# Sequency Hierarchy Truncation (SeqHT) for Adiabatic State Preparation and
Time Evolution in Quantum Simulations

Zhiyao Li  0000-0002-7614-8496<EMAIL_ADDRESS>InQubator for Quantum Simulation
(IQuS), Department of Physics, University of Washington, Seattle, WA 98195
Dorota M. Grabowska  0000-0002-0760-4734<EMAIL_ADDRESS>InQubator for Quantum
Simulation (IQuS), Department of Physics, University of Washington, Seattle,
WA 98195  Martin J. Savage  0000-0001-6502-7106<EMAIL_ADDRESS>InQubator for
Quantum Simulation (IQuS), Department of Physics, University of Washington,
Seattle, WA 98195

###### Abstract

We introduce the Sequency Hierarchy Truncation (SeqHT) scheme for reducing the
resources required for state preparation and time evolution in quantum
simulations, based upon a truncation in sequency. For the $\lambda\phi^{4}$
interaction in scalar field theory, or any interaction with a polynomial
expansion, upper bounds on the contributions of operators of a given sequency
are derived. For the systems we have examined, observables computed in
sequency-truncated wavefunctions, including quantum correlations as measured
by magic, are found to step-wise converge to their exact values with
increasing cutoff sequency. The utility of SeqHT is demonstrated in the
adiabatic state preparation of the $\lambda\phi^{4}$ anharmonic oscillator
ground state using IBM’s quantum computer ibm_sherbrooke. Using SeqHT, the
depth of the required quantum circuits is reduced by $\sim 30\%$, leading to
significantly improved determinations of observables in the quantum
simulations. More generally, SeqHT is expected to lead to a reduction in
required resources for quantum simulations of systems with a hierarchy of
length scales.

##  I Introduction

Quantum computing [1, 2, 3, 4, 5] is opening new and unexpected pathways
forward to better understand an array of quantum many-body systems that are
important for scientific and technological applications. These advances are
expected to lead to predictive capabilities for the structure and dynamics of
systems of fundamental particles that are far from equilibrium (for recent
reviews, see Refs. [6, 7, 8, 9, 10, 11]). These systems includes neutrinos in
extreme astrophysical environments, e.g., Refs. [12, 13, 14, 15, 16, 17, 18,
19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37,
38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56,
57, 58, 59, 60, 61, 62], matter in the interior of neutron stars, e.g., Refs.
[63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81,
82], and highly-inelastic collisions of hadrons created in the laboratory,
e.g., Refs. [83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98].
In recent years, there has been rapid progress in the development of quantum
simulations of these systems [45, 99, 100, 53, 101], and as they are
fundamentally quantum in nature, a quantum advantage is anticipated to be
achieved sooner rather than later. However, Noisy Intermediate-Scale Quantum
(NISQ)-era quantum computers [102] still impose significant limitations on the
scale and dimensionality of systems that can be simulated. Presently, quantum
simulations are not yet comparable with experiment, nor can they produce
results beyond the capacity of classical computing.

Despite recent demonstrations of systems with a small number of error-
corrected logical qubits [103, 104, 105] and promising results from
simulations that used more than 100 qubits [106, 107, 108, 109, 110, 111, 112,
113], NISQ-era digital quantum computers remain limited in the quality of
their entangling gates and the circuit depths that can be executed before the
quantumness of the device is effectively lost [114, 115, 116, 117]. While
current quantum computers are not yet practical for most real-world
applications, research and development are progressing rapidly. Significant
effort is being placed on creating efficient and effective quantum circuits
that provide results for the target observables within specified error
tolerances. This can include simulating systems that are nearby in theory
space (for example, a leading order Hamiltonian in a perturbative expansion
about a target Hamiltonian) [118, 119, 120], developing perturbative schemes
[121, 122, 123, 124, 125], and identifying truncations of the Hamiltonian
based upon the emergent properties of the systems such as confinement and gaps
[126, 127, 128, 129, 130]. One major recent advance in the simulation of
lattice gauge theories is the use of confinement to truncate operator
contributions based on spatial separation combined with operators that are
scalable to arbitrarily large numbers of spatial sites [112, 113]. Error
mitigation is essential for recovering meaningful results from quantum
computers due to decoherence especially for deeper circuits [131, 132, 133,
134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148,
149, 150, 151].

The formal aspects of using (ideal) quantum computers to simulate
$\lambda\phi^{4}$ theory have been established in the pioneering papers by
Jordan, Lee and Preskill (JLP) [152, 153, 154, 155]. These papers developed
all of the necessary tools for determining S-matrix elements, and have shown
that the problem is BQP-complete (when classical background sources are
included). This last point means that any system that can be simulated
efficiently on a quantum computer can be mapped (with polynomial-scaling
resources) to this particular scalar field theory. A number of subsequent
works [156, 157, 158, 159, 160, 161] further examined the theory and
identified further digitization possibilities, including using a harmonic
oscillator basis to encode the scalar field as opposed to the eigenstates of
the field operator. JLP showed that a judicious choice of conjugate-momentum
operator implemented via a local quantum Fourier transform (QFT) exponentially
suppresses digitization errors, with corrections to the continuum suppressed
by factors of $\sim e^{-\alpha/\delta_{\phi}^{2}}$ where $\alpha$ is a
digitization-independent factor and $\delta_{\phi}$ is the sampling interval
in field space. Furthermore, JLP proposed using adiabatic state preparation
(ASP) to prepare the vacuum and wavepackets in the interacting theory; this
method requires slowly evolving from the ground state of the free theory, with
$\lambda=0$, to non-zero $\lambda$ along a Trotterized trajectory in
$\lambda(s)$. Using currently available devices, implementing this algorithm
(even) on small systems results in states with relatively large errors
compared to the target state. While methods continue to become available for
preparing the ground state, including variational quantum eigensolver (VQE)
[162, 163, 164, 165, 94, 166, 167, 168], ADAPT-VQE and more (for example Refs.
[169, 170, 171, 113, 112, 172, 173, 174, 175, 176, 177]), state preparation
remains a generically challenging problem that nominally lies outside of the
BQP complexity class. This means that algorithms that can efficiently prepare
states (within a specified fidelity) are important to identify, and will
likely be of general utility.

In this work, we build upon these advances to identify a new convergent
truncation that reduces quantum resources required for state preparation and
time evolution of smooth and bounded wavefunctions. This truncation is guided
by a hierarchy in the contributions to low-energy observables from basis
operators based upon their sequency [178]. We call this scheme Sequency
Hierarchy Truncation (SeqHT), and demonstrate its utility in preparing the
ground-state wavefunction of $\lambda\phi^{4}$ interacting scalar-field
theory. Because the implementation of SeqHT will be iterative in many setting,
for example systematically increasing the cutoff sequency until changes in
observables fall below a predetermined threshold, it should be considered a
hybrid quantum-classical algorithm. We derive rigorous upper bounds on the
contributions of operators with regard to their sequency, and study the
convergence of observables and quantum computational complexity (magic) as a
function of increasing cutoff sequency. We construct the quantum circuits for
adiabatically preparing the ground state of $\lambda\phi^{4}$, utilizing
sequency truncation and JLP’s method of starting from the non-interacting
theory. An important element of the circuit development is that sequency-
ordered operators allow for (maximum) CNOT-gate cancellations in implementing
diagonal unitaries in preparing the non-interacting ground state [179, 178,
180].111For the time evolution circuit, overlapping $ZZ$ gates requires fewer
CNOT gates than arranging them in sequency order. These circuits are then
executed, along with their partner mitigation circuits, using IBM’s
superconducting-qubit quantum computer ibm_sherbrooke [114]. The results of
these simulations clearly demonstrate the utility of SeqHT for improving the
fidelity of adiabatically-prepared states (and evolution) because of the
reduced quantum resource requirements. While the results we present here are
compared with the analogous results obtained using classical computing
methods, the actual utility of the method is on preparing states that cannot
be accessed using these classical approaches.

While it is important to advance quantum simulations of scalar field theories,
a more important objective for our work is to reduce the quantum resources
required to simulate lattice gauge theories, both Abelian and non-Abelian. The
wavefunction of the gauge-field is expected to be smooth, and localized near
the scale of the physics of interest (after renormalization). As such, we
anticipate that SeqHT will also have utility in quantum simulations of (2+1) D
and (3+1) D quantum chromodynamics (QCD) [181, 182, 183, 184, 185, 186, 187,
188, 189].

##  II Sequency Hierarchy of Operators and Wavefunctions

Hermitian operators in their own eigenbasis are represented by real diagonal
matrices which can be decomposed into an orthonormal basis of tensor products
of the $2\times 2$ identity matrix, $\hat{I}$, and Pauli matrix $\hat{Z}$, in
the form of

$\hat{\mathcal{O}}=\sum_{\nu=0}^{2^{n_{q}}-1}\beta_{\nu}\hat{\mathcal{O}}_{\nu}\quad,\quad\hat{\mathcal{O}}_{\nu}=\bigotimes_{j=0}^{n_{q}-1}\hat{\sigma}_{[j]}=\hat{\sigma}_{[0]}\otimes\hat{\sigma}_{[1]}\otimes\dots\otimes\hat{\sigma}_{[n_{q}-1]}\quad,\quad$
(1)

where $n_{q}$ is the number of qubits and each $\hat{\sigma}_{[j]}$ is either
$\hat{I}$ or $\hat{Z}$ acting on qubit $j$. The subscript $\nu$ is the
sequency index that uniquely labels each basis operator, and $\beta_{\nu}$ are
the sequency coefficients (sequency and sequency indexing will be explained
later in this section). Since diagonal matrices commute, the time-evolution
operator $e^{-i\hat{\mathcal{O}}t}$ can be decomposed as

$\displaystyle
e^{-i\hat{\mathcal{O}}t}=\prod\limits_{\nu=0}^{2^{n_{q}}-1}e^{-i\beta_{\nu}\hat{\mathcal{O}}_{\nu}t}\,.$
(2)

Similarly, digital wavefunctions can be decomposed as

$|\psi\rangle=\sum_{\nu=0}^{2^{n_{q}}-1}\alpha_{\nu}|\psi_{\nu}\rangle\quad,\quad|\psi_{\nu}\rangle=\bigotimes_{j=0}^{n_{q}-1}|\ell_{[j]}\rangle=|\ell_{[0]}\rangle\otimes|\ell_{[1]}\rangle\otimes\dots\otimes|\ell_{[n_{q}-1]}\rangle\quad,$
(3)

where the normalization constant for basis states $|\psi_{\nu}\rangle$ is
omitted, $\alpha_{\nu}$ are the sequency coefficients and $|\ell_{[j]}\rangle$
is either the column vector $(1,1)$ or $(1,-1)$ (the diagonal of $\hat{I}$ or
$\hat{Z}$) for the $j$th qubit.

The diagonals of the basis operators $\hat{\mathcal{O}}_{\nu}$ in Eq. (1) and
the basis vectors $|\psi_{\nu}\rangle$ in Eq. (3) are Walsh functions [190],
which form a complete and orthogonal set of functions; the diagonal of
$\hat{\mathcal{O}}_{\nu}$ corresponds to the $\nu^{\text{th}}$ row of a
sequency-ordered Walsh-Hadamard matrix. This matrix can be constructed by
rearranging the rows of the natural-ordered Walsh-Hadamard matrix, which can
be easily constructed via a tensor product:

$\mathrm{H}_{n_{q}}=\mathrm{H}^{\otimes
n_{q}}\quad,\quad\mathrm{H}=\left(\begin{array}[]{cc}1&1\\\
1&-1\end{array}\right)\ .$ (4)

This Walsh series representation of discrete functions is the digital
counterpart of the Fourier series and have an important role in digital signal
processing [191]. The harmonics constituting a Fourier series can each be
indexed by an integer $n$ that uniquely corresponds to a frequency given by
$\omega_{n}=n$/Period. Similarly, Walsh basis operators $\mathcal{O}_{\nu}$
and basis states $|\psi_{\nu}\rangle$ have a bijection to sequency, the
digital analog of frequency, denoted as $\nu$. Sequencies of a digital
function can be obtained by counting the number of zero crossings (the number
of times the sign of a function changes) on their diagonals, so
$\nu\in\mathbb{N}$. Since there is a parallel between the sequency
decomposition of $\hat{\mathcal{O}}_{\nu}$ and $|\psi_{\nu}\rangle$, we will
focus the rest of the discussions in this section on
$\hat{\mathcal{O}}_{\nu}$, but we note that the discussed properties apply to
both (with the exception of circuit constructions for unitary evolutions
$e^{i\beta\hat{\mathcal{O}}_{\nu}}$).

In such a sequency expansion, the physics of low-energy configurations lead to
a finite range of support in frequency (sequency) space, resulting in a
hierarchy in the sequency coefficients that can be leveraged in quantum
simulation. Basis operators with coefficients below a certain threshold,
$\Lambda_{\rm cut}$, can be truncated to reduce the circuit size for
implementing a target unitary. In earlier approaches, either classical
resources were utilized to calculate all Walsh coefficients to determine which
were below the desired $\Lambda_{\rm cut}$ [180], or a mix of quantum and
classical resources were used to iteratively tune target observables to
ascertain convergence below the desired error threshold [178]. As the
classical computing resources required for decomposing target operators scale
exponentially with system size, large-system simulations remained impossible
without a effective method to select operators for truncation. Sequency
hierarchy provides insight to this problem. As will be shown later in this
paper, the value of the coefficient $\beta_{\nu}$ can be bounded from above
when considering its contribution to any polynomial operator that appears in
the Hamiltonian; this also then bounds from above $\beta_{\nu}$ for any
operator that can be written as a power expansion of polynomial operators.
This upper bound decreases for operators with increasing sequency, exhibiting
a sequency hierarchy. This allows the determination of a cutoff in sequency
space $\nu_{\rm cut}$ that respect the threshold $\Lambda_{\rm cut}$ without
calculating the whole decomposition. Thus, for large systems, it is viable to
construct only a select set of low-sequency operators, obtain their
coefficients (by projecting the target operator onto this basis), and
implement a reduced pool of quantum circuits.

Quantum circuits implementing $e^{i\beta\hat{\mathcal{O}}_{\nu}}$ are required
for time evolution induced by sequency operators. Two sequency-adjacent basis
operators $\hat{\mathcal{O}}_{\nu}$ and $\hat{\mathcal{O}}_{\nu+1}$ differ in
only one $\hat{\sigma}_{[j]}$. One can be transformed into another by flipping
the $\hat{\sigma}_{[j]}$ on the most significant qubit that was not updated in
the transformation from $\hat{\mathcal{O}}_{\nu-1}$ to
$\hat{\mathcal{O}}_{\nu}$. This resembles the Gray Code, a binary encoding
protocol where two adjacent values are only one bit flip away. In fact, the
locations of the Pauli $\hat{Z}$ operators in $\hat{\mathcal{O}}_{\nu}$
correspond to the $1$’s in the bit-reversed Gray binary encoding for its
sequency. A given sequency’s corresponding tensor-product operator can be
constructed by reversing the bits of the Gray code representation of the
sequency and mapping $0\rightarrow\hat{I}$ and $1\rightarrow\hat{Z}$. Note
that the more significant digits in the Gray code correspond to the less
significant qubits in the computational basis (the rightmost bit in the big-
Endian notation). This correlation can be leveraged to construct operators of
any desired sequency. For example, the gray code for the integer 24 is
$10100$, and the corresponding Hermitian sequency operator
$\hat{\mathcal{O}}_{24}$ is
$\hat{I}\otimes\hat{I}\otimes\hat{Z}\otimes\hat{I}\otimes\hat{Z}$; the
corresponding unitary time-evolution operator for this observable is
$e^{-i\beta_{24}t\hat{Z}_{3}\hat{Z}_{5}}$. Similarly, for the operator
$\hat{\mathcal{O}}_{10}$ implemented on five qubits, the 5-digit gray code is
$01111$, which corresponds to
$\hat{Z}\otimes\hat{Z}\otimes\hat{Z}\otimes\hat{Z}\otimes\hat{I}$. The
circuits for these examples are shown in Fig. 1, where
$R_{z}(\theta)=exp(-i\theta\hat{Z}/2)$ and $\theta=2\beta t$. Note that in
this convention, $q_{1}$ is the most significant qubit and $q_{5}$ is the
least.

$\Qcircuit@C=1em@R=.1em@!R{&\lstick{q_{1}}\qw\qw\qw\qw\\\
\lstick{q_{2}}\qw\qw\qw\qw\\\ \lstick{q_{3}}\ctrl{2}\qw\ctrl{2}\qw\\\
\lstick{q_{4}}\qw\qw\qw\qw\\\
\lstick{q_{5}}\targ\gate{R_{z}(\theta_{24})}\targ\qw}\hskip
50.0pt\Qcircuit@C=1em@R=.1em@!R{&\lstick{q_{1}}\ctrl{1}\qw\qw\qw\qw\qw\ctrl{1}\qw\\\
\lstick{q_{2}}\targ\ctrl{1}\qw\qw\qw\ctrl{1}\targ\qw\\\
\lstick{q_{3}}\qw\targ\ctrl{1}\qw\ctrl{1}\targ\qw\qw\\\
\lstick{q_{4}}\qw\qw\targ\gate{R_{z}(\theta_{10)}}\targ\qw\qw\qw\\\
\lstick{q_{5}}\qw\qw\qw\qw\qw\qw\qw\qw}$ Figure 1: Example circuits for Walsh
basis operators $\hat{\mathcal{O}}_{24}$ (left) and $\hat{\mathcal{O}}_{10}$
(right) where $24$ and $10$ are the sequency indices. Note that $q_{1}$ is the
most significant qubit and $q_{5}$ is the least; $R_{z}$ is a single-qubit
rotation gate about the Z axis.

Sequency is independent of the number of qubits from the perspective of
constructing a sequency operator for systems of different sizes. When the
number of qubits is increased, an operator with the same sequency is acquired
by adding $I$s to the less significant qubits of the operator, a procedure
analogous to adding $0$s to the bigger digits of Gray code. Since they are
merely equivalent operators in a more precise system, the few lowest sequency
operators still captures the bulk of low frequency information when the system
size increases, and higher sequency operators can be neglected when studying
low energy behaviors.

##  III Truncated Adiabatic State Preparation for $\lambda\phi^{4}$ Scalar
Field Theory

This section introduces the SeqHT scheme, which utilizes sequency and its
associated hierarchy to reduce resource requirements for quantum simulations.
To demonstrate the utility of SeqHT, it is applied to adiabatic state
preparation of $\lambda\phi^{4}$ theory. The Hamiltonian of this target
theory, a real massive scalar field with quartic self interactions ($\phi^{4}$
theory) in one spatial dimension, is given by

$\hat{H}^{\text{latt.}}=\sum_{k}\frac{1}{2}\hat{\Pi}_{k}^{2}+\frac{1}{2}m^{2}\hat{\phi}_{k}^{2}-\frac{1}{2}\hat{\phi}_{k}\nabla^{2}\hat{\phi}_{k}+\frac{\lambda}{4!}\hat{\phi}_{k}^{4}\
,$ (5)

where $\hat{\phi}$ is the field operator, $\hat{\Pi}$ the conjugate momentum
with a commutation relation
$\left[\hat{\phi}_{j},\hat{\Pi}_{k}\right]=i\delta_{jk}$, $k$ denotes the
lattice site, and $\lambda$ the coupling constant. The potential energy term
$\hat{\phi}^{4}$ generates self-interactions.

As first discussed in the papers by Jordan, Lee and Preskill [152, 153, 154,
155], state preparation of this theory on quantum devices can be accomplished
by first initializing the ground state of the exactly solvable non-interacting
theory ($\lambda=0)$ and then adiabatically turning on interactions through
unitary evolutions, with $\lambda$ slowly increasing until it reaches the
desired value. This process will evolve the free theory ground state into the
ground state of the interaction theory, providing that the theory has a non-
vanishing mass gap throughout the evolution.

For the purposes of demonstration, SeqHT is applied to adiabatic state
preparation of $\phi^{4}$ theory on one spatial site and $m$ is taken to be
$1$. Effective truncation and digitization are required to map this theory to
the registers of a digital quantum computer. In this paper, both $\hat{\phi}$
and $\hat{\Pi}$ are symmetrically digitized [157], with a field cutoff
$\phi_{\rm max}=4$, with further details on these choices explained in
Appendix A. $H_{\hat{\phi}}$ and $H_{\hat{\Pi}}$ are both diagonal matrices in
their respective eigenbasis and can both be decomposed into Walsh basis
operators. For this system, SeqHT becomes effective when the system size is
increased to five qubits per site, where the $\hat{\phi}^{4}$ term can be
truncated based on sequency hierarchy while preserving proper representation
of the system. For fewer than five qubits, the simulation is too coarse to be
truncated further.

There is a finite range of $\lambda$ for which the theory is suitable to be
simulated in this manner. When $\lambda$ becomes large, the wavefunction
becomes increasingly sharply peaked. Eventually it approaches a delta
function, which cannot be effectively represented. This could potentially be
resolved by dynamically rescaling the field cutoff and qubit mapping for
better representation of the features, in other words, zooming in on the
nonzero region of the evolved state. This paper works with $\lambda=10$, which
generates a ground-state wavefunction in the interacting theory that can be
reasonably represented without rescaling $\phi_{\rm max}$.

Figure 2:  The coefficients of Pauli strings contributing to the
$\hat{\phi}^{4}$ operator, which is digitized with increasing $n_{q}$. Each
Pauli string has a well-defined sequency, and operators with sequency below
$40$ are displayed. (Note that for $n_{q}=5$, $\hat{\phi}^{4}$ only decomposes
into operators with sequency below $32$.) The coefficients are calculated by
$Tr(\hat{\phi}^{4{\dagger}}\hat{\mathcal{O}}_{\nu})$. Values for the continuum
limit are connected by a dot-dashed line for display purposes. The numerical
values used in the subsequent analysis implement a $\nu_{\rm cut}=14$ sequency
truncation, and the values of the results displayed in this figure can be
found in Table 5.

On five qubits or more, SeqHT can be performed for the $\lambda\hat{\phi}^{4}$
interaction term. Since $\hat{\phi}^{4}$ is symmetric, it is decomposed into
symmetric Walsh basis operators, which are associated with even sequencies
because symmetric functions have an even number of zero crossings. The even
sequency terms correspond to Pauli strings with an even number of $\hat{Z}$.
Moreover, since $\hat{\phi}^{4}$ is constructed by multiplication of digitized
field operator $\hat{\phi}$, it only consists of two-body operators and four-
body operators other than identity, where an $n$-body operator refers to an
$\hat{\mathcal{O}}_{\nu}$ with $n$ appearances of $\hat{Z}$. This operator
structure, where only $n$-body operators are involved with $n$ not greater
than the order of the interaction, is a feature of the JLP basis and extends
to larger number of qubits. After SeqHT, the remaining set of operators still
construct a symmetric Hamiltonian. Because of this, the symmetries of the
ground states prepared with adiabatic evolution under the SeqHT Hamiltonian
are also preserved despite errors in adiabaticity and truncation.

Figure 2 presents the coefficients of basis operators, in sequency order,
obtained from decomposing $\hat{\phi}^{4}$. For larger numbers of qubits, the
coefficients of sequency operators start converging, and a general trend of
decreasing coefficients for operators of increasing sequencies is observed;
further analysis of this observation will be discussed in Sec. IV. The cutoff
sequency index $\nu_{\rm cut}$ is defined such that basis operators with
sequency greater than $\nu_{\rm cut}$ are truncated. Subsequent analysis
employs $\nu_{\rm cut}=14$ for $\hat{\phi}^{4}$. In quantum simulation, the
identity term with $\nu=0$ can also be ignored as it only impacts the global
phase. For a sufficiently small system, the digitized Hamiltonian can be
exactly diagonalized, with numerically determined eigenvalues and eigenvectors
that can be used to check the validity of the truncation scheme. In the five-
qubit case, when $\nu_{\rm cut}$ is set as indicated in Fig. 2, the
eigenvalues of the SeqHT Hamiltonian are found to deviate from those of the
full Hamiltonian at the $5\%$-level, shown in Fig. 3.

Figure 3: The left panel shows the eigenvalues of full $\phi^{4}$ theory
Hamiltonian and the SeqHT Hamiltonian, both of which are digitized on
$n_{q}=5$ with $\phi_{\rm max}=4$ and $\lambda=10$. The right panel shows the
percentage differences between the two sets of eigenvalues. Numerical values
for the results displayed in this figure can be found in Table 6.

Figure 4 shows the ground-state wavefunction of the free theory (the initial
state), the state prepared via the SeqHT ASP procedure with total time $t=5$
and five adiabatic steps, and the exact ground state (determined by numerical
diagonalization) of the interacting theory. For larger $\lambda$, the same
SeqHT parameters achieve a slightly worse overlap, which can be compensated
for by increasing the total time and number of steps.

Figure 4: Amplitudes of the initial ground state, the target ground state of
the interacting theory, and the state prepared via the SeqHT ASP procedure
which uses five adiabatic steps (not trotterized), with total time $t=5$. In
the left panel, the model uses $\lambda=10$ and achieves a fidelity of
$0.9999$; in the right panel, $\lambda=60$ and fidelity $0.9978$. The
$\phi^{4}$ theory Hamiltonian is digitized onto five qubits ($n_{q}=5$) with a
$\phi_{\rm max}=4$. Numerical values for the results displayed in this figure
can be found in Table 7.

The fidelity of the state preparation procedure is determined by the overlap
of the adiabatically-prepared wavefunction with the target ground-state
wavefunction. The fidelity of states prepared with full adiabatic time
evolution and SeqHT evolution are presented as a function of time in Fig. 5.

Figure 5: The fidelity of the $\lambda\phi^{4}$ ground-state wavefunction
prepared with complete adiabatic evolution (blue circles) and with SeqHT
procedure (orange crosses). The Hamiltonian is digitized on five qubits with
$\phi_{\rm max}=4$ and $\lambda=10$. Size of each time step is fixed at
$\delta t=0.1$ and number of steps is taken from $0$ to $80$ for a total time
from $0$ to $8$. The horizontal line indicates the initial overlap ($0.9729$)
of the free theory and the interacting theory ground states as a function of
time. Numerical values in this plot can be found in Table 8.

For larger systems, SeqHT on other terms in the Hamiltonian can be explored.
For example, the $\hat{\phi}^{2}$ term can also be drastically truncated in a
12-qubit system. Wavefunctions prepared for a 12-qubit system by Trotterized
ASP is shown in Fig. 6. The time scan in Appendix D guided what ASP parameters
to use. SeqHT for the $\hat{\phi}^{4}$ term is implemented with $\nu_{\rm
cut}=14$, leaving the same number of operators as for five qubits. In the left
panel of Fig. 6, no other truncations are implemented, while in the right
panel the $\hat{\phi}^{2}$ term is also truncated, with $\nu_{\rm cut}=30$,
leaving ten two-body operators.

Figure 6: Trotterized SeqHT ASP on twelve qubits ($n_{q}=12$). The Hamiltonian
is digitized in the basis of the field operator, using $\phi_{\rm max}=4$ and
$\lambda=10$. Each adiabatic step consists of one second-order Trotter step
implemented over a total evolution time of 1.6, and the system is evolved with
4 adiabatic steps. The left panel employs $\nu_{\rm cut}=14$ for the
$\hat{\phi}^{4}$ operator, and achieves a fidelity of $0.9923$. In the right
panel, $\hat{\phi}^{2}$ term is also truncated, to $\nu_{\rm cut}=30$. The
adiabatically prepared wavefunction fidelity is $0.9897$.

For a n-qubit system with $\nu_{\rm cut}=14$, the number of four-body
operators required to simulate the $\hat{\phi}^{4}$ term is reduced from
${n_{q}\choose 4}$ to one. Two-body operator resource requirements are also
reduced when truncation of the $\hat{\phi}^{2}$ term is implemented. In the
case of $n_{q}=12$, the $66$ two-qubit operators are reduced down to just $10$
for $\nu_{\rm cut}=14$.

While we have not implemented a sequency truncation in conjugate-momentum
space, we expect that such a truncation will also converge with increasing
$\nu^{(\Pi)}_{\rm cut}$. The value of $\nu^{(\Pi)}_{\rm cut}$ may or may not
coincide with its partner in $\phi$-space, as intrinsically they are
unrelated. However, such an truncation will disturb the utility of the
Nyquist-Shannon sampling theorem [192, 158] which eliminates all power-law
digitization effects. This is based upon uniform sampling in conjugate-
momentum space, and the use of the exact dispersion relation following the
local quantum Fourier transform, and not the lattice relation from the finite-
difference operator in $\phi$-space. Therefore, we anticipate larger than
naively expected errors introduced by a sequency truncation in conjugate-
momentum space. This is verified through exact calculations.

##  IV Analysis of SeqHT in the Limit of a Large Number of Qubits

This section provides an upper bound on the sequence coefficient
$\beta_{\nu}$, for an arbitrary polynomial function. This result can be used
to determine an appropriate $\nu_{\rm cut}$ for a given polynomial operator
without calculating all $\beta_{\nu}$. This upper bound is found by utilizing
the patterns in the locations of level crossings for sequence operators of
dimension $2^{n_{q}}\times 2^{n_{q}}$, with $n_{q}$ arbitrary; these patterns
can then be used to extrapolate to a large $n_{q}$ limit. This result extends
to any well-behaved function which can be carried out by utilizing an
appropriate Taylor expansion(s).

Consider the digitization of a polynomial function $F(x)=x^{p}$, where $p$ is
a positive integer. In the basis of eigenstates of the $\hat{x}$ operator
($\hat{x}|x\rangle=x|x\rangle$), this function maps to a diagonal operator
that can be decomposed into sequency operators $\hat{\mathcal{O}}_{\nu}$,
defined in Eq. (1). There exist a hierarchy in sequency such that,

$\displaystyle\beta_{\nu}\gtrsim\beta_{\nu^{\prime}}\quad\text{for}\quad\lfloor\log_{2}\nu^{\prime}\rfloor>\lfloor\log_{2}\nu\rfloor$
(6)

and therefore, in order to achieve a desired precision, it is sufficient to
only include operators with sequency up to $\nu_{\rm cut}$ with the upper
bound $B_{\nu_{\rm cut}}<\Lambda_{\rm cut}$, the coefficient threshold. This
also limits the size of operators to only $n_{\text{cut}}$-body operators with
$n_{\text{cut}}\leq\lceil\log_{2}\nu_{\text{cut}}\rceil$, hence reducing the
depth of quantum circuits required for implementation.

For sufficiently large $n_{q}$, $\beta_{\nu}$ approaches

$\displaystyle\lim_{n_{q}\rightarrow\infty}\beta_{\nu}\rightarrow\int_{-x_{M}}^{x_{M}}dx\
F(x)\ \Theta_{\nu}(x;x_{M})\ ,$ (7)

where $\Theta(x)_{\nu}$ are Heaviside step functions with $\nu$ level
crossings that occur in the same patterns as sequency operators
$\hat{\mathcal{O}}_{\nu}$, in the range $x=[-x_{M},x_{M}]$ (see App. B);
$x_{M}$ is the maximum value of $x$ used in the digitization. In the case of
the digitized $\lambda\phi^{4}$ theory, $x$ is $\phi$ and $x_{M}=\phi_{\rm
max}$. In order to better compare various sequency coefficients, it is useful
to define a normalized version of $\beta_{\nu}$ in the limit of large $n_{q}$,
222The normalization coefficient is chosen to be the integral of the
polynomial function on the interval $x_{M}>x>-x_{M}$.

$\displaystyle\tilde{\beta}_{\nu}=\frac{p+1}{2x_{M}^{p+1}}\int_{-x_{M}}^{x_{M}}dx\
F(x)\ \Theta_{\nu}(x;x_{M})\ ,$ (8)

with $\tilde{\beta}_{0}=1$, and $p$ the positive integral order of the
polynomial. Since the integrand has maximal support around $x\sim x_{M}$,
$\tilde{\beta}_{\nu}$ can be bounded from above by $\tilde{B}_{\nu}$,

$\displaystyle\tilde{B}_{\nu}\equiv\mathcal{N}_{p}\left(\int_{x_{\nu}}^{x_{M}}dx\,x^{p}+\int^{-x_{\nu}}_{-x_{M}}dx\,x^{p}\right)\
\ ,\ \ \mathcal{N}_{p}=\frac{p+1}{2x^{p+1}_{M}}\,,$ (9)

for $\nu\neq 0$, where $x_{\nu}$ is the location of the level crossing closest
to $x_{M}$ for $\hat{\mathcal{O}}_{\nu}$. For $F(x)$ an even function,
$\tilde{B}_{\nu}=0$ for odd sequency $\nu$, and for even $\nu$,

$\displaystyle\tilde{B}_{0}=1\qquad\text{and}\qquad\tilde{B}_{\nu}\equiv
1-\left(\frac{x_{\nu}}{x_{M}}\right)^{1+p}\qquad\nu\in\text{positive even
integers}.$ (10)

It is important to note that this is an upper bound on the value of
$\tilde{\beta}_{\nu}$ because $\Theta_{\nu}(x;x_{M})$ becomes negative at
$x<x_{\nu}$; while it can become positive at even smaller values of $x$, the
next largest (and negative) contribution to the integrand comes from
$x_{\nu}>x>x^{\prime}_{\nu}$, with $x^{\prime}_{\nu}$ being the location of
the second level crossing.

The expression for $x_{\nu}$ can be derived by utilizing the aforementioned
pattern in the level crossings for sequency operators
$\hat{\mathcal{O}}_{\mu}$ of dimension $n_{q}$. This is performed explicitly
in App. B, with the result that

$\displaystyle
x_{\nu}=x_{M}\left(1-\frac{1}{2^{\lfloor\log_{2}\nu\rfloor}}\right)\ ,$ (11)

Therefore, an upper bound on the (normalized) sequency coefficient is

$\displaystyle\tilde{B}_{\nu}=1-\left(1-\frac{1}{2^{\lfloor\log_{2}\nu\rfloor}}\right)^{1+p}\
,$ (12)

for positive, even sequency index. This upper bound significantly
overestimates $\beta_{\nu}$ for $\nu$ not close to an integer power of two,
but it gives an intuitive argument for the number of operators to include in
the truncated sum. The bound could be further refined by including additional
contributions to the integral.

(a)

(b)

Figure 7: Comparison between calculated coefficients, $\tilde{\beta}_{\nu}$,
and upper bounds, $\tilde{B}_{\nu}$, for a $x^{4}$ potential. The blue dashed
lines are the $\tilde{B}_{\nu}$, and the orange dotted lines are the
$\tilde{\beta}_{\nu}$. The left panel shows the results for eight qubits
linearly, while right panel displays the results in a log-log scale. The
vertical axis is the value of the normalized coefficients while the horizontal
axis is $\nu$. Numerical values of the results displayed in this figure are
included in Table 9.

Figure 7 shows the values of the coefficients $\tilde{\beta}_{\nu}$, along
with the upper bound, $\tilde{B}_{\nu}$ for digitization using eight qubits.
The upper bounds are respected by the computed $\tilde{\beta}_{\nu}$, as
expected. With an increasing number of qubits, and associated digitization of
a smooth function in the Hilbert space, the range of values for the various
coefficients can be substantial, with the largest values approaching the
bound. An important aspect of this is that the bound decreases with increasing
sequency, allowing for a sequency truncation, even in the “worst” case
function. This analysis can be repeated for odd polynomial functions, which
result in an analogous result. Namely for $F(x)$ an odd polynomial,
$\tilde{B}_{\nu}=0$ for even sequency and $\tilde{B}_{\nu}$ is as given in Eq.
(12), except with $\nu$ being positive and odd.

The approach developed above can be generalized to arbitrary functions. In the
case of an even function with support near $x_{M}$, Eq. (9) can be explicitly
evaluated. For example, for $F(x)=1-\cos x$, the upper bound on the
(normalized) sequency coefficients, for $x_{M}<\pi/2$, is given by

$\displaystyle\tilde{B}_{\nu}^{(F)}$ $\displaystyle=$ $\displaystyle
1-{x_{\nu}-\sin x_{\nu}\over x_{M}-\sin x_{M}}.$ (13)

Lastly, the results for polynomials of definite parity can be combined to
derive an upper bound on sequency coefficients of any well-behaved function.
In particular, for a function $F(x)$ with a polynomial expansion, an upper
bound for the sequency coefficients is

$\displaystyle B_{\nu}^{(F)}$ $\displaystyle=$
$\displaystyle\sum^{\infty}_{p=0}|a_{p}|B_{\nu}^{(p)}\ \ {\rm for}\ \ F(x)\ =\
\sum^{\infty}_{p=0}a_{p}x^{p}\,.$ (14)

Note that $B_{\nu}^{(p)}$ is the non-normalized upper bound
$B_{\nu}^{(p)}=\tilde{B}_{\nu}^{[p]}/\mathcal{N}_{p}$.

##  V Computational Complexity: The Behavior of Magic with SeqHT

It is interesting to understand the behavior of the computational complexity
with increasing levels of sequency truncation. In particular, we focus on the
quantum computational complexity required to establish a given wavefunction as
measured by its magic (non-stabilizerness) [193, 194, 195, 196, 197, 198],
reflecting the number of T-gates in the quantum circuit. For a given
wavefunction on a register of qubits, the magic in the state can be determined
by the matrix elements of the complete set of $n$-qubit Pauli strings,

$\displaystyle c_{P}$ $\displaystyle=$
$\displaystyle\langle\psi|\hat{P}|\psi\rangle\ \ ,$ (15)

where
$\hat{P}\in\\{\hat{P}_{1}\otimes\hat{P}_{2}\otimes...\otimes\hat{P}_{n}\\}$
and $\hat{P}_{i}\in\\{\hat{I},\hat{X},\hat{Y},\hat{Z}\\}$. For a stabilizer
state (a state that can be prepared efficiently using classical resources with
a Clifford gate set), only $d=2^{n_{q}}$ of the $d^{2}=4^{n_{q}}$ matrix
elements are non-zero, and take the values of $\pm 1$. It is useful to define
$\Xi_{P}\equiv c_{P}^{2}/d$ which can be identified as probabilities that
satisfy $\sum\limits_{P}\Xi_{P}=1$. Using $\Xi_{P}$, there are a number of
ways that are used to quantify the non-stabilizerness of the state, including
the Renyi-entropy and linear magic. We use the latter, defined by

$\displaystyle{\cal M}_{\rm lin}(|\psi\rangle)$ $\displaystyle=$
$\displaystyle 1-d\sum_{P}\Xi_{P}^{2}\ \ ,$ (16)

which vanishes for stabilizer states.

We consider the convergence properties of magic in the digitization and
sequency truncation of a one-dimensional Gaussian wavefunction centered in the
middle of the Hilbert space. Specifically, we calculate the magic of a
digitized Gaussian wavepacket of width $\sigma=1/\sqrt{2}$, with $\phi_{\rm
max}=4.0$, scanning over a range of $n_{q}$ for which the wavefunction itself
is well-contained on the register. As described above, the wavefunction
$|\psi\rangle$ is digitized onto the $n_{q}$ qubits, to give
$|\psi\rangle_{d}$ and renormalized such that
${}_{d}\langle\psi|\psi\rangle_{d}=1$.

One interesting property to investigate is the magic in the digitized Gaussian
wavefunctions $|\psi\rangle_{d}$ as a function of the square of the number of
qubits used for digitization with fixed $\phi_{max}$. Figure 8 shows the
convergence of magic to its asymptotic value in the Gaussian wavefunction as a
function of the number of qubits on a log-linear scale.

Figure 8:  The convergence of magic in a Gaussian wavefunction with
$\sigma=1/\sqrt{2}$ centered in the middle of the $n$-qubit Hilbert spaces
with $\phi_{max}=4$. The left panel shows the deviation from the asymptotic
value as a function of the number of qubits, while the right panel shows the
deviation from the asymptotic value as a function of $\nu_{\rm cut}$ for
$n_{q}=9$ using a log-log scale. Numerical values for the results displayed in
this figure can be found in Table 3 and Table 2.

The magic is found to converge to a fixed value, ${\cal M}_{\rm lin}=0.362007$
(for these parameters), with an error, $\epsilon$, scaling in a way that is
consistent with $\log{1\over|\epsilon|}\sim{\cal O}(n_{q}^{2})$. This result
demonstrates that, for the Gaussian wavefunction, and more generally
continuous functions with bounded support, its magic can be defined by the
limit of the digitized wavefunction.

Another property to study is the magic in the digitized Gaussian wavefunctions
$|\psi\rangle_{d}$ as a function of the cutoff sequency $\nu_{cut}$. The
digitized state is projected onto Hadamard-Walsh functions, which are then
used to construct a truncated wavefunction including sequencies up to
$\nu_{\rm cut}$, $|\psi\rangle_{d}^{(\nu_{\rm cut})}$, which are also
renormalized to give $\prescript{(\nu_{\rm
cut})}{d}{\langle}\psi|\psi\rangle_{d}^{(\nu_{\rm cut})}=1$. As shown in the
right panel of Fig. 8, the magic of the Gaussian wavefunction prepared on
$n_{q}=9$ qubits approaches its asymptotic value with increasing precision as
$\nu_{\rm cut}$ is increased, but in a step-wise fashion. It is observed to be
scaling approximately linearly on the log-log plot.

##  VI Quantum Simulations

Generally, implementing adiabatic time evolution using a digital quantum
computer requires that the evolution operator is Trotterized. Each adiabatic
step is realized using a fixed number of Trotter steps, and each Trotter step
consists of the same set of gates with different rotation angles to capture
the time dependence of the Hamiltonian. Due to limitations of NISQ-era quantum
devices, such as imperfect gate operations and limited coherence time,
resources need to be optimally distributed to minimize the overall error in a
simulation.

From a physics perspective, the fidelity of wavefunctions prepared using ASP
improves with increasing time intervals over which the Hamiltonian is evolved.
On the other hand, the fidelity of a Trotterized evolved wavefunction
decreases with increasing time intervals. Therefere, there is an intrinsic
tension in using Trotterized evolution to implement ASP; while this is of no
consequence for ideal quantum computers, it is important for realistic quantum
computers, particularly NISQ-era devices. In practice, there is a tuning, or
optimization, that has to be performed to use these combined methods on a
real-world device.

To focus on the study of ASP, we use one Trotter step for each adiabatic step.
Figure 9 presents (tuning) scans of state fidelity of the ground state of the
$\lambda\phi^{4}$ theory as a function of time-step size $\delta t$ and the
number of adiabatic steps; the ground state has been adiabatically prepared on
a classical noiseless quantum simulator. Both the full and SeqHT adiabatic
evolution with $\nu_{\rm cut}=14$ for both first-order and second-order
Trotterization are displayed. SeqHT evolution performs comparably well to the
full evolution in both cases. Second-order Trotterization has a larger region
of parameters that generate effective results compared to first-order
Trotterization. Therefore, we choose to work with two adiabatic steps, each
with one second-order Trotterized evolution, in the analysis that follows.

Figure 9: The fidelity $\mathcal{F}$ of the $\lambda\phi^{4}$ ground state
adiabatically prepared using a noiseless quantum simulator as a function of
time-step size $\delta t$ and the number of adiabatic steps, for a five-qubit
system ($n_{q}=5$) with $\phi_{\rm max}=4$ and $\lambda=10$. Each full (left)
or SeqHT (right) adiabatic step is comprised of one first-order (upper) or one
second-order (lower) Trotter step. The initial overlap of the free-theory
ground state ($\lambda=0$) and target ground state is $0.9729$. Basis
operators of the interacting term with $\nu>14$ are truncated. Numerical
values for the results displayed in this figure can be found in Table 10 and
Table 11.

###  VI.1 Quantum Circuits and Resource Requirements

The first step of the adiabatic evolution to $\lambda\phi^{4}$ from the non-
interacting theory is to prepare the ground state of the non-interacting
theory on the quantum register. The circuit that prepares a general unitary
transformation on $n_{q}$ qubits with all-to-all connectivity is presented in
Ref. [199]. Since the ground state of the non-interacting theory can be
defined to have positive and real amplitudes, all rotations in the circuit are
about the $y$-axis. The rotation angles for this sequence can either be
directly calculated [200, 199, 201] or obtained via a variational minimization
procedure, for instance via a Variational Quantum Eigensolver (VQE). Symmetry
of the free theory ground state can be exploited by preparing half of the
state on all but the most significant qubit, then reflecting it to form the
full state [201]. In the circuit shown in Fig. 10, $F$ denotes the operators
used to reflect about the midpoint of the Hilbert space, and the other circuit
elements constitute a general state preparation circuit. This initialization
requires $2^{n_{q}-1}+n_{q}-3$ CNOT gates for $n_{q}$ qubits with all-to-all
connectivity. Figure 11 shows the general reflection circuit $F$ (left), and a
nearest-neighbor connectivity adaptation (right) that is based on the
discussions in Ref. [202].

$\Qcircuit@C=0.6em@R=.1em@!R{&\lstick{q_{1}}\qw\qw\qw\qw\qw\qw\qw\qw\qw\qw\qw\qw\qw\qw\qw\qw\qw\qw\qw\qw\qw\qw\qw\qw\qw\qw\multigate{4}{F}\qw\\\
\lstick{q_{2}}\gate{R}\ctrl{1}\qw\ctrl{1}\qw\qw\ctrl{2}\qw\qw\qw\ctrl{2}\qw\qw\qw\qw\qw\qw\ctrl{3}\qw\qw\qw\qw\qw\qw\qw\ctrl{3}\ghost{F}\qw\\\
\lstick{q_{3}}\gate{R}\targ\gate{R}\targ\ctrl{1}\qw\qw\qw\ctrl{1}\qw\qw\qw\qw\ctrl{2}\qw\qw\qw\qw\qw\qw\qw\ctrl{2}\qw\qw\qw\qw\ghost{F}\qw\\\
\lstick{q_{4}}\qw\qw\qw\gate{R}\targ\gate{R}\targ\gate{R}\targ\gate{R}\targ\ctrl{1}\qw\qw\qw\ctrl{1}\qw\qw\qw\ctrl{1}\qw\qw\qw\ctrl{1}\qw\qw\ghost{F}\qw\\\
\lstick{q_{5}}\qw\qw\qw\qw\qw\qw\qw\qw\qw\qw\gate{R}\targ\gate{R}\targ\gate{R}\targ\gate{R}\targ\gate{R}\targ\gate{R}\targ\gate{R}\targ\gate{R}\targ\ghost{F}\qw}$
Figure 10: Quantum circuit for preparing the ground state of the non-
interacting theory on a quantum computer adapted from Ref. [201]. For a
positive and real wavefunction, all of the rotations are about the $y$-axis.
$F$ denotes the circuitry to reflect the wavefunction about the mid-point of
the Hilbert space, and is given in Fig. 11.
$\Qcircuit@C=.9em@R=.1em@!R{&\lstick{q_{1}}\gate{H}\ctrl{4}\ctrl{3}\qw\ctrl{1}\qw\\\
\lstick{q_{2}}\qw\qw\qw\qw\targ\qw\\\ \lstick{q_{3}}\cdots\cdots\\\
\lstick{q_{4}}\qw\qw\targ\qw\qw\qw\\\
\lstick{q_{5}}\qw\targ\qw\qw\qw\qw}\hskip
50.0pt\Qcircuit@C=.9em@R=.1em@!R{&\lstick{q_{1}}\gate{H}\qw\qw\ctrl{1}\qw\qw\qw\qw\\\
\lstick{q_{2}}\qw\qw\ctrl{1}\targ\ctrl{1}\qw\qw\qw\\\
\lstick{q_{3}}\qw\ctrl{1}\targ\qw\targ\ctrl{1}\qw\qw\\\
\lstick{q_{4}}\ctrl{1}\targ\qw\qw\qw\targ\ctrl{1}\qw\\\
\lstick{q_{5}}\targ\qw\qw\qw\qw\qw\targ\qw}$ Figure 11: Quantum circuits for
reflecting a wavefunction about the mid-point of the Hilbert space on a device
with all-to-all connectivity (left) as discussed in Ref. [201], and on a
device with nearest-neighbor connectivity (right).

For the time evolution generated by the Hamiltonian given in Eq. (5), since
the mass term and the interaction term commute, the circuit can be Trotterized
into $\tilde{\Phi}(s,t)$ and $\tilde{\Pi}(t)$ contributions; these operators
are defined to be

$\tilde{\Phi}(s,t)\equiv
e^{-i(\frac{1}{2}\hat{\phi}^{2}+\frac{\lambda(s)}{4!}\hat{\phi}^{4})t}\qquad\text{and}\qquad\tilde{\Pi}(t)\equiv
e^{-i\frac{1}{2}\hat{\Pi}^{2}t}.$ (17)

Detailed construction of the quantum circuits for $\tilde{\Phi}(s,t)$ and
$\tilde{\Pi}(t)$ can be found later in this section and in Appendix C. Each
adiabatic step in the evolution from the non-interacting ground state to the
$\lambda\phi^{4}$ ground state contains one Trotter step. In a second-order
Trotter step, the circuit implements the unitary operator,

$\hat{U}_{\rm 2nd}^{(1)}\ =\
\tilde{\Phi}\left(s,t/2\right)\,\tilde{\Pi}(t)\,\tilde{\Phi}\left(s,t/2\right)\
.$ (18)

Between adjacent second-order Trotterized adiabatic steps, the last term of
the previous step and the first term of the current step are combined (because
$\phi^{2}$ and $\phi^{4}$ commute) and therefore

$\hat{U}_{\rm 2nd}\ =\
\tilde{\Phi}(s_{1},t/2)\,\tilde{\Pi}(t)\,\tilde{\Phi}(s_{1}+s_{2},t/2)\,\tilde{\Pi}(t)\,\tilde{\Phi}(s_{2},t/2)\
.$ (19)

The quantum circuits employed to implement the error-mitigation strategy of
decoherence renormalization (DR) and operator-DR (ODR) are designed such that
they share similar structure with the physics circuit, but can be efficiently
simulated classically. The corresponding circuits are implemented as

$\hat{U}_{\rm 2nd}^{(2,\text{DR})}\ =\
\tilde{\Phi}(s_{1},t/2)\,\tilde{\Pi}(t)\,\tilde{\Phi}(s_{1}+s_{2},0)\,\tilde{\Pi}(-t)\,\tilde{\Phi}(s_{2},-t/2)\
\ ,$ (20)

corresponding to forward evolution for half of the time and backward evolution
for the other half.

Nearest-neighbor connectivity is considered as the following quantum
simulations are carried out on IBM’s superconducting-qubit quantum computers.
Two-body operators $R_{Z_{a}Z_{b}}=e^{i\theta\hat{Z}_{a}\hat{Z}_{b}}$ with the
same most significant qubit can be overlaid to improve efficiency [113], shown
in Fig. 12. Additional CNOT cancellations from neighboring blocks further
reduces the depth. As derived in Ref. [113], the total number of CNOTs $N$ and
circuit depth $D$ for the construction of all $R_{ZZ}$ gates with regard to
the number of qubits $n_{q}$ is

$N=2\binom{n_{q}}{2},\quad D=n_{q}\left(n_{q}-2\right)+3.$ (21)

$\Qcircuit@C=.7em@R=.1em@!R{&\lstick{q_{1}}\qw\qw\qw\ctrl{1}\qw\qw\qw\qw\qw\qw\qw\ctrl{1}\qw\qw\qw\qw\\\
\lstick{q_{2}}\qw\qw\ctrl{1}\targ\ctrl{1}\qw\qw\gate{R_{Z}(\theta_{12})}\qw\qw\ctrl{1}\targ\ctrl{1}\qw\qw\qw\\\
\lstick{q_{3}}\qw\ctrl{1}\targ\qw\targ\ctrl{1}\qw\gate{R_{Z}(\theta_{13})}\qw\ctrl{1}\targ\qw\targ\ctrl{1}\qw\qw\\\
\lstick{q_{4}}\ctrl{1}\targ\qw\qw\qw\targ\ctrl{1}\gate{R_{Z}(\theta_{14})}\ctrl{1}\targ\qw\qw\qw\targ\ctrl{1}\qw\\\
\lstick{q_{5}}\targ\qw\qw\qw\qw\qw\targ\gate{R_{Z}(\theta_{15})}\targ\qw\qw\qw\qw\qw\targ\qw}$
Figure 12: Quantum circuit for implementing multiple two-body $R_{ZZ}$
operators on a quantum computer from Ref. [113].

In conjugate-momentum space, $\hat{\Pi}^{2}$ can be implemented with the same
set of basis operators as $\hat{\phi}^{2}$, but with different coefficients
[152]. The Quantum Fourier Transform (QFT) is used to take these basis
operators to the $\phi$-space, with a CNOT number and gate depth that scales
as $n_{q}^{2}+n_{q}-4$ [202] (with nearest neighbor connectivity for each QFT
circuit). The resource requirements are calculated in the case where the
controlled-phase gate is not a native gate of the quantum device, and is
implemented using CNOTs and single-qubit rotations.

For a n-qubit system, when the interacting term $\hat{\phi}^{4}$ is
implemented with $\nu_{cut}=14$, the number of four-body operators required to
simulate the $\hat{\phi}^{4}$ term is reduced from ${n_{q}\choose 4}$ to one.
Each four-body operator is implemented with six CNOTs and total CNOT depth
six, shown in Fig. 1. Two-body operators in $\hat{\phi}^{4}$ can be absorbed
into the two body operators of $\hat{\phi}^{2}$ and pose no additional cost.

Table 1 presents the two-qubit gate counts and depths for implementing a
single adiabatic step (left panel) and the entire simulation with free theory
initialization (right panel) before and after SeqHT. We find SeqHT results in
a $\sim 35\%$ reduction in depth for a single step and $\sim 29\%$ reduction
for the complete simulation. Note that performing QFT and inverse QFT wth
nearest-neighbor connectivity constitutes more than half of the two-qubit gate
depth required for a single adiabatic step, which could potentially be
improved by running on a quantum computer with all-to-all connectivity, such
as a trapped-ion system; in such systems, the two-qubit gate depth of QFT
scales as $4n_{q}-6$ instead.

| Two-qubit Gate Depth | Two-qubit Gate Count
---|---|---
Full | 156 | 173
Truncated | 101 | 117

Two-qubit Gate Depth | Two-qubit Gate Count
---|---
291 | 321
208 | 237

Table 1: The entangling-gate resources required to perform a single adiabatic
step time evolution not including free theory initialization (left panel) and
the complete simulation with two adiabatic steps each containing one second-
order Trotter step (right panel) using ibm_sherbrooke before and after SeqHT
with $\nu_{\rm cut}=14$.

###  VI.2 Quantum Simulations using IBMs Quantum Computers

The ASP algorithms, with and without SeqHT, were run on IBM’s superconducting
qubit quantum computer ibm_sherbrooke with an Eagle r3 processor and ECR gates
as native two-qubit operations. For both simulations, qubits with longer
coherence time are selected, but this approach is only necessary for the full
evolution as the significantly longer circuit approaches the decoherence
boundary. The error mitigation techniques used were read-out error mitigation
[203] through the qiskit [204] Runtime Sampler primitive, Dynamic Decoupling
[132, 137, 205] via qiskit Transformation Pass [206], Pauli Twirling [207],
and Operator Decoherence Renormalization (ODR) [112, 113]. Employing Pauli-
twirling in both sets of circuits converts coherent errors into incoherent
errors, which can be suppressed by large ensemble sizes. This approach
substantially reduces the error introduced into observables due to the quantum
device’s decoherence. In ODR, results from the physical circuits are
renormalized by results from the mitigation circuits, as described in Eq.
(20). Both the physical and mitigation circuits employ 80 Pauli-twirled
instances with 8000 shots per twirl, with corresponding physical circuit and
mitigation circuit twirled the same way. The central values and uncertainties
are obtained from bootstrap resampling over the 80 twirls with 1000 resamples.

Fig. 13 shows a comparison between the expectation values
$\langle\hat{Z}\hat{Z}\rangle$ obtained using the SeqHT adiabatic evolution
and using the full adiabatic evolution. Encouragingly, the results from the
SeqHT evolution are found to be superior to those of the full evolution, with
smaller errors introduced by fewer gates in the shallower circuits and a
reduced decoherence due to a shorter circuit run-time.

Figure 13: Expectation values of $\hat{Z}_{i}\hat{Z}_{j}$ operators in the
ground state of $\lambda\phi^{4}$ theory with $\phi_{\rm max}=4$ and
$\lambda=10$ measured from quantum simulations using IBM’s ibm_sherbrooke. The
left panel computes the expectation values in the ground state prepared using
SeqHT adiabatic evolution with $\nu_{\rm cut}=14$, while the right panel uses
the full adiabatic evolution. Both evolutions involve two adiabatic steps,
each consisting of one second-order Trotter step. Raw results and error-
mitigated results are displayed, along the corresponding results from
classical computations (from the numerically solved target state). The
$x$-axis is $\\{i,j\\}$. Both the physical and mitigation circuits employ 80
Pauli twirls with 8000 shots per twirl. The SeqHT circuit employs 237 two-
qubit gates with depth 208 while the full circuit uses 321 two-qubit gates
with depth 291. The points are joined for display purposes. Numerical values
for the results displayed in this figure can be found in Table 4.

The full circuit implementations display larger deviations of central values
of observables from the known results, and also substantially larger
uncertainties in some instances. The truncated circuit is able to reproduce
$\langle\hat{Z}\hat{Z}\rangle$s close to the values calculated from the
exactly diagonalized interacting wavefunctions. As anticipated, the raw
results obtained from the full circuit exhibit more sensitivity to device
decoherence than those from the truncated circuit, as can be seen in Fig. 13
and in the numerical values displayed in Table 4. Note that we studied the
$\langle\hat{Z}\hat{Z}\rangle$ observables instead of state fidelity or
amplitudes since recovering states and state fidelities from a quantum
computer requires methods such as quantum-state tomography [208, 209] or
classical shadows [210, 211, 212, 213]; additionally, ODR has not yet been
shown to be effective for these methods.

##  VII Summary and Outlook

We have introduced a systematically-improvable truncation scheme that makes
use of the sequency hierarchies present in many physical systems. The Sequency
Hierarchy Truncation (SeqHT) scheme reduces the quantum resources needed to
prepare and time-evolve states within a target fidelity for a given
simulation. We have derived upper bounds on the sequency coefficients of
polynomial interactions, allowing for a convergent sequency truncation; these
results are generalized to provide upper bounds on the sequency coefficients
of any well-behaved function. SeqHT is expected to be particularly effective
in improving results obtained in NISQ-era quantum simulations, where device
decoherence remains a limitation.

As a demonstration of the potential of SeqHT, and using IBM’s quantum computer
ibm_sherbrooke, we have performed quantum simulations of the adiabatic
preparation of the ground state of $\lambda\phi^{4}$ from that of the harmonic
oscillator using a time-ordered Trotterization evolution. Specifically, we
have computed the expectation values of Pauli strings of two $\hat{Z}$
operators in these prepared state, and compared with exact results from
classical computations. These results, with cutoff sequency tuned to remain
within overall error tolerances, generally have smaller uncertainties than
those obtained with the full evolution. Further, they are in better agreement
with the classically-computed results.

As part of the study of the convergence of SeqHT, we considered the
convergence of the measurement of quantum computational complexity using magic
(non-stabilizerness) in a single-site Gaussian wavefunction with sequency
truncation. The magic in a digitized Gaussian wavefunction is found to
converge to a fixed value, with an error $\epsilon$ scaling as
$\log{1\over|\epsilon|}\sim{\cal O}(n_{q}^{2})$ with increasing number of
qubits used in its digitization. Further, the magic is found to step-wise
converge with increasing sequency cutoff, $\nu_{\rm cut}$, towards the exact
value.

SeqHT is an organizational method based on the structure of wavefunctions and
probes. We have demonstrated its utility in the context of adiabatic state
preparation of the $\lambda\phi^{4}$ ground state using Trotterized evolution.
However, it is more general applicability as it does not depend on the choice
of methods or algorithms. Consequently, we anticipate that it can be used in
simulations of systems exhibiting hierarchies of scales, which encompass many
systems of physical interest. In the context of nuclear physics applications,
SeqHT may, for example, accelerate quantum simulations of mean-field
descriptions of nuclear matter and relativistic hydrodynamics describing
heavy-ion collisions. Further, because its efficacy is related to the
structure and dynamics of physical systems, we anticipate that SeqHT can be
fruitfully implemented in digital, analog and hybrid quantum simulations.

###### Acknowledgements.

The authors would like to thank Nikita Zemlevskiy, Roland Farrell, Marc Illa,
and Anthony Ciavarella for valuable discussions. This work was supported in
part by the U.S. Department of Energy, Office of Science, Office of Nuclear
Physics, InQubator for Quantum Simulation (IQuS) (https://iqus.uw.edu) under
Award Number DOE (NP) Award DE-SC0020970 via the program on Quantum Horizons:
QIS Research and Innovation for Nuclear Science. This work was supported, in
part, through the Department of Physics333https://phys.washington.edu and the
College of Arts and Sciences444https://www.artsci.washington.edu at the
University of Washington. We acknowledge the use of IBM Quantum services for
this work. The views expressed are those of the authors, and do not reflect
the official policy or position of IBM or the IBM Quantum team. We have made
extensive use of IBM’s qiskit [204] and Wolfram Mathematica [214].

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##  Appendix A Digitization

The scalar field theory is digitized following Ref. [215, 192, 157, 158] which
is rooted in the JLP Formalism [152, 153, 154, 155]. A scalar field can be
mapped to the $2^{n_{q}}$ states in the Hilbert space, in uniform intervals,
defined by a selected maximum value of the field, Specifically,

$\displaystyle\phi$ $\displaystyle\rightarrow$ $\displaystyle\\{\ -\phi_{\rm
max}+j\delta\phi\ \ \\}\ \ ,\ \ \delta\phi\ =\ {2\phi_{\rm max}\over
2^{n_{q}}-1}\ \ ,\ \ j\ \in\ [0,2^{n_{q}}-1]\ .$ (22)

The $\hat{\Pi}^{2}$ term contributing to the Hamiltonian describing the
dynamics of the scalar field (Eq. (5)) can be constructed with twisted
boundary condition [216, 217, 218, 219] to preserve its symmetry.
Symmetrically digitizing both $\phi$ and $\Pi$ preserves the discrete
symmetries for the Hamiltonian and improves efficiency of quantum simulation
[157]. For the sake of having enough support for the evolved interacting
theory wavefunctions in our digitization, the maximum field value $\phi_{\rm
max}$ is taken as $4$ instead of its optimal value in relation to the number
of qubits $n_{q}$ defined in [220, 180],

$\phi_{\max}=\frac{2^{n_{q}}}{2}\sqrt{\frac{\sqrt{8}\pi}{2^{n_{q}}}}.$ (23)

The non-interacting scalar field theory (free theory) on one site can be
viewed as a quantum harmonic oscillator (QHO). As is well known, analytic
solutions to the eigenstates of the QHO in the continuum are the Hermite
polynomials. With mass $\hbar=m=1$, $n$ being the $n^{\rm th}$ energy level,
$H_{n}$ the Hermite Polynomial, these eigenstates and associated energy
eigenvalues are

$\displaystyle E_{n}$ $\displaystyle=n+\frac{1}{2}\ \ \ ,\ \ \ \left\langle
x\mid\psi_{n}\right\rangle$
$\displaystyle=\sqrt{\frac{1}{\sqrt{\pi}2^{n}n!}}e^{-\frac{x^{2}}{2}}H_{n}\left[x\right]\
\ \ ,\ \ \ n=0,1,2,...$ (24)

This analytical wavefunction in the continuum is digitized by first sampling
on the selected set of field values, and then appropriately renormalizing,

$\displaystyle\psi$ $\displaystyle\rightarrow$
$\displaystyle\\{\,\psi(\phi_{j})\,\\}\ \equiv\
\\{\,\tilde{\psi}_{d}(\phi_{j})\,\\}\quad,\quad\sum_{j}|A\tilde{\psi}_{d}(\phi_{j})|^{2}=1\
,$ (25)

which defines the digitized wavefunction,

$\\{\,\psi_{d}(\phi_{j})\,\\}\equiv\\{\,A\tilde{\psi}_{d}(\phi_{j})\,\\}\ .$
(26)

For a wavefunction with support within the register, which becomes
increasingly densely sampled with increasing numbers of qubits, the digitized
wavefunction amplitudes $\psi_{d}(\phi_{j})$ approaches
$\psi(\phi_{j})\sqrt{\delta\phi}$. A quantum simulation involving such a
scalar field will requiring using appropriate $\phi_{\rm max}$ to optimize the
use of quantum resources.

To check that the one-site free theory Hamiltonian is digitized appropriately,
we compare the eigenvalues and eigenstates that are numerically solved from
the digitized Hamiltonian with the analytical solutions. The analytical
eigenvalues, the eigenvalues of a digitized Hamiltonian with optimal max field
value defined in 23, and with $\phi_{\rm max}=4$ are shown in the left panel
of Fig. 14. The eigenvalues start to deviate at higher energies but faithfully
reproduce low-energy behaviors. The ground states acquired from directly
digitizing the analytical solution and from numerically solving the digitized
Hamiltonian are shown in the right panel of Fig. 14. It can be observed that
they have good agreement.

Figure 14: Comparison of analytic and digitized eigenvalues and wavefunctions
for the non-interacting theory. Five qubits, $n_{q}=5$, are used for the
digitization. In the left plot, the blue circular points are the analytical
eigenvalues; orange square points are the eigenvalues of a digitized QHO
Hamiltonian with optimal maximum field value defined in Eq. (23), while the
green triangular points are the corresponding eigenvalues with $\phi_{\rm
max}=4$. The right plot shows the ground-state wavefunctions. Blue circular
points are the ground state obtained from normalizing the analytical solution,
connected with dashed line for better visual representation. Orange triangles
are the numerically solved ground state from the digitized Hamiltonian with
$\phi_{\rm max}=4$ using $n_{q}=5$. Numerical values for the results displayed
in this figure can be found in Table 7 and Table 6.

##  Appendix B Details about the Sequency Hierarchy and the Hadamard-Walsh
Basis

The diagonals of the basis operator $\hat{\mathcal{O}}_{\nu}$, defined in Eq.
(1), are given by the entries in the $\nu^{\text{th}}$ row of the sequency-
ordered Walsh-Hadamard matrix, $H_{n_{q}}$, where the entries of $H_{n_{q}}$
are given by

$\displaystyle H^{[n_{q}]}_{ij}$
$\displaystyle=\left(-1\right)^{\nu^{[n_{q}]}_{ij}}\ ,$ (27)

with $i,j$ indexed from 1. Here, the integer $\nu$ is given by

$\displaystyle\nu^{[n_{q}]}_{ij}$
$\displaystyle=\sum_{k=0}^{n_{q}-1}b_{k,i-1}p_{k,j-1}\,,$ (28)

where $b_{i,j}$ is the $i^{\text{th}}$ bit of the binary representation of the
integer $j$,

$\displaystyle p_{i,j}=\begin{cases}b_{n_{q}-1,j}\quad&i=0\\\
b_{n_{q}-i,j}+b_{n_{q}-i-1,j}&i=1,\dots n_{q}\end{cases}\,.$ (29)

Note that the sequency-ordered Walsh-Hadamard matrix is a reordering of the
recursively-defined natural-ordered Hadamard matrix in Eq. (4). Using the
orthogonality of the sequency operators, the value of sequency coefficients
$\beta_{\nu}$ is given by

$\displaystyle\beta_{\nu}=\text{Tr}f(x_{j})\hat{\mathcal{O}}_{\nu}\ ,$ (30)

where $f(x_{j})$ is the digitization of the function $F(x)$, written as a
diagonal matrix. Assuming an equipartitioning of the variable $x$ (see App.
A), $x_{j}$ is given by

$\displaystyle x_{j}=-x_{M}+j\,\delta x\ \ ,\ \ \delta
x=\frac{2x_{M}}{2^{n_{q}}-1}\ \ {\rm with}\ \ x_{M}>0\ ,$ (31)

where $n_{q}$ is the number of qubits used to digitize $x$ and $x_{M}$ is the
maximum value of $x$ used in the simulation.

If $n_{q}$ is sufficiently large, the trace can be approximated by an integral
with computable corrections using the Poisson resummation formula, and
$\beta_{\nu}$ becomes

$\displaystyle\lim_{n_{q}\rightarrow\infty}\beta_{\nu}\rightarrow\int_{-x_{M}}^{x_{M}}dx\
F(x)\ \Theta_{\nu}(x;x_{M})\ ,$ (32)

The Heaviside step functions with $\nu$ level crossings are given by

$\displaystyle\Theta_{0}(x;x_{M})$ $\displaystyle=1,$
$\displaystyle\Theta_{1}(x;x_{M})$ $\displaystyle=2H(x)-1,$
$\displaystyle\Theta_{2}(x;x_{M})$
$\displaystyle=2\left[H(x-x_{M}/2)-H(x+x_{M}/2)\right]+1,$
$\displaystyle\Theta_{3}(x;x_{M})$
$\displaystyle=2\left[H(x)-H(x-x_{M}/2)-H(x+x_{M}/2)\right]+1\,,$ (33)

where $H(x)$ is the Heaviside step function.

In order to derive an explicit form of $x_{\nu}$, it is helpful to understand
how to generate sequency operators recursively. Let us assume that the
sequency operators $\hat{\mathcal{O}}^{[n_{q}]}_{\nu}$, which are matrices of
dimension $2^{n_{q}}\times 2^{n_{q}}$, have already been constructed, and for
this derivation, a superscript is added to the operators in order to denote
their dimension. The sequency operators $\hat{\mathcal{O}}^{[n_{q}+1]}_{\nu}$
are then given by

$\displaystyle\hat{\mathcal{O}}^{[n_{q}+1]}_{\nu}$
$\displaystyle=\hat{\mathcal{O}}^{[n_{q}]}_{\nu}\otimes\hat{I},$
$\displaystyle\hat{\mathcal{O}}^{[n_{q}+1]}_{2^{n_{q}+1}-1-\nu}$
$\displaystyle=\hat{\mathcal{O}}^{[n_{q}]}_{\nu}\otimes\hat{Z}\qquad
2^{n_{q}}-1\geq\nu\geq 0\,.$ (34)

This is already sufficient to derive an explicit expression for $x_{\nu}$.
Starting with $n_{q}=2$, the Walsh-Hadamard matrix is given by

$\displaystyle H_{2}$
$\displaystyle=\left(\begin{array}[]{@{}cc!{\color{blue}\vline width
0.6pt}cc@{}}1&1\hfil\lx@intercol\color[rgb]{0,0,1&1&1\\\
1&1\hfil\lx@intercol\color[rgb]{0,0,1&-1&-1\\\
1&-1\hfil\lx@intercol\color[rgb]{0,0,1&-1&1\\\
1&-1\hfil\lx@intercol\color[rgb]{0,0,1&1&-1\end{array}\right)\qquad\text{with}\qquad\nu=\left(\begin{array}[]{cccc}0\\\
1\\\ 2\\\ 3\end{array}\right)\,,}}}}$ (43)

where the rows have been labeled by sequency index and the vertical line
between column two and three marks the location of the origin when converting
the discrete entries in each row to the continuous function
$\Theta_{\nu}(x;x_{M})$. In particular, note that for $\nu=0$, both entries to
the left of the line are $+1$, while for $\nu=2$, one entry is $+1$ while the
other is $-1$. Therefore,

$\displaystyle n^{D}_{0}=2\qquad n^{D}_{1}=2\qquad
n_{2}^{D}=1=\frac{1}{2}n^{D}_{0}\qquad x_{3}^{D}=1=\frac{1}{2}n^{D}_{1}\,$
(44)

where $n^{D}_{\nu}$ is defined to be the number of entries, reading left to
right, before a level crossing occurs OR the ‘origin’ is reached; in this
convention $n^{D}_{0}=2^{n-1}$. To construct the sequency operators for
$n_{q}=3$, each entry of $H_{2}$ is expanded into a 2-component vector, either
$\\{1\\}\rightarrow\\{1,1\\}$, or $\\{1\\}\rightarrow\\{1,-1\\}$, depending on
whether the desired sequency index is less than eight or greater. In
particular

$\displaystyle H_{3}$
$\displaystyle=\left(\begin{array}[]{cccc!{\color{blue}\vline width
0.6pt}cccc}1&1&1&1\hfil\lx@intercol\color[rgb]{0,0,1&1&1&1&1\\\
1&-1&1&-1\hfil\lx@intercol\color[rgb]{0,0,1&1&-1&1&-1\\\ \hline\cr
1&1&1&1\hfil\lx@intercol\color[rgb]{0,0,1&-1&-1&-1&-1\\\
1&-1&1&-1\hfil\lx@intercol\color[rgb]{0,0,1&-1&1&-1&1\\\ \hline\cr
1&1&-1&-1\hfil\lx@intercol\color[rgb]{0,0,1&-1&-1&1&1\\\
1&-1&-1&1\hfil\lx@intercol\color[rgb]{0,0,1&-1&1&1&-1\\\ \hline\cr
1&1&-1&-1\hfil\lx@intercol\color[rgb]{0,0,1&1&1&-1&1\\\
1&-1&-1&1\hfil\lx@intercol\color[rgb]{0,0,1&1&-1&-1&-1\end{array}\right)\qquad\text{with}\qquad\nu=\left(\begin{array}[]{cccc}0\\\
7\\\ \hline\cr 1\\\ 6\\\ \hline\cr 2\\\ 5\\\ \hline\cr 3\\\
4\end{array}\right)\,,}}}}}}}}$ (61)

where the vertical line still marks the origin and the horizontal lines denote
sequency operators $\hat{\mathcal{O}}^{[n_{q}+1]}_{\nu}$ that are constructed
out of the same $\hat{\mathcal{O}}^{[n_{q}+1]}_{\nu}$. For example,
$\hat{\mathcal{O}}^{[3]}_{0}$ and $\hat{\mathcal{O}}^{[3]}_{7}$ are created by
taking the tensor product of $\hat{\mathcal{O}}^{[2]}_{0}$ with $\hat{I}$ and
$\hat{Z}$, respectively. The key thing to notice is that all the sequency
operators with $\nu\geq 4$ are constructed by taking
$\\{1\\}\rightarrow\\{1,-1\\}$ and therefore for these four operators, there
is only a single $+1$ entry before a level crossing to $-1$. Therefore, in
this case,

$\displaystyle n^{D}_{0}$ $\displaystyle=4\ ,\
n^{D}_{2}=2=\frac{1}{2}n^{D}_{0}\ ,\
n^{D}_{4,6}=\frac{1}{2}n^{D}_{2}=\frac{1}{4}n^{D}_{0},$ $\displaystyle
n^{D}_{1}$ $\displaystyle=4\ ,\ n^{D}_{3}=2=\frac{1}{2}n^{D}_{1}\ ,\
n^{D}_{5,7}=\frac{1}{2}n^{D}_{3}=\frac{1}{4}n^{D}_{1}\ .$ (62)

From this, a clear pattern emerges and it is simple enough to convince
oneself, using recursive arguments, that

$\displaystyle n^{D}_{\nu}$
$\displaystyle=\frac{1}{2}n^{D}_{\lfloor\log_{2}\nu\rfloor-1}\ =\
\frac{1}{2^{2}}n^{D}_{\lfloor\log_{2}\nu\rfloor-2}\ =\
\frac{1}{2^{\lfloor\log_{2}\nu\rfloor}}n_{0}^{D}\ ,\ \nu>0\,.$ (63)

Recalling that $x=x_{\nu}$ is the position of the last level crossing before
$x=x_{M}$, the parameter $n_{\nu}^{D}$ is related to $x_{\nu}$ via

$\displaystyle
x_{\nu}=x_{M}\left(1-\frac{1}{2^{\lfloor\log_{2}\nu\rfloor}}\right)\ ,$ (64)

and inserting this into Eq. (9), the upper bound on the sequency coefficient
is given by

$\displaystyle\tilde{B}_{\nu}=1-\left(1-\frac{1}{2^{\lfloor\log_{2}\nu\rfloor}}\right)^{1+p}\
,$ (65)

for positive, even sequency index.

Because $x_{\nu}$ is known and the integral that gives $\tilde{B}_{\nu}$ can
be evaluated, the derivation of $\tilde{B}_{\nu}$ for functions that are odd
polynomial powers is straightforward. Thus, the upper bound for any polynomial
functions can be shown to be

$\displaystyle\tilde{\beta}_{\nu}^{\text{B}}=\begin{cases}1-\left(1-\frac{1}{2^{\lfloor\text{Log}_{2}\nu\rfloor}}\right)^{p+1}&\qquad\nu,p\in\text{Evens}\quad\text{OR}\quad\nu,p\in\text{Odds}\\\
0&\qquad\text{all other cases}\end{cases}\ .$ (66)

While this upper bound does not capture the finer structure of the
coefficients $\tilde{\beta}_{\nu}$, as is clear from Fig. 7, it is useful in
estimating a reasonable truncation scale in achieving a target precision.

##  Appendix C Additional Details about the Quantum Circuits used for
simulation

For the time evolution discussed around Eq. (19), the circuit contains two
segments $\tilde{\Phi}(s,t)$ and $\tilde{\Pi}(t)$, with

$\tilde{\Phi}(s,t)\equiv
e^{-i(\frac{1}{2}\hat{\phi}^{2}+\frac{\lambda(s)}{4!}\hat{\phi}^{4})t}\ \ ,\ \
\tilde{\Pi}(t)\equiv e^{-i\frac{1}{2}\hat{\Pi}^{2}t}\ .$ (67)

In their own eigenbasis,

$\hat{\Pi}^{2}=\left(\frac{(2^{n_{q}}-1)\pi}{2^{n_{q}}\phi_{\rm
max}}\right)^{2}\hat{\mathcal{O}}^{(2,n_{q})}\
,\quad\hat{\phi}^{2}=\left(\frac{2\phi_{\rm
max}}{2^{n_{q}}-1}\right)^{2}\hat{\mathcal{O}}^{(2,n_{q})}\
,\quad\hat{\phi}^{4}=\left(\frac{2\phi_{\rm
max}}{2^{n_{q}}-1}\right)^{4}\hat{\mathcal{O}}^{(4,n_{q})}\ ,$ (68)

where

$\displaystyle\hat{\mathcal{O}}^{(2,n_{q}=5)}=$ $\displaystyle\
\sum\beta_{\mu}\hat{\mathcal{O}}_{\mu}$ (69) $\displaystyle=$ $\displaystyle\
64\,\hat{Z}\otimes\hat{Z}\otimes\hat{I}\otimes\hat{I}\otimes\hat{I}+32\,\hat{Z}\otimes\hat{I}\otimes\hat{Z}\otimes\hat{I}\otimes\hat{I}+16\,\hat{Z}\otimes\hat{I}\otimes\hat{I}\otimes\hat{Z}\otimes\hat{I}$
$\displaystyle+8\,\hat{Z}\otimes\hat{I}\otimes\hat{I}\otimes\hat{I}\otimes\hat{Z}+16\,\hat{I}\otimes\hat{Z}\otimes\hat{Z}\otimes\hat{I}\otimes\hat{I}+8\,\hat{I}\otimes\hat{Z}\otimes\hat{I}\otimes\hat{Z}\otimes\hat{I}$
$\displaystyle+4\,\hat{I}\otimes\hat{Z}\otimes\hat{I}\otimes\hat{I}\otimes\hat{Z}+4\,\hat{I}\otimes\hat{I}\otimes\hat{Z}\otimes\hat{Z}\otimes\hat{I}+2\,\hat{I}\otimes\hat{I}\otimes\hat{Z}\otimes\hat{I}\otimes\hat{Z}$
$\displaystyle+\hat{I}\otimes\hat{I}\otimes\hat{I}\otimes\hat{Z}\otimes\hat{Z}+\frac{341}{4}\hat{\mathds{I}}.$

The decomposition of the $\lambda\hat{\phi}^{4}$ interaction term for
$n_{q}=5$ is presented in Table 5.

Since $\hat{\mathcal{O}}^{(2,n_{q})}$ consists of all combinations of
$\hat{Z}_{a}\hat{Z}_{b}$ pairs and all these pairs commute,
$\Phi^{2}_{n_{q}}(\theta)=e^{i\theta\hat{\mathcal{O}}^{(2,n_{q})}}$ can be
implemented by a circuit of a complete set of two-qubit rotation gates
$e^{i\theta_{ab}\hat{Z}_{a}\hat{Z}_{b}}$. Ref. [113] presents an efficient way
to apply these rotations. A circuit for $\Phi^{2}_{n_{q}=5}(\theta)$ is shown
in Fig. 15. For the complete implementation of $\tilde{\Phi}(s,t)$, four-body
operators $R_{ZZZZ}$ are also included, and a circuit for implementing
$R_{ZZZZ}$ is shown in Fig. 1.

$\Qcircuit@C=.7em@R=.9em{&\multigate{4}{\Phi^{2}_{5}(\theta)}\qw\\\
\ghost{\Phi^{2}_{5}(\theta)}\qw\\\ \ghost{\Phi^{2}_{5}(\theta)}\qw=\\\
\ghost{\Phi^{2}_{5}(\theta)}\qw\\\
\ghost{\Phi^{2}_{5}(\theta)}\qw}\Qcircuit@C=.7em@R=.3em@!R{&\qw\qw\qw\ctrl{1}\qw\qw\qw\qw\qw\qw\qw\ctrl{1}\qw\qw\qw\qw\qw\qw\qw\qw\qw\qw\\\
\qw\qw\ctrl{1}\targ\ctrl{1}\qw\qw\gate{R_{Z}(\theta_{12})}\qw\qw\ctrl{1}\targ\qw\qw\qw\qw\ctrl{1}\qw\qw\qw\qw\qw\\\
\qw\ctrl{1}\targ\qw\targ\ctrl{1}\qw\gate{R_{Z}(\theta_{13})}\qw\ctrl{1}\targ\ctrl{1}\qw\gate{R_{Z}(\theta_{23})}\qw\ctrl{1}\targ\qw\qw\ctrl{1}\qw\qw\\\
\ctrl{1}\targ\qw\qw\qw\targ\ctrl{1}\gate{R_{Z}(\theta_{14})}\ctrl{1}\targ\qw\targ\ctrl{1}\gate{R_{Z}(\theta_{24})}\ctrl{1}\targ\ctrl{1}\gate{R_{Z}(\theta_{34})}\ctrl{1}\targ\ctrl{1}\qw\\\
\targ\qw\qw\qw\qw\qw\targ\gate{R_{Z}(\theta_{15})}\targ\qw\qw\qw\targ\gate{R_{Z}(\theta_{25})}\targ\qw\targ\gate{R_{Z}(\theta_{35})}\targ\gate{R_{Z}(\theta_{45})}\targ\qw}$
Figure 15: A quantum circuit for implementing all two-body $R_{ZZ}$ operators
for $n_{q}=5$ on a quantum computer.

A circuit for implementing $\tilde{\Pi}(t)$ in field space using the Symmetric
QFT [157] is shown in Fig. 16, where the phase gate is defined as
$P(\theta)=\left(\begin{array}[]{cc}1&0\\\ 0&e^{i\theta}\end{array}\right)\ $
and $M=\sum\limits_{j=0}^{n-1}2^{j}$.

$\Qcircuit@C=1em@R=.9em@!R{&\lstick{q_{1}}\gate{P(-\frac{M}{2}\pi)}\multigate{3}{QFT}\multigate{3}{\Phi^{2}_{n_{q}}(\theta_{\Pi})}\multigate{3}{QFT^{\dagger}}\gate{P(\frac{M}{2}\pi)}\qw\\\
\lstick{q_{2}}\gate{P(-\frac{M}{4}\pi)}\ghost{QFT}\ghost{\Phi^{2}_{n_{q}}(\theta_{\Pi})}\ghost{QFT^{\dagger}}\gate{P(\frac{M}{4}\pi)}\qw\\\
\gate{\vdots}\ghost{QFT}\ghost{\Phi^{2}_{n_{q}}(\theta_{\Pi})}\ghost{QFT^{\dagger}}\gate{\vdots}\qw\\\
\lstick{q_{n_{q}}}\gate{P(-\frac{M}{2^{n_{q}}}\pi)}\ghost{QFT}\ghost{\Phi^{2}_{n_{q}}(\theta_{\Pi})}\ghost{QFT^{\dagger}}\gate{P(\frac{M}{2^{n_{q}}}\pi)}\qw}$
Figure 16: Circuit for implementing $\tilde{\Pi}(t)$ in $\phi$-space with
Symmetric QFT, where $M=\sum\limits_{j=0}^{n-1}2^{j}$.

An efficient circuit to implement the QFT with nearest-neighbor connectivity,
as is relevant for implementing using IBM’s quantum computers, can be found in
Fig. 6 of Ref. [202].

##  Appendix D Additional Fidelity Scans

Fig. 17 presents scans of state fidelity of the $\lambda\phi^{4}$ ground state
prepared using a noiseless quantum simulator as a function of time-step size
$\delta t$ and the number of adiabatic steps for a 12-qubit system. Adiabatic
evolutions with $\hat{\phi}^{4}$ truncation and additional $\hat{\phi}^{2}$
truncation are studied with second-order Trotterization. The evolution with
additional $\hat{\phi}^{2}$ truncation performs comparably well. Scans of full
adiabatic evolution (without SeqHT) is not performed for twelve qubits because
completely decomposing $\hat{\phi}^{4}$ and executing all the four-body
operators is too costly.

Figure 17:  The fidelity of the $\lambda\phi^{4}$ ground state prepared using
a noiselss quantum simulator as a function of time-step size and the number of
adiabatic steps. The left panel employs $\nu_{\rm cut}=14$ for the
$\hat{\phi}^{4}$ operator while in the right panel, $\hat{\phi}^{2}$ term is
also truncated, to $\nu_{\rm cut}=30$. The Hamiltonian is digitized on 12
qubits with a maximum field value $\phi_{\rm max}=4$ and $\lambda=10$. Each
adiabatic step consists of one second-order Trotter step. Numerical data used
to plot this scan can be found in Table 12.

##  Appendix E Magic Results

In this appendix, we present tables with the results shown in Fig. 8 in Sec.
V. The (undigitized) Gaussian wavefunction that we examine is given by

$\displaystyle\psi(\phi)$ $\displaystyle=$
$\displaystyle{1\over\sqrt{\sigma\sqrt{2\pi}}}\
e^{-(\phi-\phi_{0})^{2}/(4\sigma^{2})}\ ,$ (70)

where $\phi_{0}=0$ and $\sigma=1/\sqrt{2}$ has been used in our numerical
analysis. The magic in the digitized Gaussian wavefunction as a function of
the number of qubits are given in Table 2.

$n_{q}$ | $M_{lin}$
---|---
3 | 0.19103
4 | 0.329949
5 | 0.355307
6 | 0.360661
7 | 0.361788
8 | 0.361992
9 | 0.362007
Table 2: Linear magic in the digitized Gaussian wavefunction described in the
main text as a function of the number of qubits, $n_{q}$. The wavefunction is
centered in the Hilbert space with a width of $\sigma=1/\sqrt{2}$. These
results, after subtraction of the asymptotic value, are shown in the left
panel of Fig. 8.

Table 3 shows the results for the magic in the reconstructed sequency-
truncated digital wavefunction described in the main text using $n_{q}=9$
qubits as a function of $\nu_{\rm cut}$.

$\nu_{\rm cut}$ | $M_{lin}$ | $\nu_{\rm cut}$ | $M_{lin}$ | $\nu_{\rm cut}$ | $M_{lin}$ | $\nu_{\rm cut}$ | $M_{lin}$ | $\nu_{\rm cut}$ | $M_{lin}$ | $\nu_{\rm cut}$ | $M_{lin}$ | $\nu_{\rm cut}$ | $M_{lin}$ | $\nu_{\rm cut}$ | $M_{lin}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
0 | 0 | 64 | 0.360366 | 128 | 0.361616 | 192 | 0.361617 | 256 | 0.361929 | 320 | 0.361929 | 384 | 0.361929 | 448 | 0.361929
2 | 0.00877133 | 66 | 0.360366 | 130 | 0.361616 | 194 | 0.361617 | 258 | 0.361929 | 322 | 0.361929 | 386 | 0.361929 | 450 | 0.361929
4 | 0.229904 | 68 | 0.36037 | 132 | 0.361617 | 196 | 0.361618 | 260 | 0.361929 | 324 | 0.361929 | 388 | 0.361929 | 452 | 0.361929
6 | 0.25894 | 70 | 0.360372 | 134 | 0.361617 | 198 | 0.361618 | 262 | 0.361929 | 326 | 0.361929 | 390 | 0.361929 | 454 | 0.361929
8 | 0.259058 | 72 | 0.360372 | 136 | 0.361617 | 200 | 0.361618 | 264 | 0.361929 | 328 | 0.361929 | 392 | 0.361929 | 456 | 0.361929
10 | 0.260592 | 74 | 0.360372 | 138 | 0.361617 | 202 | 0.361618 | 266 | 0.361929 | 330 | 0.361929 | 394 | 0.361929 | 458 | 0.361929
12 | 0.291475 | 76 | 0.360373 | 140 | 0.361617 | 204 | 0.361618 | 268 | 0.361929 | 332 | 0.361929 | 396 | 0.361929 | 460 | 0.36193
14 | 0.335475 | 78 | 0.360373 | 142 | 0.361617 | 206 | 0.361618 | 270 | 0.361929 | 334 | 0.361929 | 398 | 0.361929 | 462 | 0.36193
16 | 0.335475 | 80 | 0.360373 | 144 | 0.361617 | 208 | 0.361618 | 272 | 0.361929 | 336 | 0.361929 | 400 | 0.361929 | 464 | 0.36193
18 | 0.335676 | 82 | 0.360373 | 146 | 0.361617 | 210 | 0.361618 | 274 | 0.361929 | 338 | 0.361929 | 402 | 0.361929 | 466 | 0.36193
20 | 0.336793 | 84 | 0.360373 | 148 | 0.361617 | 212 | 0.361618 | 276 | 0.361929 | 340 | 0.361929 | 404 | 0.361929 | 468 | 0.36193
22 | 0.337285 | 86 | 0.360373 | 150 | 0.361617 | 214 | 0.361618 | 278 | 0.361929 | 342 | 0.361929 | 406 | 0.361929 | 470 | 0.36193
24 | 0.337302 | 88 | 0.360373 | 152 | 0.361617 | 216 | 0.361618 | 280 | 0.361929 | 344 | 0.361929 | 408 | 0.361929 | 472 | 0.36193
26 | 0.337705 | 90 | 0.360373 | 154 | 0.361617 | 218 | 0.361618 | 282 | 0.361929 | 346 | 0.361929 | 410 | 0.361929 | 474 | 0.36193
28 | 0.344123 | 92 | 0.360373 | 156 | 0.361617 | 220 | 0.361619 | 284 | 0.361929 | 348 | 0.361929 | 412 | 0.361929 | 476 | 0.36193
30 | 0.355368 | 94 | 0.360373 | 158 | 0.361617 | 222 | 0.361619 | 286 | 0.361929 | 350 | 0.361929 | 414 | 0.361929 | 478 | 0.36193
32 | 0.355368 | 96 | 0.360373 | 160 | 0.361617 | 224 | 0.361619 | 288 | 0.361929 | 352 | 0.361929 | 416 | 0.361929 | 480 | 0.36193
34 | 0.355381 | 98 | 0.360376 | 162 | 0.361617 | 226 | 0.361619 | 290 | 0.361929 | 354 | 0.361929 | 418 | 0.361929 | 482 | 0.36193
36 | 0.355442 | 100 | 0.360391 | 164 | 0.361617 | 228 | 0.361623 | 292 | 0.361929 | 356 | 0.361929 | 420 | 0.361929 | 484 | 0.361931
38 | 0.355467 | 102 | 0.360397 | 166 | 0.361617 | 230 | 0.361625 | 294 | 0.361929 | 358 | 0.361929 | 422 | 0.361929 | 486 | 0.361931
40 | 0.355468 | 104 | 0.360398 | 168 | 0.361617 | 232 | 0.361625 | 296 | 0.361929 | 360 | 0.361929 | 424 | 0.361929 | 488 | 0.361931
42 | 0.355471 | 106 | 0.360398 | 170 | 0.361617 | 234 | 0.361625 | 298 | 0.361929 | 362 | 0.361929 | 426 | 0.361929 | 490 | 0.361931
44 | 0.355482 | 108 | 0.360401 | 172 | 0.361617 | 236 | 0.361625 | 300 | 0.361929 | 364 | 0.361929 | 428 | 0.361929 | 492 | 0.361931
46 | 0.355485 | 110 | 0.360402 | 174 | 0.361617 | 238 | 0.361626 | 302 | 0.361929 | 366 | 0.361929 | 430 | 0.361929 | 494 | 0.361931
48 | 0.355485 | 112 | 0.360402 | 176 | 0.361617 | 240 | 0.361626 | 304 | 0.361929 | 368 | 0.361929 | 432 | 0.361929 | 496 | 0.361931
50 | 0.355534 | 114 | 0.360414 | 178 | 0.361617 | 242 | 0.361629 | 306 | 0.361929 | 370 | 0.361929 | 434 | 0.361929 | 498 | 0.361932
52 | 0.355798 | 116 | 0.360479 | 180 | 0.361617 | 244 | 0.361645 | 308 | 0.361929 | 372 | 0.361929 | 436 | 0.361929 | 500 | 0.361936
54 | 0.355914 | 118 | 0.360508 | 182 | 0.361617 | 246 | 0.361652 | 310 | 0.361929 | 374 | 0.361929 | 438 | 0.361929 | 502 | 0.361938
56 | 0.355917 | 120 | 0.360508 | 184 | 0.361617 | 248 | 0.361652 | 312 | 0.361929 | 376 | 0.361929 | 440 | 0.361929 | 504 | 0.361938
58 | 0.356018 | 122 | 0.360534 | 186 | 0.361617 | 250 | 0.361659 | 314 | 0.361929 | 378 | 0.361929 | 442 | 0.361929 | 506 | 0.36194
60 | 0.35755 | 124 | 0.360912 | 188 | 0.361617 | 252 | 0.361753 | 316 | 0.361929 | 380 | 0.361929 | 444 | 0.361929 | 508 | 0.361963
62 | 0.360366 | 126 | 0.361616 | 190 | 0.361617 | 254 | 0.361929 | 318 | 0.361929 | 382 | 0.361929 | 446 | 0.361929 | 510 | 0.362007

Table 3: Linear magic as a function of $\nu_{\rm cut}$ for the Gaussian
wavefunction described in the main text with $n_{q}=9$ qubits. Results are
shown in the right panel of Fig. 8. As the odd-sequency coefficients vanish
for this wavefunction, corresponding results for the magic are not shown.

##  Appendix F Tables of Results

The numerical data used to plot the figures in the text is provided in this
appendix. Table 4 corresponds to Fig. 13 in the main text. It gives the
expectation values of the Pauli strings in the $\lambda\phi^{4}$ ground-state
wavefunction prepared on ibm_sherbrooke using the full ASP and SeqHT ASP, both
error-mitigated and unmitigated, along with exact results. The uncertainties
are obtained form bootstrap resampling using 100 resamples. Table 5 gives the
coefficients of the Pauli strings contributing to the $\hat{\phi}^{4}$
operator as a function of $\nu$ for increasing $n_{q}$ and in the continuum
limit, as shown in Fig 2. Table 6 gives the $\hat{\phi}^{4}$ eigenvalues from
exactly solving a digitized interacting Hamiltonian with and without SeqHT, as
shown in Fig. 3, and the analytical eigenvalues of the free theory and the
eigenvalues from a QHO Hamiltonian digitized with optimal maximum field value
defined in Eq. (23) and with $\phi_{\rm max}=4$, as shown in the left panel of
Fig. 14. Table 7 gives the amplitudes of the SeqHT evolved and target ground
states of the interacting theory with $\lambda=10$ and $\lambda=60$, and the
ground state wavefunction of the free theory obtained analytically and
numerically, shown in Fig. 4 and the right panel of Fig. 14. Table 8 shows the
fidelity of the $\lambda\phi^{4}$ ground-state wavefunction prepared with
complete adiabatic evolution and with the SeqHT procedure as a function of
time, corresponding to Fig. 5. Table 9 shows the calculated sequency
coefficients, $\tilde{\beta}_{\nu}$, and upper bounds, $\tilde{B}_{\nu}$, for
a $x^{4}$ potential, as displayed in Fig. 7. Table 10, Table 11, and Table 12
shows the fidelity of the adiabatically prepared $\lambda\phi^{4}$ ground
state as a function of $\delta t$ and the number of adiabatic steps with
$\phi_{\rm max}=4$ and $\lambda=10$, for $n_{q}=5$ and $n_{q}=12$; each
adiabatic step consists of one first-order or second-order Trotter step.
Results are shown in Fig. 9 and Fig. 17. Data for Fig. 6 can be provided upon
request.

|  |  | Truncated | Full
---|---|---|---|---
Sequency | Operator | Ideal | Raw | Mitigated | Raw | Mitigated
2 | $Z_{1}Z_{2}$ | -0.9999 | -0.092 (03) | -0.890 (35) | -0.053 (02) | -0.879 (42)
4 | $Z_{2}Z_{3}$ | 0.8737 | 0.060 (03) | 0.800 (52) | 0.026 (03) | 0.743 (80)
6 | $Z_{1}Z_{3}$ | -0.8738 | -0.058 (03) | -0.794 (53) | -0.036 (02) | -1.29 (12)
8 | $Z_{3}Z_{4}$ | 0.2718 | 0.022 (04) | 0.227 (42) | 0.009 (02) | 0.229 (65)
12 | $Z_{2}Z_{4}$ | 0.3839 | 0.028 (03) | 0.514 (59) | 0.011 (02) | 0.59 (12)
14 | $Z_{1}Z_{4}$ | -0.3839 | -0.033 (03) | -0.511 (48) | -0.014 (02) | -0.437 (77)
16 | $Z_{4}Z_{5}$ | -0.0027 | 0.008 (03) | 0.063 (27) | -0.001 (02) | -0.016 (44)
24 | $Z_{3}Z_{5}$ | 0.1126 | 0.003 (03) | 0.035 (34) | 0.007 (03) | 0.192 (76)
28 | $Z_{2}Z_{5}$ | 0.1804 | 0.007 (03) | 0.142 (51) | 0.006 (02) | 0.300 (95)
30 | $Z_{1}Z_{5}$ | -0.1805 | -0.008 (03) | -0.145 (50) | -0.007 (02) | -0.263 (70)
Table 4:  The results obtained for strings of Pauli operators evaluated in the $\lambda\phi^{4}$ ground-state wavefunction prepared on ibm_sherbrooke using SeqHT ASP, both error-mitigated and unmitigated, along with exact results, as displayed in Fig. 13. The uncertainties are obtained from bootstrap resampling using 1000 samples.  |  | Coefficients in $\hat{\phi}^{4}$ Expansion
---|---|---
Sequency | Operator | $n_{q}=5$ | $n_{q}=6$ | $n_{q}=7$ | $n_{q}=8$ | $n_{q}=10$ | $n_{q}=11$ | $n_{q}=12$ | Continuum
0 | I | 57.94 | 54.48 | 52.82 | 52.01 | 51.40 | 51.30 | 51.25 | 51.20
2 | $Z_{1}Z_{2}$ | 54.36 | 51.09 | 49.52 | 48.76 | 48.19 | 48.09 | 48.05 | 48.00
4 | $Z_{2}Z_{3}$ | 30.62 | 28.75 | 27.86 | 27.43 | 27.11 | 27.05 | 27.03 | 27.00
6 | $Z_{1}Z_{3}$ | 33.99 | 31.93 | 30.95 | 30.47 | 30.12 | 30.06 | 30.03 | 30.00
8 | $Z_{3}Z_{4}$ | 8.720 | 8.185 | 7.932 | 7.809 | 7.718 | 7.703 | 7.695 | 7.687
10 | $Z_{1}Z_{2}Z_{3}Z_{4}$ | 6.812 | 6.390 | 6.191 | 6.095 | 6.023 | 6.012 | 6.006 | 6.000
12 | $Z_{2}Z_{4}$ | 15.74 | 14.77 | 14.32 | 14.09 | 13.93 | 13.90 | 13.89 | 13.88
14 | $Z_{1}Z_{4}$ | 17.85 | 16.77 | 16.25 | 16.00 | 15.81 | 15.78 | 15.77 | 15.75
16 | $Z_{4}Z_{5}$ | 2.246 | 2.109 | 2.043 | 2.012 | 1.988 | 1.984 | 1.982 | 1.980
18 | $Z_{1}Z_{2}Z_{4}Z_{5}$ | 1.703 | 1.598 | 1.548 | 1.524 | 1.506 | 1.503 | 1.501 | 1.500
20 | $Z_{2}Z_{3}Z_{4}Z_{5}$ | 0.4258 | 0.3994 | 0.3870 | 0.3809 | 0.3765 | 0.3757 | 0.3754 | 0.3750
22 | $Z_{1}Z_{3}Z_{4}Z_{5}$ | 0.8516 | 0.7988 | 0.7739 | 0.7618 | 0.7529 | 0.7515 | 0.7507 | 0.7500
24 | $Z_{3}Z_{5}$ | 4.386 | 4.118 | 3.990 | 3.928 | 3.882 | 3.875 | 3.871 | 3.867
26 | $Z_{1}Z_{2}Z_{3}Z_{5}$ | 3.406 | 3.195 | 3.096 | 3.047 | 3.012 | 3.006 | 3.003 | 3.000
28 | $Z_{2}Z_{5}$ | 7.921 | 7.436 | 7.206 | 7.094 | 7.012 | 6.998 | 6.991 | 6.984
30 | $Z_{1}Z_{5}$ | 9.030 | 8.483 | 8.222 | 8.094 | 8.000 | 7.984 | 7.977 | 7.969
32 | $Z_{5}Z_{6}$ | – | 0.5311 | 0.5146 | 0.5066 | 0.5007 | 0.4998 | 0.4993 | 0.4988
34 | $Z_{1}Z_{2}Z_{5}Z_{6}$ | – | 0.3994 | 0.3870 | 0.3809 | 0.3765 | 0.3757 | 0.3754 | 0.3750
36 | $Z_{2}Z_{3}Z_{5}Z_{6}$ | – | 0.09985 | 0.09674 | 0.09523 | 0.09412 | 0.09393 | 0.09384 | 0.09375
38 | $Z_{1}Z_{3}Z_{5}Z_{5}$ | – | 0.1997 | 0.1935 | 0.1905 | 0.1882 | 0.1879 | 0.1877 | 0.1875
Table 5: The coefficients of the Pauli strings contributing to the $\hat{\phi}^{4}$ operator as a function of $\nu$ for increasing $n_{q}$ and in the continuum limit, as shown in Fig 2. The points shown in Fig. 2 are normalized by $\phi_{\rm max}^{4}=256$.  |  $\hat{\phi}^{4}$ theory | QHO
---|---|---
Index | Truncated | Original | $\phi_{\rm max}=4$ | Optimal $\phi_{\rm max}$ | Analytical
0 | 0.7003 | 0.6735 | 0.500 | 0.500 | 1.5
1 | 2.304 | 2.236 | 1.500 | 1.500 | 2.5
2 | 4.239 | 4.142 | 2.500 | 2.500 | 3.5
3 | 6.399 | 6.279 | 3.499 | 3.500 | 4.5
4 | 8.739 | 8.603 | 4.505 | 4.500 | 5.5
5 | 11.25 | 11.08 | 5.472 | 5.500 | 6.5
6 | 13.82 | 13.70 | 6.573 | 6.500 | 7.5
7 | 16.53 | 16.45 | 7.276 | 7.500 | 8.5
8 | 19.68 | 19.31 | 8.916 | 8.500 | 9.5
9 | 22.60 | 22.28 | 9.188 | 9.500 | 10.5
10 | 25.02 | 25.34 | 11.76 | 10.50 | 11.5
11 | 28.16 | 28.50 | 11.85 | 11.49 | 12.5
12 | 32.18 | 31.74 | 15.19 | 12.52 | 13.5
13 | 36.11 | 35.06 | 15.23 | 13.44 | 14.5
14 | 39.13 | 38.46 | 19.22 | 14.60 | 15.5
15 | 41.49 | 41.94 | 19.23 | 15.25 | 16.5
16 | 44.51 | 45.49 | 23.83 | 16.87 | 17.5
17 | 48.55 | 49.11 | 23.83 | 17.15 | 18.5
18 | 53.03 | 52.79 | 29.02 | 19.41 | 19.5
19 | 57.64 | 56.55 | 29.03 | 19.52 | 20.5
20 | 61.90 | 60.27 | 34.80 | 22.28 | 21.5
21 | 65.71 | 64.29 | 34.81 | 22.32 | 22.5
22 | 68.68 | 68.39 | 41.15 | 25.47 | 23.5
23 | 70.00 | 70.49 | 41.17 | 25.47 | 24.5
24 | 77.60 | 79.72 | 48.08 | 28.95 | 25.5
25 | 78.01 | 79.86 | 48.12 | 28.97 | 26.5
26 | 98.89 | 95.09 | 55.58 | 32.68 | 27.5
27 | 99.99 | 95.60 | 55.66 | 32.83 | 28.5
28 | 107.2 | 113.3 | 63.61 | 36.19 | 29.5
29 | 118.8 | 118.9 | 63.83 | 37.63 | 30.5
30 | 135.6 | 133.0 | 70.90 | 40.70 | 31.5
31 | 158.5 | 164.3 | 74.60 | 47.08 | 32.5
Table 6: The $\hat{\phi}^{4}$ columns show the eigenvalues from exactly solving a digitized interacting Hamiltonian with and without SeqHT, as shown in Fig. 3. The QHO columns show the analytical eigenvalues of the free theory and the eigenvalues from a QHO Hamiltonian digitized with optimal maximum field value defined in Eq. (23) and with $\phi_{\rm max}=4$, as shown in the left panel of Fig. 14.  $\lambda=10$ | $\lambda=60$ | Free
---|---|---
Evolved | Target | Evolved | Target | Digitized | Analytical
3.572$\times 10^{-4}$ | 9.630$\times 10^{-11}$ | 1.191$\times 10^{-4}$ | 3.370$\times 10^{-10}$ | 8.155$\times 10^{-5}$ | 1.280$\times 10^{-4}$
1.053$\times 10^{-4}$ | 4.029$\times 10^{-9}$ | 1.662$\times 10^{-4}$ | -4.556$\times 10^{-10}$ | 3.292$\times 10^{-4}$ | 3.476$\times 10^{-4}$
2.473$\times 10^{-4}$ | 1.058$\times 10^{-7}$ | 6.891$\times 10^{-4}$ | 6.488$\times 10^{-10}$ | 8.752$\times 10^{-4}$ | 8.830$\times 10^{-4}$
1.886$\times 10^{-4}$ | 1.824$\times 10^{-6}$ | 1.740$\times 10^{-3}$ | -9.599$\times 10^{-10}$ | 2.095$\times 10^{-3}$ | 2.099$\times 10^{-3}$
2.277$\times 10^{-4}$ | 2.127$\times 10^{-5}$ | 4.153$\times 10^{-3}$ | 1.759$\times 10^{-9}$ | 4.665$\times 10^{-3}$ | 4.667$\times 10^{-3}$
5.275$\times 10^{-4}$ | 1.733$\times 10^{-4}$ | 2.375$\times 10^{-3}$ | 3.498$\times 10^{-8}$ | 9.709$\times 10^{-3}$ | 9.710$\times 10^{-3}$
2.305$\times 10^{-3}$ | 1.017$\times 10^{-3}$ | 6.063$\times 10^{-3}$ | 1.997$\times 10^{-6}$ | 1.890$\times 10^{-2}$ | 1.890$\times 10^{-2}$
6.672$\times 10^{-3}$ | 4.436$\times 10^{-3}$ | 3.768$\times 10^{-3}$ | 5.189$\times 10^{-5}$ | 3.441$\times 10^{-2}$ | 3.441$\times 10^{-2}$
1.593$\times 10^{-2}$ | 1.483$\times 10^{-2}$ | 1.950$\times 10^{-2}$ | 7.073$\times 10^{-4}$ | 5.863$\times 10^{-2}$ | 5.863$\times 10^{-2}$
3.498$\times 10^{-2}$ | 3.914$\times 10^{-2}$ | 3.802$\times 10^{-3}$ | 5.454$\times 10^{-3}$ | 9.345$\times 10^{-2}$ | 9.345$\times 10^{-2}$
7.829$\times 10^{-2}$ | 8.402$\times 10^{-2}$ | 3.901$\times 10^{-2}$ | 2.570$\times 10^{-2}$ | 1.394$\times 10^{-1}$ | 1.394$\times 10^{-1}$
1.450$\times 10^{-1}$ | 1.508$\times 10^{-1}$ | 1.089$\times 10^{-1}$ | 7.987$\times 10^{-2}$ | 1.944$\times 10^{-1}$ | 1.944$\times 10^{-1}$
2.315$\times 10^{-1}$ | 2.321$\times 10^{-1}$ | 1.999$\times 10^{-1}$ | 1.762$\times 10^{-1}$ | 2.538$\times 10^{-1}$ | 2.538$\times 10^{-1}$
3.141$\times 10^{-1}$ | 3.137$\times 10^{-1}$ | 2.927$\times 10^{-1}$ | 2.957$\times 10^{-1}$ | 3.099$\times 10^{-1}$ | 3.099$\times 10^{-1}$
3.807$\times 10^{-1}$ | 3.791$\times 10^{-1}$ | 3.883$\times 10^{-1}$ | 4.015$\times 10^{-1}$ | 3.540$\times 10^{-1}$ | 3.540$\times 10^{-1}$
4.173$\times 10^{-1}$ | 4.153$\times 10^{-1}$ | 4.580$\times 10^{-1}$ | 4.618$\times 10^{-1}$ | 3.784$\times 10^{-1}$ | 3.784$\times 10^{-1}$
4.173$\times 10^{-1}$ | 4.153$\times 10^{-1}$ | 4.580$\times 10^{-1}$ | 4.618$\times 10^{-1}$ | 3.784$\times 10^{-1}$ | 3.784$\times 10^{-1}$
3.807$\times 10^{-1}$ | 3.791$\times 10^{-1}$ | 3.883$\times 10^{-1}$ | 4.015$\times 10^{-1}$ | 3.540$\times 10^{-1}$ | 3.540$\times 10^{-1}$
3.141$\times 10^{-1}$ | 3.137$\times 10^{-1}$ | 2.927$\times 10^{-1}$ | 2.957$\times 10^{-1}$ | 3.099$\times 10^{-1}$ | 3.099$\times 10^{-1}$
2.315$\times 10^{-1}$ | 2.321$\times 10^{-1}$ | 1.999$\times 10^{-1}$ | 1.762$\times 10^{-1}$ | 2.538$\times 10^{-1}$ | 2.538$\times 10^{-1}$
1.450$\times 10^{-1}$ | 1.508$\times 10^{-1}$ | 1.089$\times 10^{-1}$ | 7.987$\times 10^{-2}$ | 1.944$\times 10^{-1}$ | 1.944$\times 10^{-1}$
7.829$\times 10^{-2}$ | 8.402$\times 10^{-2}$ | 3.901$\times 10^{-2}$ | 2.570$\times 10^{-2}$ | 1.394$\times 10^{-1}$ | 1.394$\times 10^{-1}$
3.498$\times 10^{-2}$ | 3.914$\times 10^{-2}$ | 3.802$\times 10^{-3}$ | 5.454$\times 10^{-3}$ | 9.345$\times 10^{-2}$ | 9.345$\times 10^{-2}$
1.593$\times 10^{-2}$ | 1.483$\times 10^{-2}$ | 1.950$\times 10^{-2}$ | 7.073$\times 10^{-4}$ | 5.863$\times 10^{-2}$ | 5.863$\times 10^{-2}$
6.672$\times 10^{-3}$ | 4.436$\times 10^{-3}$ | 3.768$\times 10^{-3}$ | 5.189$\times 10^{-5}$ | 3.441$\times 10^{-2}$ | 3.441$\times 10^{-2}$
2.305$\times 10^{-3}$ | 1.017$\times 10^{-3}$ | 6.063$\times 10^{-3}$ | 1.997$\times 10^{-6}$ | 1.890$\times 10^{-2}$ | 1.890$\times 10^{-2}$
5.275$\times 10^{-4}$ | 1.733$\times 10^{-4}$ | 2.375$\times 10^{-3}$ | 3.498$\times 10^{-8}$ | 9.709$\times 10^{-3}$ | 9.710$\times 10^{-3}$
2.277$\times 10^{-4}$ | 2.127$\times 10^{-5}$ | 4.153$\times 10^{-3}$ | 1.759$\times 10^{-9}$ | 4.665$\times 10^{-3}$ | 4.667$\times 10^{-3}$
1.886$\times 10^{-4}$ | 1.824$\times 10^{-6}$ | 1.740$\times 10^{-3}$ | -9.599$\times 10^{-10}$ | 2.095$\times 10^{-3}$ | 2.099$\times 10^{-3}$
2.473$\times 10^{-4}$ | 1.058$\times 10^{-7}$ | 6.891$\times 10^{-4}$ | 6.488$\times 10^{-10}$ | 8.752$\times 10^{-4}$ | 8.830$\times 10^{-4}$
1.053$\times 10^{-4}$ | 4.029$\times 10^{-9}$ | 1.662$\times 10^{-4}$ | -4.556$\times 10^{-10}$ | 3.292$\times 10^{-4}$ | 3.476$\times 10^{-4}$
3.572$\times 10^{-4}$ | 9.630$\times 10^{-11}$ | 1.191$\times 10^{-4}$ | 3.370$\times 10^{-10}$ | 8.155$\times 10^{-5}$ | 1.280$\times 10^{-4}$
Table 7: Amplitudes of the SeqHT evolved and target ground states of the interacting theory with $\lambda=10$ and $\lambda=60$, and the ground state wavefunction of the free theory obtained analytically and numerically, shown in Fig. 4 and the right panel of Fig. 14.  $t$ | Full | Trunc | $t$ | Full | Trunc | $t$ | Full | Trunc
---|---|---|---|---|---|---|---|---
0.0 | 0.973 | 0.973 | 2.7 | 0.997 | 0.997 | 5.4 | 0.999 | 0.998
0.1 | 0.974 | 0.974 | 2.8 | 0.997 | 0.997 | 5.5 | 0.999 | 0.998
0.2 | 0.975 | 0.974 | 2.9 | 0.997 | 0.997 | 5.6 | 0.999 | 0.998
0.3 | 0.976 | 0.976 | 3.0 | 0.997 | 0.996 | 5.7 | 0.999 | 0.998
0.4 | 0.977 | 0.977 | 3.1 | 0.997 | 0.996 | 5.8 | 0.999 | 0.998
0.5 | 0.979 | 0.978 | 3.2 | 0.997 | 0.996 | 5.9 | 0.999 | 0.998
0.6 | 0.981 | 0.980 | 3.3 | 0.997 | 0.996 | 6.0 | 0.999 | 0.999
0.7 | 0.983 | 0.981 | 3.4 | 0.998 | 0.997 | 6.1 | 0.999 | 0.999
0.8 | 0.985 | 0.983 | 3.5 | 0.998 | 0.997 | 6.2 | 0.999 | 0.999
0.9 | 0.986 | 0.984 | 3.6 | 0.998 | 0.997 | 6.3 | 0.999 | 0.999
1.0 | 0.988 | 0.986 | 3.7 | 0.998 | 0.997 | 6.4 | 0.999 | 0.999
1.1 | 0.990 | 0.987 | 3.8 | 0.999 | 0.998 | 6.5 | 0.999 | 1.000
1.2 | 0.991 | 0.989 | 3.9 | 0.999 | 0.998 | 6.6 | 0.999 | 1.000
1.3 | 0.992 | 0.990 | 4.0 | 0.999 | 0.998 | 6.7 | 0.999 | 1.000
1.4 | 0.994 | 0.992 | 4.1 | 0.999 | 0.999 | 6.8 | 0.999 | 1.000
1.5 | 0.995 | 0.993 | 4.2 | 0.999 | 0.999 | 6.9 | 0.999 | 1.000
1.6 | 0.996 | 0.994 | 4.3 | 0.999 | 0.999 | 7.0 | 0.999 | 0.999
1.7 | 0.996 | 0.995 | 4.4 | 0.999 | 0.999 | 7.1 | 0.999 | 0.999
1.8 | 0.997 | 0.996 | 4.5 | 0.999 | 0.999 | 7.2 | 0.999 | 0.999
1.9 | 0.997 | 0.997 | 4.6 | 0.999 | 0.999 | 7.3 | 0.999 | 0.999
2.0 | 0.997 | 0.997 | 4.7 | 0.999 | 0.999 | 7.4 | 0.999 | 0.999
2.1 | 0.997 | 0.997 | 4.8 | 0.999 | 0.999 | 7.5 | 0.999 | 0.999
2.2 | 0.997 | 0.997 | 4.9 | 0.999 | 0.999 | 7.6 | 0.999 | 0.999
2.3 | 0.997 | 0.998 | 5.0 | 0.999 | 0.999 | 7.7 | 0.999 | 0.999
2.4 | 0.997 | 0.997 | 5.1 | 0.999 | 0.998 | 7.8 | 0.999 | 0.999
2.5 | 0.997 | 0.997 | 5.2 | 0.999 | 0.998 | 7.9 | 0.999 | 0.999
2.6 | 0.997 | 0.997 | 5.3 | 0.999 | 0.998 | 8.0 | 0.999 | 0.999
Table 8: The fidelity of the $\lambda\phi^{4}$ ground-state wavefunction prepared with complete adiabatic evolution and with SeqHT procedure as a function of time $t$, as described in the main text and shown in Fig. 5. $\nu$ | Coeff | Bound | $\nu$ | Coeff | Bound | $\nu$ | Coeff | Bound | $\nu$ | Coeff | Bound
---|---|---|---|---|---|---|---|---|---|---|---
0 | 1.0000 | 1.0000 | 64 | 0.0024 | 0.0757 | 128 | 0.0006 | 0.0385 | 192 | 0.0012 | 0.0385
2 | 0.9375 | 0.9688 | 66 | 0.0018 | 0.0757 | 130 | 0.0005 | 0.0385 | 194 | 0.0009 | 0.0385
4 | 0.5274 | 0.7627 | 68 | 0.0005 | 0.0757 | 132 | 0.0001 | 0.0385 | 196 | 0.0002 | 0.0385
6 | 0.5859 | 0.7627 | 70 | 0.0009 | 0.0757 | 134 | 0.0002 | 0.0385 | 198 | 0.0005 | 0.0385