Huang  _et al._ (2020). In practice, we must restrict $V,W$ to a few possible
values in numerical simulations. Specifically, we take
$V=W\in\\{\sigma_{x},\sigma_{z}\\}$ for TOCs, and $V=W\in\\{\sigma_{z}\\}$ for
OTOCs. The OTOC is observed to be relatively insensitive to basis of $V$ and
$W$, hence our choice to restrict to a single operator, $\sigma_{z}$ (further,
we note that adding $\sigma_{x}$ OTOCs could only improve the relative
advantage of OTOCs compared to TOCs). More broadly, we do not expect that
adding additional pairs of $\\{V,W\\}$ will change the qualitative behavior of
learning via TOCs and OTOCs. Specifically, we have seen that the learning
advantage of OTOCs arises from their ability to detect highly non-local
correlations in the system (i.e. large-weight components of the time-evolved
operator $V_{p}(t)$, see Appendix B for more detailed phenomenological
estimates). These correlations are not detectable by any time-ordered
correlator involving only few-body operators; indeed, in ergodic systems we
generically expect that they are not efficiently detectable by _any_ time-
ordered measurement.

For Figs. 3, 4 of the main text, we utilize two-point correlation functions
between pairs of local operators:

$\begin{split}C_{\text{TOC}}&=\left\langle
V_{x}(t)\,W_{x^{\prime}}(0)\right\rangle,\\\ C_{\text{OTOC}}&=\left\langle
V_{x}(t)\,W_{x^{\prime}}(0)\,V_{x}(t)\,W_{x^{\prime}}(0)\right\rangle.\end{split}$
(5)

We again take $V=W\in\\{\sigma_{x},\sigma_{z}\\}$ for TOCs and
$V=W\in\\{\sigma_{z}\\}$ for OTOCs. We allow $x,x^{\prime}$ to span all qubits
within a distance 2 of the link—this consists of 6 possible values for each of
$x,x^{\prime}$, corresponding to distances 0, 1, and 2 to both the left and
right of the link. In principle, we would like $x,x^{\prime}$ to run over the
entire lattice; however, in practice we observe that correlation functions
involving qubits distant from the link provide little information, and so can
be safely neglected.

We now briefly comment on our numerical methods for computing the above
correlation functions and the Fisher information [Figs. 2(b), 3(b)]. We
compute the infinite temperature average in the correlation functions by
sampling over Haar-random initial states $\ket{\psi}$. To motivate this, we
can insert a resolution of the identity,
$\mathbbm{1}=\frac{1}{2^{L}}\sum_{\psi}\outerproduct{\psi}{\psi}$ into the
correlation functions Eq. (5) to obtain:

$\begin{split}C_{\text{TOC}}&=\frac{1}{2^{L}}\sum_{\psi}\bra{\psi}V_{i}(t)\,W_{j}(0)\ket{\psi}\,,\\\
\end{split}$ (6)

and similarly for the OTOC. In numerics, we approximate this sum by sampling a
finite number $N_{\psi}$ of states $|\psi\rangle$ drawn from the Haar
distribution; errors in this approximation will scale as $\sim
1/\sqrt{N_{\psi}2^{L}}$.

Figure 6:  Depiction of the extrapolation method used to calculate the maximum
Fisher information over time-ordered correlators [Fig. 2(b)]. Each correlation
function is computed for 25 Haar-random values of the state $\ket{\psi}$ [Eq.
(6)]. For each value of $N_{\psi}$ between 1 and 25, we choose a random subset
of $N_{\psi}$ values of $\ket{\psi}$ and compute the average correlation
function over the subset. (Left) For each value of $N_{\psi}$, we then compute
the maximum Fisher information over all correlation functions, $\text{max
FI}(N_{\psi})$ (solid red lines, darker lines corresponds to higher
$N_{\psi}$). (Right) Our estimate of the maximum Fisher information at
infinite temperature (dotted lines, both plots) is obtained by fitting
$\text{max FI}(N_{\psi})=\text{max FI}(\infty)+A/N_{\psi}$ and taking
$N_{\psi}\rightarrow\infty$ (points denote data, solid lines denote
$1/N_{\psi}$-fit).

In the learning problems considered in the main text [Fig. 2(a), 3(a), 4], we
take $N_{\psi}=25,25,10,1,1$ for system sizes $L=6,8,10,12,14$, respectively.
In contrast, when estimating the Fisher information [Fig. 2(b), 3(b)], we
perform a large-$N_{\psi}$ extrapolation to improve precision. This is
required in order to establish the asymptotic scaling of the Fisher
information at large $d$ [Fig. 2(b)] and small $J_{\ell}$ [Fig. 3(b)].
Specifically, we compute the estimated correlation function $C_{N_{\psi}}$
averaged over $N_{\psi}=1,\ldots,25$ Haar-random initial states, as well as
the resultant Fisher information $\text{max FI}(N_{\psi})\equiv|\partial
C_{N_{\psi}}/\partial J|^{2}$, maximized over all relevant correlation
functions. We then perform a linear fit $\text{max FI}(N_{\psi})=\text{max
FI}(\infty)+\frac{A}{N_{\psi}}$, where $\text{max FI}(\infty)$ and $A$ are
fitting parameters. Finally, the fitting parameter $\text{max FI}(\infty)$
represents our estimation of the Fisher information at
$N_{\psi}\rightarrow\infty$, which we plot in Figs. 2(b), 3(b). We illustrate
this procedure in Fig. 6, using the data for Fig. 2(b). On the left of Fig. 6,
we plot the maximum Fisher information, $\text{max FI}(N_{\psi})$, for each
$N_{\psi}$, as a function of the distance $d$. We observe that in regions
where $C$ is relatively large (i.e. small $d$), the estimates are quite
accurate even for $N_{\psi}=1$, while in regions where $C$ is small (i.e.
large $d$) the Fisher information becomes successively smaller as the number
of sampled states $N_{\psi}$ increases. On the right of Fig. 6, we re-plot the
Fisher information for each $d$ as a function of $N_{\psi}$. Solid lines
represent the results of the linear fit, which we observe to fit the
$N_{\psi}$-dependence of the data quite well. The extrapolated Fisher
information [as displayed in Fig. 2(b)] is shown in Fig. 6 as a dashed line.

###  A.2 Imperfect time-reversal via cavity mode

In Fig. 4(a), we benchmark the effects of decoherence on learning by coupling
the spin system to a single cavity mode. Our motivation for studying this
model is two-fold. First, in ergodic many-body systems the effect of local
errors on OTOCs is expected to be independent of the precise microscopic form
of the error Schuster and Yao (2022). We therefore expect the spin-cavity
system to display similar OTOC physics to more generic local error models.
Second, for $L=10$ spins the spin-cavity system can be exactly simulated in a
Hilbert space of size $2^{L}\times L$ (we assume the cavity initially has zero
occupation number; since the sum of spin magnetization and the cavity
occupation is conserved, the cavity occupation is upper bounded by $L$). This
is substantially smaller than the requirements to exactly simulate a mixed
state quantum system, $~{}2^{2L}$.

More specifically, the spin-cavity Hamiltonian is as follows. We modify the
Floquet time-evolution described in the main text to alternate between the
following two Hamiltonians:

$\begin{split}H_{1}&=\pm
H_{f}+g\sum_{i}\left(a^{\dagger}\sigma_{i}^{-}+a\sigma_{i}^{+}\right)+\omega
a^{\dagger}a\\\ H_{2}&=\pm
H_{c}+g\sum_{i}\left(a^{\dagger}\sigma_{i}^{-}+a\sigma_{i}^{+}\right)+\omega
a^{\dagger}a\end{split}$ (7)

where $H_{f},H_{c}$ are the field and coupling Hamiltonians written in the
main text, $a,a^{\dagger}$ are lowering/raising operators for a bosonic cavity
mode, $g=\\{0.0,0.25,0.5\\}$ is the spin-cavity interaction strength, and
$\omega=1.7$ is the cavity frequency. Here, the $\pm$ denote values during
forwards/backwards time-evolution; note that we do _not_ reverse the spin-
cavity interaction or the cavity frequency during backwards time-evolution.

###  A.3 Learning model

We now detail the machine learning techniques used in Figs. 2(a), 3(a), 4 of
the main text, and Fig. 8(b) of the Appendix. Throughout, read-out error is
mimicked by adding a random Gaussian variable with mean zero and standard
deviation $\delta$ to the exact correlation functions.

We begin with Fig. 2(a). Our goal is to predict the value of $d$ [which
specifies the geometry of the spin system, see Fig. 2(a)] from the correlation
functions of the system, Eq. (4). To do so, we train a learning model on 3000
randomly drawn disorder realizations of the Hamiltonian, consisting of 300
realizations each for $d=0,1,\ldots,9$. We test model performance on 2000
additional disorder realizations, again consisting of 200 realizations each
for $d=0,1,\ldots,9$. For each disorder realization, the input to our learning
model consists of the correlation functions Eq. (4), evaluated at
$x=2,\ldots,L$ and $30$ evenly spaced times between $0$ and $12$. We apply
Gaussian distributed read-out error $\delta=3\%$ to each correlation function.
We repeat this procedure, as well as the model training and evaluation that
follows, first using only TOCs as input to the learning algorithm, and second
using both TOCs and OTOCs.

Next, we input these correlation functions into a support vector regression
(SVR) model with radial basis functions Buitinck  _et al._ (2013). The radial
SVR contains two hyperparameters: $C$, the regularization parameter, and
$\gamma$, which controls the width of the radial basis functions. We choose
$C$ and $\gamma$ by performing five-fold cross-validation over the sets
$C=\\{10,30,100\\}$, $\gamma=\\{0.03,0.06,0.1,0.3,1,3,6,10\\}$. We obtain
$C=10,\gamma=1$ for learning via TOCs, and $C=10,\gamma=0.03$ for learning via
both TOCs and OTOCs. In the identical learning task for Hamiltonian evolution
[Fig. 7(a)], we obtain $C=10,\gamma=1$ for learning via TOCs, and
$C=100,\gamma=0.03$ for learning via both TOCs and OTOCs.

We now turn to Fig. 3(a) and 4. Our goal is to perform binary classification
using the correlation functions Eq. (5) to distinguish whether the link
interaction strength is zero or nonzero. To do so, we simulate the correlation
functions of 300 randomly drawn disorder realizations of the Hamiltonian for
each link strength,
$J_{\ell}=\\{0,0.01,0.017,0.03,0.06,0.1,0.17,0.3,0.6,1.0\\}$. For each nonzero
$J_{\ell}$, we train a learning model to perform binary classification between
link strength $0$ and $J_{\ell}$. We test model performance on 400 additional
disorder realizations, again consisting of 200 realizations each for link
strength $0$ and $J_{\ell}$.

The first step of our learning model is to prune the correlation functions
used as input. We do so by estimating the mutual information between each
individual correlation function and the link interaction strength, and
selecting the $K$ correlation functions with the highest mutual information.
Here $K$ is a hyperparameter that will ultimately be chosen via cross-
validation. To estimate the mutual information, we fit the distribution of
correlation functions values over disorder realizations to a Gaussian for each
link strength, and compute the Jensen-Shannon divergence between the Gaussian
distributions. The Jensen-Shannon divergence is equal to the desired mutual
information Lin (1991).

As before, we input the selected correlation functions into a support vector
machine (SVM) with radial basis functions Buitinck  _et al._ (2013). We now
have three hyperparameters: the SVM hyperparameters $C$ and $\gamma$ and the
number of selected correlation functions $K$. We choose $C$, $\gamma$, and $K$
by performing five-fold cross-validation over each value
$C=\\{0.1,1,10,100,1000\\}$, $\gamma=\\{0.1,0.3,1,3,10,30\\}$. We obtain Figs.
4(b), by repeating this procedure for various simulated read-out errors,
$\delta=\\{0.0001,0.0003,0.001,0.003,0.01,0.03,0.1\\}$. At each read-out
error, we perform a linear interpolation of the classification accuracy as a
function of $J_{\ell}$ [as shown in Fig. 3(a) for $\delta=0.03$]. The minimum
detectable link strength $J^{*}_{\ell}$ occurs at the intersection of this
interpolation with a horizontal line (not depicted) corresponding to a
classification accuracy of $90\%$.

Finally, we turn to Fig. 8(b) in Appendix C. In the probe qubit scenario,
training and testing are performed on $300$ and $200$ samples respectively for
each geometry and each value of $d$. In the global state preparation and read-
out scenario, we instead use $60$ and $40$ samples respectively for each
geometry. Our learning model consists of a support vector machine with
hyperparameters chosen via 4-fold cross-validation from the sets
$C=\\{0.1,1,10\\}$, $\gamma=\\{0.3,3,30\\}$. As in our previous learning
tasks, we apply a read-out error $\delta=3\%$ to each correlation function
before use in learning.

##  Appendix B Phenomenological estimates

In this section, we provide more detailed reasoning behind the
phenomenological estimates of the Fisher information presented in the main
text. We begin with the Fisher information under unitary dynamics with
restricted access and then turn to the effects of imperfect time-reversal and
decoherence.

###  B.1 Fisher information in restricted access scenario

At sufficiently large times and distances, we expect the profile of
correlation functions in ergodic many-body systems to be described by only a
few phenomenological parameters. For instance, in one-dimensional systems the
out-of-time-order correlator is predicted to take the following functional
form Aleiner  _et al._ (2016); Nahum  _et al._ (2017),

$C_{\text{OTOC}}(x,t)\approx f\left(\frac{x/v_{B}-t}{A\sqrt{t}}\right)$ (8)

where the phenomenological parameters $v_{B}$ and $A$ describe the butterfly
velocity and the width of the OTOC wavefront, respectively. Here $f$ is a
compactly supported bump function which interpolates between zero and one and
then zero again within an $O(1)$-sized region about the origin. Meanwhile, in
systems with a local conservation law, we expect time-ordered correlators to
be dominated by diffusion of the conserved quantity. This leads to the
following profile for the auto-correlation function,

$C_{\text{TOC}}(t)\sim\frac{1}{\sqrt{Dt}},$ (9)

where $D$ is a diffusion constant. In the absence of conserved quantities, one
expects time-ordered correlation functions to instead decay exponentially in
time,

$C_{\text{TOC}}(t)\sim\exp(-\gamma t),$ (10)

parameterized by a decay rate $\gamma$.

To obtain the Fisher information, $\text{FI}(J|C)=\left|\partial C/\partial
J\right|^{2}$, we must compute the derivative of the correlation functions
with respect to a local coupling strength, $J_{y}$. To do so while leveraging
the above phenomenological predictions, we must first recognize that the
phenomenological parameters are themselves dependent on the local coupling
strengths of the system, e.g. $v_{B}\rightarrow v_{B}(\\{J_{y}\\})$. We expand
on this in further detail for each case below. The resultant scaling of the
Fisher information in various physical regimes is summarized in Table 2.

Table 1: Maximum Fisher information in restricted access scenarios Correlation function |  Probe qubit with | Probe qubit with
---|---|---
|  local unitary control | global unitary control
TOC without |  $O(\exp(-d))$, |  $O(\exp(-d))$,
conserved quantity | [Fig. 2(b)] |
TOC with | $O(1/d^{4})$ | $O(1/d^{4})$
conserved quantity |  |
OTOC |  $O(1/d)$, |  $O(1/d^{2})$,
| [Fig. 2(b)] |
Table 2: Phenomenological estimates of the scaling of the Fisher information
in the restricted access scenario, for learning an interaction that lies a
distance $d$ from the probe qubit.

####  B.1.1 Fisher information of OTOCs

We begin with the Fisher information of OTOCs. Our treatment is broken into
two parts, corresponding to the scenarios where the experimenter has either
local or global unitary control over the larger system. The former scenario is
simulated numerically in Fig. 2 of the main text.

_Local unitary control._ —We consider local OTOCs [Eq. (4)] and are interested
in the dependence of the OTOC on the local coupling strengths, $\\{J_{y}\\}$.
To approach this, we will assume that OTOC takes the same functional form as
in Eq. (8),

$C_{\text{OTOC}}(x,t)=\left\langle
V_{p}(t)\,W_{x}(0)\,V_{p}(t)\,W_{x}(0)\right\rangle\approx
f\left(\frac{x/v_{B}(x)-t}{A\sqrt{t}}\right),$ (11)

but now with a _position-dependent_ butterfly velocity, $v_{B}(x)$.
Specifically, we assume that the effective butterfly velocity at time $t$
receives contributions from all couplings that have been visited thus far,
i.e. all $J_{y}$ with $y\lesssim x$. Since the time to traverse a single
coupling is proportional to the inverse coupling strength $1/J_{y}$, we expect
the time to traverse all couplings up to a distance $x$ to be proportional to
the sum $\sum_{y=0}^{x}1/J_{y}$. Equating this time to the distance divided by
the effective butterfly velocity $x/v_{B}(x)$, we have:

$v_{B}(x)\approx\left[\frac{1}{x}\sum_{x=0}^{x}\frac{1}{J_{x}}\right]^{-1}.$
(12)

If each coupling strength is drawn independently from some disorder
realization, then at large times the butterfly velocity will be close to its
typical value, $v_{B}=\overline{1/J}$.

We can now compute derivatives of the correlation function with respect to a
given coupling strength via the chain rule. The derivative of the butterfly
velocity is

$\partial_{J_{d}}v_{B}(x)\approx\frac{v_{B}(x)^{2}}{J_{d}^{2}x}\cdot\delta_{d\leq
x},$ (13)

which yields the following for the OTOC:

$\begin{split}\partial_{J_{d}}C_{\text{OTOC}}(x,t)&\approx-\frac{x}{A\sqrt{t}v_{B}(x)^{2}}\cdot
f^{\prime}\left(\frac{x/v_{B}(x)-t}{A\sqrt{t}}\right)\cdot\partial_{J_{d}}v_{B}(x)\\\
&\approx-\frac{1}{AJ_{d}^{2}\sqrt{t}}\cdot\delta_{d\leq x}\cdot
f^{\prime}\left(\frac{x/v_{B}(x)-t}{A\sqrt{t}}\right).\\\ \end{split}$ (14)

There are two parameters of the local OTOC chosen by a potential
experimentalist: the position $x$ of the local perturbation, and the evolution
time $t$. We are interested in the maximum Fisher information given an optimal
choice of $x$ and $t$. Observing Eq. (14), we see that the derivative
$f^{\prime}$ is maximized by the choice $x=v_{B}t$, while the delta function
then sets $v_{B}t=d$. Plugging these values in, we find the Fisher information

$\max_{C}\text{FI}(J|C_{\text{OTOC}})\approx\left|\frac{f^{\prime}(0)}{AJ_{y}^{2}\sqrt{v_{B}d}}\right|^{2}\sim\frac{1}{d}.$
(15)

_Global unitary control._ —We now turn to an alternate experimental scenario,
where one has only global unitary control over the larger system. In this
scenario, the natural generalization of the correlation functions Eq. (4) is
the following:

$\begin{split}C_{\text{TOC}}&=\left\langle V_{p}(t)\,V_{p}(0)\right\rangle\\\
C_{\text{TOC}}&=\left\langle
V_{p}(t)\,e^{i\phi\sum_{x}W_{x}(t/2)}\,V_{p}(0)\,e^{-i\phi\sum_{x}W_{x}(t/2)}\right\rangle\\\
C_{\text{OTOC}}&=\left\langle
V_{p}(t)\,e^{i\phi\sum_{x}W_{x}(0)}\,V_{p}(t)\,e^{-i\phi\sum_{x}W_{x}(0)}.\right\rangle\end{split}$
(16)

Here we replace the local unitary operations of Eq. (4) with global spin
rotations, $e^{i\phi\sum_{x}W_{x}}$, by an angle $\phi$ (here, $W_{x}$ is a
local Hermitian operator on qubit $x$).

We expect the behavior of the OTOC under global control to be governed by the
“size” of time-evolved operators Sánchez  _et al._ (2019, 2021); Domínguez
_et al._ (2021). The size corresponds to the average of local OTOCs over all
qubits in the system Qi and Streicher (2019). In one-dimensional ergodic
systems the size grows linearly $\sim v_{B}t$, which yields the following
phenomenological expectation for the global OTOC Domínguez  _et al._ (2021):

$C^{\text{glob}}_{\text{OTOC}}=\exp(-\phi^{2}v_{B}(t)t).$ (17)

Here we have made the butterfly velocity time-dependent to capture its
dependence on the local coupling strengths,

$v_{B}(t)\approx\left[\frac{1}{v_{B}t}\sum_{y=0}^{v_{B}t}\frac{1}{J_{y}}\right]^{-1},$
(18)

where $v_{B}=\overline{1/J}$ is the typical butterfly velocity.

Taking the derivative of the OTOC via the chain rule, we have:

$\begin{split}\partial_{J_{d}}C^{\text{glob}}_{\text{OTOC}}&\approx-\phi^{2}t\cdot\partial_{J_{d}}v_{B}(t)\cdot\exp(-\phi^{2}v_{B}(t)t)\\\
&=-\phi^{2}t\cdot\frac{v_{B}(t)^{2}}{v_{B}J_{d}^{2}t}\cdot\delta_{d\leq
v_{B}(t)t}\cdot\exp(-\phi^{2}v_{B}(t)t)\\\
&\approx-\phi^{2}\cdot\frac{v_{B}}{J_{d}^{2}}\cdot\delta_{d\leq
v_{B}t}\cdot\exp(-\phi^{2}v_{B}t)\,.\\\ \end{split}$ (19)

We would like to maximize the Fisher information over the parameters
$(\phi,t)$. This entails taking the time $t$ to be as early as allowed by the
delta function, $t\approx d/v_{B}$, in order to minimize the exponential. The
correlator is then maximized by choosing $\phi$ such that $\phi^{2}v_{B}t\sim
1$. This gives a Fisher information:

$\max_{C}\text{FI}(J|C_{\text{OTOC}})\approx\left|\frac{v_{B}e^{-1}}{J_{y}^{2}d}\right|^{2}\sim\frac{1}{d^{2}},$
(20)

which decays algebraically, with an additional factor of $d$ compared to the
local unitary control scenario.

Before moving on, we briefly summarize the intuition behind the two above
estimates. In both cases, an $O(1)$ perturbation in a local coupling strength
produces an $O(1)$ shift in the location of the OTOC wavefront. With local
control, this shift produces an $O(1/\sqrt{d})$ change in the OTOC, since the
OTOC wavefront is spread across a width $\sim\\!\sqrt{d}$ by the time it
reaches the coupling. With global control, this produces an $O(1/d)$ change in
the OTOC, since the global OTOC depends on the average of $\sim\\!d$
individual coupling strengths. Since the Fisher information involves the
square of the OTOC derivative, these lead to an $O(1/d)$ and $O(1/d^{2})$
Fisher information, respectively.

####  B.1.2 Fisher information of TOCs in absence of conserved quantities

We now turn to a simpler case, the Fisher information of time-ordered
correlators in the absence of conserved quantities [Fig. 2(b)]. Under ergodic
dynamics, we expect such correlation functions to decay exponentially in time
at sufficiently large times, see Eq. (10). Now, consider the derivative of the
correlation function with respect to a local coupling strength at a distance
$d$ away from the probe qubit. By causality, this derivative can only be non-
zero after a time $t\gtrsim d/v_{B}$. However, at such times the magnitude of
the correlation function has already decayed by a factor of $e^{-\gamma t}$.
This suggests that the Fisher information will decay exponentially in the
distance $d$,

$\max_{C}\text{FI}(J|C_{\text{TOC}})\lesssim\exp(-2\gamma x/v_{B}),$ (21)

as observed numerically in Fig. 2(b).

####  B.1.3 Fisher information of TOCs in presence of conserved quantities

The scaling of the Fisher information for TOCs is modified in the presence of
a conserved quantities. In this case, one expects the TOC at sufficiently
large times to be dominated by slow diffusive dynamics of the conserved
quantity. This lies in contrast to the exponential decay expected in the
absence of conserved quantities.

To study this, we begin with the auto-correlation function [first line of Eq.
(4)]. Recall that the auto-correlation function can be measured with access
solely to the probe qubit, and is thus accessible in both the local and global
control scenarios. Similar to the case of OTOCs, we will assume that the
dependence of the correlation function on the local coupling strengths is
captured by replacing the diffusion constant, $D$, with a _time-dependent_
value,

$C_{\text{TOC}}(t)=\frac{1}{\sqrt{D(t)t}}.$ (22)

Following the logic of the previous section, we assume the effective diffusion
constant takes the form Revathi and Balakrishnan (1993),

$D(t)\sim\left[\frac{1}{\sqrt{Dt}}\sum_{y=0}^{\sqrt{Dt}}\frac{1}{J_{y}}\right]^{-1},$
(23)

where $D=\overline{1/J}$ is the diffusion constant’s typical value.
Differentiating with respect to the local coupling strength gives

$\partial_{J_{d}}D(t)\sim\frac{D(t)^{2}}{J_{d}^{2}\sqrt{Dt}}\cdot\delta_{d<\sqrt{Dt}}\,.$
(24)

Computing the derivative of the auto-correlation function, we have:

$\begin{split}\partial_{J_{d}}C_{\text{TOC}}(t)&=-\frac{1}{2D(t)^{3/2}t^{1/2}}\cdot\partial_{J_{d}}D(t)\\\
&=-\frac{D(t)^{1/2}}{2J_{d}^{2}t\sqrt{D}}\cdot\delta_{d<\sqrt{Dt}}\\\
&\approx-\frac{1}{2J_{d}^{2}t}\cdot\delta_{d<\sqrt{Dt}}\,.\\\ \end{split}$
(25)

The magnitude of the derivative is maximized by taking the minimum possible
time, $t\approx d^{2}/D$, which yields a Fisher information,

$\max_{C}\text{FI}(J|C_{\text{TOC}})\approx\left|\frac{D}{2J_{x}^{2}d^{2}}\right|^{2}\sim\frac{1}{d^{4}}.$
(26)

This can be understood intuitively as follows. For the auto-correlation
function to be sensitive to the coupling strength $J_{d}$, the conserved
quantity must have spread to at least distance $d$. At such a distance the
magnitude of the auto-correlation function is $O(1/d)$, since the conserved
quantity has spread over $\sim d$ sites. In addition, the derivative with
respect to an individual coupling strength is suppressed by an additional
factor $O(1/d)$, since the auto-correlator depends only on the average
(inverse) coupling strength over $\sim d$ sites. Combining these two factors
and squaring leads to an $O(1/d^{4})$ Fisher information.

We now turn to the remaining time-ordered correlation functions in Eqs. (4,
16), which require either local or global unitary control over the non-probe
qubits. We will find that such correlators provide no scaling advantage beyond
the auto-correlator.

We first consider the case of local unitary control [Eq. (4)]. Physically,
these correlation functions correspond to preparing an amount of the conserved
quantity (e.g. a spin polarization) at the probe qubit, letting it diffuse for
a time $t/2$, flipping the spin polarization at a qubit $x$, and measuring the
polarization at the probe qubit after an additional time $t/2$. We thus expect
the TOC behave as follows,

$C_{\text{TOC}}(x,t)=\left\langle
V_{p}(t)\,W_{x}(0)\,V_{p}(t)\,W_{x}(0)\right\rangle\approx
q(0,t)-2q(x,t/2)\cdot q(x,t/2),$ (27)

where $q(x,t)\approx(2\pi D(t)t)^{-1/2}\exp(-x^{2}/(2D(t)t))$ is the
propagator of the conserved quantity from position $0$ to position $x$ (or
vice versa). The first term is equal to the auto-correlation function. The
second term arises from the spin flip at position $x$ and time $t/2$. The spin
flip effectively inserts a negative polarization $-2q(x,t/2)$ on the qubit
$x$, which propagates back to the probe qubit with amplitude $q(x,t/2)$.

The derivative of the second term is as follows,

$\begin{split}\partial_{J_{d}}\left[q(x,t/2)^{2}\right]&=\partial_{J_{d}}\left[\frac{1}{\pi
D(t)t}\exp(-\frac{2x^{2}}{D(t)t})\right]\\\
&\approx\partial_{J_{d}}\left[\frac{-1}{\pi
J_{y}^{2}D^{1/2}t^{3/2}}\exp(-\frac{2x^{2}}{D(t)t})+\frac{2x^{2}}{\pi
J_{y}^{2}D^{3/2}t^{5/2}}\exp(-\frac{2x^{2}}{D(t)t})\right]\cdot\delta_{d<\sqrt{Dt}}\,.\\\
\end{split}$ (28)

The magnitude of the derivative is maximized at $Dt\sim x^{2},x\sim d$, and is
of order $O(1/d^{3})$. This is subleading compared to the auto-correlation
function, of order $O(1/d^{2})$, and thus does not affect the asymptotic
scaling of the Fisher information with $d$.

The case of a global control [Eq. (16)] is even simpler. A global spin
rotation about the $x$-axis by an angle $\phi$ multiplies the conserved
quantity at each site by a factor of $\cos(\phi)$. Here we assume that the
$x$\- and $y$-components of spin that are generated by the rotation quickly
decay in time if they are not conserved by the ergodic dynamics. The resulting
correlation function is then given by the auto-correlation multiplied by
$\cos(\phi)$. Again, this provides no scaling advantage in the Fisher
information.

###  B.2 Effect of imperfect time-reversal and decoherence on Fisher
information

We now incorporate imperfect time-reversal dynamics into our estimates of the
Fisher information of OTOCs. Previous works have been found that a wide range
of experimental errors (e.g. extrinsic decoherence, coherent errors in time-
reversal) have a similar effect on OTOC measurements, as long as the relevant
errors are local and the dynamics are ergodic Swingle and Halpern (2018); Mi
_et al._ (2021); Sánchez  _et al._ (2021); Domínguez  _et al._ (2021);
Schuster and Yao (2022).

Specifically, in one-dimensional systems, one expects that the OTOC under
open-system dynamics, $\tilde{C}_{\text{OTOC}}$, is equal to the same OTOC
under unitary dynamics, $C_{\text{OTOC}}$, multiplied by an overall Gaussian
decay in time Schuster and Yao (2022):

$\tilde{C}_{\text{OTOC}}\approx\exp(-a\varepsilon v_{B}t^{2})\times
C_{\text{OTOC}}\,.$ (29)

Here $\varepsilon$ is an effective local error rate, $v_{B}$ is the butterfly
velocity, and $a$ is an order one constant. The argument of the above
exponential is proportional the volume of the time-evolved operator’s light
cone. Intuitively, Eq. (29) states that each error in the causal past of an
operator contributes a roughly equal amount to the decay of the OTOC. We note
that in finite-size systems we do not expect Eq. (29) to precisely hold,
however, corrections are expected to be suppressed by $\sim\\!\varepsilon/J$
where $J$ is the local interaction strength Schuster and Yao (2022), so we
neglect them here.

Substituting Eq. (29) into our estimate for the Fisher information [Eq. (20)]
and setting $v_{B}t\approx d$, we find:

$\max_{C}\,\text{FI}(J_{d}|\tilde{C}_{\text{OTOC}})\sim\frac{1}{d^{2}}\exp(-a\varepsilon
d^{2}/v_{B})\,.$ (30)

Meanwhile, we assume that the Fisher information with respect to TOCs is
comparatively unaffected by error, and thus once again follows a linear
exponential decay in $d$:

$\max_{C}\,\text{FI}(J_{d}|\tilde{C}_{\text{TOC}})\sim\exp(-\gamma d)\,.$ (31)

Setting the two exponentials to be equal,
$\max_{C}\,\text{FI}(J_{d}|\tilde{C}_{\text{OTOC}})\sim\max_{C}\,\text{FI}(J_{d}|\tilde{C}_{\text{TOC}})$,
we find that the OTOC continues to provide an advantage over the TOC up to

$d\lesssim\frac{\gamma v_{B}}{\varepsilon}\,,$ (32)

as quoted in the main text.

We now apply the same analysis to our second learning regime. Let us set
$v_{B}\sim J$ for consistency with the main text. The Fisher information of an
OTOC between operators on either side of the link with respect to the link
interaction strength is now modified to

$\max_{C}\,\text{FI}(J_{\ell}|\tilde{C}_{\text{OTOC}})\sim\frac{J_{\ell}^{4}t^{2}}{J^{2}}\exp(-a\varepsilon
Jt^{2}),\,\,\,\,\,\,\,Jt\lesssim L\,.$ (33)

The maximum of the Fisher information as a function of time now occurs at

$t^{*}\sim\min\left\\{\sqrt{\frac{1}{\varepsilon J}},\frac{L}{J}\right\\},$
(34)

with value

$\max_{C}\,\text{FI}(J_{\ell}|\tilde{C}_{\text{OTOC}})\sim\min\left\\{\frac{J_{\ell}^{4}}{\varepsilon
J^{3}},\frac{J_{\ell}^{4}L^{2}}{J^{4}}\right\\},$ (35)

which differs from the unitary OTOC for sufficiently high error rates. Again,
we assume that the Fisher information of TOC is not affected by error to
leading order. Taking the square root of Eq. (35), we thus find the $L$-fold
advantage of the OTOC is replaced by a $\sqrt{J/\varepsilon}$-fold advantage
at error rates $\sqrt{\varepsilon/J}\gtrsim 1/L$, as quoted in the main text.

##  Appendix C Additional numerics

Figure 7:  Learning in the restricted access scenario under Hamiltonian
evolution. Numerical simulations are performed identically to Fig. 2 but now
with Hamiltonian evolution under $(H_{c}+H_{f})/2$ instead of Floquet
evolution. In both (a) the learning task and (b) the Fisher information, the
results for learning Hamiltonian dynamics are qualitatively similar to the
results for learning Floquet dynamics [Fig. 2]. In (b), the maximum Fisher
information is averaged over 100 disorder realizations for both TOCs and
OTOCs. At large $d$, we expect the Fisher information for Hamiltonian
evolution to approach a power law decay $\sim\\!1/d^{4}$ (see Appendix B), but
this cannot be observed in our finite-size numerics.

In this section we provide numerical results in two additional learning
scenarios. We begin by repeating the simulations leading to Fig. 2 for time-
independent Hamiltonian evolution instead of Floquet evolution. We then
discuss learning in the restricted access scenario with global unitary
control, in contrast to local unitary control as considered in the main text).

###  C.1 Learning with time-independent Hamiltonian evolution

In the main text numerical simulations, we utilized Floquet time-evolution in
which the spin interactions and local fields were applied in a stroboscopic
fashion. Our motivations for using Floquet time-evolution instead of
Hamiltonian time-evolution were three-fold: First, Floquet dynamics are
prevalent in a variety of quantum systems that one might wish to learn, e.g.
in digital quantum simulators, and NMR or solid-state defect setups with
optical driving. Second, the Floquet dynamics considered are moderately faster
to simulate via Krylov subspace methods than Hamiltonian dynamics, since the
Hamiltonian of the former contains fewer terms at a given instant in time.
Third, we do not expect the behavior of learning via TOCs or OTOCs under the
two dynamics to qualitatively differ at moderate times and distances (although
at large distances they may, see Appendix B).

Here, we check the latter assumption by repeating the numerical analysis of
Fig. 2 using time-independent Hamiltonian dynamics. As shown in Fig. 7(a), we
find that the results of the learning task of Fig. 2(a) behave quite similarly
for Hamiltonian and Floquet dynamics. In particular, access to OTOCs continues
to enable substantially more accurate predictions for the crossing distance
$d$ for all $d\gtrsim 3$. In Fig. 7(b), we turn to the behavior of the Fisher
information as a function of a coupling’s distance from the probe qubit.
Unfortunately, we are not able to discern the $\sim\\!1/d^{4}$ scaling
predicted in Appendix B in our finite-size numerics. Instead, the Fisher
information behaves qualitatively similar to that of Floquet dynamics [Fig.
2(b)]. We anticipate that at sufficiently large distances the Fisher
information of Hamiltonian dynamics will indeed asymptote to the expected
power law decay. However, at such distances the Fisher information will likely
already be too small to be useful for most practical purposes.

###  C.2 Learning under restricted access with global unitary control

We now turn to learning when one has only global unitary control over the
system of interest. We consider a learning task where one wishes to classify
the geometry of an unknown spin system, which we assume is drawn with equal
probability from the three geometries shown in Fig. 8(a). We find that access
to OTOCs provides a substantial advantage in this classification task.
Notably, we find that OTOCs continue to improve learning even when one has
only global state preparation, control, and read-out (i.e. even in the absence
of a probe qubit).

Figure 8:  (a) The three spin geometries considered in the learning task
defined in the text. Each geometry consists of $L=14$ spins. The probe qubit
(purple) is located along a subset of the system that is identical between the
three geometries up to a distance $d$ away from the probe. (b) Accuracy of
classification, using correlation functions that can be measured with (left)
state preparation and read-out on the probe qubit and global unitary control
over the remaining system, and (right) global state preparation, unitary
control, and read-out. For the former, the accuracy is plotted as a function
of the distance $d$ of the probe qubit from the geometric feature of interest.
In both scenarios, access to OTOCs (blue) substantially improves the
classification accuracy compare solely accessing TOCs (red).

The classification problem we consider is a close variant of those introduced
in the main text. We suppose that one has access to the correlation functions
of an unknown Hamiltonian whose connectivity corresponds to one of the three
geometries shown in Fig. 8(a). The goal is to distinguish which geometry
describes the Hamiltonian. We again approach this task by training and testing
a support vector machine on samples of disorder realizations, see Section A.3
for details.

We consider learning in two different experimental access scenarios. First, we
consider the scenario where one has state preparation and read-out from a
single probe qubit, and global control over the remainder of the system. In
this case, we take the probe qubit to be a distance $d$ away from any
distinguishing features of the geometry (see Fig. 8), and study the
learnability as a function of $d$. Note that we are restricted to relatively
small distances, $d\leq 4$, owing to the particular form of the three
geometries considered. We find that access to OTOCs increases the
classification accuracy between $10\%$ and $35\%$ for all values of $d$ [Fig.
8(b)]. For instance, OTOCs allow classification with accuracy $\sim 65\%$ at
$d=3$, at which learning via TOCs has nearly trivial accuracy.

Our second scenario is even more restrictive: we suppose that one has only
global state preparation, control and read-out over the entire system. Despite
being commonplace in experiments such as NMR spectroscopy Laws  _et al._
(2002), learning in this scenario remains quite difficult in strongly-
interacting systems, due to the combination of time-ordered correlators
decaying quickly and local information being averaged out by global control
and measurement. Indeed, in our learning task, we find that learning via TOCs
features a classification accuracy of only $\sim\\!55\%$. Intuitively, we
expect access to global OTOCs to improve learning, as operator spreading at
late times is dependent on global geometric features of the system. In keeping
with this intuition, we find that learning via both TOCs and OTOCs improves
the classification accuracy to $\sim\\!80\%$.