perspective of non-equilibrium EFT may have important applications in many
areas of physics including understanding vortices in certain dark matter
models FDM .

Acknowledgments: I would like to thank Alberto Nicolis and Lam Hui for their
wonderful mentorship and Sašo Grozdanov and Matteo Baggioli for insightful
conversations. This work was partially supported by the US Department of
Energy grant DE-SC0011941.

## References

  * (1) S. Dubovsky, L. Hui, A. Nicolis and D. T. Son, _Effective field theory for hydrodynamics: thermodynamics, and the derivative expansion,_ Phys. Rev. D 85, 085029 (2012) doi:10.1103/PhysRevD.85.085029 [arXiv:1107.0731 [hep-th]].
  * (2) A. Nicolis, R. Penco, F. Piazza and R. Rattazzi, _Zoology of condensed matter: Framids, ordinary stuff, extra-ordinary stuff,_ JHEP 1506, 155 (2015) doi:10.1007/JHEP06(2015)155 [arXiv:1501.03845 [hep-th]].
  * (3) A. Nicolis, R. Penco, F. Piazza and R. A. Rosen, _More on gapped Goldstones at finite density: More gapped Goldstones,_ JHEP 1311, 055 (2013) doi:10.1007/JHEP11(2013)055 [arXiv:1306.1240 [hep-th]].
  * (4) A. Nicolis, R. Penco and R. A. Rosen, _Relativistic Fluids, Superfluids, Solids and Supersolids from a Coset Construction,_ Phys. Rev. D 89, no. 4, 045002 (2014) doi:10.1103/PhysRevD.89.045002 [arXiv:1307.0517 [hep-th]].
  * (5) A. Nicolis, _Low-energy effective field theory for finite-temperature relativistic superfluids,_ arXiv:1108.2513 [hep-th].
  * (6) S. Grozdanov and J. Polonyi, “Viscosity and dissipative hydrodynamics from effective field theory,” Phys. Rev. D 91, no.10, 105031 (2015) doi:10.1103/PhysRevD.91.105031 [arXiv:1305.3670 [hep-th]].
  * (7) H. Liu and P. Glorioso, _Lectures on non-equilibrium effective field theories and fluctuating hydrodynamics,_ PoS TASI 2017, 008 (2018) doi:10.22323/1.305.0008 [arXiv:1805.09331 [hep-th]].
  * (8) P. Glorioso, M. Crossley and H. Liu, _Effective field theory of dissipative fluids (II): classical limit, dynamical KMS symmetry and entropy current,_ JHEP 1709, 096 (2017) doi:10.1007/JHEP09(2017)096 [arXiv:1701.07817 [hep-th]].
  * (9) M. Crossley, P. Glorioso and H. Liu, _Effective field theory of dissipative fluids,_ JHEP 1709, 095 (2017) doi:10.1007/JHEP09(2017)095 [arXiv:1511.03646 [hep-th]].
  * (10) P. Glorioso, H. Liu and S. Rajagopal, _Global Anomalies, Discrete Symmetries, and Hydrodynamic Effective Actions,_ JHEP 1901, 043 (2019) doi:10.1007/JHEP01(2019)043 [arXiv:1710.03768 [hep-th]].
  * (11) P. Gao, P. Glorioso and H. Liu, _Ghostbusters: Unitarity and Causality of Non-equilibrium Effective Field Theories,_ arXiv:1803.10778 [hep-th].
  * (12) P. Glorioso and H. Liu, _The second law of thermodynamics from symmetry and unitarity,_ arXiv:1612.07705 [hep-th].
  * (13) M. Blake, H. Lee and H. Liu, _A quantum hydrodynamical description for scrambling and many-body chaos,_ JHEP 1810, 127 (2018) doi:10.1007/JHEP10(2018)127 [arXiv:1801.00010 [hep-th]].
  * (14) M. Harder, P. Kovtun and A. Ritz, _On thermal fluctuations and the generating functional in relativistic hydrodynamics,_ JHEP 1507, 025 (2015) doi:10.1007/JHEP07(2015)025 [arXiv:1502.03076 [hep-th]].
  * (15) N. Banerjee, J. Bhattacharya, S. Bhattacharyya, S. Jain, S. Minwalla and T. Sharma, _Constraints on Fluid Dynamics from Equilibrium Partition Functions,_ JHEP 1209, 046 (2012) doi:10.1007/JHEP09(2012)046 [arXiv:1203.3544 [hep-th]].
  * (16) K. Jensen, N. Pinzani-Fokeeva and A. Yarom, _Dissipative hydrodynamics in superspace,_ JHEP 1809, 127 (2018) doi:10.1007/JHEP09(2018)127 [arXiv:1701.07436 [hep-th]].
  * (17) K. Jensen, R. Marjieh, N. Pinzani-Fokeeva and A. Yarom, _A panoply of Schwinger-Keldysh transport,_ SciPost Phys. 5, no. 5, 053 (2018) doi:10.21468/SciPostPhys.5.5.053 [arXiv:1804.04654 [hep-th]].
  * (18) K. Jensen, R. Marjieh, N. Pinzani-Fokeeva and A. Yarom, _An entropy current in superspace,_ JHEP 1901, 061 (2019) doi:10.1007/JHEP01(2019)061 [arXiv:1803.07070 [hep-th]].
  * (19) K. Jensen, M. Kaminski, P. Kovtun, R. Meyer, A. Ritz and A. Yarom, _Towards hydrodynamics without an entropy current,_ Phys. Rev. Lett. 109, 101601 (2012) doi:10.1103/PhysRevLett.109.101601 [arXiv:1203.3556 [hep-th]].
  * (20) P. Kovtun, G. D. Moore and P. Romatschke, _Towards an effective action for relativistic dissipative hydrodynamics,_ JHEP 1407, 123 (2014) doi:10.1007/JHEP07(2014)123 [arXiv:1405.3967 [hep-ph]].
  * (21) F. M. Haehl, R. Loganayagam and M. Rangamani, _Two roads to hydrodynamic effective actions: a comparison,_ arXiv:1701.07896 [hep-th].
  * (22) F. M. Haehl, R. Loganayagam and M. Rangamani, _Effective Action for Relativistic Hydrodynamics: Fluctuations, Dissipation, and Entropy Inflow,_ JHEP 1810, 194 (2018) doi:10.1007/JHEP10(2018)194 [arXiv:1803.11155 [hep-th]].
  * (23) F. M. Haehl, R. Loganayagam and M. Rangamani, _The Fluid Manifesto: Emergent symmetries, hydrodynamics, and black holes,_ JHEP 1601, 184 (2016) doi:10.1007/JHEP01(2016)184 [arXiv:1510.02494 [hep-th]].
  * (24) M. Hongo, S. Kim, T. Noumi and A. Ota, _Effective Lagrangian for Nambu-Goldstone modes in nonequilibrium open systems,_ arXiv:1907.08609 [hep-th].
  * (25) D.V. Volkov, Phenomenological Lagrangians, Fiz.Elem.Chast.Atom.Yadra 4 (1973) 3–41.
  * (26) T. Hayata, Y. Hidaka, T. Noumi and M. Hongo, _Relativistic hydrodynamics from quantum field theory on the basis of the generalized Gibbs ensemble method,_ Phys. Rev. D 92, no. 6, 065008 (2015) doi:10.1103/PhysRevD.92.065008 [arXiv:1503.04535 [hep-ph]].
  * (27) M. Hongo, _Path-integral formula for local thermal equilibrium,_ Annals Phys. 383, 1 (2017) doi:10.1016/j.aop.2017.04.004 [arXiv:1611.07074 [hep-th]].
  * (28) M. Hongo, _Nonrelativistic hydrodynamics from quantum field theory: (I) Normal fluid composed of spinless Schrödinger fields,_ doi:10.1007/s10955-019-02224-4 arXiv:1801.06520 [cond-mat.stat-mech].
  * (29) M. J. Landry, “Dynamical chemistry: non-equilibrium effective actions for reactive fluids,” [arXiv:2006.13220 [hep-th]].
  * (30) M. Baggioli and M. Landry, “Effective Field Theory for Quasicrystals and Phasons Dynamics,” SciPost Phys. 9, no.5, 062 (2020) doi:10.21468/SciPostPhys.9.5.062 [arXiv:2008.05339 [hep-th]].
  * (31) M. J. Landry, “Second sound and non-equilibrium effective field theory,” [arXiv:2008.11725 [hep-th]].
  * (32) M. J. Landry, “The coset construction for non-equilibrium systems,” JHEP 07, 200 (2020) doi:10.1007/JHEP07(2020)200 [arXiv:1912.12301 [hep-th]].
  * (33) D. Gaiotto, A. Kapustin, N. Seiberg and B. Willett, “Generalized Global Symmetries,” JHEP 02, 172 (2015) doi:10.1007/JHEP02(2015)172 [arXiv:1412.5148 [hep-th]].
  * (34) J. Zhao, Z. Yan, M. Cheng and Z. Y. Meng, “Higher-form symmetry breaking at Ising transitions,” [arXiv:2011.12543 [cond-mat.str-el]].
  * (35) E. Lake, “Higher-form symmetries and spontaneous symmetry breaking,” [arXiv:1802.07747 [hep-th]].
  * (36) Y. Hidaka, Y. Hirono and R. Yokokura, “Counting Nambu-Goldstone modes of higher-form global symmetries,” [arXiv:2007.15901 [hep-th]].
  * (37) D. M. Hofman and N. Iqbal, “Goldstone modes and photonization for higher form symmetries,” SciPost Phys. 6, no.1, 006 (2019) doi:10.21468/SciPostPhys.6.1.006 [arXiv:1802.09512 [hep-th]].
  * (38) L. V. Delacrétaz, D. M. Hofman and G. Mathys, “Superfluids as Higher-form Anomalies,” SciPost Phys. 8, 047 (2020) doi:10.21468/SciPostPhys.8.3.047 [arXiv:1908.06977 [hep-th]].
  * (39) D. Harlow and H. Ooguri, “Symmetries in quantum field theory and quantum gravity,” [arXiv:1810.05338 [hep-th]].
  * (40) S. Grozdanov, D. M. Hofman and N. Iqbal, “Generalized global symmetries and dissipative magnetohydrodynamics,” Phys. Rev. D 95, no.9, 096003 (2017) doi:10.1103/PhysRevD.95.096003 [arXiv:1610.07392 [hep-th]].
  * (41) S. Grozdanov and N. Poovuttikul, “Generalized global symmetries in states with dynamical defects: The case of the transverse sound in field theory and holography,” Phys. Rev. D 97, no.10, 106005 (2018) doi:10.1103/PhysRevD.97.106005 [arXiv:1801.03199 [hep-th]].
  * (42) P. Glorioso and D. T. Son, “Effective field theory of magnetohydrodynamics from generalized global symmetries,” [arXiv:1811.04879 [hep-th]].
  * (43) S. Grozdanov, A. Lucas and N. Poovuttikul, “Holography and hydrodynamics with weakly broken symmetries,” Phys. Rev. D 99, no.8, 086012 (2019) doi:10.1103/PhysRevD.99.086012 [arXiv:1810.10016 [hep-th]].
  * (44) S. Grozdanov and N. Poovuttikul, “Generalised global symmetries in holography: magnetohydrodynamic waves in a strongly interacting plasma,” JHEP 04, 141 (2019) doi:10.1007/JHEP04(2019)141 [arXiv:1707.04182 [hep-th]].
  * (45) N. Iqbal and N. Poovuttikul, “2-group global symmetries, hydrodynamics and holography,” [arXiv:2010.00320 [hep-th]].
  * (46) V. I. Ogievetsky, Nonlinear realizations of internal and space-time symmetries, in X-th winter school of theoretical physics in Karpacz, Poland. 1974.
  * (47) E. Ivanov and V. Ogievetsky, The Inverse Higgs Phenomenon in Nonlinear Realizations, Teor.Mat.Fiz. 25 no. 2, (1975) 1050–1059.
  * (48) I. Low and A. V. Manohar, _Spontaneously broken space-time symmetries and Goldstone’s theorem,_ Phys. Rev. Lett. 88, 101602 (2002) doi:10.1103/PhysRevLett.88.101602 [hep-th/0110285].
  * (49) Steven Weinberg, The quantum theory of fields. Vol. 2: Modern applications, (Cambridge University Press, 1996).
  * (50) L. V. Delacrétaz, S. Endlich, A. Monin, R. Penco and F. Riva, _(Re-)Inventing the Relativistic Wheel: Gravity, Cosets, and Spinning Objects,_ JHEP 1411, 008 (2014) doi:10.1007/JHEP11(2014)008 [arXiv:1405.7384 [hep-th]].
  * (51) M. J. Landry and G. Sun, “The coset construction for particles of arbitrary spin,” [arXiv:2010.11191 [hep-th]].
  * (52) S. Endlich, A. Nicolis and R. Penco, _Ultraviolet completion without symmetry restoration,_ Phys. Rev. D 89, no. 6, 065006 (2014) doi:10.1103/PhysRevD.89.065006 [arXiv:1311.6491 [hep-th]].
  * (53) M. Greiter, F. Wilczek and E. Witten, _Hydrodynamic Relations in Superconductivity,_ Mod. Phys. Lett. B 3, 903 (1989). doi:10.1142/S0217984989001400
  * (54) D. T. Son, _Low-energy quantum effective action for relativistic superfluids,_ hep-ph/0204199.
  * (55) B. Horn, A. Nicolis and R. Penco, “Effective string theory for vortex lines in fluids and superfluids,” JHEP 10, 153 (2015) doi:10.1007/JHEP10(2015)153 [arXiv:1507.05635 [hep-th]].
  * (56) J. Armas and A. Jain, “Hydrodynamics for charge density waves and their holographic duals,” Phys. Rev. D 101, no.12, 121901 (2020) doi:10.1103/PhysRevD.101.121901 [arXiv:2001.07357 [hep-th]].
  * (57) J. Armas and A. Jain, “Viscoelastic hydrodynamics and holography,” JHEP 01, 126 (2020) doi:10.1007/JHEP01(2020)126 [arXiv:1908.01175 [hep-th]].
  * (58) L. P. Pitaevskiĭ, _Second Sound in Solids,_ Soviet Physics Uspekhi 342, 3 1968 doi:10.1070/pu1968v011n03abeh003839
  * (59) A. Cepellotti, G. Fugallo, L. Paulatto, M. Lazzeri, F. Mauri, N. Marzari, “Phonon hydrodynamics in two-dimensional materials,” Nature Communications 6, no. 1, 6400 (2015) DOI: 10.1038/ncomms7400
  * (60) A. J. Beekman, J. Nissinen, K. Wu, K. Liu, R. Slager, Z. Nussinov, V. Cvetkovic, J. Zaanen, “Dual gauge field theory of quantum liquid crystals in two dimensions” [arXiv:1603.04254 [cond-mat.str-el]].
  * (61) L. Hui, A. Joyce, M. J. Landry and X. Li, “Vortices and waves in light dark matter,” [arXiv:2004.01188 [astro-ph.CO]].


#  Classical surrogate simulation of quantum systems with LOWESA

Manuel S. Rudolph<EMAIL_ADDRESS>Institute of Physics, Ecole
Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland
Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico
87545, USA  Enrico Fontana  Department of Computer and Information Sciences,
University of Strathclyde, 26 Richmond Street, Glasgow G1 1XH, UK  Quantinuum,
Terrington House, 13-15 Hills Road, Cambridge CB2 1NL, UK  National Physical
Laboratory, Hampton Road, Teddington TW11 0LW, UK  Global Technologies Applied
Research, JPMorgan Chase, 25 Bank St, London E14 5JP, UK  Zoë Holmes
Institute of Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL), CH-1015
Lausanne, Switzerland  Lukasz Cincio  Theoretical Division, Los Alamos
National Laboratory, Los Alamos, New Mexico 87545, USA

###### Abstract

We introduce LOWESA as a classical algorithm for faithfully simulating quantum
systems via a classically constructed surrogate expectation landscape. After
an initial overhead to build the surrogate landscape, one can rapidly study
entire families of Hamiltonians, initial states and target observables. As a
case study, we simulate the 127-qubit transverse-field Ising quantum system on
a heavy-hexagon lattice with up to 20 Trotter steps which was recently
presented in Nature 618, 500-505 (2023). Specifically, we approximately
reconstruct (in minutes to hours on a laptop) the entire expectation landscape
spanned by the heavy-hex Ising model. The expectation of a given observable
can then be evaluated at different parameter values, i.e. with different
onsite magnetic fields and coupling strengths, in fractions of a second on a
laptop. This highlights that LOWESA can attain state-of-the-art performance in
quantum simulation tasks, with the potential to become the algorithm of choice
for scanning a wide range of systems quickly.

##  I Introduction

It is widely believed that quantum hardware, such as analog quantum simulators
or digital quantum computers, will be able to tackle certain tasks better than
classical hardware. While usually the focus is on total runtime, quantum
algorithms could exceed classical capabilities on a wider variety of measures,
such as financial cost, energy efficiency or simply reliability. On the flip
side, there exist a broad spectrum of classical methods that have their own
particular niches where they excel over other state-of-the-art classical or
quantum methods. Examples include tensor network (TN) methods Orús (2014,
2019); Ayral  _et al._ (2023); Pan and Zhang (2022), neural quantum states
Carleo and Troyer (2017); Torlai  _et al._ (2018); Schmitt and Heyl (2020),
and path-based Schrödinger Bernstein and Vazirani (1997); Markov  _et al._
(2018); Häner and Steiger (2017) or Heisenberg-picture Gottesman (1998); Rall
_et al._ (2019) propagation. These classical simulation methods commonly focus
on approximating the ideal solutions or dynamics that are expected given the
task at hand.

Recently, the low-weight simulation algorithm LOWESA Fontana  _et al._ (2023)
was introduced to approximate expectation values of Pauli-observables when the
quantum circuit is affected by noise. Under the assumption of single-qubit
Pauli noise channels, this algorithm is provably efficient (i.e. polynomial
time) in both the number of qubits and circuit depth when the error rate is
kept constant. This highlights that under these conditions, quantum algorithms
executed on noisy quantum hardware cannot be expected to exhibit an
exponential advantage. However, there are no guarantees for this algorithm to
accurately reproduce the ideal expectation values in the absence of noise.
This is of course expected due to the complexity of simulating quantum
systems, and is a property that is shared by all classical simulation methods.

In the noiseless case the main strength of LOWESA, over most other simulation
methods, is that the algorithm reconstructs a classical representation of the
full expectation landscape that is spanned by the quantum circuit parameters.
More precisely, LOWESA classically constructs a surrogate landscape Schreiber
_et al._ (2022); Jerbi  _et al._ (2023); Landman  _et al._ (2022) without
requiring any reference expectation estimations, which can then be evaluated
not just efficiently, but truly quickly. In the case of Variational Quantum
Algorithms (VQAs) Cerezo  _et al._ (2021), this implies that one could
optimize the parametrized quantum circuit fully classically on the
approximated landscape.

The possibilities for simulating physical quantum systems are equally
intriguing. For example, if simulating Hamiltonian dynamics using Trotter-
circuits Trotter (1959); Lloyd (1996); Sornborger and Stewart (1999), the
parameters of the circuits now include the coefficients of the individual
Hamiltonian terms. Thus, with a single surrogate landscape, one can simulate a
family of Hamiltonians that share the same operators. An additional strength
of LOWESA is that its performance is not directly limited by the entangling
topology of the employed quantum circuit.

Figure 1: Framework schematic of LOWESA for classically constructing surrogate
expectation landscapes. The task is to simulate the expectation value of an
observable $O$ for an initial state $\psi_{0}$ evolved under a unitary quantum
circuit $U$ with parameters $\bm{\theta}$. Typically, such parameters are only
introduced in the context of Variational Quantum Algorithms. However,
depending on the particular task, they can also be viewed as parametrizing a
change of the observable or the initial state. Another possibility is that
they can indirectly be defined by the coefficients of a Hamiltonian for the
purposes of studying time evolution dynamics. While simulation methods
conventionally assign numerical values to the parameters $\bm{\theta}$ before
simulation, LOWESA instead aims to build a classical surrogate of the entire
expectation landscape $f(\bm{\theta})$. The initial overhead of this approach
may be high, but re-evaluation of the surrogate landscape for different
parameter values can be exceptionally fast.

With the aim of demonstrating the practical utility of quantum computers, Ref.
Kim  _et al._ (2023) presented state-of-the-art experiments for gate-based
devices using 127 superconducting qubits. The task was to estimate certain
Pauli expectation values of states evolved under a transverse-field Ising
(TFI) Hamiltonian for 5, 6 or 20 Trotter steps and with varying on-site
magnetization. The connectivity of the TFI Hamiltonian was chosen to reflect
the 2D topology of the quantum device. Using sophisticated noise-
characterization techniques Berg  _et al._ (2022); Bennett  _et al._ (1996);
Knill (2004) and zero-noise extrapolation (ZNE) Temme  _et al._ (2017);
Kandala  _et al._ (2019); Giurgica-Tiron  _et al._ (2020), the authors were
able to obtain expectations that generally agreed with the true values in the
cases where exact calculation was possible. In the cases where such
verification was not possible, the authors argued that the experimentally
observed trends were more plausible than those obtained with the employed
tensor network methods, which had struggled throughout their work to reproduce
the exact solutions. However, in a quick turn of events, three works Tindall
_et al._ (2023); Kechedzhi  _et al._ (2023); Begušić and Chan (2023), and
later three more Torre and Roses (2023); Liao  _et al._ (2023); Begušić  _et
al._ (2023a) appeared that were able to classically reproduce expectation
values of similar or better quality, and in less time, using distinct
classical simulation techniques.

In this work, we showcase the noise-free version of the LOWESA algorithm for
the example of the TFI model studied in Ref. Kim  _et al._ (2023). To do so,
we construct a surrogate for the entire expectation landscape spanned by the
circuit parameters, which can be done in minutes to hours (depending on the
task) on a single laptop. The surrogate can then be evaluated at different
parameter values in fractions of a second to reproduce the results of Ref. Kim
_et al._ (2023) to high accuracy and with many more evaluations than Refs.
Tindall  _et al._ (2023); Kechedzhi  _et al._ (2023); Begušić and Chan (2023);
Torre and Roses (2023); Liao  _et al._ (2023); Begušić  _et al._ (2023a). See
Fig. 1 for a schematic depiction of the algorithm. Interestingly, with our
approach, we find that two of the expectation curves can be reproduced to high
accuracy with simple trigonometric functions, namely $\sin^{25}$ and
$\sin^{34}$.

To highlight the advantages of LOWESA, we generate additional expectation
curves and high-resolution expectation surfaces with low computational cost,
establishing that the algorithm has a non-neglible initial overhead, but can
then freely be used to probe global characteristics using the surrogate
landscape. Furthermore, we go beyond the case study in Ref. Kim  _et al._
(2023) by employing quantum circuits with more than double the number of
parameters, which allows us to highlight the full potential of using LOWESA
for quantum simulation tasks.

Finally, we acknowledge that our algorithm, analogous to all other classical
methods that aim to reproduce exact quantum dynamics, has no guarantee to
produce high-quality results within a fixed computational budget. We merely
highlight its strengths in the milieu of varied and specialized classical
algorithms, and advocate for a constructive interplay between quantum and
classical methods to produce reliable results at large scales that can be
trusted.

##  II The LOWESA algorithm

LOWESA is a classical simulation algorithm to reconstruct the parametrized
expectation landscape $f(\bm{\theta})$ formed by the expression

$f(\bm{\theta})=\langle 0|U^{\dagger}(\bm{\theta})OU(\bm{\theta})|0\rangle,$
(1)

where $O$ is a normalized Pauli operator and $U(\bm{\theta})$ is a quantum
circuit parametrized by the vector of rotation angles $\bm{\theta}$. LOWESA
assumes a circuit structure composed of arbitrary Clifford operations $C_{i}$
and $m$ parameterized single-qubit Pauli Z-rotation gates $Rz(\theta_{i})$,
i.e.,

$U(\bm{\theta})=\prod_{i=1}^{m}\big{(}C_{i}\cdot Rz(\theta_{i})\big{)}\cdot
C_{0}\,.$ (2)

Any quantum circuit can be represented in this form.

Originally, LOWESA was introduced for simulating variational quantum
algorithms Fontana  _et al._ (2023). Here we consider its application to
dynamical simulation. For a given dynamical simulation task composed of a set
of initial states, Hamiltonians and measurements of interest, one can write
down a parameterized quantum circuit that could be used to implement each of
these simulations. After compiling these circuits into the Clifford + RZ
structure in Eq. (2), LOWESA can be used to construct a surrogate for the
expectation landscape. This surrogate can then be used to quickly simulate the
evolution of any state, Hamiltonian and observable captured by the
parameterized circuit.

The LOWESA algorithm works in the Pauli Transfer Matrix (PTM) formalism Chow
_et al._ (2012), which is commonly applied in the Heisenberg-picture Gottesman
(1998); Rall  _et al._ (2019), and to simulate open quantum systems Wood  _et
al._ (2011). In this formalism, operators are defined by their decomposition
in the Pauli basis. This means that observables and quantum states (via their
density operator formulation) are treated entirely equivalently.

Specifically, an $n$-qubit operator can be represented as a
$4^{n}$-dimensional vector, where the $i$’th entry is defined as the
corresponding coefficient for each Pauli operator $P_{i}\in\\{I,X,Y,Z\\}^{n}$
with $|\cdot\rrangle_{i}=\text{Tr}[(\cdot)P_{i}]$. For example, for a Pauli
target operator $P_{j}$ we have
$[|P_{j}\rrangle]_{i}=\text{Tr}[P_{j}P_{i}]=\delta_{ij}$. This implies that
the representation of Pauli operators is sparse, with exactly one entry being
non-zero. In contrast, quantum states are usually very dense. For example
$\rho_{0}=|0\rangle\langle 0|=\left((I+Z)/2\right)^{\otimes n}$. Thus, using
the shorthand notation $|0\rrangle:=|\rho_{0}\rrangle$, we have that
$[|0\rrangle]_{i}=\text{Tr}[|0\rangle\langle 0|P_{i}]$ has $2^{n}$ non-zero
values.

A unitary $U$ is represented in the PTM formalism as U where

$[\textbf{U}]_{ij}=\llangle
P_{i}|\textbf{U}|P_{j}\rrangle=\text{Tr}[P_{i}UP_{j}U^{\dagger}]\,.$ (3)

The expectation function in Eq. (1) with a parametrized unitary and a target
operator $O$ represented in Pauli basis can consequently be written as

$f_{\text{PTM}}(\bm{\theta})=\llangle
O|\textbf{U}(\bm{\theta})|0\rrangle=\llangle
0|\textbf{U}^{\dagger}(\bm{\theta})|O\rrangle\,.$ (4)

Since Pauli operators are sparse in the PTM formalism, it is generally easier
to start the simulation with the final measurement operator $|O\rrangle$ and
propagate it backwards through the quantum circuit
$\textbf{U}^{\dagger}(\bm{\theta})$, and then calculate the overlap with the
$|0\rrangle$ state. Thus LOWESA adopts this approach.

As described in Eq. (2), the algorithm assumes that the circuit
$U(\bm{\theta})$ has been decomposed into Clifford operations and parametrized
single-qubit Z-rotations. Clifford operations are by definition those that map
one $n$-qubit Pauli operator to one other Pauli operator, i.e., they are
permutations of the entries of the PTM vector $|\cdot\rrangle$. In our work,
we have a well-defined gate set and thus all Clifford operations are known a
priori. This allows us to pre-compute how each operation acting on $k$ qubits
acts on all possible combinations of $k$-qubit Pauli operators, and store
these relations as a look-up table of size $4^{k}$. This time-memory trade-off
allows Clifford circuits to be simulated in a time that scales exclusively
linear in the number of gates. For example, the Clifford circuits that
represent the endpoints $\theta_{h}=0,\pi/2$ of the 127-qubit system in Fig. 2
can be calculated in less than a millisecond.

On the other hand, the PTM simulation of non-Clifford operations is
exponential in the number of such operations. This is because non-Clifford
gates “mix” entries in $|\cdot\rrangle$ and thus drastically increase the
number of non-zero entries one needs to keep track of. For a relevant example,
consider the parametrized single-qubit Z-rotation gate, which, in the PTM
formalism, is represented by the $4\times 4$\- dimensional matrix,

$\textbf{Rz}(\theta)=\begin{bmatrix}1&0&0&0\\\
0&\cos(\theta)&-\sin(\theta)&0\\\ 0&\sin(\theta)&\cos(\theta)&0\\\
0&0&0&1\end{bmatrix}.$ (5)

A common strategy in this case is to use path-based methods that decompose
inefficient components (here the RZ-gate) into a sum of efficient components.
This transforms one exponentially costly calculation into exponentially many
efficient calculations. While still exponential in nature, path-based methods
allow for finite-fidelity approximations using a non-exponential amount of
computational resources. This is due to the ability to neglect some (or most)
terms in the decomposition, which can lead to efficient trade-offs. In our
case, the RZ-gate admits the decomposition into three operations

$\textbf{Rz}(\theta)=\textbf{D}_{0}+\cos(\theta)\textbf{D}_{1}+\sin(\theta)\textbf{D}_{-1}\,,$
(6)

with

$\displaystyle\textbf{D}_{0}$ $\displaystyle=\begin{bmatrix}1&0&0&0\\\
0&0&0&0\\\ 0&0&0&0\\\ 0&0&0&1\end{bmatrix}$ (7) $\displaystyle\textbf{D}_{1}$
$\displaystyle=\begin{bmatrix}0&0&0&0\\\ 0&1&0&0\\\ 0&0&1&0\\\
0&0&0&0\end{bmatrix}$ (8) $\displaystyle\textbf{D}_{-1}$
$\displaystyle=\begin{bmatrix}0&0&0&0\\\ 0&0&-1&0\\\ 0&1&0&0\\\
0&0&0&0\end{bmatrix}.$ (9)

Since in the PTM formalism the four-dimensional vector representing any
single-qubit operator contains the coefficients of $I,X,Y,Z$ Pauli operators
(by convention in this order), the $D_{0}$ operator projects any $X,Y$
contribution to zero (we call this annihilation), whereas the $D_{1},D_{-1}$
operators annihilate any contribution of $I,Z$. That is, a path either does
not split, which means that the $D_{0}$ operator is applied, or it does split
into two paths, where the $D_{1}$ or $D_{-1}$ operator are applied to each
path, respectively. On the subspace that these operators do not annihilate,
they act like Clifford gates, i.e. they permute incoming vectors. Using this
decomposition, a worst-case simulation of the expectation function in Eq. (4)
would require the calculation of $2^{m}$ so-called paths where every
application of an RZ-gate splits the incoming operator into two, where $D_{1}$
is applied to one and $D_{-1}$ to the other.

By linearity, the final expectation value is a straight sum of all paths.
Specifically, for any path that is defined by its coordinate vector
$\bm{\omega}=\\{0,1,-1\\}^{m}$, the expectation function $f$ can be written as

$f(\bm{\theta})=\sum_{\bm{\omega}}\Phi_{\bm{\omega}}(\bm{\theta})\llangle
0|\textbf{U}^{\dagger}_{\bm{\omega}}|O\rrangle\,,$ (10)

where

$\Phi_{\bm{\omega}}(\bm{\theta})=\prod_{i=1}^{m}\begin{cases}1&\mbox{if
}\omega_{i}=0\\\ \cos(\theta_{i})&\mbox{if }\omega_{i}=1\\\
\sin(\theta_{i})&\mbox{if }\omega_{i}=-1\end{cases}$ (11)

is a parameter-dependent trigonometric monomial, and $\llangle
0|\textbf{U}^{\dagger}_{\bm{\omega}}|O\rrangle\in\\{-1,0,1\\}$ is the
expectation value of the purely-Clifford circuit
$\textbf{U}^{\dagger}_{\bm{\omega}}$ for the path indexed by $\bm{\omega}$.
Here, each RZ-gate at position $i$ is replaced by one of the operators
$\textbf{D}_{0},\textbf{D}_{1},\textbf{D}_{-1}$ depending on the value of
$\bm{\omega}$ at position $i$.

From another perspective, this formulation in Eq. (10) can be understood as a
Fourier-series representation of $f$, where $\Phi_{\bm{\omega}}(\bm{\theta})$
are trigonometric basis functions, and $\llangle
0|\textbf{U}^{\dagger}_{\bm{\omega}}|O\rrangle$ are the Fourier coefficients.
In this picture, $\bm{\omega}$ can be seen as the frequency vector, as via Eq.
(11), one can directly connect the 1-norm $|\bm{\omega}|_{1}$ of this vector
to the frequency with which this path contributes to the landscape. Thus,
paths with many splits generally contribute highly oscillatory features, while
paths with fewer splits determine the coarse-grained outlines of the
landscape.

In summary, LOWESA is an algorithm to construct a surrogate function
$\tilde{f}$ for the exact function in Eq. (10), using different forms of
truncation to keep the computation tractable. Due to its Fourier-series
nature, this results in forming a surrogate for the entire expectation
landscape defined by $f(\bm{\theta})$.

In the context of quantum simulation, there might not appear to be a concept
of a landscape, as the goal is often to run a fixed quantum circuit derived
from exponentiating a Hamiltonian. If the circuit, for example, approximates
time evolution under a Hamiltonian until time $t$ via the Trotter-Suzuki
decomposition, the effective “angles” of the circuit are directly proportional
to Hamiltonian coefficients and the time step $\Delta t$. However, by adapting
the view of these values as parameters, one can use LOWESA to simulate time
evolution under the entire family of Hamiltonians that share the same
structure, while allowing complete freedom to change the coefficients or
$\Delta t$ afterward. We note that, if the goal is to simulate only a single
circuit instance, LOWESA in its general form is unlikely to be the best
technique.

The critical insight in Ref. Fontana  _et al._ (2023) is that LOWESA is
provably efficient at simulating quantum circuits affected by single-qubit
Pauli noise, as paths with many splits, i.e. those with large 1-norm
$|\bm{\omega}|_{1}$, are suppressed. In other words, high-frequency
contributions to the expectation landscape are suppressed most by noise. Here
we however aim to simulate the exact expectation function $f$. In this case,
the error one makes by truncating is unbounded in general– a property shared
by all classical simulation methods that aim to simulate exact quantum
circuits. Nonetheless, as demonstrated in Section IV, in many cases our
algorithm can be used for sufficiently accurate simulations, even for the
127-qubit systems considered here.

Shortly before the original publication of LOWESA Fontana  _et al._ (2023),
the authors in Ref. Nemkov  _et al._ (2023) showcased a similar algorithm and
presented a study of the Fourier series properties of VQAs. Among other
insights, they found that strongly-contributing frequencies clustered and
generally were lower than one might expect. These findings help explain
LOWESA’s proficiency at simulating quantum systems at scale. Soon after the
publication of the 127-qubit experiments in Ref. Kim  _et al._ (2023), a
similar approach under the name of Clifford perturbation theory (CPT) was used
in Ref. Begušić and Chan (2023) to simulate single expectation values from the
time-evolved state (see Ref. Begušić  _et al._ (2023b) for more details about
CPT). While sharing similarities with our line of research, neither work
developed a surrogate-based algorithm for simulating the dynamics of wide
families of Hamiltonians, initial states and observables.

##  III Approximation and Tailoring Techniques

In this section, we introduce a selection of empirical truncation or
computational tree-pruning techniques that we use to apply LOWESA to large-
scale problems with hundreds or thousands of parametrized gates. These methods
assume that finding all paths is unfeasible or unnecessary. Thus, we introduce
a sense of prioritization in the tree search for contributing paths. We
summarize these methods below and discuss them in more detail in Appendix A.

_Maximal frequency $\ell$:_ The simplest truncation criterion is to drop all
paths with $|\bm{\omega}|_{1}>\ell$ for a truncation parameter $\ell$. This
reduces the absolutely worst-case scaling of the tree search for valid paths
from $2^{m}$ paths to $2^{\ell}$. This truncation scheme is particularly
justified for simulating noisy circuits since, as discussed above and as shown
in Ref. Fontana  _et al._ (2023), high $|\bm{\omega}|_{1}$ paths are most
suppressed by noise. In the noise-free case, the error one makes with any
$\ell<m$ is trivially bounded by $1$. However, via the interpretation of
$\bm{\omega}$ as the frequency of a path, truncation by $\ell$ can be seen as
enforcing a low-frequency approximation of the expectation landscape.

_Truncation probability $p$:_ This parameter provides a way to preemptively
truncate paths based on an estimate of the likelihood that it will split more
often than $\ell$. To do so, we crudely assume that the probability of
splitting is uniform across circuit layers and extrapolate from the history of
the number of splits so far to estimate the probability that a path will
remain with $|\bm{\omega}|_{1}\leq\ell$. Once that probability goes below $p$,
we preemptively truncate that path and do not explore it any further. This
method of truncation can save a lot of computational effort by neglecting
unpromising paths early, but it may introduce a bias against paths that split
often early in the circuit, and generally those with frequencies close to the
cutoff $\ell$. We have found that there exist values for $p$ that are small
enough to not neglect many valuable paths that would have stayed below the
threshold $\ell$ but still save computational time; however, this optimal
value cannot be known in general.

_Maximal operator weight $W$:_ This parameter provides a way to preemptively
truncate paths based on the current weight of the back-propagating Pauli
operator. The weight of an operator is defined as the number of qubits that it
acts on non-trivially, i.e., with $X$, $Y$, or $Z$ (not $I$). We truncate a
path early if the back-propagated operator surpasses a weight of $W$ at any
stage in the circuit. The weight of the operator can be understood as an
effective lightcone per path, which is closely related to the practice of
using effective entanglement lightcones for lowly-entangling circuit layers
(see Ref. Kechedzhi  _et al._ (2023) for a relevant example). Furthermore, the
weight of the operator makes it exponentially more likely that a path contains
$X$ or $Y$ operators on any qubit at the end of the circuit and therefore not
contribute to the landscape as $\llangle
0|\textbf{U}^{\dagger}_{\bm{\omega}}|O\rrangle=0$. Thus, this truncation
method significantly increases the odds that a path eventually contributes to
the surrogate landscape formed by LOWESA.

Figure 2: Simulating 127-qubit TFI dynamics. The observables measured
correspond to Figs. 3a, 3b, 3c and 4a from Ref. Kim  _et al._ (2023), which
also provides the exact curves and the mitigated quantum computer (QC)
results. The observables are the magnetization
$M_{z}=\frac{1}{127}\sum_{i}\langle Z_{i}\rangle$, $\langle
X_{13,29,31}Y_{9,30}Z_{8,12,17,28,32}\rangle$, $\langle
X_{37,41,52,56,57,58,62,79}Y_{75}Z_{38,40,42,63,72,80,90,91}\rangle$, and
$\langle X_{37,41,52,56,57,58,62,79}Y_{38,40,42,63,72,80,90,91}Z_{75}\rangle$,
respectively. After reconstructing the expectation landscape of these
observables with $L=5$ (a-c) or $L=6$ (d) trotter steps, we are able to
replicate close to exact expectation curves for the correlated angle case
$\theta_{h}^{(i)}=\theta_{h}$. Note that the exact curves are mostly hidden
because of the high-quality approximation. In panel a) we use $\ell=40,W=8$,
and in panel b) $\ell=32,p=0.05$. For this restricted 1D “slice” of the
expectation landscape, the trivial paths corresponding to the trigonometric
monomials $\sin^{25}$ and $\sin^{34}$ were sufficient to provide near-exact
dynamics in panel c) and d), respectively. The 31-qubit lightcone simulations
were generated with the code accompanying Ref. Kechedzhi  _et al._ (2023).

_Trivial paths:_ Trivial paths are what we call paths that satisfy the
boundary conditions of the parameter domain of interest. For the simulation
tasks considered here, they tend to be the biggest contributors to the
qualitative shape of the expectation landscape, i.e., among the lowest-
frequency contributions, and are exceptionally easy to find. Additional paths
then form corrections that eventually reveal the true dynamics, but we show
that the trivial paths alone can replicate two of the expectation curves in
Ref. Kim  _et al._ (2023) (and can be found in less than a millisecond). For
example, consider the case of a single RZ-gate with angle $\theta$ in an
otherwise Clifford circuit. We can then verify the boundary cases of
$\theta=0$ and $\theta=\pi/2$. If any of the expectations at the boundaries is
1, we can be sure that either the path that always goes down the $D_{1}$ or
$D_{-1}$ channel (depending on whether $\theta=0$ or $\theta=\pi/2$ has
expectation 1, respectively) is valid and satisfies this boundary value
because these are the only paths that can possibly be non-zero at these
angles.

_Sine or cosine biasing:_ Similarly to the trivial path method above, we can
use information about verifiable domains of the expectation landscape to
inform us which types of paths are most likely to contribute significantly.
That is to say, we gain information on whether amplitudes of relevant paths
are expected to have a larger contribution of $\sin$ or $\cos$ on the
parameter domain of interest. For correlated angles, as is the case in Figs. 2
and 3a, we have found that contributing paths tend to have a strong tendency
to mostly split into $D_{1}$ or $D_{-1}$ in exact correspondence with the
dominant trivial path.

##  IV Results

We consider the time-evolution dynamics of a 127-spin system governed by the
transverse-field Ising (TFI) Hamiltonian

$H=-\sum_{\langle i,j\rangle}J^{(i,j)}Z_{i}Z_{j}+\sum_{i}h^{(i)}X_{i}\,,$ (12)

where $X$ and $Z$ represent X- and Z-Pauli operators, respectively, and
$\langle i,j\rangle$ denotes neighboring spin indices on the so-called heavy-
hex topology (also shown in Fig. 4). See Ref. Kim  _et al._ (2023) for
details.

The evolution operator under this Hamiltonian for time $t$ can be written as
$U(t)=e^{-iHt}$. This cannot be implemented directly on digital quantum
devices due to non-commuting operators in $H$. Instead, one can use the
Trotter-Suzuki decomposition Trotter (1959); Lloyd (1996); Sornborger and
Stewart (1999),

$U(\bm{\theta})=\prod_{l=1}^{L}\left[\prod_{\langle
i,j\rangle}\left(R_{ZZ}(\theta_{J}^{(i,j)})\right)\prod_{i}\left(R_{X}(\theta_{h}^{(i)})\right)\right]\,,$
(13)

which breaks up the evolution into $L$ discrete time steps of length $\Delta
t=t/L$. Here $R_{ZZ}(\theta)$ and $R_{X}(\theta)$ are Pauli-rotation gates
with rotation angles $\theta_{J}^{(i,j)}=-J^{(i,j)}\Delta t$ and
$\theta_{h}^{(i)}=h^{(i)}\Delta t$. With $\Delta t\xrightarrow[]{}0$, this
approximate time evolution circuit becomes exact. For LOWESA, the parameters
$\theta$ are freely adjustable after computing the surrogate which allows us
to reconstruct expectation landscapes of the TFI dynamics for arbitrary
individual coupling strengths $J^{(i,j)}$, magnetic fields $h^{(i)}$ and time
steps $\Delta t$.

_Reproducing the simpler experiments:_ In a first experiment, we verify that
we can reconstruct the expectation values obtained on a 127-qubit quantum
computer in Ref. Kim  _et al._ (2023). For these cases, the coupling
coefficients are all fixed at $J^{(i,j)}=J=-\pi/2\;\forall\;i,j$, such that
$R_{ZZ}(\theta_{J}^{(i,j)})$ are Clifford gates. The local fields are also all
identical, i.e. $h^{(i)}=h\;\forall\;i$, and are tuned in unison such that
$\theta_{h}^{(i)}=\theta_{h}\in[0,\pi/2]$. Our results are shown in Fig. 2. We
note that panels a) to c) use a circuit with $L=5$ Trotter layers, whereas
panel d) uses a circuit with an additional layer of $R_{X}$ rotations, which
effectively corresponds to $L=6$ (as the $R_{ZZ}$ layer acts trivially on the
initial state $|0\rangle$).

For the cases where exact verification was possible, we show that LOWESA
recovers the dynamics with remarkably high precision, and is always in good
agreement with the error-mitigated experimental results from Ref. Kim  _et
al._ (2023). The surrogate construction took under an hour per observable
(sometimes significantly less), and then it took fractions of a second to
evaluate each of the 158 values of $\theta_{h}$ per curve. As such, our
approach is competitive with the methods in Refs. Tindall  _et al._ (2023) and
Begušić and Chan (2023), which took of the order of minutes per $\theta_{h}$
value, for reproducing the full curves. However, our surrogate landscapes can
then be used to quickly probe significantly more parameter values, including
uncorrelated ones with $\theta_{h}^{(i)}$ acting independently on each qubit
$i$.

Figure 3: Simulating 127-qubit TFI dynamics with $L=20$ Trotter steps. Panel
a) showcases the example of Fig. 4b from Ref. Kim  _et al._ (2023), for which
we are able to generate very plausible results that are broadly within error
bars of the quantum hardware results and close to the approximate results
demonstrated in Ref. Kechedzhi  _et al._ (2023). Since we approximately
reconstructed the entire expectation landscape spanned by the Hamiltonian
coefficients $h^{(i)}$ in Eq. (12), we are able to generate additional
$158\times 158$ pixel expectation surfaces measuring $\langle Z_{62}\rangle$
for alternating magnetic field values on the qubits (b) and for random noise
on the magnetic fields $h^{(i)}$, drawn from a normal distribution with
standard distribution $\sigma/\Delta t$ on each qubit (c). Both panels b) and
c) contain the curve in panel a) as special case, which is along the diagonal
of panel b), and the x-axis of panel c) with $\sigma=0$.

Our results in Fig. 2 highlight four of the approximation and tailoring
techniques outlined in Sec. III. Firstly, we always use a frequency truncation
$\ell<m$ where $m=635$ for panels a)-c), and $m=762$ in panel d). This is
motivated by lower-frequency trigonometric functions generally being more
important for revealing general trends. Furthermore, because these cases have
correlated angles $\theta_{h}$, one can see that
$\sin(\theta_{h})^{q}\cdot\cos(\theta_{h})^{r}$ is exponentially suppressed to
zero for large integer values $q,r$ for all $\theta_{h}$. Thus, one would
expect that only cases where $q\ll r$ or $r\ll q$ have significant amplitudes
on this parameter range. Depending on the verifiable boundary cases (i.e. the
magnetization values at $0$ and $\pi/2$), one may therefore additionally bias
the tree search to predominantly focus on the $D_{1}$ or $D_{-1}$ channels.
For uncorrelated parameters, these latter arguments do not hold.

Secondly, in panel a) we utilize truncation based on the weight of the back-
propagated observable with $W=8$, and in panel b) we utilize a modest
truncation probability of $p=0.05$. These values may not be optimal to
reproduce the entire uncorrelated parameter landscape, but they suffice for
the cases presented here. It is likely that there exist stronger truncations
which heavily reduce computational time but still accurately approximate the
exact dynamics for the chosen parameter range. However, since the optimal
hyperparameter values are likely to be highly problem-specific, and for non-
trivial problems one is unlikely to have access to the exact dynamics to
compare to, we were not interested in fine-tuning the result with the lowest
possible computational resources for these cases.

And finally, under our approach, panels c) and d) can be reconstructed in
milliseconds because the trivial paths strongly dominate the expectation
values of the target operators. As mentioned in Sec. III, from the verifiable
Clifford circuit cases at $\theta_{h}=0$ and $\theta_{h}=\pi/2$, we can infer
that the paths that always take the $D_{-1}$ branch, i.e., where
$\Phi_{\bm{\omega}}$ consist only of $\sin$-functions, must exist and have
non-zero amplitudes. That is, this is the only path that can take an
expectation value of $1$ at $\theta_{h}=\pi/2$. Applying LOWESA to find these
paths is effectively instantaneous and results in the amplitudes
$\sin(\theta_{h})^{25}$ and $\sin(\theta_{h})^{34}$ for panels c) and d),
respectively. Additional paths would then form corrections to these trivial
paths, but in this work, we did not find any additional paths with sizeable
amplitudes in this parameter range, suggesting even without exact verification
to compare to that the trivial reconstruction is of high quality.

Figure 4: Simulating 127-qubit TFI dynamics with individually controllable
angles $\theta_{J}^{(i,j)},\theta_{h}^{(i)}$. Extending the case of Fig. 2a,
we now employ the circuit in Eq. (13) with fully flexible angles, i.e., non-
Clifford entangling gates. Here we show how each single-site magnetization
evolves in time given the initial condition labeled by $\Delta t=0$, and
either $J^{(i,j)}=1$ between all qubits (a), or a diagonal linear ramp (b)
from $J^{(0,1)}=0$ at the top left qubit to $J^{(125,126)}=-3$ at the lower
right (indicated by the thickness of the connecting edges). Having
reconstructed the expectation landscapes for all 127 single-qubit $Z$
operators, recalculation of a snapshot in time can be done in seconds.

_Reproducing the hardest experiments:_ We now move on to reproducing Fig. 4b
from Ref. Kim  _et al._ (2023). Here, the circuit is significantly deeper with
$L=20$ Trotter layers, such that even with a single-qubit Z-observable, the
entanglement lightcone of the circuit includes all qubits. This case seems to
be significantly harder to reproduce using LOWESA, mostly because we can
quickly find millions of paths, many of them contributing significantly, and
in turn fill up the RAM of a laptop. Using a range of moderate truncation
values, we do however find trends that are generally in good agreement with
the experimentally observed values, as well as with the 31-qubit effective
lightcone simulations in Ref. Kechedzhi  _et al._ (2023). In Appendix A, we
provide a detailed study of how the expectation curves behave for different
truncation parameters, and differences in convergence behavior that we expect
compared to, for example, tensor network simulations. One of the curves that
we found to be converging in Fig. 5 in Appendix A is depicted in Fig. 3a.

Having built the surrogate landscape and verified its quality against other
methods, we can now depart from the correlated-angle case. We highlight this
in Fig. 3b and c, which contain Fig. 3a (i.e. Fig. 4b from Ref. Kim  _et al._
(2023)) as a special case. First, in panel b) we show a 2D expectation surface
spanned by two parameters $\theta_{h}^{(1)}$ and $\theta_{h}^{(2)}$, which
control all odd and all even qubit indices, respectively. The diagonal,
$\theta_{h}^{(1)}=\theta_{h}^{(2)}$, contains the data shown in panel a).
Second, we allow each of the $2540$ single-qubit RZ-gates to vary
independently by adding Gaussian random noise with zero mean and standard
deviation $\sigma/\Delta t$ to the fields $h^{(i)}$ on each qubit $i$.

We stress that the surface plots shown in Fig. 3 are only illustrative
examples: there are a myriad of different cuts of the expectation landscape
one could study with our surrogate. Similarly, with this surrogate one could
simulate any product initial state with $R_{X}$ rotations. Or, more generally,
by shorterning the circuit depth one could simulate entangled initial states
and more complex measurements. This demonstrates the flexibility of LOWESA-
one constructs the surrogate, and then can flexibly explore different
simulation regimes afterward.

We note that most other classical simulation techniques would take
significantly longer than the approximately two hours it took us to evaluate
the $158\times 158$ surface on a 128 CPU-core machine. This wall time
advantage may potentially even hold when including the construction of the
high-quality surrogate landscape (see convergence in Fig. 5), which took on
the order of one day. Furthermore, with every additional surface or initial
state simulated via the surrogate, the time saved using LOWESA, as compared to
other established approaches, only increases.

Recent results using tensor network techniques Anand  _et al._ (2023); Liao
_et al._ (2023) found that the true expectation values in the middle section
of the curve in Fig. 3a may be slightly larger than those predicted by the
non-tensor network methods Begušić and Chan (2023); Kechedzhi  _et al._
(2023), those obtained on noisy quantum hardware Kim  _et al._ (2023), or
recent large-scale simulations in Ref. Begušić  _et al._ (2023a), which were
claimed to be converged. This raises the interesting topic of verification
beyond the exactly verifiable regime. Each simulation method likely has its
niche where they are most accurate, and we show that LOWESA can contribute to
that joint effort of simulating quantum systems. In particular, other
simulation methods can be used to verify certain corners of LOWESA’s
expectation landscape so that it can be trusted when quickly observing novel
sections.

_Beyond Clifford entangling gates:_ To go beyond prior results on this TFI
model, we now employ Trotter time evolution circuits where all parameters
$\theta_{J}^{(i,j)}$ and $\theta_{h}^{(i)}$ are free parameters. While this
makes the simulation significantly harder by more than doubling the number of
free parameters and making all gates in Eq. (13) non-Clifford, it gives us
complete control over all Hamiltonian coefficients in Eq. (12) as well as
$\Delta t$. Previously, this value was not independently tunable, due to the
constraint that $\theta_{J}=-J\Delta t=-\pi/2$ had to remain constant to make
the entangling gate Clifford.

Applying this circuit to studying the magnetization
$M_{z}=\frac{1}{127}\sum_{i}\langle Z_{i}\rangle$, as in Fig. 2a, we can
investigate how each qubit’s magnetization evolves in time depending on the
initial condition and coupling strengths $J^{(i,j)}$. While LOWESA generically
assumes the all-zero initial state $|0\rangle^{\otimes n}$, the existing first
layer of RX-rotations can naturally be used to create a boundary wall with
individual qubits starting in the $|1\rangle$ state. Furthermore, we can
impose a gradient in the coupling strengths on the topology of the qubits, and
then evolve the system in time by changing $\Delta t$. These are all
capabilities of LOWESA that are enabled by the fast re-evaluation of the
reconstructed surrogate landscape and highlight how circuit parameters can be
interpreted in several ways with distinct applications.

Fig. 4 depicts the single-qubit magnetization on the 127-qubit heavy-hex
topology evolving in time for $h^{(i)}=h=1$ and $J^{(i,j)}=J=1$ in panel a)
and for a linear ramp in the couplings from the top left to the lower right in
panel b). This results in $J^{(0,1)}=0$ at the top left qubit to
$J^{(125,126)}=-3$ at the lower right. While the homogeneous coupling case
exhibits a steady decay of magnetization over time, the non-homogeneous case
displays more intricate dynamics where large couplings initially block the
oscillation induced by the local magnetic fields $h$. To be able to trust
these results, we tested the quality of this more flexible circuit on the
restricted 1-dimensional slice of the landscape, which is depicted in Fig. 2a
(see Fig. 6 for that comparison).

While the joint memory requirements for all 127 individually reconstructed
landscapes (in our implementation) exceed the RAM capabilities of conventional
laptops, re-evaluation of one snapshot in time only takes a few seconds if all
landscapes are loaded into a memory. Using distributed computing resources
paired with LOWESA’s natural parallelizable workflow, we believe that the size
and complexity of systems that can realistically be simulated could be
substantially increased.

Finally, we push our algorithm to its current limit by opening up all angles
for the $L=20$ Trotter steps case in Fig. 3. This circuit contains 5420 free
parameters, which for us results in $m=5420$ RZ-gates and 5760 CNOT gates in
the circuit in a 2D entangling topology (which are more than twice as many RZ
gates and CNOT gates as in Fig. 3). Fig. 6 in Appendix B estimates the quality
of the reconstructed landscape by comparing to the correlated-angle slice of
the landscape with $J^{(i,j)}=J=-\pi/2$. While our results for the fully open-
angle circuit are arguably of equal (or higher quality) than some of the
methods presented in Refs. Kim  _et al._ (2023) and Anand  _et al._ (2023), we
do not feel confident that finer details in the full landscape can be trusted.
Even so, LOWESA’s performance on this quantum circuit of elevated difficulty
highlights its scalability relative to more established methods such as tensor
networks. More computing resources, smarter truncation, or a better tree-
search algorithm than the parallelized depth-first search Rao and Kumar (1987)
employed here would further benefit our method.

##  V Discussion

In this work, we present LOWESA as an algorithm to classically construct a
surrogate for the expectation landscapes in quantum simulation tasks. While
this algorithm was initially introduced for simulating noisy circuits (where
it enjoys efficiency guarantees), we show that it is fully capable of
predicting close to exact expectation values in 127-qubit systems with
thousands of entangling gates. While conventional classical algorithms
simulate quantum circuits with one fixed parameter vector $\bm{\theta}$ at a
time, LOWESA approximately reconstructs the entire expectation landscape
spanned by the circuit parameters. This process is more costly than
calculating a single expectation value, but the surrogate landscape can then
be re-evaluated at different parameters at a significantly lower computational
cost than existing methods.

We apply this algorithm to the heavy-hex TFI system studied in Ref. Kim  _et
al._ (2023) and find that it produces competitive results on a laptop. Where
exact calculation is possible, LOWESA demonstrably gives high-quality
predictions. Even beyond exact verification, our results are in good agreement
with the state-of-the-art classical simulation methods presented in Refs.
Tindall  _et al._ (2023); Kechedzhi  _et al._ (2023); Begušić and Chan (2023);
Torre and Roses (2023); Liao  _et al._ (2023); Begušić  _et al._ (2023a).
Interestingly, by use of the efficiently calculable Clifford circuit boundary
cases, we find that two of the five observables considered allow for trivial
solutions with $\sin^{25}$ and $\sin^{34}$ for Figs. 2c and 3a, respectively.
These trivial solutions are however not sufficient to construct an accurate
surrogate outside of the presented parameter domain. This highlights that full
reconstruction of the expectation landscape, as done in our work, is vastly
more challenging than the setup considered in Ref. Kim  _et al._ (2023) and
follow-up works Tindall  _et al._ (2023); Kechedzhi  _et al._ (2023); Begušić
and Chan (2023); Torre and Roses (2023); Liao  _et al._ (2023); Begušić  _et
al._ (2023a).

Having created the surrogate expectation landscape for the 127-qubit examples
with fully opened-up angles (which in its current form took two days on a
laptop), we can freely tune the initial state of the simulation, the
Hamiltonian coefficients, and even the target observable. This allows us to
study single-site magnetization in the system evolving in time with
homogeneous or in-homogeneous couplings between sites. Evaluating one snapshot
in time for all 127 observables and a new initial state only takes a few
seconds on a laptop once the surrogate is loaded into memory. We stress that
as well as providing a powerful tool to quickly simulate a range of physical
problems, LOWESA (in contrast to many other simulation methods) is not
directly limited by circuit topology.

Going forward, we believe that LOWESA can fulfill a valuable function that is
complementary to existing classical simulation methods as well as quantum
hardware experiments. Its ability to quickly and broadly scan expectation
landscapes is a feature that no other popular simulation method shares. The
surrogate landscape can, for example, be used to verify the existence and
narrow down the location of critical points, to perform meta-learning of
Hamiltonians with certain time evolution properties, or it can be used for
tasks where explicit access to the Fourier spectrum of an observable is
required (for example, phase estimation Somma  _et al._ (2002), or calculating
Green’s functions Gomes  _et al._ (2023)).

How far this method can be pushed to larger and more complicated systems
remains to be studied and likely requires further development of truncation
and approximation techniques. We also expect symmetries in the system or
quantum circuit to help find or trim paths Fontana  _et al._ (2022), whose
contribution can be predicted with prior knowledge. However, even in its
current form presented in this work, we show that LOWESA is competitive with
leading classical simulation methods in producing high-quality expectation
estimates, with a likely computational scaling advantage as the number of
expectation function evaluations increases.

##  VI Acknowledgements

The authors would like to acknowledge Cristina Cîrstoiu for helpful
discussions. The authors would like to acknowledge Zach Morell and Carleton
Coffrin for their contribution and valuable suggestions on the current
implementation of the LOWESA algorithm. This work was supported by the U.S.
Department of Energy (DOE) through a quantum computing program sponsored by
the Los Alamos National Laboratory (LANL) Information Science & Technology
Institute. M.S.R. was partially supported by the U.S. DOE, Office of Science,
Office of Advanced Scientific Computing Research, under the Accelerated
Research in Quantum Computing (ARQC) program. L.C. acknowledges support by the
Laboratory Directed Research and Development (LDRD) program of LANL under
project number 20230049DR. E.F. acknowledges support from the UK government
department for Business, Energy and Industrial Strategy through the UK
National Quantum Technologies Programme, and of an industrial CASE
studentship, funded by the UK Engineering and Physical Sciences Research
Council (grant EP/T517665/1), in collaboration with the University of
Strathclyde, the National Physical Laboratory, and Quantinuum. E.F. also
acknowledges support by JPMorgan Chase & Co. via the Quantum Computing Summer
Associate Program. Z.H. acknowledges support from the Sandoz Family
Foundation-Monique de Meuron program for Academic Promotion.

## References

  * Orús (2014) Román Orús, “A practical introduction to tensor networks: Matrix product states and projected entangled pair states,” Annals of Physics 349, 117–158 (2014).
  * Orús (2019) Román Orús, “Tensor networks for complex quantum systems,” Nature Reviews Physics 1, 538–550 (2019).
  * Ayral  _et al._ (2023) Thomas Ayral, Thibaud Louvet, Yiqing Zhou, Cyprien Lambert, E Miles Stoudenmire, and Xavier Waintal, “Density-matrix renormalization group algorithm for simulating quantum circuits with a finite fidelity,” PRX Quantum 4, 020304 (2023).
  * Pan and Zhang (2022) Feng Pan and Pan Zhang, “Simulation of quantum circuits using the big-batch tensor network method,” Physical Review Letters 128, 030501 (2022).
  * Carleo and Troyer (2017) Giuseppe Carleo and Matthias Troyer, “Solving the quantum many-body problem with artificial neural networks,” Science 355, 602–606 (2017).
  * Torlai  _et al._ (2018) Giacomo Torlai, Guglielmo Mazzola, Juan Carrasquilla, Matthias Troyer, Roger Melko, and Giuseppe Carleo, “Neural-network quantum state tomography,” Nature Physics 14, 447–450 (2018).
  * Schmitt and Heyl (2020) Markus Schmitt and Markus Heyl, “Quantum many-body dynamics in two dimensions with artificial neural networks,” Physical Review Letters 125, 100503 (2020).
  * Bernstein and Vazirani (1997) Ethan Bernstein and Umesh Vazirani, “Quantum complexity theory,” SIAM Journal on computing 26, 1411–1473 (1997).
  * Markov  _et al._ (2018) Igor L Markov, Aneeqa Fatima, Sergei V Isakov, and Sergio Boixo, “Quantum supremacy is both closer and farther than it appears,” arXiv preprint arXiv:1807.10749 (2018).
  * Häner and Steiger (2017) Thomas Häner and Damian S Steiger, “5 petabyte simulation of a 45-qubit quantum circuit,” in  _Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis_ (2017) pp. 1–10.
  * Gottesman (1998) Daniel Gottesman, “The heisenberg representation of quantum computers, talk at,” in  _International Conference on Group Theoretic Methods in Physics_ (Citeseer, 1998).
  * Rall  _et al._ (2019) Patrick Rall, Daniel Liang, Jeremy Cook, and William Kretschmer, “Simulation of qubit quantum circuits via pauli propagation,” Physical Review A 99, 062337 (2019).
  * Fontana  _et al._ (2023) Enrico Fontana, Manuel S Rudolph, Ross Duncan, Ivan Rungger, and Cristina Cîrstoiu, “Classical simulations of noisy variational quantum circuits,” arXiv preprint arXiv:2306.05400 (2023).
  * Schreiber  _et al._ (2022) Franz J Schreiber, Jens Eisert, and Johannes Jakob Meyer, “Classical surrogates for quantum learning models,” arXiv preprint arXiv:2206.11740 (2022).
  * Jerbi  _et al._ (2023) Sofiene Jerbi, Casper Gyurik, Simon C Marshall, Riccardo Molteni, and Vedran Dunjko, “Shadows of quantum machine learning,” arXiv preprint arXiv:2306.00061 (2023).
  * Landman  _et al._ (2022) Jonas Landman, Slimane Thabet, Constantin Dalyac, Hela Mhiri, and Elham Kashefi, “Classically approximating variational quantum machine learning with random fourier features,” arXiv preprint arXiv:2210.13200 (2022).
  * Cerezo  _et al._ (2021) M. Cerezo, Andrew Arrasmith, Ryan Babbush, Simon C Benjamin, Suguru Endo, Keisuke Fujii, Jarrod R McClean, Kosuke Mitarai, Xiao Yuan, Lukasz Cincio, and Patrick J. Coles, “Variational quantum algorithms,” Nature Reviews Physics 3, 625–644 (2021).
  * Trotter (1959) H. F. Trotter, “On the product of semi-groups of operators,” Proceedings of the American Mathematical Society 10, 545–551 (1959).
  * Lloyd (1996) Seth Lloyd, “Universal quantum simulators,” Science , 1073–1078 (1996).
  * Sornborger and Stewart (1999) AT Sornborger and Ewan D Stewart, “Higher-order methods for simulations on quantum computers,” Physical Review A 60, 1956 (1999).
  * Kim  _et al._ (2023) Youngseok Kim, Andrew Eddins, Sajant Anand, Ken Xuan Wei, Ewout Van Den Berg, Sami Rosenblatt, Hasan Nayfeh, Yantao Wu, Michael Zaletel, Kristan Temme, _et al._ , “Evidence for the utility of quantum computing before fault tolerance,” Nature 618, 500–505 (2023).
  * Berg  _et al._ (2022) Ewout van den Berg, Zlatko K Minev, Abhinav Kandala, and Kristan Temme, “Probabilistic error cancellation with sparse pauli-lindblad models on noisy quantum processors,” arXiv preprint arXiv:2201.09866 (2022).
  * Bennett  _et al._ (1996) Charles H Bennett, Gilles Brassard, Sandu Popescu, Benjamin Schumacher, John A Smolin, and William K Wootters, “Purification of noisy entanglement and faithful teleportation via noisy channels,” Physical review letters 76, 722 (1996).
  * Knill (2004) Emanuel Knill, “Fault-tolerant postselected quantum computation: Threshold analysis,” arXiv preprint quant-ph/0404104 (2004).
  * Temme  _et al._ (2017) Kristan Temme, Sergey Bravyi, and Jay M. Gambetta, “Error mitigation for short-depth quantum circuits,” Phys. Rev. Lett. 119, 180509 (2017).
  * Kandala  _et al._ (2019) Abhinav Kandala, Kristan Temme, Antonio D. Córcoles, Antonio Mezzacapo, Jerry M. Chow, and Jay M. Gambetta, “Error mitigation extends the computational reach of a noisy quantum processor,” Nature 567, 491–495 (2019).
  * Giurgica-Tiron  _et al._ (2020) Tudor Giurgica-Tiron, Yousef Hindy, Ryan LaRose, Andrea Mari, and William J Zeng, “Digital zero noise extrapolation for quantum error mitigation,” 2020 IEEE International Conference on Quantum Computing and Engineering (QCE) , 306–316 (2020).
  * Tindall  _et al._ (2023) Joseph Tindall, Matt Fishman, Miles Stoudenmire, and Dries Sels, “Efficient tensor network simulation of IBM’s kicked ising experiment,” arXiv preprint arXiv:2306.14887 (2023).
  * Kechedzhi  _et al._ (2023) K Kechedzhi, SV Isakov, S Mandrà, B Villalonga, X Mi, S Boixo, and V Smelyanskiy, “Effective quantum volume, fidelity and computational cost of noisy quantum processing experiments,” arXiv preprint arXiv:2306.15970 (2023), https://doi.org/10.48550/arXiv.2306.15970.
  * Begušić and Chan (2023) Tomislav Begušić and Garnet Kin Chan, “Fast classical simulation of evidence for the utility of quantum computing before fault tolerance,” arXiv preprint arXiv:2306.16372 (2023).
  * Torre and Roses (2023) Emanuele G Dalla Torre and Mor M Roses, “Dissipative mean-field theory of ibm utility experiment,” arXiv preprint arXiv:2308.01339 (2023).
  * Liao  _et al._ (2023) Hai-Jun Liao, Kang Wang, Zong-Sheng Zhou, Pan Zhang, and Tao Xiang, “Simulation of IBM’s kicked ising experiment with projected entangled pair operator,” arXiv preprint arXiv:2308.03082 (2023).
  * Begušić  _et al._ (2023a) Tomislav Begušić, Johnnie Gray, and Garnet Kin Chan, “Fast and converged classical simulations of evidence for the utility of quantum computing before fault tolerance,” arXiv preprint arXiv:2308.05077 (2023a).
  * Chow  _et al._ (2012) Jerry M Chow, Jay M Gambetta, Antonio D Corcoles, Seth T Merkel, John A Smolin, Chad Rigetti, S Poletto, George A Keefe, Mary B Rothwell, John R Rozen, _et al._ , “Universal quantum gate set approaching fault-tolerant thresholds with superconducting qubits,” Physical review letters 109, 060501 (2012).
  * Wood  _et al._ (2011) Christopher J. Wood, Jacob D. Biamonte, and David G. Cory, “Tensor networks and graphical calculus for open quantum systems,” arXiv preprint arXiv:1111.6950 (2011).
  * Nemkov  _et al._ (2023) Nikita A Nemkov, Evgeniy O Kiktenko, and Aleksey K Fedorov, “Fourier expansion in variational quantum algorithms,” arXiv preprint arXiv:2304.03787 (2023).
  * Begušić  _et al._ (2023b) Tomislav Begušić, Kasra Hejazi, and Garnet Kin Chan, “Simulating quantum circuit expectation values by clifford perturbation theory,” arXiv preprint arXiv:2306.04797 (2023b).
  * Anand  _et al._ (2023) Sajant Anand, Kristan Temme, Abhinav Kandala, and Michael Zaletel, “Classical benchmarking of zero noise extrapolation beyond the exactly-verifiable regime,” arXiv preprint arXiv:2306.17839 (2023).
  * Rao and Kumar (1987) V Nageshwara Rao and Vipin Kumar, “Parallel depth first search. part i. implementation,” International Journal of Parallel Programming 16, 479–499 (1987).
  * Somma  _et al._ (2002) Rolando Somma, Gerardo Ortiz, James E Gubernatis, Emanuel Knill, and Raymond Laflamme, “Simulating physical phenomena by quantum networks,” Physical Review A 65, 042323 (2002).
  * Gomes  _et al._ (2023) Niladri Gomes, David B Williams-Young, and Wibe A de Jong, “Computing the many-body green’s function with adaptive variational quantum dynamics,” Journal of Chemical Theory and Computation (2023).
  * Fontana  _et al._ (2022) Enrico Fontana, M Cerezo, Andrew Arrasmith, Ivan Rungger, and Patrick J Coles, “Non-trivial symmetries in quantum landscapes and their resilience to quantum noise,” Quantum 6, 804 (2022).

##  Appendix A Convergence with truncation parameters

Figure 5: Improving approximation with higher truncation weight $W$. For the
system considered in Fig. 3a, we show the convergence of the expectation
curve, and thus the surrogate landscape, with operator weight truncation $W$,
given fixed values for the cutoff frequency $\ell=36$ and a truncation
probability of $p=0.05$.

The truncation and tailoring techniques outlined in Sec. III can be understood
as hyperparameters that control both the computational resource requirements
and the quality of the surrogate landscape. Similarly to the bond dimension
hyperparameter in tensor network simulations, reduction in computational cost
may not come with a reduction in quality, and a more expensive simulation may
not yield better results. That is, there exists a physical maximal bond
dimension that allows one to capture all entanglement in the system, below
which quality degrades, but above which only computational cost increases at
the same quality. In that case, the simulation is said to be converged. The
same can be observed for the truncations used in this work, where, at a
certain point, relaxing the truncations does not further improve performance,
and one may have achieved close-to-exact results at a tractable computational
cost.

There are however two significant differences between the truncations employed
in this work to the well-understood bond dimension in tensor networks.

First, the computational resource scaling of LOWESA is generally exponential
in some of the current truncation parameters. Increasing the maximum frequency
$\ell$ by one up to doubles the number of paths explored, and similarly
increasing the maximum operator weight $W$ by one roughly doubles the chance
of an operator $\textbf{U}^{\dagger}_{\bm{\omega}}|O\rrangle$ to annihilate
against the initial state, i.e., $\llangle
0|\textbf{U}^{\dagger}_{\bm{\omega}}|O\rrangle=0$. This is in contrast to the
bond dimension, which generally needs to scale exponentially to faithfully
simulate a quantum system, but the computational scaling is polynomial. This
tends to cause discrete jumps in LOWESA’s quality for every increase in $\ell$
and $W$, but convergence is still possible.

As an example, in Fig. 5 we show how the curve corresponding to the $L=20$
Trotter step system in Fig. 3 converges with increasing $W$. We recall that
the operator weight truncation can be viewed as enforcing an effective
entanglement lightcone per path. Thus the convergence can be attributed to the
truncated lighcone approaching the true physical lightcone. On a side note,
LOWESA naturally respects the entanglement lightcone induced by the circuit
topology, because RZ-gates outside the entanglement lightcone never cause the
paths to split.

In the version of LOWESA presented in this work, we utilized two main
truncation parameters, $\ell$ and $W$. This multitude of truncation parameters
is a second key difference between LOWESA and tensor networks, where only one
truncation (i.e, bond dimension) is generally used. When LOWESA is used with
multiple truncation parameters, each parameter can only converge with respect
to the limits imposed by the other truncations. While the curves in Fig. 5
appear to be converging with $W$, they are still not exact because the chosen
value of $l$ is too low. In such cases it may occasionally turn out that more
restrictive values of a given truncation parameter (given constraints from
other fixed truncation parameters) are actually closer to the true dynamics.
This is seen in Fig. 5 where $W=9$ appears slightly closer to the true
dynamics than $W=10$.

One may wonder why the truncation probability parameter $p$ is not mentioned
above as one of the main truncation parameters. That is because in effect it
acts as a relaxation of $\ell$. Assuming that the splitting probability is
constant along any given path, the probability of eventual truncation can be
estimated as
$P(|\bm{\omega}|_{1}\leq\ell\,|\,l,k)=\sum_{i=0}^{m-k}\binom{m-k}{i}(\frac{l}{k})^{i}(1-\frac{l}{k})^{m-k-i}$,
where $l$ is the number of splits so far when encountering the
$(k+1)^{\text{th}}$ RZ-gate out of $m$ in total. $P$ tends to be highest for
the paths that eventually end up with close to $\ell$ splits. Thus $p$
effectively reduces the contribution of frequencies close to the cutoff which
are most likely to have $P(|\bm{\omega}|_{1}\leq\ell\,|\,l,k)<p$.

It is vital to the reliability of LOWESA that future work investigates and
quantifies the precise impact of individual truncations on the surrogate
landscape, and potentially reveals novel truncations which have a more
predictable computational cost to quality trade-off. While that is plausibly
impossible for the simulation of noise-free systems in the worst case,
practical cases may allow powerful heuristics.

##  Appendix B Verification of the free-angle surrogate

Figure 6: Verifying the open-angle quantum circuits. We depict the expectation
curves corresponding to Figs. 2a (left) and 3a (right), as well as the
expectations produced by the surrogate using the significantly more
challenging circuit in Eq. (13) with fully individually controllable angles.
The truncation for the open-angle cases are $\ell=30,p=0.05,W=8$ (left) and
$\ell=40,p=0.05,W=8$ (right). The MPS and isoTNS simulations of the simpler
restricted-angle presented in Ref. Kim  _et al._ (2023) are depicted in pink
and red.

In this section, we validate the performance of the fully open-angle quantum
circuits discussed towards the end of Sec. IV and presented in Fig. 4. As a
reminder, while the quantum circuit for the Trotter time evolution of the TFI
Hamiltonian is given in Eq. (13), the example benchmarks introduced in Ref.
Kim  _et al._ (2023) fixed all coupling strengths
$J^{(i,j)}=J=-\frac{\pi}{2\Delta t}$, such that
$\theta_{J}^{(i,j)}=\theta_{J}=-\frac{\pi}{2}$. This makes the $R_{ZZ}$
entangling gates Clifford-gates (and so more efficient to classically
simulate) and only require one CNOT gate to implement on IBM’s quantum device
Kim  _et al._ (2023).

The quantum circuit used in Fig. 4 on the other hand employs a more
challenging version of the circuit where all gates are freely parametrized,
i.e. the circuit is actually that in Eq. (13). Seeing as the surrogate
landscape constructed by this circuit includes the circuit for Figs. 2 and 3
as a special case, we verify the quality of the landscape on those restricted
slices of the landscape.

The results for the magnetization $M_{z}=\frac{1}{127}\sum_{i}\langle
Z_{i}\rangle$ (see Fig 2a) and the most challenging case of the $\langle
Z_{62}\rangle$ observable with $L=20$ Trotter-layers (see Fig. 3) are shown in
Fig. 6. We see that the expectation curve for the magnetization is nearly
exact, which suggests that the surrogate landscape is overall of high quality.
On the other hand, the surrogate for the $Z_{62}$ observable cannot be
expected to be highly accurate for all angles since it does not match the
experimental and simulation data for the simplified circuit. However, the
general trend of the curve can still be argued to be of equal or better
quality than some of the methods presented in Refs. Kim  _et al._ (2023) and
Anand  _et al._ (2023), which highlights the scalability of the LOWESA
algorithm, even on this significantly harder quantum simulation task.

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