# Quantum simulation of fermionic systems using hybrid digital-analog quantum
computing approach

N. M. Guseynov1,2, W. V. Pogosov1,3,4 1Dukhov Research Institute of Automatics
(VNIIA), Moscow, Russia  2Moscow State University, Moscow, Russia  3Institute
for Theoretical and Applied Electrodynamics, Russian Academy of Sciences,
Moscow, Russia  4HSE University, Moscow, Russia

###### Abstract

We consider a hybrid digital-analog quantum computing approach, which allows
implementing any quantum algorithm without standard two-qubit gates. This
approach is based on the always-on interaction between qubits, which can
provide an alternative to such gates. We show how digital-analog approach can
be applied to simulate the dynamics of fermionic systems, in particular, the
Fermi-Hubbard model, using fermionic SWAP network and refocusing technique. We
concentrate on the effects of connectivity topology, the spread of interaction
constants as well as on errors of entangling operations. We find that an
optimal connectivity topology of qubits for the digital-analog simulation of
fermionic systems of arbitrary dimensionality is a chain for spinless fermions
and a ladder for spin 1/2 particles. Such a simple connectivity topology makes
digital-analog approach attractive for the simulation of quantum materials and
molecules.

August 28, 2024

##  I Introduction

Quantum computing is a new computing paradigm, which is based on principles of
quantum mechanics Feynman (1982). Various physical platforms can be utilized
to construct quantum computers: superconducting quantum circuits Córcoles et
al. (2015), linear optical quantum systems Lund et al. (2017); Zhong et al.
(2020), trapped ions Blatt and Roos (2012); Harty et al. (2014), neutral atoms
Maller et al. (2015), etc. The advantage of quantum computing is that it
allows one to tackle some problems that are hard to solve using conventional
computing systems. Quantum machines can be used for mathematical problems,
such as quantum search algorithm Grover (1996); Ambainis (2005), factoring
integers by Shor’s algorithm Shor (1999, 1994), solving systems of linear
equations Harrow et al. (2009), etc. Another promising application of quantum
computing is a quantum simulation, as proposed originally by R. Feynman
Feynman (1982); Kandala et al. (2017); Ortiz et al. (2001); Lee et al.
(2006a); Jiang et al. (2018). For example, the number of sampling points
required for a classical simulation of the Fermi-Hubbard model at low
temperatures using Monte Carlo methods rises exponentially as the system
volume grows, but quantum computing can provide tools to determine its zero-
temperature phase diagram. Note that the Fermi-Hubbard model plays a central
role in theoretical studies of strongly correlated materials Lee et al.
(2006b); Voit (1995); Imada et al. (1998). Particularly, this model is of
importance to understand their magnetic and superconducting properties as well
as a complex interplay between them. Therefore, the development of methods of
quantum simulation for fermionic systems is an extremely important task.

The ability of quantum computers to outperform classical supercomputers has
recently been demonstrated for some special tasks. This regime of functioning
of quantum computers corresponds to the so-called quantum supremacy Lund et
al. (2017); Zhong et al. (2020); Li et al. (2019); Arute et al. (2019).
Conventional quantum computers operate in accordance with the gate model of
quantum computation which is based on a universal set of quantum gates. The
minimal set of required gates Vatan and Williams (2004) includes arbitrary
single-qubit gates and two-qubit operations such as controlled-NOT (CNOT) or
some equivalent gate. Such quantum devices are usually referred to as digital
quantum computers. However, control of all relevant degrees of freedom becomes
much more difficult when the system is scaled up. Quantum error correction
codes Devitt et al. (2013) in combination with optimization protocols Kandala
et al. (2017); Gidney (2018); Häner et al. (2020) constitute a promising
approach, but their effective implementation on modern quantum devices is
challenging due to the relatively small number of qubits as well as existing
gates errors. As a result of hardware imperfections, the Fermi-Hubbard model
was simulated using digital quantum computers only for small systems with
limited numbers of fermionic lattice sites Arute et al. (2020). Note that the
main contribution to the gates error rate is due to the two-qubit gates.
Thereby, it is interesting to consider alternative approaches, which are not
based on the execution of the standard two-qubit gates.

One of the alternatives is associated with the idea of a hybrid digital-analog
quantum computing Benjamin and Bose (2003); Hu et al. (2007); Parra-Rodriguez
et al. (2020); Babukhin et al. (2020); Céleri et al. (2021); Ying et al.
(2021); Martin et al. (2020); Headley et al. (2020); Yu et al. (2021);
Arrazola et al. (2016); Lamata (2017); Gonzalez-Raya et al. (2021). In the
simplest realization of such a strategy, single-qubit gates, as well as
individual readouts of qubits, must be accessible similarly to the case of
digital quantum devices, but instead of conventional two-qubit gates quantum
entanglement is created using always-on native interaction of qubits. This
requires a certain trade-off between the fidelities of single-qubit gates
diminished by the always-on interaction between qubits and fidelities of
entangling operations Babukhin et al. (2020). A more sophisticated approach is
possible which requires switchable interaction between qubits that can be
turned off during the execution of single-qubit gates. Particularly, we would
like to mention a recent paper Yu et al. (2021), which suggests a specialized
superconducting circuit architecture for the digital-analog quantum
computation and simulation.

In this paper, we treat a native interaction of Ising type as a source of
entangling operations and analyze the digital-analog implementation of quantum
simulation of fermionic systems. Particularly, we address a method described
in Ref. Kivlichan et al. (2018) for digital quantum computers, which is based
on the fermionic SWAP network. The fact that the interaction is always on
implies that quantum evolution is governed by operators of pairwise
interaction between qubits coupled according to the qubit network topology.
Moreover, each qubit pair is, in general, characterized by its own coupling
constant. Despite this fact, it turns out that using sequences of single-qubit
gates interleaved with idling time intervals it is possible to compensate
undesirable interactions and to engineer a needed global unitary
transformation. The basic ideas behind this technique were developed in
quantum computing studies based on nuclear magnetic resonance (NMR) Jones and
Knill (1999); Leung et al. (2000); Dodd et al. (2002). We incorporate these
refocusing operations into the fermionic SWAP network. We then extend the
concept of the refocusing operation to the situation, when the spread in the
interaction constants is present. We also focus on the impact of different
connectivity topologies of qubit networks and the effect of errors in the
implementation of entangling operations.

This paper is organized as follows. We formulate the problem in Section II.
Section III deals with the fermionic SWAP network implemented using the
refocusing technique. This Section, in particular, focuses on the impact of
connectivity topology of the qubit network. Section IV deals with the qubit
networks having a spread in interaction constants. In Section V we discuss the
effect of errors of entangling operations. We conclude in Section VI.

##  II Preliminaries

Two-qubit operations between qubits $p$ and $q$ in a digital-analog device
correspond to the action of a quantum operator of the form
$J_{pq}=e^{-iH_{int}^{p,q}t}$, where $H_{int}^{p,q}$ is the interaction
Hamiltonian of two qubits. In the present paper, we consider Ising coupling
between qubits, which is relevant for various physical implementations of
quantum devices. For example, superconducting qubits can be coupled directly
by $XX$ dipole-dipole interaction. Such coupling can be also mediated by some
bosonic mode, such as a quantized photon field of the cavity coupled to both
qubits. In the latter case, it is possible to eliminate perturbatively the
boson mode from the effective description, provided the system is in a
dispersive regime, which again results in the $XX$ interaction between the
qubits. In Ref. Babukhin et al. (2020) $ZZ$-crosstalks between superconducting
transmons-qubits were utilized as a source of native interaction used to
implement a quantum simulation of spin models as well as the quantum Fourier
transform. Ising coupling is also natural to some ion-trapped implementations
of quantum computers Debnath et al. (2016).

The evolution operator in the appropriate basis takes the form
$J_{pq}=e^{-iZ_{p}Z_{q}t}$, where $t$ is a dimensionless time measured in
terms of the interaction constant. This time is a free parameter for the
entangling operation. Hereafter, $X_{j}$, $Y_{j}$, $Z_{j}$ refer to the Pauli
operators acting in the space of $j$’th qubit. Of course, a direct application
of operators $J_{pq}$ is possible only for those qubits which can interact
according to the connectivity topology of the qubit network. The full
interaction Hamiltonian, in this case, has the Ising-type form

$\displaystyle H_{int}=\hbar\sum_{p>q}\epsilon_{pq}Z_{p}Z_{q},$ (1)

where $\epsilon_{pq}$ is 1 or 0 depending on connectivity topology. We
hereafter denote $J(t)=\prod_{p,q}J_{pq}(t)$ a global evolution operator
corresponding to the physical interaction between the qubits of the network
according to its topology.

In this article, we discuss a method of quantum simulation of fermionic
systems, which uses a concept of the fermionic SWAP network. This smart method
was developed in Ref Kivlichan et al. (2018). It allows one to circumvent
difficulties associated with the mismatch between the fermion statistics of
the simulated system and paulion statistics relevant for qubits. Our aim is to
adapt this approach to the digital-analog quantum computing strategy based on
always-on interaction.

We begin our analysis with the case of spinless fermions and then extend it to
the case of spin-$1/2$ fermions. We consider a system with a general
Hamiltonian

$\displaystyle\\!\\!H\\!=\\!\\!\sum_{n}\\!U_{n}a_{n}^{\dagger}\\!a_{n}\\!+\\!\\!\\!\sum_{m\neq
n}\\!\\!T_{nm}a_{n}^{\dagger}\\!a_{m}\\!+\\!\\!\\!\sum_{m\neq
n}\\!\\!V_{nm}a_{n}^{\dagger}\\!a_{n}a_{m}^{\dagger}\\!a_{m},$ (2)

where $a_{n}$ and $a_{n}^{\dagger}$ are fermionic creation and annihilation
operators corresponding to different orbitals or fermionic sites in lattice
models. This Hamiltonian generalizes many models: the Hubbard model, finite-
difference discretization in quantum chemistry, double-basis coding, etc.

The method of Ref. Kivlichan et al. (2018) uses the well known Jordan-Wigner
transformation Somma et al. (2002) as well as the Trotter-Suzuki decomposition
Trotter (1959); Suzuki (1993) of the evolution operator. A key idea is that
qubit numbering in Jordan-Wigner transformation can be changed on-fly and this
operation can be combined with the action of the evolution operator. The
resulting operation is represented in terms of the so-called two-qubit
fermionic simulation gate Kivlichan et al. (2018) (FSG) parameterized by the
Hamiltonian constants. In Appendix A we show how FSG can be represented in
terms of the native gates. We also discuss the general structure of the
quantum circuit for both spinless and spin-$1/2$ fermions.

##  III Fermionic SWAP network for different connectivity topologies

A quantum circuit presented in terms of standard quantum gates (for example,
CNOTs) can be recast through the native operators (Appendix A). However, these
operators must be executed only between qubits affected by CNOTs. Thus, an
important task is to disable the interaction between certain qubits, which can
be achieved using the so-called refocusing operations Jones and Knill (1999);
Leung et al. (2000); Dodd et al. (2002). The need for a refocusing operation
occurs each time a set of two-qubit gates must be applied simultaneously to
different pairs of qubits (for example, CNOTs).

We now explain the general principle of engineering the refocusing operations
when implementing a fermionic SWAP network. A refocusing operation is based on
a partition of the total interaction time. The total time is determined by the
desired operators we wish to realize. For example, for CNOT gates this
dimensionless time is $\pi/4$ (Appendix A). For Cphase gates applied
simultaneously, such a duration is determined by the maximum single-qubit
rotation, which characterizes the given set of Cphase gates. The refocusing
operation also involves the action of the Pauli operators $X$ between some of
the idling times. Any refocusing operation can be understood in terms of the
action or non-action of $X$ between the times of the $t_{i}$ interaction, as
shown in Fig. 1. In our work, $\sum_{i=0}^{k}t_{i}=T$ and $t_{i}=T/k$, where
$T$ is the total interaction time. The rule of application of $X$ and $I$
gates will be explained in the next Subsection.

Figure 1: General layout of the refocusing operation. It contains layers of
single-qubit operations $X$ or $I$ interleaved with the action of multi-qubit
entangler $J$. Each qubit is characterized by its sequence of operations $X$
and $I$.

The unitary transformation with the operator $X$ in Fig. 1 corresponds to the
reversed time for a given qubit:

$\displaystyle X_{p}\exp(-iZ_{p}Z_{q}t)X_{p}=\exp(iZ_{p}Z_{q}t).$ (3)

Thus, using $X$ gates during the refocusing operation allows the
implementation of a desired multi-qubit gate. However, if there are several
such transformations in a row, there are double applications of $X$ gates,
which can be replaced by the action of the operator $I=XX$. For the sake of
convenience, we will say that within a given refocusing operation, the qubit
is in an even-parity (odd-parity) state if the operator $X$ acted on the qubit
an even (odd) number of times. Thus, the description of odd- and even-parity
states is equivalent to the description with the time-reversed.

###  III.1 All-to-all connectivity

In this Subsection, we consider such quantum networks for which any qubit
interacts with all other qubits. Suppose that our goal is the implementation
of effective interaction between only qubits $i$ and $j$ over time $T$. The
minimum time of the refocusing operation coincides with $T$. In order to
achieve the goal, both qubits $i$ and $j$ must be either in the even-parity
state, or in the odd-parity state at all steps of the refocusing operation.
For the sake of convenience, we will refer to such pairs of qubits as
interaction pairs. For example, the number of such qubits pairs for the
fermionic SWAP network is $\frac{n}{2}$ for an odd layer of FSG operators and
$\frac{n}{2}-1$ for an even layer, where $n$ is the number of qubits (see,
e.g., Fig. 11 of Appendix A). It is necessary not only to implement the
desired operation within each interaction pair. It is also necessary to
disable interaction between qubits from different pairs, as well as between
all qubits and those qubits that are not included in any of the interaction
pairs. We will refer to such qubits as single qubits. An interaction pair is
in an odd-parity (even-parity) state if each qubit of the pair is in an odd-
parity (even-parity) state.

In our approach, the number of time intervals $t_{i}$ in the refocusing
operation is always $m$, and all these intervals are equal, where $m$ is a
natural number satisfying a condition

$\displaystyle k\leq m<2k,$ (4)

where $k$ is the total number of interaction pairs and single qubits,
determined by the type of quantum operation we wish to accomplish. Let us
stress that Eq. (4) is correct only for the case of all-to-all connectivity.

Appendix B presents the mathematical formalism behind the concept of the
refocusing operation including the Hadamard operator $H(m)$, which plays a
crucial role Jones and Knill (1999). It is defined in a matrix form as

$\displaystyle H(m)H(m)^{T}=mI,$ (5)

where $I$ is identity matrix and $H(m)$ is the square matrix $m\times m$, with
elements $\pm 1$. For instance, $H(2)=\left(\begin{array}[]{cc}1&1\\\ 1&-1\\\
\end{array}\right).$

An important property of the operator $H(m)$ is the fact that any two columns
of this operator in a matrix form, if they are considered as two vectors, are
orthogonal to each other, and absolute values of any element are the same.
This property is used for the construction of the refocusing operation Jones
and Knill (1999); Leung et al. (2000); Dodd et al. (2002): the operator $H(m)$
defines the sequences of states (even- or odd-parities) for every single qubit
or interaction pair, which they must follow. Every single qubit or interaction
pair can be associated with an arbitrary column up to the mutual permutations.
The element $i$ of column $j$ determines the state (even or odd parity) of an
interaction pair or single qubit with index $j$ during $t_{i}$. If this
element is equal to $1$, then the qubit pair or single qubit must be in the
even-parity state, and if $-1$, then it must be in the odd-parity state. As a
result, each matrix column defines a set of states. If the qubit follows the
sequence of these states during the refocusing operation, it does not
effectively experience interaction with the qubits of other groups. If our
goal is to implement CNOT operations simultaneously between certain qubits,
then, in the beginning, we must find an appropriate $m$ for a given
connectivity topology. In the case of all-to-all connectivity, $m$ is defined
according to Eq. (4). The algorithm for determining $m$ in the case of a
general connectivity topology is presented in Appendix C.

Let us estimate a circuit depth, i.e., the number of multi-qubit gate
applications per single Trotter step per qubit. We start with the special
case, when qubit number $n$ can be represented as
$\frac{n}{2}+2=2^{\mathcal{N}}$, where $\mathcal{N}$ is a natural number. From
the viewpoint of resources, this is an ideal case for a digital-analog quantum
device with full connectivity applied for the simulation of the quantum system
described by the Hamiltonian (2). It is known that $H(m)$ exists for
$m=\frac{n}{2}+2$. The number of rows of the square matrix $H(m)$ corresponds
to the number of $t_{i}$ for an individual refocusing operation. In this case,
the number of applications of interaction gates $J$ is $\frac{3n^{2}}{2}+6n$
per Trotter step. The number of CNOTs needed to implement the same Trotter
step on a digital quantum computer is $\frac{3n^{2}-3n}{2}$ (the number of
FSGs is $\frac{n^{2}-n}{2}$, while each FSG is represented in terms of three
CNOTs). In both cases, the number of multi-qubit gates is
$\mathcal{O}(n^{2})$, which is preserved for any $n$. However, the circuit
depth for a digital-analog device is $\mathcal{O}(n^{2})$, and for a digital
one, it is $\mathcal{O}(n)$. This difference is significant, so the use of a
digital-analog quantum device is not optimal for implementing the fermionic
SWAP network in the case of the full connectivity of qubits. Next, we will
consider the connectivity, for which the circuit depth will be the same for
digital and digital-analog quantum devices.

###  III.2 Two-dimensional topology

In this Subsection, we discuss two-dimensional connectivity topologies. An
example, represented by the square lattice with nearest-neighbor interaction,
is shown schematically in Fig. 2.

Figure 2:  Two-dimensional topology of qubit connectivity: square lattice with
nearest-neighbor interaction. Lines connecting qubits show the action of the
operator $J_{pq}$ between them.

Consider, for example, the qubits $1$ and $n+2$ in Fig. 2. Since the native
interaction does not act between them, these two qubits will not interact
during the refocusing operation, no matter what sequences they follow. The
fact that the qubit interaction is limited reduces the number of iterations
$t_{i}$ required for the refocusing operation. We will demonstrate this fact
using the example shown in Fig. 3. The qubits included in the interaction pair
are circled. The odd stage of the procedure is used for the odd layers of FSG
operators, and even stage for the even layers. These stages correspond to the
single Trotter step for simulating the system described by the Hamiltonian
(2). Note that the qubit numbering, as indicated in Fig. 3, and the fermionic
site numbering generally do not match. For an odd stage, the refocusing
operation can be performed using only a two-qubit behavior sequence defined by
the matrix $H(2)$ that dramatically simplifies the problem compared to the
case of all-to-all connectivity. In order to implement the refocusing
operation at odd stages, interaction pairs must be positioned in a staggered
pattern order, as depicted in Fig. 3. In this case, qubit pairs indicated by
”I” follow the first sequence, while qubit pairs indicated by ”II” follow the
second sequence. Since only two-qubit behavior sequences are used, the number
of time intervals $t_{i}$ is two, where $t_{i}=\frac{T}{2}$.

Figure 3:  General layout for the implementation of a single Trotter step (see
in the text).

In the case of even stages, also shown in Fig. 3, the sequences are determined
by $H(4)$, which encodes four possible sequences of states. Two sequences, in
this case, are not sufficient. Indeed, let us consider the interaction pairs
formed by qubits $1-6$ at the even stage in Fig. 3. Two sequences are not
sufficient since there are triples of interaction pairs, where all three
interact with each other. According to Appendix C, four sequences of states
are enough for this connectivity topology. Moreover, three sequences can be
used in the even stage, and the association between the qubits and the three
sequences is shown in Fig. 3 by ”I”, ”II”, and ”III”. However, $H(3)$ does not
exist, which leads us to use the three sequence of states generated by $H(4)$.

The circuit depth for the implementation of a single Trotter step grows as
$12n$, i.e., it scales as $\mathcal{O}(n)$. This estimate coincides with a
similar result for a digital quantum device. In the latter case, the total
number of CNOT gates is $\mathcal{O}(n^{2})$. We conclude that a digital-
analog quantum device with a two-dimensional topology is prospective for the
quantum simulation of fermionic models.

###  III.3 One-dimensional topology

The one-dimensional connectivity topology is the simplest case of qubit
network connectivity. It turns out that this is an ideal case of using the
fermionic SWAP network. For the one-dimensional topology, two sequences of
states are sufficient so that the appropriate sequences of states are
determined by the matrix $H(2)$. The association between qubits of the chip
and interaction pairs at even and odd stages of the single Trotter step is
shown in Fig. 4. This association takes an elementary form. We remind that a
single Trotter step consists of $n/2$ odd stages and $n/2$ even stages. The
circuit depth grows as $6n$, and this value is twice smaller than that for the
case of the two-dimensional square qubit array due to the number of $t_{i}$
used in the refocusing operation. In the case of one- and two-dimensional
qubit networks, the circuit depth required to implement a single layer of FSGs
is $\mathcal{O}(1)$, since the number of sequences of states required to
implement one layer of FSGs is constant for any qubit number $n$, according to
Appendix C.

Figure 4: Scheme of linear connectivity of qubits. The interaction pairs at
even and odd stages of a single Trotter step are shown.

Note that using quantum chips with one-dimensional connectivity usually
dramatically increases the two-qubit gate number due to the necessity to
utilize multiple SWAP operations. Surprisingly, quantum simulations of
fermionic models in any dimension does not belong to this case due to the one-
dimensional character of Jordan-Wigner transformation and despite the non-
locality of the fermionic statistics. However, it turns out that for
spin-$1/2$ fermions, the optimal connectivity topology of the quantum network
is a ladder, as shown in Appendix D.

We also wish to stress that the general structure of the fermionic SWAP
network is appropriate for the digital-analog strategy since all qubits of the
device are used uniformly in time. The quantum circuit is represented by
periodically accomplished layers of FSGs, which are executed each time for
nearly half of qubits. Thus, multiple two-qubit entangling gates must be
implemented simultaneously. This structure of the quantum circuit maximizes
the usage of global entangling gates.

##  IV Systems with a spread in interaction constants

In this Section, we consider digital-analog quantum devices, which have some
spread in interaction constants of qubits, and extend the refocusing technique
to this important case that can be relevant for solid-state qubits, such as
superconducting macroscopic artificial atoms, for which a precise control of
chip parameters is difficult. The operator $e^{-i\alpha_{pq}Z_{p}Z_{q}t}$ acts
on pairs of qubits with different coefficients $\alpha_{pq}$ for each qubit
pair ($p$, $q$), as imposed by the topology. We assume that all these
interaction constant are known.

We introduce a notion of a global refocusing operation, which represents a
sequence of individual refocusing operations, described in the preceding
Section. The result of such an operation is assumed to be the action of the
operator $e^{-i\alpha_{pq}Z_{p}Z_{q}T_{pq}}$ only between pre-selected qubits,
while for any two interaction pairs $\alpha_{pq}T_{pq}=\alpha_{km}T_{km}$. We
are interested in implementation of such an operator for each interaction
pair, since when executing CNOT gates simultaneously between certain qubits,
our goal is the same two-qubit operator $e^{-iZ_{p}Z_{q}T}$ with a fixed time
$T$ for each pair $(p,q)$.

Without loss of generality, we assume that $\alpha_{01}$ is minimum among
coupling constants, then we reassign the time $t^{\prime}=t\alpha_{01}$. The
operator $e^{-i\beta_{pq}Z_{p}Z_{q}t^{\prime}}=e^{-i\alpha_{pq}Z_{p}Z_{q}t}$
acts on pairs of qubits according to the connectivity, where
$\beta_{pq}=\alpha_{pq}/\alpha_{01}\geq 1$ so that $\beta_{01}=1$. From the
condition $\alpha_{pq}T_{pq}=\alpha_{km}T_{km}$, it follows that $T_{01}\geq
T_{pq}$ for any ($p$, $q$). We construct a global refocusing operation with
the duration $T=T_{01}$. Then the qubits $0$ and $1$ must interact during the
time $T$. For all other interaction pairs, the same result is achieved in
shorter time. Thus, for each interaction pair, there is an excess time
interval $\tau_{pq}=T-T/\beta_{pq}$. This means that in the global refocusing
operation, there is a time interval $\tau_{pq}$ for which the interaction
between the qubits $p$ and $q$ must be disabled. This can be achieved by
constructing a special refocusing operation, which is a part of the global
refocusing operation.

Note that, within each individual refocusing operation, qubits characterized
by different sequences of states do not interact with each other, since all
time intervals $t_{i}$ are equal to each other. Hereafter, for the sake of
simplicity, we use indices for numbering interaction pairs and single qubits
(not individual qubits as before). Index $i=1$ will be used for the
interaction pair with minimum interaction constant.

###  IV.1 General strategy

We start our consideration with a relatively simple case, when the difference
between $\alpha_{i}$ is small enough, so that $\sum_{i}\tau_{i}<T$. Then the
total interaction time can be divided into intervals $T_{i}=\tau_{i}$, $i\neq
1$, and $T_{1}=T-\sum_{k=2}^{n_{p}}\tau_{k}$, where $n_{p}$ is the number of
interaction pairs. During each interval $T_{i}$, different interaction pairs
must correspond to different sequences, then after this time, the interaction
between qubits of different sequences is disabled, so they remain effectively
non-interacting. For each interaction pair $i$ (except $i=1$), there must be a
time interval $T_{i}$ when qubits from this interaction pair must remain non-
interacting in order to compensate a difference in interaction constants.
Hence, qubits of pair $i$ must follow different sequences of states during
$T_{i}$ (except $i=1$) and the algorithm presented in Appendix B is again
applicable for each $T_{i}$.

The condition $\sum_{i}\tau_{i}<T$ is quite difficult to be fulfilled for a
large number of qubits in the system. However, this condition is used for
demonstration purposes only. Several interaction pairs can be destroyed during
some of the time intervals $T_{i}$. Consider, for example, the case of an
overlap of two times $\tau_{i}$ and $\tau_{j}$. Suppose that
$\tau_{i}>\tau_{j}$, then in the global refocusing operation $T_{j}=\tau_{j}$,
and $T_{i}=\tau_{i}-\tau_{j}$. This means that during $T_{j}$, the refocusing
operation is constructed so that both interaction pairs $i$ and $j$ are
destroyed (all four qubits become single), and during $T_{i}$, only the pair
$i$ is destroyed (pair of qubits becomes single). Thus, the time intervals
$\tau_{i}$ can overlap, which means that the quantum chip can be characterized
by any coefficients $\alpha_{pq}$.

###  IV.2 Different topologies

Let us consider a digital-analog quantum device with qubits positioned in the
chain (linear connectivity). As an example, we analyze the system of six
qubits that can be readily extended to the general case. We show how to
construct an odd stage, shown in Fig. 4. The even stage is constructed
similarly. The coupling between qubits 0 and 1 is again assumed to be minimum
among all pairs of qubits. As it was shown earlier, two sequences are
sufficient to implement the refocusing operation. There are three interaction
pairs, and to achieve the result, we divide the time $T$ into three intervals
as $T=T_{1}+T_{2}+T_{3}$. During each interval $T_{1}$, $T_{2}$, $T_{3}$ one
of the interaction pairs is destroyed, except for $T_{1}$.

Table 1 shows a sequence selection for six qubits. This table demonstrates
what sequence each qubit must follow, according to the general algorithm
presented in the preceding Subsection. In total, two sequences are used, and
therefore, for each $T_{i}$, the refocusing operation contains only two
intervals $t_{i}=T_{i}/2$. The result of such a global refocusing operation
allows the implementation of a set of CNOTs between interaction pairs shown in
Fig. 4 for six qubits. Our general algorithm allows extending this approach to
any qubit number for the one-dimensional topology.

Q | 0 | 1 | 2 | 3 | 4 | 5
---|---|---|---|---|---|---
$T_{1}$ | 1 | 1 | 2 | 2 | 1 | 1
$T_{2}$ | 1 | 1 | 2 | 1 | 2 | 2
$T_{3}$ | 1 | 1 | 2 | 2 | 1 | 2
Table 1. A sequence of states to achieve a fixed action of the operator
$e^{-iZ_{p}Z_{q}T}$ between all interaction pairs for the case of linear
connectivity of qubits with the spread in interaction constants. The number in
the table shows which sequence, given by the matrix $H(m)$, the qubit is
associated.

We now consider a digital-analog quantum device with two-dimensional
connectivity of qubits, as shown in Fig. 5. In this case, each qubit can
interact with no more than four neighboring qubits. Therefore, four sequences
defined by the matrix $H(4)$ are sufficient for refocusing. They are shown in
Table 2. This example is interesting because the qubits 5 and 6, which form
the interaction pair, interact with five other interaction pairs at once.
However, according to our general algorithm, four sequences are sufficient to
implement the desired operator between the selected qubits circled in Fig. 5.
Note that the selection of sequences is not unique up to permutations in the
same way as the system with identical interaction constants.

Figure 5: Schematic layout of the two-dimensional square qubit network. Q | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11
---|---|---|---|---|---|---|---|---|---|---|---|---
$T_{1}$ | 1 | 1 | 2 | 2 | 4 | 3 | 3 | 2 | 2 | 1 | 1 | 4
$T_{2}$ | 1 | 1 | 2 | 1 | 4 | 3 | 3 | 2 | 2 | 1 | 1 | 4
$T_{3}$ | 1 | 1 | 2 | 2 | 4 | 3 | 4 | 2 | 2 | 1 | 1 | 4
$T_{4}$ | 1 | 1 | 2 | 2 | 4 | 3 | 3 | 2 | 4 | 1 | 1 | 4
$T_{5}$ | 1 | 1 | 2 | 2 | 4 | 3 | 3 | 2 | 2 | 1 | 2 | 4
$T_{6}$ | 1 | 1 | 2 | 2 | 4 | 3 | 3 | 2 | 2 | 1 | 1 | 2
Table 2. A sequence of states needed to achieve a fixed action of the operator
$e^{-iZ_{p}Z_{q}T}$ between all interaction pairs for the case of two-
dimensional connectivity of qubits with a spread in the interaction constants.

###  IV.3 Remarks on scaling

When the spread of parameters is considered, the circuit depth for a single
Trotter step increases by a factor of $\mathcal{O}(n)$ and becomes equal to
$\mathcal{O}(n^{2})$. This is due to the fact that the refocusing operations
have to be replaced by global refocusing operations. Each global operation
consists of $\mathcal{O}(n)$ individual refocusing operations. The number of
multi-qubit operations also grows as $\mathcal{O}(n^{2})$.

The general idea of engineering a global refocusing operation is also valid
for the all-to-all connectivity topology of qubits. However, as noted earlier,
this connectivity is not effective for the implementation of the fermionic
SWAP network since each refocusing operation contains $\mathcal{O}(n)$ time
intervals $t_{i}$. Thus, the number of multi-qubit gates for the single
Trotter step grows as $\mathcal{O}(n^{3})$.

##  V Error analysis

In this Section, we consider the impact of two types of errors, which are
important for digital-analog quantum computing. One of these errors is due to
the the inaccurate determination of $\alpha_{pq}$, which differs from the
actual value, for the entangling operator $e^{-i\alpha_{pq}Z_{p}Z_{q}t}$
between qubits $p$ and $q$. Another error we consider is the depolarizing
noise, which is here described in terms of the action of Kraus operators after
each application of $J(t)$ gate.

To describe the effect of errors, we use the concept of fidelity of quantum
processes defined as

$\displaystyle
F_{proc}(A,B)=\frac{|Col(A)^{\dagger}Col(B)|^{2}}{Col(A)^{\dagger}Col(A)Col(B)^{\dagger}Col(B)},$
(6)

where $Col$ is the operation of converting a matrix into a column:

$\displaystyle Col\left(\begin{array}[]{cc}a&b\\\ c&d\\\
\end{array}\right)=\left(\begin{array}[]{c}a\\\ c\\\ b\\\ d\\\
\end{array}\right).$ (13)

In order to estimate the difference between the results of the ideal and not
ideal simulations, we use fidelity of quantum states defined as

$\displaystyle
F(\sigma,\rho)=\left(tr(\sqrt{\sqrt{\rho}\sigma\sqrt{\rho}})\right)^{2},$ (14)

where $\sigma$ and $\rho$ are the density matrices of the quantum states.

###  V.1 Error of inaccurate determination of the coupling constant

In this Subsection, we analyze an impact of the error of inaccurately defining
the constant $\alpha_{pq}$ for the entanglement operator
$e^{-i\alpha_{pq}Z_{p}Z_{q}t}$ between qubits $p$ and $q$. Suppose we were
assuming that the operator $e^{-i\alpha_{pq}Z_{p}Z_{q}t}$ acts between them,
but in reality the interaction constant was
$\alpha_{pq}+\alpha_{pq}^{\prime}$. Thus, an error occurs associated with
nonzero $\alpha^{\prime}_{pq}$.

We start our consideration with an estimate of $F_{proc}$ for the CNOT
execution for two isolated qubits for which the actual interaction constant is
$\alpha+\alpha^{\prime}$ instead of $\alpha$. The fidelity of quantum process
value introduced in Eq. (6) can be expressed explicitly as

$\displaystyle
F_{\text{proc}}(CX,CX_{\text{ideal}})=\cos^{2}\left(\frac{\pi}{4}\frac{\alpha^{\prime}}{\alpha}\right).$
(15)

The infidelity $1-F_{\text{proc}}(CX,CX_{\text{ideal}})$ scales as
$\frac{1}{2}(\frac{\pi}{4}\frac{\alpha^{\prime}}{\alpha})^{2}$ at
$\alpha^{\prime}\ll\alpha$. Thus, only a weak quadratic growth of infidelity
is expected, as $\alpha^{\prime}$ grows. Figure 6 presents the numerically
obtained dependence (dots) of $F_{\text{proc}}(CX,CX_{\text{ideal}})$ on
$\alpha^{\prime}/\alpha$, which shows a full agreement with the explicit
result (solid line). A weak dependence of infidelity on $\alpha^{\prime}$ is a
positive fact in the view of perspectives of the digital-analog approach.

Figure 6: The dependence of $F_{\text{proc}}(CX,CX_{\text{ideal}})$ on
$\alpha^{\prime}/\alpha$.

Our next step is a consideration of the impact of an error in the definition
of coupling constants for the single Trotter step for the Hamiltonian (2). We
consider the case when a deviation of the coupling constant is random for any
pair of neighboring qubits. The most optimal linear connectivity topology of
the qubit network is assumed. We assume they are randomly selected from a
uniform distribution from the segment $[-\omega,\omega]$. Figure 7 shows the
dependence of $F$ of an ideal Trotter step simulation and a simulation with
errors on $\omega/\alpha$ for the chain of twelve qubits. This dependence was
found numerically for different values of $\omega/\alpha$, and each value
corresponds to a dot in Fig. 7. According to the Haar measure, a random state
is selected as the initial state, and then averaging is performed for 100
initial states. The energy parameters $V_{nm}$ and $T_{nm}$ of the Hamiltonian
(2) are randomly selected from a uniform distribution so that $V_{nm}\delta
t$, $T_{nm}\delta t\in[-0.1,0.1]$ for any $n,m$ (all-to-all connectivity of
the fermionic cluster). The results are presented for a particular realization
of disorder. However, we found that they are essentially insensitive to the
realizations. The coefficients $U_{n}$ are assumed to be zero since single-
qubit gates simulate the corresponding evolution operator. We again see a weak
quadratic growth of infidelity, now for the whole Trotter step, as a function
of $\omega$.

Figure 7: The dependence of $F$ on $\omega/\alpha$ for a single Trotter step
and 12 qubits.

Note that the time determination error is equivalent to the error in the
determination of $\alpha$ since $\alpha$ and $t$ are symmetric for the
entanglement operator $J(t)$.

###  V.2 The effects of depolarizing noise, amplitude damping, and phase
damping

In this Subsection, we consider the impacts of three types of uncorrelated
noise Nielsen and Chuang (2002) in the case of the quantum circuit
corresponding to the single Trotter step. In our model, the density matrix of
each qubit is transformed according to the noise model after each two-qubit
operation, which is $J(t)$ and CNOT in the case of digital-analog and digital
quantum computing, respectively. The transformations correspond to the action
of the Kraus operators on the density matrix, given by

$\displaystyle\mathcal{E}^{d}_{p}(\rho)=(1-p)\rho+\frac{p}{3}X\rho
X+\frac{p}{3}Y\rho Y+\frac{p}{3}Z\rho Z,$ (16)

$\displaystyle\mathcal{E}^{a}_{\gamma}(\rho)=E_{0}^{a}\rho
E_{0}^{a}+E_{1}^{a}\rho E_{1}^{a\dagger},\qquad
E_{0}^{a}=\left(\begin{array}[]{cc}1&0\\\ 0&\sqrt{1-\gamma}\\\
\end{array}\right),\qquad
E_{1}^{a}=\left(\begin{array}[]{cc}0&\sqrt{\gamma}\\\ 0&0\\\
\end{array}\right),$ (21)

$\displaystyle\mathcal{E}^{p}_{\lambda}(\rho)=E_{0}^{p}\rho
E_{0}^{p}+E_{1}^{p}\rho E_{1}^{p},\qquad
E_{0}^{p}=\left(\begin{array}[]{cc}1&0\\\ 0&\sqrt{1-\lambda}\\\
\end{array}\right),\qquad E_{1}^{p}=\left(\begin{array}[]{cc}0&0\\\
0&\sqrt{\lambda}\\\ \end{array}\right),$ (26)

In the model corresponding to Eq. (16) one of the Pauli errors occurs with
probability $p$ giving rise to a completely mixed qubit state. The model (21)
corresponds to the effect of energy dissipation, while the relaxation time is
characterized by the parameter $\gamma$. The noise that describes the loss of
information without energy loss (phase damping) is represented by the model
(26).

Figure 8: The dependence of $F$ for a single Trotter step on error parameters
for the quantum processor consisting of six qubits positioned in a chain. Two
types of quantum devices are considered: digital-analog quantum computer (a)
and digital quantum computer (b). Three types of errors are analyzed: phase
damping (curves 1), amplitude damping (curves 2), depolarizing noise (curves
3).

We again consider the single Trotter step of evolution for the Hamiltonian (2)
from the initial random state selected according to the Haar measure and then
average the fidelity for 100 different initial states. The parameters of the
fermionic model are the same as in the preceding Subsection. A linear
connectivity topology of the quantum chip is assumed both in the digital-
analog and digital cases. In the latter case, execution of CNOTs is assumed.
Figure 8 shows the dependence of the fidelity on the noise parameters for the
chain of six qubits computed numerically for the digital-analog (a) and the
digital (b) models. In all cases considered, we see a pronounced linear
decrease of fidelity as noise parameter grows. Thus, these noises are
generally more dangerous for the simulation of fermionic systems than the
error of inaccurate determination of the coupling constant within the digital-
analog strategy. In addition, for each type of error, the value of infidelity
in the case of digital-analog approach is twice as large as in the case of
digital computation for the same values of noise parameters. This is directly
related to the fact that two native entangling operations are used to
implement a single refocusing operation according to linear connectivity
topology. We, however, stress that such a direct comparison of fidelities
without specifying the implementation and architecture of a quantum computer
is not reasonable, since actual values of noise parameters for both strategies
can be very different even for the same physical realization of the quantum
device and are generally determined by specific error budgets. However, in all
cases, we observe a linear fidelity decrease, while the unique error for
digital-analog strategy has only a weak quadratic effect on infidelity. At the
same time, a key advantage of the digital-analog strategy is that it does not
require two-qubit gates implementation that is associated with complicated
engineering and wiring issues. This supports our vision that digital-analog
quantum computing is attractive for quantum simulation of fermionic systems.

We also would like to mention that Ref. Babukhin et al. (2020) presents simple
estimates for the optimal coupling strength for a digital-analog strategy
based on always-on interaction, which appears as a result of a balance between
the fidelities of single-qubit gates diminished by always-on interaction and
fidelities of entangling operations decreased due to the decoherence. Such a
consideration can be helpful in construction of digital-analog devices based
on different physical realizations.

##  VI Conclusions

This article pointed out that digital-analog quantum devices are prospective
for the quantum simulation of fermionic models in different dimensions. The
approach we proposed is based on the application of fermionic SWAP network
Kivlichan et al. (2018) implemented through always-on interaction between
qubits of the device. The fermionic simulation gate Kivlichan et al. (2018)
can be used to circumvent the non-locality problem of the Hamiltonian (37)
under the Jordan-Wigner transformation. This gate is represented through the
native interaction of qubits and single-qubit gates. We suggested using a
refocusing technique when implementing a fermionic SWAP network to compensate
for undesirable interactions between qubits at each simulation step. We argued
that the digital-analog strategy is appropriate for the fermionic simulation
since all qubits of the chip can be used uniformly in time, enabling it to
perform multiple entangling operations simultaneously.

We concentrated mainly on the impact of the connectivity topology of the qubit
network, its spread in interaction constants, and the effect of errors in
entangling operations on the efficiency of quantum algorithm implementation.
We found that an optimal topology for the digital-analog chip in the context
of fermionic simulation is a one-dimensional chain for any dimensionality of
the simulated model, in the case of spinless fermions, and a ladder, in the
case of spin-1/2 fermions. In this case, the overhead associated with
compensating undesired interactions between qubits is minimum. Such a simple
connectivity topology makes it easier to probe our findings in experiments.

For optimal topology, the simulation of the single Trotter step of the
evolution requires $\mathcal{O}(n)$ applications of multi-qubit gates, where
$n$ is the qubit number. The method of refocusing has been extended to the
case when there is a spread of interaction constants. It is shown that the
circuit depth increases by a factor of $\mathcal{O}(n)$. We also found that
errors associated with the inaccuracy in determining interaction constants
lead to the weak quadratic growth of infidelity as a function of the deviation
of the actual coupling constants from the assumed values, which is a positive
fact in the view of the perspectives of the digital-analog approach. The
effects of the depolarizing noise, phase damping and amplitude damping are
more harmful since they lead to the linear decrease of fidelity as a function
of error parameter both for digital-analog and digital strategies.

## Acknowledgements

W. V. P. acknowledges a support from RFBR (project no. 19-02-00421).

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##  Appendix A Fermionic SWAP network under always-on interaction

###  A.1 Spinless fermions

Under Jordan-Wigner transformation for any numbering of spins, the first term
of the Hamiltonian (2) takes the form $\sum_{p}U_{p}\sigma_{p}^{+}\sigma_{p}$,
where $\sigma_{p}^{+}$ and $\sigma_{p}$ are the spin creation and annihilation
operators. The evolution operator corresponding to this term can be modeled by
the action of a gate $R_{z}$ on each qubit and we do not consider this
contribution to the Hamiltonian.

Our goal thus is to simulate the evolution during the time interval $\delta t$
under the Hamiltonian

$\displaystyle H^{\prime}=\sum_{m\neq n}T_{nm}a_{n}^{\dagger}a_{m}+\sum_{m\neq
n}V_{nm}a_{n}^{\dagger}a_{n}a_{m}^{\dagger}a_{m}.$ (27)

As a result of the Jordan-Wigner transformation, the Hamiltonian (27) takes an
essentially non-local form. In order to circumvent this problem, FSG gate can
be utilized Kivlichan et al. (2018); Arute et al. (2020), which is defined as

$\displaystyle\begin{gathered}\\!\\!\\!\\!F_{\delta
t}^{nm}\\!\\!=\\!e^{-iV_{pq}a^{\dagger}_{n}a_{n}a^{\dagger}_{m}a_{m}\delta
t}e^{-iT_{nm}(a^{\dagger}_{n}a_{m}+a^{\dagger}_{m}a_{n})\delta
t}f_{swap}^{nm},\end{gathered}$ (29)

where
$f_{swap}^{nm}=1+a_{n}^{\dagger}a_{m}+a_{m}^{\dagger}a_{n}-a_{n}^{\dagger}a_{n}-a_{m}^{\dagger}a_{m}$
is a fermionic orbital change operation that preserves the correct
antisymmetrization. FSG is applied only for $m=n+1$, i.e., for neighboring
qubits. It can be represented using the matrix $F(\phi,\theta)$ defined as

$\displaystyle F(\phi,\theta)=\left(\begin{array}[]{cccc}1&0&0&0\\\
0&-i\sin(\frac{\phi}{2})&\cos(\frac{\phi}{2})&0\\\
0&\cos(\frac{\phi}{2})&-i\sin(\frac{\phi}{2})&0\\\ 0&0&0&e^{i\theta}\\\
\end{array}\right),$ (34)

where $\phi/2=T_{nm}\delta t$ and $\theta=-V_{nm}\delta t+\pi$. Our strategy
is to represent FSG in terms of single-qubit gates and CNOTs and then express
CNOT through the native two-qubit gate. The implementation of FSG using CNOTs
gates is shown in Fig. 9.

Figure 9: Quantum circuit for FSG implementation in terms of single-qubit
gates and CNOTs.

The implementation of CNOT gate on a digital-analog device is visualized in
Fig. 10: we consider an _isolated_ couple of qubits that experience native
interaction and express CNOT in terms of the quantum gate associated with this
interaction. A duration of always-on interaction is $t_{CNOT}=\frac{\pi}{4}$.
The refocusing technique used to effectively implement the required isolation
of a selected set of couples is discussed in Sections III and IV of the main
text.

Figure 10:  Implementation of CNOT gate on a digital-analog quantum device.
Control qubit corresponds to the upper line, while target qubit corresponds to
the lower line.

The resulting quantum scheme for a single Trotter step is represented by
layers of FSGs, as shown in Fig. 11. It is worth noting that this quantum
circuit corresponds to the evolution of the Hamiltonian $H^{\prime}$ for any
values of $T_{pq}$, $V_{pq}$. This means that this quantum circuit is
appropriate for systems of any dimension described by the Hamiltonian (27),
since the parameters responsible for the dimension, as well as for hopping and
interaction are $V_{pq}$ and $T_{pq}$. Thus, models of different dimensions
are distinct only by the value of the parameters in the circuit shown in Fig.
9. Note that all FSGs in Fig. 10 are shown using the same quantum
representation diagrams, but for each FSG, the internal parameters $\phi$ and
$\theta$ differ, in general.

Figure 11:  Quantum circuit for a single Trotter step of the Hamiltonian
$H^{\prime}$ for an even number of qubits, as a result of which the nodes have
the reverse order.

Each FSG can be accomplished in a digital-analog quantum device with always-on
interaction. The quantum circuit thus consists of layers of CNOTs between
certain qubits interleaved by layers of single-qubit gates. Moreover, each
layer of CNOTs can be executed with a digital-analog strategy that maximizes
the usage of global entangling gates. It is of importance to note that the
duration of the interaction between qubits in a digital-analog quantum device
do not depend on the parameters $\phi$ and $\theta$, since these parameters
influence only single-qubit rotations. The CNOT layer is implemented via a
simultaneous pairwise interaction during the same dimensionless time
$\frac{\pi}{4}$.

###  A.2 Spin 1/2 fermionic Hamiltonians

In this Subsection, we consider spin-1/2 fermionic systems described by the
Hamiltonian

$\displaystyle\begin{gathered}H=\sum_{n\neq
m}T_{nm}a^{\dagger}_{n\uparrow}a_{m\uparrow}+\sum_{n\neq
m}U_{nm}a^{\dagger}_{n\downarrow}a_{m\downarrow}+\\\
\sum_{n}V_{n}a^{\dagger}_{n\uparrow}a_{n\uparrow}a_{n\downarrow}^{\dagger}a_{n\downarrow}.\end{gathered}$
(37)

Antisymmetrization must be taken into account only for fermions within each
sort. Each qubit of the chip corresponds to the fermions of one type. Thus,
the number of qubits required to simulate evolution is twice as large as the
number of fermionic sites.

Consider the term of the Hamiltonian $\sum_{n\neq
m}T_{nm}a^{\dagger}_{n\uparrow}a_{m\uparrow}$. The evolution operator for this
contribution, as described above, can be simulated by layers of FSG operators
with $V_{nm}=0$. For the part of the Hamiltonian $\sum_{n\neq
m}U_{nm}a^{\dagger}_{n\downarrow}a_{m\downarrow}$, the above statement is also
correct. Note that these contributions to the Hamiltonian belong to different
types of fermions, therefore, due to the commutation relations, they can be
simulated independently, as shown in Fig. 12: the FSG layers for different
types of fermions do not intersect. Thus, for the part of the Hamiltonian (37)
$\sum_{n\neq m}T_{nm}a^{\dagger}_{n\uparrow}a_{m\uparrow}+\sum_{n\neq
m}U_{nm}a^{\dagger}_{n\downarrow}a_{m\downarrow}$ is simulated using FSG, and
the evolution of the part
$\sum_{n}V_{n}a^{\dagger}_{n\uparrow}a_{n\uparrow}a_{n\downarrow}^{\dagger}a_{n\downarrow}$
is implemented using the Cphase gates and the Jordan-Wigner transformation.
Indeed, the part of the Hamiltonian
$\sum_{n}V_{n}a^{\dagger}_{n\uparrow}a_{n\uparrow}a_{n\downarrow}^{\dagger}a_{n\downarrow}$
under the Jordan-Wigner transformation takes the form
$\sum_{n}V_{n}\sigma^{+}_{n\uparrow}\sigma^{-}_{n\uparrow}\sigma_{n\downarrow}^{+}\sigma^{-}_{n\downarrow}$.
The Cphase operator, up to the total phase, is described by the operator
$e^{i\phi(Z_{\uparrow}+Z_{\downarrow}-Z_{\uparrow}Z_{\downarrow})}$. Evolution
operator for
$V_{n}\sigma^{+}_{n\uparrow}\sigma^{-}_{n\uparrow}\sigma_{n\downarrow}^{+}\sigma^{-}_{n\downarrow}$
when decomposing the operators $\sigma^{\pm}=\frac{X\pm iY}{2}$, has the form
$e^{iV_{n}(Z_{\uparrow}+Z_{\downarrow}-Z_{\uparrow}Z_{\downarrow})t/\hbar}$.
Thus, the Cphase operator in Fig. 12 is used to simulate the evolution under
the action of
$V_{n}\sigma^{+}_{n\uparrow}\sigma^{-}_{n\uparrow}\sigma_{n\downarrow}^{+}\sigma^{-}_{n\downarrow}$
with $\Phi=V_{n}t/\hbar$.

Figure 12:  Quantum circuit for a single Trotter step of the Hubbard
Hamiltonian with two electron spins for an even number of fermionic nodes.

Note that again the parameters of the Cphase and FSG operators are determined
by the coefficients $V_{n}$, $T_{nm}$, and $U_{nm}$ and are not equal to each
other. For example, the parameter of the Cphase operator acting on the qubits
$1\uparrow$ and $1\downarrow$ is determined by the coefficient $V_{1}$ of the
Hamiltonian (37), since the FSG operator also performs a fermionic SWAP
operation.

Figure 13: Implementation of Cphase($\phi$) gate on a digital-analog quantum
device. When constructing this circuit, it is assumed that the interaction
with other qubits is disabled. The interaction time cannot be negative, but
adding $2\pi$ to $t$ does not change the result. Hence the interaction time
can always be made positive.

The difference in rotation angles in Cphase gates leads to the difference in
interaction times needed to implement such operations using a digital-analog
strategy. This complicates the construction of the refocusing operation. In
order to describe the needed algorithm for this case, we have first to present
a similar algorithm for a set of CNOTs between chosen qubits. So, we will
discuss the construction of the quantum circuit for the simultaneous
implementation of the Cphase gates in Appendix D.

##  Appendix B Matrix H(m) as a generator of sequences of states

The matrix $H(m)$ provides instruction on implementing the refocusing
operation. Let us consider two qubits $p$ and $q$ that follow different
sequences of states corresponding to the refocusing operation having $m$ time
intervals. Also, consider the two columns $A_{p}$ and $A_{q}$ of the matrix
$A=H(m)$ that define these two distinct sequences. The elements of columns
$A_{p}$ and $A_{q}$ can be considered as elements of two orthogonal vectors

$\displaystyle\sum^{2^{m}}_{n=1}a_{np}a_{nq}=0;\qquad\forall
u,v\quad|a_{uv}|=1,$ (38)

where $a_{np}$ is the $n$ element of the vector $A_{p}$. The pairwise
orthogonality of the vectors $A_{p}$ follows from the properties of the matrix
of the Hadamard operator $H(m)$.

The action of $Z$ on a qubit in the odd state is equivalent to the action of
$-Z$ on a qubit in the even state: $Z=-XZX$. Then switching the qubit $i$ to
the odd state can be interpreted as changing the sign in the exponent in all
the operators $e^{-iZ_{p}Z_{q}}$ during the subsequent application of the
operator $J$, where $q$ is an index of other qubits participating in the
refocusing operation. Since all operators $J_{pq}$ of $J$ commute, the action
of the entire refocusing operation is equivalent to the action of the unitary
operator $e^{B}$, where the operator $B$ is the sum of all exponents of the
operators $J_{pq}$ participating in the refocusing operation:

$\displaystyle B=-\frac{i}{2}\sum_{p\neq q;i}b_{pqi}Z_{p}Z_{q}t_{i},$ (39)

where $b_{pqi}$ is determined by the state of the qubits $p$ and $q$ (odd or
even) on the interval $t_{i}$ of the refocusing operation. Namely, the
individual terms of this sum $B$ are the operators $iZ_{p}Z_{q}$ if one of the
qubits is in a odd state ($b_{pqi}=-1$), or $-iZ_{p}Z_{q}$ ($b_{pqi}=1$) for
all other cases. Now consider only the connection between qubits $p$ and $q$.
That is, consider the contribution to the operator $B$ defined as

$\displaystyle B_{pq}=-\frac{i}{2}\sum_{i}b_{pqi}Z_{p}Z_{q}t_{i}.$ (40)

Suppose that the qubits $p$ and $q$ follow different sequences of states that
define the vectors $A_{p}$ and $A_{q}$. Then $b_{pqi}=a_{ip}a_{iq}$, which
implies that

$\displaystyle\begin{gathered}B_{pq}=-\frac{i}{2}\sum_{i}a_{ip}a_{iq}Z_{p}Z_{q}t_{i}=\\\
-\frac{iT}{2^{m+1}}Z_{p}Z_{q}\sum_{i=1}^{2^{m}}a_{ip}a_{iq}=0,\end{gathered}$
(43)

where $T$ is the total refocusing operation time. Equation (43) is valid,
because $t_{i}=t_{j}$ for any $i$, $j$. The number of $t_{i}$ is even,
coinciding with the column length of this matrix. As a result, the interaction
during the above period will be compensated during all remaining time
intervals $t_{i}$. Thus, since any two vectors formed by the columns of the
matrix $H(m)$ are orthogonal, the interaction between all the qubits that
follow different sequences is compensated. However, if two qubits follow the
same sequence, they will be affected by the desired operator
$e^{-iZ_{p}Z_{q}T}$, where $T$ is the total time of the refocusing operation,
$p$ and $q$ are the indices of qubits that follow the same sequence.

##  Appendix C Number of sequences for an arbitrary connectivity topology

We have previously defined in Eq. (4) the minimum number of state sequences
for the all-to-all connectivity topology. This Appendix shows how to determine
this number for the general case.

First, let us construct the refocusing operation, which is equivalent to the
action of the identity operator $I$. In this case, no interaction pair exists.
This example is fundamental because it forms a basis for constructing any
refocusing operation. Consider the all-to-all connectivity topology of $k=m$
qubits. For implementing the refocusing operation, it is necessary to have $m$
sequences of states, according to Eq. (4). If we add an extra qubit to the
network, i.e., the qubit $k+1$, which has a connection to all qubits except
for the $1$’st qubit, then the number of sequences of states for implementing
the refocusing operation does not change. This is due to the fact that the
$k+1$’th qubit can repeat the parity states of the $1$’st qubit during all
time intervals of the refocusing operation. Let us add one extra qubit, which
again has connections to all qubits except the first one. To construct a
refocusing operation, it is already necessary to have $m+1$ sequences of
states, since the qubits from $2$ to $k+2$ form an all-to-all topology of
$k+1$ qubits, according to Eq. (4). From this analysis, we draw an important
conclusion that adding a qubit that does not change the size of the largest
graph with all-to-all connectivity does not increase the number of necessary
sequences of states.

Let us now consider a network of $k$ qubits with an arbitrary connectivity
topology. Without reducing the generality, we assume that the qubits from $1$
to $k^{\prime}$ form the largest all-to-all connectivity cluster. In order to
find the necessary number of sequences, we first consider this cluster and
then start adding the remaining qubits one by one according to the given
connectivity topology. It directly follows from our analysis that the number
of sequences does not change after each step. Thus, in the case of an
arbitrary connectivity topology, the required number of sequences of states is
determined by the size of the largest all-to-all connectivity cluster that it
contains.

Figure 14: Layout of the deformation of the connectivity topology due to the
interaction pair formed by the qubits $6$ and $7$. For clarity, this layout
does not show the scheme of interaction between the qubits $0$ – $5$, since it
does not change.

Next, we must consider interaction pairs. Since the qubits forming the
interaction pair follow the same sequence of states, each qubit from this pair
must have a different sequence of states, with all qubits interacting with the
qubits of the pair. The latter fact can be understood in terms of the
deformation of the initial topology: qubits belonging to the same pair must be
merged. Such a new node (a combined pair) has connections with all qubits with
which at least one qubit forming an interaction pair had a connection. The
layout of the deformation process is shown in Fig. 14. As a result of the
deformation, a new connectivity topology is formed. Determining the required
number of sequences of states for such systems without interaction pairs has
already been demonstrated above (a refocusing operation which is equivalent to
the action of the identity operator $I$). Thus we explained how to find a
required number of sequences of states to construct the refocusing operation
for the arbitrary connectivity topology.

In general, the process of simultaneous implementation of local entangling
gates (CNOTs, SWAPs, Cz, Cphase) consists of the following steps:

(i) identification of interaction pairs depending on the general algorithm we
wish to implement,

(ii) determination of the required number of sequences of states according to
the general rules of Appendix B (matrix $H(m)$), which depends on the
connectivity (dimensionality) of the problem,

(iii) construction of a quantum scheme implementing a refocusing operation
based on $H(m)$.

##  Appendix D Spin-1/2 fermionic Hamiltonian: Cphase gates with always-on
interaction

This Appendix demonstrates the possibility of implementing the Trotter step
for the Fermi-Hubbard Hamiltonian with spin $1/2$. According to the quantum
circuit shown in Fig. 12, for that, it is necessary to implement Cphase gates
as well as FSG layers. In Fig. 12, the parameters of the Cphase gate are
determined by the coefficients $V_{n}$ in the Hamiltonian (37), which in
general differ from each other. Hence for each interaction pair, the required
duration of the action of $H_{int}^{p,q}$, in the general case, is different
according to the circuit shown in Fig. 12. Note that this condition is similar
to the requirement in the previously described case of systems with the spread
of interaction parameters (Section IV), where each interaction pair existed
for a different time. In the case of the Cphase implementation, for each
interaction pair formed by the qubits that the Cphase gate acts on,
$H_{int}^{p,q}$ acts for a different time. However, the previously described
approach is also applicable in this case.

The parameters of each Cphase gate are determined by the Hamiltonian (37).
Hence, there is a Cphase operator for which the required interaction time is
maximum. Similar to the case of the chip with the spread of interaction
parameters, the total time of simultaneous implementation of all Cphase gates
is exactly this maximum time. The interaction pair with the minimum
interaction constant exists during all time intervals. Interaction pairs for
other Cphase gates require less time, so there is an excess time for each of
them. During this time, the interaction pair is destroyed because the qubits
form and follow different states’ sequences.

All the arguments described above are correct in the case of simultaneous
implementation of Cphase gates on a digital-analog quantum device with the
spread of interaction parameters. However, for each interaction pair, the
duration of interaction depends not only on the coefficients of the
Hamiltonian (37) but also on the parameter $\alpha_{pq}$, where $p$ and $q$
are the qubits forming the interaction pair.

Figure 15:  Two-dimensional qubit connectivity scheme for the quantum
simulation of spin-1/2 Fermi-Hubbard model in any dimension.

Let us consider the connectivity scheme shown in Fig. 15. Such a topology is
most suitable for simulating the evolution of the system described by the
Hamiltonian (37) according to the quantum scheme shown in Fig. 12. To
implement the algorithm described above, two sequences determined by the
matrix $H(2)$ are sufficient. This result is similar to the case of the odd
step shown in Fig. 3, where the interaction pair had a staggered pattern
order.

We now estimate the complexity of the implementation of the quantum circuit
shown in Fig. 12 in view of the connectivity scheme shown in Fig. 15. In the
case of digital-analog quantum devices with identical interaction parameters,
the Cphase gates implementation requires $\mathcal{O}(n)$ multi-qubit
operations. Implementing FSGs layers also requires $\mathcal{O}(n)$ multi-
qubit operations. Hence the circuit depth is $\mathcal{O}(n)$ gates. In the
case of a chip with the spread in the interaction parameters, the complexity
of the realization of Cphase gates does not change, as described above. The
implementation of FSGs layers requires $\mathcal{O}(n^{2})$ multi-qubit gates.
Hence the circuit depth, in this case, is $\mathcal{O}(n^{2})$.

In a particular case, when all coefficients $V_{n}$ are the same, there
appears a significant simplification in the case of a chip with identical
interaction parameters. The equality of all the coefficients of $V_{n}$
implies the same desired interaction time within each interaction pair. Thus,
the number of multi-quit operations for implementing Cphase gates grows as
$\mathcal{O}(1)$.