Dual Exponential Coupled Cluster Theory: Unitary Adaptation, Implementation in the Variational Quantum Eigensolver Framework and Pilot Applications
Dipanjali Halder^1, V. S. Prasannaa^2, Rahul Maitra^1,†
$^1$ Department of Chemistry,
Indian Institute of Technology Bombay, Powai, Mumbai 400076, India

$^2$ Centre for Quantum Engineering, Research and Education, TCG CREST, Salt Lake, Kolkata 700091, India

$^\dagger$<EMAIL_ADDRESS>
In this paper, we have developed a unitary variant of a double exponential coupled cluster
theory, which is capable of mimicking the effects of connected
excitations of arbitrarily high rank, using only  rank-one and rank-two parametrization of the wavefunction ansätz.
While its implementation in a classical computer necessitates the construction of an
effective Hamiltonian which involves infinite number of terms with arbitrarily high
many-body rank, the same can easily be implemented in the hybrid quantum-classical
variational quantum eigensolver framework with a reasonably shallow quantum circuit.
The method relies upon the nontrivial action of a unitary, containing a set of rank-two
scattering operators, on entangled states generated via cluster operators. We have
further introduced a number of variants of the ansätz with different degrees of
expressibility by judiciously approximating the scattering operators. With a number of applications on strongly correlated molecules, we have shown that all our schemes can
perform uniformly well throughout the molecular potential energy surface without
significant additional implementation cost and quantum complexity over the unitary
coupled cluster approach with single and double excitations.

§ INTRODUCTION

The emerging field of quantum computation has gained substantial attention due to its promise to solve certain computational problems that are difficult to handle with classical computers [1, 2, 3, 4]. Simulation of many body quantum systems is considered to be one such strenuous task owing to the exponential overhead of computational resources, thus limiting its application to small chemical systems. Unlike exact classical methods like full configuration interaction (FCI) that scales exponentially with system size, quantum algorithms such as quantum phase estimation (QPE) can tackle such problems with polynomial overhead. An overview of the developments in this field can be found in relevant review articles  [5, 6].

Earlier quantum computational algorithms to simulate molecular energies were based on the QPE  approach as envisioned by Abrams and Lloyd [7, 8]. The objective of the phase estimation approach is to extract eigenvalues of a unitary operator by projecting the input state onto an eigenstate, provided the input state has substantial overlap with the exact eigenstate of the unitary operator. In 2005, Aspuru Guzik et al. [9] first employed the phase estimation algorithm to simulate ground state energies of a few molecular systems. Eventually, Hefeng Wang and co-workers estimated excited state energies employing multiconfigurational self-consistent field (MCSCF) wavefunction in the framework of QPE [10]. Apart from these papers, there have been numerous works in the QPE literature which deal with techniques for preparation of initial states suitable for the phase estimation approach, for example, see Ref. [11, 12, 13]. Although it offers an exponential speedup over the exact classical algorithms, QPE fails to meet the expectations of the near term noisy intermediate scale quantum (NISQ) devices owing to the requirement of long coherence times.

To circumvent this problem, a hybrid classical-quantum approach,
namely variational quantum eigensolver (VQE) was proposed by
Peruzzo and co-workers [14]. The idea stems from
the Rayleigh-Ritz variational principle [15], which dates back to early
1900s. VQE utilizes a combination of both
classical and quantum architectures to find the best variational
approximation to the ground state of a given Hamiltonian
corresponding to a chosen trial wavefunction ansätz. Till date, VQE has been
experimentally implemented in various quantum hardware
architectures, viz. photonic quantum
processors [14],
superconducting quantum processors [16, 17],
trapped ion architectures [18, 19, 20] etc. It is worth
mentioning at this point that the selection of a suitable
wavefunction ansätz plays a pivotal role in the performance
of VQE. In recent times, there have been a number of developments
to formulate an expressive and cost efficient ansätz for VQE. An extensively used ansätz is the chemically
motivated unitary coupled cluster ansätz, abbreviated as UCC [21, 22, 14, 23, 24, 25]. A generalized extension of the
unitary coupled cluster ansätz truncated at singles and
doubles excitations (UCCGSD), inspired from the earlier works
of Nakatsuji [26] and Nooijen [27], has been studied by Van Voorhis and Head-Gordon [28], and further implemented in the VQE framework by Lee et
al. [29]. UCCGSD is considered to be the most expressive ansätz
to date, but the expressibility comes at the expense of much higher scaling in terms of
the number of parameters and the circuit depth (in comparison to
UCCSD). In the same paper, Lee and co-workers have further
proposed an ansätz, termed as UpCCGSD, which is analogous
to the UCCGSD ansätz, except for an added constraint of
inclusion of only generalized paired excitations in
the definition of two-body operators. UpCCGSD has been found to
scale linearly against the system size, thus leading to a synergy
between efficiency (in terms of cost of implementation),
expressibility and the accuracy. The authors further showed that
one may reach chemical accuracy across the potential
energy surface of strongly correlated molecules with UpCCGSD
ansätz by repeating the UpCCGSD circuit $k$ number of times.
This leads to $k$-fold increase of the circuit depth which
anyway scales linearly against the system size. The same train of thought has led to the development of multireference unitary coupled cluster with partially generalized singles and doubles (MR-UCCpGSD) ansätz [30] by Sugisaki and co-workers. Based on a
slightly different idea, Grimsley et al. [31] introduced a systematically expandable  ansätz, the adaptive derivative-assembled pseudo-Trotter variational quantum eigensolver, abbreviated as ADAPT-VQE. The underlying idea behind such an ansätz is to extract most of the correlation energy with the lowest possible number of variational parameters (or fermionic excitation operators).
A complementary approach for enhancing  the efficiency (from the quantum complexity reduction perspective) of the VQE algorithm involves the implementation of a downfolded Hamiltonian [32, 33] or a rotated Hamiltonian [34, 35]. Few more notable works in this direction includes Unitary Cluster Jastrow (uCJ) ansätz [36], low rank representations [37], which are known to have a quadratic scaling of the parameters against the system size.
Apart from the aforementioned developments in the context of traditional VQE, Stair et. al. [38] introduced a variant of VQE, namely Projective Eigensolver algorithm (abbreviated as PQE). Based on the projective techniques used for solving traditional coupled cluster equations, PQE is known to carry out optimization through the residue calculations (on a quantum hardware), unlike the energy gradient calculations employed in traditional VQE.

While the success of VQE depends on the choice
of parameters in the UCC ansätz, in recent times, there is an increasing interest in the
many-body electronic structure community to explore the
possibility of writing an $N-$electron correlated
wavefunction with lower rank parametrization. One of the present authors had worked in this direction, and introduced a double exponential
waveoperator based coupled cluster theory, which implicitly
incorporates the effects of connected rank-three and rank-four
excitations with only rank-one and -two parametrization,
in a computationally inexpensive manner [39, 40]. In the said theory,
termed as iterative n-body excitation inclusive CCSD (iCCSDn),
the authors had introduced a set of rank-two scattering
operators on top of the conventional single and double
excitation operators. iCCSDn relies heavily on
the non-commutativity of these two sets of operators,
and the connected high rank excitations are simulated through
the tensor contractions between them.

With the advent of VQE algorithm for the calculations
of atomic and molecular ground state energetics with a variety
of unitary CC ansätze, it is compelling to verify the efficacy
and accuracy of the iCCSDn methodology on a quantum architecture.
As the first step
towards achieving this, we develop a unitarized version of
iCCSDn (UiCCSDn) theory and a number of variants thereof,
and implement those on a quantum
simulator for the molecular ground state energy evaluation.
The various UiCCSDn variants seem to be extremely precise
over the entire range of potential energy surface without a significant increase in the gate count as compared to the conventional UCCSD. This
point will be elaborated towards the end of the theory section
and also in the results and discussion
section in the later part of the manuscript.

The paper is organized as follows: we discuss in section
<ref> the general VQE algorithm, with emphasis on the
chemically motivated unitary (generalized) coupled cluster
ansätz. This would be followed by a brief discussion on
the conventional double exponential ansätz, iCCSDn, in
section  <ref>. In section  <ref>, we
introduce the unitarized version of the aforementioned ansätz
and its various approximations along with their efficiency of
implementation in the context of VQE. We will compare
and contrast various aspects of (U)iCCSDn in its classical and
quantum implementation. Finally, we present our results in
section  <ref>, where we would study the potential
energy surface of a few strongly correlated molecular systems
and demonstrate that our methods can handle electronic states
with varied complexity with uniform accuracy. Finally, we
conclude in section <ref> along with a road-map to
future directions.

§ THEORY

§.§ The variational quantum eigensolver algorithm with unitary (generalized) coupled cluster ansätz

The hybrid classical-quantum VQE algorithm uses a combination of both classical and quantum computation to find an upper bound to the ground state energy associated with a given
Hamiltonian, a problem central to the field of quantum chemistry. It is based on the
variational principle, and hence the ground state energy functional can be written as,

\begin{eqnarray}
E(\theta) &=& \frac{\langle \Psi(\theta)|H|\Psi(\theta) \rangle}{\langle \Psi(\theta)|\Psi(\theta) \rangle} \nonumber \\
&=& \langle \Psi_{HF}|U(\theta)^\dag HU(\theta)|\Psi_{HF} \rangle, \label{eqE}
\end{eqnarray}

where $|\Psi(\theta) \rangle$ is a parameterized trial wavefunction, $|\Psi_{HF} \rangle$ is a suitable many-body reference state, which is usually the Hartree-Fock (HF) state, and H is the electronic Hamiltonian.
The algorithm is divided into two subparts:
the first subpart uses a quantum computer for preparation of a parameterized wavefunction or
the ansätz and measurement of the expectation value of the Hamiltonian. And the second
subpart comprises of a classical optimization algorithm that minimizes the measured energy
and updates the variational parameters for ansätz preparation. Thus, we can write,

\begin{eqnarray}
E_{0} = \mathrm{min}_{\theta} \langle \Psi_{HF} | U(\theta)^\dag H U(\theta) | \Psi_{HF} \rangle, \label{eqE1}
\end{eqnarray}

where $U(\theta)$ is a unitary parametrized operator.
The many-body molecular Hamiltonian, H, can be represented in second quantized form as shown in Eq. (<ref>),

\begin{eqnarray}
\label{hamiltonian}
\end{eqnarray}

One of the crucial components of VQE is the choice of the parametrized ansätz.
Till date, various types of ansätze have been explored in the VQE literature. A few important things to keep in mind while selecting an ansätz are:
it should be able to span the N-electron Hilbert space,  and the circuit depth
corresponding to the ansätz should not be beyond the reach of near term NISQ devices.
One widely used ansätz is the chemically motivated UCC
ansätz. It uses an unitary exponential parametrized wave operator, $e^X$ to span the
$N$-electron Hilbert space. There have been a number of developments along this line to
judiciously choose $X$. In the traditional UCC method, the operator is
chosen to be the sum of an excitation operator, X (defined with respect to a
Fermi vacuum, usually chosen as the HF determinant) and its de-excitation counterpart, $X^{\dag}$.
The operator $X$ (and $X^\dagger$) is usually truncated with rank-one and rank-two
operators, giving rise to the well known UCCSD ansätz:
On the other hand, the operator $X$ may be chosen to be agnostic to the hole and
particle orbitals, and one may define $X$ to be an anti-hermitian sum of
generalized rank-one and rank-two operators. This is known as the UCCGSD ansätz in the

In the most general form,

\begin{eqnarray}\label{uccop}
\ket{\Psi(\theta)}_{UCC(G)SD}&=&e^{{{X}({\theta})}-{X}^{\dagger}({\theta})}\ket{\Psi_{HF}}; \label{excop}
\end{eqnarray}

where $X$ is the cluster operator,

\begin{eqnarray}
X &=& X_{1}+X_{2},
\end{eqnarray}

\begin{eqnarray}
{X_{1}}({\theta})&=&\sum \limits_{p;q}\theta_{p}^{q}a_{q}^{\dagger}a_{p};\label{t1}\\
{X_{2}}({\theta})&=&\sum \limits_{p,r;q,s}\theta_{pr}^{qs}a_{s}^{\dagger}a_{q}^{\dagger}a_{r}a_{p}. \label{t2}
\end{eqnarray}

Here, the subscripts $p,q,r,s$ refer to general spinorbital indices for UCCGSD. If one
restricts the orbitals $p,r$ to occupied orbitals $i,j$ and $q,s$ to the unoccupied
orbitals $a,b$, the UCCGSD ansätz reduces to  the UCCSD  one, with $X_1(\theta) \equiv T_1$, $X_2(\theta) \equiv T_2$, and $T = T_1 + T_2$. An efficient variant of the UCCGSD
asnsatz was proposed by Lee et. al. where the authors had included only generalized
pair double excitations where two electrons are “excited" from a given spatial orbital to
another spatial orbital. That implies that any combination $p,r$ from Eq. (7) have the same spatial orbital and so do their corresponding
$q,s$. The resulting Up-CCGSD ansätz has linear circuit depth with respect to the number
of spinorbitals, and is thus considered to be the current state of the art. Note that the
$\theta_{p}^{q}$ and $\theta_{pr}^{qs}$ are the VQE parameters, which are iteratively
updated through an appropriate optimizer.

In the following sections, we first present the parent iCCSDn theory from a many-body theoretic
perspective and discuss its various advantages and disadvantages in its implementation on a
classical computer. This would set the stage for the next section,  where we
would develop the unitary version of the theory and show that the unitarized version
is perfectly suited to be implemented on a quantum computer.

§.§ The double exponential ansätz based conventional iCCSDn

In the conventional iCCSDn theory, one aims to simulate the high rank correlation
effects (dominantly triples and quadruples) through a sequential similarity transformation
with the exponentiated rank one and rank two operators.
Towards this, one introduces a set of rank-two scattering operators, $S$, whose structure
contains a true excitation in one of the vertices and a hole-hole/particle-particle
scattering in another vertex. The scattering vertex contains a single (quasi-) hole or
(quasi-) particle type one-electron state destruction operator. Depending on whether a hole type
or a particle type of orbital that appears as the destruction operator in the scattering
vertex of $S$, one may classify it as $S_h$ and $S_p$.

\begin{eqnarray}
S_h &=& \frac{1}{2} \sum_{amij} s^{am}_{ij} \{a^\dagger m^\dagger ji\}; \nonumber \\
S_p &=& \frac{1}{2} \sum_{abie} s^{ab}_{ie} \{a^\dagger b^\dagger ei\} \mathrm{\ \ and} \nonumber \\
S &=& S_h + S_p.
\label{eq4}
\end{eqnarray}

Here, a, b, c, ... etc. denote the particle orbitals
and i, j, k, ..., etc. are the hole orbitals with
respect to HF determinant taken as the Fermi vacuum. In the second quantized
representation of the $S$ operators, the hole state $m$ in $S_h$ and
the particle state $e$ in $S_p$ are the ones that appear as the
destruction operators. Note that in the conventional CC theory with HF determinant
taken as the Fermi vacuum, all the quasi-particle orbitals (e.g. a, b, c, ...) and quasi-hole orbitals (e.g. i, j, k, ...) appear
as the creation operators. In order
to distinguish the hole/particle states that appear as the destruction operator
in $S$, we have used different symbols for them.

The $S$ operators, due to the presence of the destruction operators, annihilate the
HF reference determinant: $S|\Psi_{HF}\rangle=0$. This is to be referred as the
killer condition. Furthermore, the $S$ operators do not commute among
themselves, neither do they commute with the cluster operators $T$. However, they
have non-trivial action on an excited determinant where a given hole or particle
state is created by the preceding action of an $T$ operator. In fact, the non-commutativity
between the $S$ and the $T$ operators are the key to simulate the connected
high-rank excitations. In order to construct a terminating series of the effective
Hamiltonian, in the conventional many-body formalism of iCCSDn, the $S$ operators are
not allowed to contract among themselves.

Due to the killer condition and the non-commutativity of the $S$ and $T$ operators,
one may write down a double exponential parametrized ansätz:

\begin{equation}
%\Omega = \{\exp(S)\}\exp(T_1 + T_2)
\Omega = \{e^S\}e^{T_1 + T_2},
\label{ansatzeq}
\end{equation}

where the action of the two sets of exponential operators are fixed, This
is to be distinguished with an ansätz like $e^{\{S\}+T}$, where the terms appearing
in the exponent can appear in any order. Note that the $\{...\}$ denotes
the normal ordering to avoid the $S-S$ contraction to ensure a naturally terminating
effective Hamiltonian structure. In order to have a non-trivial coupling among $S$ and $T$ operators,
the destruction operator in $S$ should necessarily get
contracted with one of the creation operators present in $T$.

The solutions for the optimized $s$ and $t$ amplitudes (corresponding to $S$ and $T$
operators, respectively) are done in a sequentially coupled manner. One defines the
first similarity transformed effective Hamiltonian, $W$ such that

\begin{equation}
%\{\exp(S)\} W = H \{\exp(S)\}
\{e^S\} W = H \{e^S\}
\label{eqxx4}
\end{equation}

is satisfied. Since the inverse operation of a normal ordered ansätz is not
explicitly defined, one may determine $W$ with a recursive substitution technique.
The resulting structure of $W$ takes the form:

\begin{eqnarray}
W=\{\contraction{}{H}{}{\exp(S)} H \exp(S)\}  - \{\contraction[2ex]
{}{(\exp(S)-1)}{}{\contraction{}{H}{}{\exp(S)} H \exp(S)} (\exp(S)-1)
\contraction{}{H}{}{\exp(S)} H \exp(S)\} + \nonumber \\
\{\contraction[3ex]{}{(\exp(S)-1)}{}{\contraction[2ex]
{}{(\exp(S)-1)}{}{\contraction{}{H}{}{\exp(S)} H \exp(S)} (\exp(S)-1)
\contraction{}{H}{}{\exp(S)} H \exp(S)} (\exp(S)-1) \contraction[2ex]
{}{(\exp(S)-1)}{}{\contraction{}{H}{}{\exp(S)} H \exp(S)} (\exp(S)-1)
\contraction{}{H}{}{\exp(S)} H \exp(S) \}  - \cdots .
\label{eqxx6}
\end{eqnarray}

This is an effective operator containing many-body terms, where usually
only the rank-one and rank-two terms are retained and the series is restricted
till the second term in the right hand side. With a judiciously truncated $W$, one
may employ the second set of similarity transformation to construct double
similarity transformed effective Hamiltonian $H^{eff}=e^{-T}We^T$. Note that
in the double similarity transformed effective Hamiltonian, $H^{eff}$, both the
$T$ and $S$ operators get coupled. Their amplitudes are determined by
employing a many-body expansion and demanding that the corresponding amplitudes
of $H^{eff}$ vanish.

\begin{equation}
(h^{eff})^{a}_{i} = (h^{eff})^{ab}_{ij} = (h^{eff})^{am}_{ij} = (h^{eff})^{ab}_{ie}
= 0.
\label{eqxx18}
\end{equation}

Note that in the construction of $H^{eff}$, the $s$ and the $t$ amplitudes are
iteratively optimized in a coupled manner, and as such the high rank correlation
effects are simulated through the contraction between them: $T_{ijk}^{abc} \leftarrow S_{ij}^{am}T_{mk}^{bc} + S_{ie}^{ab}T_{jk}^{ec}$.

§.§ Unitary version of double exponential CC ansätz

§.§.§ Motivation towards the development of the unitarized ansätz

While the iCCSDn methodology is shown to be quite accurate, particularly for weakly
to moderately strong correlated systems, there are a few drawbacks that it suffers
from in its implementation on a classical computer.

Schematic illustration of the UiCCSDn scheme for an $n$-qubit example system. Note that each of the two circuits shown in the figure is a representative illustration of an excitation (the first block) and scattering (the second block), and does not reflect an actual full circuit, which is much deeper.

* The expansion in Eq. (<ref>) (with a normal ordered ansätz as shown) is exact only up to
leading order, but is naturally terminating. However, in the practical implementation, the series is usually further truncated retaining only a subset of the leading terms. That leaves out a large number of terms that otherwise could effectively contribute to the amplitude equations, Eq. (<ref>). Furthermore, the rank of the terms appearing in the effective Hamiltonian, $W$
is truncated to one and two-body that essentially leaves out the effects of high
rank (beyond triples) excitations. Thus the ansätz with a normal ordered exponential
structure has less span of the $N-$electron Hilbert space, compared to the
one without the normal ordering. We must emphasise that any such
theory based on many-body transformation would suffer from the unavoidable drawback.

* While it is absolutely necessary that one adopts normal ordering and eliminates the $S-S$ contractions
in order to generate a terminating series of the effective Hamiltonian, this leaves
out the clustering effects, which would otherwise be present in a conventional (full) exponential structure with $S$.

* The conventional iCCSDn, in practice, relies on the
choice of a set of chemically active orbitals. The contractible (destruction)
operators ($m$ and $e$) are restricted to only those
active orbitals. The cost associated with the determination of the $s-$amplitudes is usually dictated by the construction of the $(h^{eff})_{ie}^{ab}$ residue, which formally scales as $n_on_v^4n_v^{act}$. Here, $n_o$ is the number of electrons, $n_v=(N-n_o)$ is the number of virtual spinorbitals and $N$ is the total number of spinorbitals.
designate the number of chemically active virtual spinorbitals (the dimension of $e$).
This, in practice, is less than the $n_o^2n_v^4$ which is the typical scaling
of CCSD, and thus the most expensive step is still dictated by the conventional CCSD
terms. One may, however, choose to expand the choice of the active space to include
more occupied and/or virtual orbitals to include more complete description of
the connected triple excitations at the cost of an increased computational scaling.
Note that in iCCSDn,
the $T$ and the $S$ operators are simulated through a set of local denominators, which can be perturbatively estimated by solving $[H_0, S^{(1)}] + V = 0$. Solution to this results in the following perturbative structure of the $s-$amplitudes:

\begin{equation}
\begin{split}
(s_p)^{ab}_{ie} + v^{ab}_{ie} & =0 \\
\Rightarrow (s_p)^{ab}_{ie} =
v^{ab}_{ie}/(\epsilon_{i}+\epsilon_{e}-\epsilon_{a}- \epsilon_{b})
\label{eqxx19}
\end{split}
\end{equation}

\begin{equation}
\epsilon_{a}-\epsilon_{m}),
\label{eqyy19}
\end{equation}

where $\epsilon_p$ is the orbital energy of the $p^{th}$ spinorbital.
Thus the local energy denominator involves energies of only those orbitals
that appear in a given $S$ operator, and is equal to the energy difference between the
doubly excited determinant and the triply excited determinant. While inclusion of a
large number of `active' orbitals generally improves the quality of the
results, it also makes the theory more prone to intruders, particularly in the
regions of molecular strong correlation. Thus, there is a delicate balance
between the accuracy and computational scaling that one needs to maintain
in order to solve the energetics of molecular systems in an uniform manner.

In order to overcome the potential drawbacks in its implementation in a classical
computer as described above, it is warranted that
the full exponential structure of $\exp(S)$ is utilized such that the $S$ operators
are allowed to contract among themselves. However, in such a case, the many-body
first similarity transformed effective Hamiltonian $W$ would
generate exponentially large number of algebraic terms with all possible many-body ranks,
which are intractable to handle on a classical computer. Also the amplitudes during the
optimization process are likely to be highly
plagued by the intruders, as described in point 3 above. Thus, rather
than taking the many-body route, one may evolve starting from
the HF reference determinant in a quantum device by means of
a sequential evolution through a fixed structure double unitary
ansätz and variationally optimize the energy to get the updated set of parameters.

§.§.§ Development of the unitarized iCCSDn ansätz and implementation

Towards the development of a unitary variant of iCCSDn (to be referred to as UiCCSDn
henceforth) for its implementation in the VQE framework, the normal ordering of
$e^S$ is removed by the conventional exponential structure and the $S$ operators are
replaced by an anti-Hermitian sum of $S$ and $S^\dagger$. Similarly, the $T$ operators are
replaced by the anti-Hermitian sum of the excitation
operator ($T$ itself) and the de-excitation operators ($T^\dagger$):

\begin{eqnarray}
& S_h \rightarrow S_h - S_h^\dagger = \sigma_h : {(\sigma_h)}_{ij}^{am} = {(S_h)}_{ij}^{am} - {(S_h)}_{am}^{ij} \nonumber \\
& S_p \rightarrow S_p - S_p^\dagger = \sigma_p : {(\sigma_p)}_{ie}^{ab} = {(S_p)}_{ie}^{ab} - {(S_p)}_{ab}^{ie} \nonumber \\
& T_2 \rightarrow T_2 - T_2^\dagger = \tau_2 : \tau_{ij}^{ab} = T_{ij}^{ab} - T_{ab}^{ij} \nonumber \\
& T_1 \rightarrow T_1 - T_1^\dagger = \tau_1 : \tau_{i}^{a} = T_{i}^{a} - T_{a}^{i}.
\end{eqnarray}

Thus, in Eq. (<ref>), both the exponential terms can be replaced by their
unitary counterparts:

\begin{equation}
U = U_1 U_2 = \exp{(\sigma)} \exp{(\tau)},
\end{equation}

where $\sigma = \sigma_h + \sigma_p$ and $\tau=\tau_1 + \tau_2$. Note that due to the
exponential structure, the effective Hamiltonian can in principle be expanded in the
Baker-Campbell-Hausdorff expansion to obtain normalized Hamiltonian expectation value
with variational optimization. However, due to the contraction among various components
of the ansätz, the BCH expansion is non-terminating, making it intractable on a  classical computer.

The unitary version of iCCSDn may be employed to study the coherent time evolution
in a quantum computer. Towards this, the ansätz is broken down into several time
ordered sequences of few-particle operators using Trotter expansion. One may note
that the $S$ and the $T$ operators do not commute in general. Since we have started
with a double unitary structure and the $\sigma$ and $\tau$ operators do not commute,
$e^\sigma e^\tau \ne e^{\sigma+\tau}$. However, from each
set of operators, one may group terms which are mutually commuting such that

\begin{eqnarray}
%&    e^{\sigma_A + \sigma_B} e^{\tau_A + \tau_B} = \nonumber \\ & \left(\lim_{\rho\to \infty} (e^{\sigma_A/\rho}e^{\sigma_B/\rho})^\rho\right) \left( \lim_{\lambda\to \infty} (e^{\tau_A/\lambda}e^{\tau_B/\lambda})^\lambda\right)
e^{\sigma_A + \sigma_B} e^{\tau_A + \tau_B} =   \left(\lim_{\rho\to \infty} (e^{\sigma_A/\rho}e^{\sigma_B/\rho})^\rho\right)\times \nonumber \\ \left( \lim_{\lambda\to \infty} (e^{\tau_A/\lambda}e^{\tau_B/\lambda})^\lambda\right).
\end{eqnarray}

Although a truncated Trotter expansion of the set of non-commuting operators
is an approximation in the UiCCSDn wavefunction, the choice of the operators
provide us with enough variational flexibility to simulate strongly correlated
systems. Even with $\rho=\lambda=1$
and a suitably approximated $\sigma$ operator, it can reproduce results which
are well within the limit of chemical accuracy, irrespective of the degree
of correlation of the system. The Trotterized ansätz thus takes a form:

\begin{equation}
| \Psi_{UiCCSDn}\rangle = \left( \prod_{q\in\{q_h, q_p\}} e^{\sigma_q} \prod_{r \in \{q_1, q_2\}} e^{\tau_r}\right)|\Psi_{HF}\rangle.
\end{equation}

Here, $q_h$ and $q_p$ are the unique quartet of all $(i,j,a,m)$ and $(i,e,a,b)$
respectively, and $r$ takes the unique combinations of all singles and doubles.
This is realized through a concatenated quantum circuit where $e^\tau$ (with
selection of $\tau$ operators in lexical ordering) acts on the HF state first,
followed by $e^{\sigma_h}$ and $e^{\sigma_p}$
(with selection of the $\sigma_h$ and $\sigma_p$ operators in lexical
ordering among themselves). One may further increase the variational flexibility
of the ansätz by independently taking a product of $k$ and $l$ number of $e^{\sigma}$
and $e^{\tau}$ unitary circuits, respectively. The ansätz thus takes a generalized form:

\begin{eqnarray}
%&   |\Psi_{UiCCSDn}\rangle =\nonumber \\
%&\left(\prod_{\alpha=1}^k \prod_{q\in\{q_h, q_p\}} e^{\sigma_q^{(\alpha)}}\right)\left( \prod_{\beta=1}^l \prod_{r \in \{q_1, q_2\}} e^{\tau_r^{(\beta)}}\right)|\psi_{HF}\rangle. \nonumber
\prod_{q\in\{q_h, q_p\}} e^{\sigma_q^{(\alpha)}}\right)\times \nonumber\\
\left( \prod_{\beta=1}^l \prod_{r \in \{q_1, q_2\}} e^{\tau_r^{(\beta)}}\right)|\Psi_{HF}\rangle.
\end{eqnarray}

Here $\sigma_q^{(\alpha)}$ and $\tau_r^{(\beta)}$ are treated as independent
parameters. We will show that for a few specific variants, the circuit for
$e^{\sigma}$ is quite shallow (vide infra), allowing us to
increase the value of $k$ (keeping $l=1$) without significant increase in the
circuit depth. A schematic representation of the quantum circuit is shown in Fig.
<ref> below.

One may note that choosing the ansätz through a double unitary has an added
advantage over a single unitary where one may write the unitary evolution
operator as $e^{\sigma + \tau}$. In the first order Trotter
decomposition, although both ways of writing the ansätz may superficially
look similar, in our double unitary case, the order of the action of
$e^\sigma$ and $e^\tau$ are fixed, contrary to a
single unitary. For the single unitary ansätz like $e^{\sigma + \tau}$,
one needs to optimize the order of appearance of various terms in the circuit.
Typically, those operators should be allowed to act on the HF state first which do not
annihilate the reference. In our case, $e^\sigma$, by construction, is allowed
to act on the entangled state, $e^\tau | \Psi_{HF} \rangle$, It was found that the energy depends on the ordering of the operators even among the two separate circuits; however, in
the current study both $\tau$ and $\sigma$ operators are
taken in their lexical ordering.

One may also note that our ansätz allows us to repeat a
part of the whole circuit to improve accuracy in a
systematic manner.

Variants      Choice of $\sigma$     Description

UiCCSDn      $\{{(\sigma_h)}_{ij}^{am}; (\sigma_p)_{ie}^{ab}\}$
1. Untruncated.

UiCCSDn-A      $\{{(\sigma_h)}_{ij}^{au}; (\sigma_p)_{iv}^{ab}\}$
1. Active orbitals.

UiCCSDn-A-op      $\{{(\sigma_h)}_{i_{\alpha}j_{\beta}}^{a_{\alpha}u_{\beta}},
\}$

1. Active orbitals.
2. Opposite spin in scattering and excitation vertex of $\sigma$.

UiCCSDn-A-diag     $ \{{(\sigma_h)}_{iu}^{au}; {(\sigma_p)}_{iv}^{av}\} $

1. Active orbitals.
2. Diagonal terms in scattering vertex of $\sigma$.

\{{(\sigma_h)}_{i_{\alpha}i_{\beta}}^{a_{\alpha}u_{\beta}},
\}$

1. Active orbitals.
2. Same spatial orbitals for quasi-hole (in $\sigma_h$) and quasi-particle (in $\sigma_p$).

Summary of various approximate UiCCSDn schemes depending on the choice of the $\sigma$ operators. The cluster operator $\tau$ is taken to be all $1h-1p$ and $2h-2p$ excitations without any further truncation.

§.§.§ Various approximate UiCCSDn schemes and the associated resource estimation

Ansätz      $T_{1}$     $T_{2}$     $T_{3}$     $S_{h}$     $S_{p}$     Gate depth

UCCSD     $\mathcal{O}(n_{o}n_{v})$     lightgray$\mathcal{O}(n_{o}^{2}n_{v}^{2})$     -     -     -     $\mathcal{O}(n_{o}n_{v}^{2})$

UCCSDT     $\mathcal{O}(n_{o}n_{v})$     $\mathcal{O}(n_{o}^{2}n_{v}^{2})$     lightgray$\mathcal{O}(n_{o}^{3}n_{v}^{3})$     -     -      $\mathcal{O}(n_{o}^{2}n_{v}^{3})$

UiCCSDn     $\mathcal{O}(n_{o}n_{v})$     $\mathcal{O}(n_{o}^{2}n_{v}^{2})$     -     $\mathcal{O}(n_{o}^{3}n_{v})$      lightgray$\mathcal{O}(n_{o}n_{v}^{3})$     $\mathcal{O}(n_{v}^{3})$

UiCCSDn-A     $\mathcal{O}(n_{o}n_{v})$     lightgray$\mathcal{O}(n_{o}^{2}n_{v}^{2})$     -     $\mathcal{O}(n_{o}^{2}n_{v}n_{o}^{act})$      $\mathcal{O}(n_{o}n_{v}^{2}n_{v}^{act})$     $\mathcal{O}(n_{o}n_{v}^{2})$

UiCCSDn-A-op     $\mathcal{O}(n_{o}n_{v})$     lightgray$\mathcal{O}(n_{o}^{2}n_{v}^{2})$     -     $\mathcal{O}(\frac{1}{16}n_{o}^{2}n_{v}n_{o}^{act})$      $\mathcal{O}(\frac{1}{16}n_{o}n_{v}^{2}n_{v}^{act})$     $\mathcal{O}(n_{o}n_{v}^{2})$

UiCCSDn-A-diag     $\mathcal{O}(n_{o}n_{v})$     lightgray$\mathcal{O}(n_{o}^{2}n_{v}^{2})$     -     $\mathcal{O}(n_{o}n_{v}n_{o}^{act})$      $\mathcal{O}(n_{o}n_{v}n_{v}^{act})$     $\mathcal{O}(n_{o}n_{v}^{2})$

UiCCSDn-A-pp     $\mathcal{O}(n_{o}n_{v})$     lightgray$\mathcal{O}(n_{o}^{2}n_{v}^{2})$     -     $\mathcal{O}(\frac{1}{8}n_{o}n_{v}n_{o}^{act})$      $\mathcal{O}(\frac{1}{8}n_{o}n_{v}n_{v}^{act})$       $\mathcal{O}(n_{o}n_{v}^{2})$

The number of parameters (and gates) required for each class of operators for the implementation of UCCSD, UCCSDT and various UiCCSDn variants. The leading gate count for each of the ansätz are highlighted and put in the bold text. Also the corresponding leading gate depth is reported.
Note that UiCCSDn variants have much fewer number of gates required than UCCSDT.

In this section, we will introduce a number of variants of UiCCSDn ansätz, in which we
will judiciously approximate the $S$ operators such that the resulting ansätze
strikes the right balance between expressibility, accuracy, number of
parameters and circuit depth. We will also justify our choice from the
perspective of many-body perturbation theory. Noting the fact that our scheme is
not agnostic to the choice of holes and particles, we comment that the introduction
of an operator with structures like that of $S$ is the minimal
requirement to introduce connected high-rank correlation effects beyond singles
and doubles, and thus the resulting ansätz is expected to bring in better span of the Hilbert space. Furthermore,
one may significantly cut down the number of free parameters (and gate count)
by restricting the contractible orbitals (which appears as the quasi-hole/particle
destruction operator) to some chemically active ones. We introduce the
variants below which differ from each other in the
choice of the scattering component in $\sigma$. Please note that in all the
variants, the hole-particle structure of $\tau$ is retained in full.

* UiCCSDn: In the parent UiCCSDn theory, both the $\sigma$ and $\tau$
operators are retained in full and there is no further approximation involved.
That implies that the contractible set of orbitals
$m$ and $e$ in $\sigma_h$ and $\sigma_p$ respectively are taken to be all
the occupied and unoccupied orbitals. Implementation of the parent (untruncated)
UiCCSDn requires $\mathcal{O}(n_{o}^{3}n_{v})$ $\sigma_h$ and
$\mathcal{O}(n_{o}n_{v}^{3})$ $\sigma_p$ parameters on top of the conventional
$\mathcal{O}(n_{o}^2n_{v}^{2})$ elements of $\tau$. Note that the leading order of
parameter count and gate count are dictated by the number of $\sigma_p$ elements,
while the circuit depth scales as $\mathcal{O}(n_{v}^{3})$.

* UiCCSDn-A: In this variant, the contractible (hole and particle)
orbital indices are restricted to the chemically active orbitals. Since in our ansätz,
the high-rank correlation is simulated through the contraction of the $S$ and the $T$
operators, we span the full $N-$electron Hilbert space through only those $S$ operators
which involve contractible orbitals near the Fermi level. Note that in this variant,
there is no loss in the expressibility of the ansätz as it spans the full $N-$electron
Hilbert space; however, they are generated through a selected contraction of the most
dominant $\sigma$ with the $\tau$ amplitudes.

\begin{eqnarray}\label{Sactive}
\sigma_{\scaleto{UiCCSDn-A}{3.6pt}} \in \{{(\sigma_h)}_{ij}^{au}; (\sigma_p)_{iv}^{ab}\}
\end{eqnarray}

where $u$ and $v$ are the “active" hole and particle orbital lines, respectively.
Noting the fact that $n_o^{act}, n_v^{act} << n_o, n_v$, the parameter count is
$\mathcal{O}(n_{o}^2n_{v}^{2})$ due to $\tau$. The circuit depth scales

* UiCCSDn-A-op: On top of the selection of the active orbitals as
the contractible indices, we only incorporate those $\sigma$ operators in which
the spins in the excitation vertex and the scattering vertex of $\sigma$ are
different. This implies that some of the high-spin correlation channels are
switched off.

\begin{eqnarray}\label{S_no_same_spin}
\sigma_{\scaleto{UiCCSDn-A-op}{3.6pt}} \in \{{(\sigma_h)}_{i_{\alpha}j_{\beta}}^{a_{\alpha}u_{\beta}},
{(\sigma_h)}_{i_{\beta}j_{\alpha}}^{a_{\beta}u_{\alpha}}; \nonumber \\
\}
\end{eqnarray}

$u$ and $v$ denote active spatial occupied and virtual orbitals, and
$\alpha$ and $\beta$ denote the spin-up and spin-down electrons, respectively.
While the number of $\sigma_h$ and $\sigma_p$ parameters in the UiCCSDn-A-op
variant scale similar to UiCCSDn-A, there is a small pre-factor of $1/16$ in the
former case. Thus for practical applications of UiCCSDn-A-op, there is a
tremendous reduction in the number of the free parameters over the UiCCSDn-A
variant. While this requires $\mathcal{O}(\frac{1}{16}n_{o}^{2}n_{v}n_{o}^{act})$
number of $\sigma_h$ and $\mathcal{O}(\frac{1}{16}n_{o}n_{v}^{2}n_{v}^{act})$
number of $\sigma_p$ parameters, the parameter count is, however, dictated
by $\mathcal{O}(n_{o}^2n_{v}^{2})$ number of $\tau$ amplitudes with a circuit
depth of $\mathcal{O}(n_{o}n_{v}^{2})$.

* UiCCSDn-A-diag: Note that the orbitals involved in the scattering
vertex of $S$ appear with opposite signs in their perturbative energy denominator
(see Eqs. (<ref>) and (<ref>)).
One may thus decipher that the most dominant terms in $S$ (or $\sigma$) are those
which have diagonal scattering. In the UiCCSDn-A-diag variant, we thus
restrict the orbitals involved in the scattering vertex of $\sigma$ to be the same.
Inclusion of diagonal terms in the scattering vertex reduces the computation
cost of Jordan-Wigner overhead significantly.

\begin{eqnarray}\label{Sdiag}
\sigma_{\scaleto{UiCCSDn-A-diag}{3.6pt}} \in \{{(\sigma_h)}_{iu}^{au}; {(\sigma_p)}_{iv}^{av}\}
\end{eqnarray}

The number of $\sigma_h$ and $\sigma_p$ parameters are further reduced to
$\mathcal{O}(n_{o}n_{v}n_{o}^{act})$ number of $\sigma_h$ and
$\mathcal{O}(n_{o}n_{v}n_{v}^{act})$ number of $\sigma_p$ elements. The leading
parameter count thus still scales as $\mathcal{O}(n_{o}^2n_{v}^{2})$ with a
circuit depth of $\mathcal{O}(n_{o}n_{v}^{2})$.

* UiCCSDn-A-pp: In the UiCCSDn-A-pp variant (where “pp" stands for
partial pairing), in addition to the
active orbital selection in the destruction component of $S$, we include
(i) only those quasi-hole creation operators which originate from the same
spatial orbitals (for $\sigma_h$) and (ii) only those quasi-particle creation
operators which have the same spatial orbitals (for $\sigma_h$).

\begin{eqnarray}
\sigma_{\scaleto{UiCCSDn-A-pp}{3.6pt}} \in
\{{(\sigma_h)}_{i_{\alpha}i_{\beta}}^{a_{\alpha}u_{\beta}},
{(\sigma_h)}_{i_{\beta}i_{\alpha}}^{a_{\beta}u_{\alpha}}; \nonumber \\
\}
\end{eqnarray}

Note that there is only partial pairing either among the quasi-hole or quasi-particle
orbitals and the structure of the $\sigma$ operators does not allow to have complete
This variant further lowers the number of $\sigma_h$ and $\sigma_p$ elements
over UiCCSDn-A-diag and the gate count and the circuit depth scales similar to

One may note that in all the variants, except the parent (untruncated)
UiCCSDn, there is no increase in the order of the gate count and circuit depth
over the conventional UCCSD. However, we will demonstrate that all our variants
predict energy well within the chemical accuracy.
A succinct summary of the choice of the various operators and the associated parameter
count in different approximate schemes are given in Table. <ref> and <ref>

§ RESULTS AND DISCUSSIONS

§.§ Methodology

Difference between VQE energies corresponding to various ansätze and FCI energies for $H_{2}O$ at symmetric stretching in STO-3G basis.

Non-parallelity error (red triangles) and parameters count (blue dots) corresponding to various ansätze for the symmetric stretching of $H_{2}O$ in STO-3G basis.

2Method     4c|Energy ($E_{h}$)

d=1Å     d=2Å     d=3Å    d=4Å

FCI     -7.78446028     -7.86108777     -7.79884316    -7.78427817

UCCSD      -7.78445508(0.0052)     -7.86106905(0.0187)     -7.79875361(0.0895)    -7.78414449(0.1336)

UiCCSDn-A-pp     -7.78445982(0.0004)     -7.86108722(0.0005)     -7.79884258(0.0005)    -7.78427777(0.0004)

UiCCSDn-A-diag     -7.78446017(0.0001)     -7.86108736(0.0004)     -7.79884162(0.0015)    -7.78427446(0.0037)

UiCCSDn-A-op     -7.78446000(0.0002)     -7.86108766(0.0001)     -7.79884256(0.0006)    -7.78427780(0.0003)

UiCCSDn-A     -7.78446025(0.0000)     -7.86108772(0.0000)     -7.79884302(0.0001)    -7.78427781(0.0003)

The energy values (in $E_h$) for some selected geometries of $LiH$ molecule using UCCSD and different UiCCSDn variants. The corresponding FCI energy values are also reported to demonstrate the accuracy of our schemes. The quantities in parenthesis denote the energy difference (in $mE_h$) from the FCI values. Note that all the energy values reported here used $k=l=1$.

We have chosen to study the potential energy surface for symmetric stretching of $H_{2}O$
and LiH to assess the performance of the newly developed UiCCSDn ansätz and the subsequent
variants in the context of VQE. For all our computations, we have chosen the contracted STO-3G
basis [41].
The one- and two- electron integrals were taken from PySCF [42],
and the UiCCSDn ansätz has been implemented in Qiskit
0.26 [43]. Also, the UCCSD calculations were done
using the same version of Qiskit whereas the UCCSDT calculations were
carried out in Qiskit nature. The Jordan-Wigner scheme [44]
was chosen for encoding the creation and annihilation operators into the
qubit operators. Also, we had employed the direct spin-orbital to qubit
mapping: that is each spin-orbital is represented as one qubit,
independent of the occupancy of the
spin-orbitals. For the classical subpart of the VQE, we chose L_BFGS_B optimizer [45] throughout. For all the calculations, the initial values of all the
parameters, $\theta_{init}$, were set to zero. Further, in the present study, we have worked with the statevector backend simulator. The
Trotter number for all our calculations was set to be one.

§.§ Symmetric stretching of $H_{2}O$

§.§.§ The potential energy surface and the non-parallelity error

We determined the VQE energies employing our ansätze for all the geometries along the potential energy surface. We have varied the O-H bond length from around 0.75 times the equilibrium distance to 2.5 times the equilibrium distance in order to generate the potential energy surface. For the purpose of comparison, we have also computed the VQE energies using the conventional UCCSD ansätz. As shown in Fig.  <ref>, we have plotted the differences between the VQE energies computed using the aforementioned ansätze and the FCI energies. It is quite evident from the plot that VQE with the conventional UCCSD ansätz performs quite well and agrees to 0.18 milliHartree ($mE_{h}$) with respect to FCI, for geometries below 1.75 times the equilibrium length, but starts deviating beyond that point. This trend shows that for the domains where strong correlation effects come into play, UCCSD fails to estimate the ground state energies. Whereas the energies obtained using the UiCCSDn-A-pp ansätz (which is the least parametrized ansätz amongst our other ansätze) are precise up to tens of microHartree ($\sim 10 \mu E_h$) throughout the potential energy surface. In order to explicate our observations, we have further calculated the corresponding non-parallelity error, abbreviated as NPE. NPE is defined as the difference between the maximum and minimum deviations from the exact ground state energy, i.e., FCI in our case. We observe that the NPE corresponding to the conventional UCCSD and UCCSDT ansätz is 4.74 $mE_{h}$ and 2.169 $mE_{h}$ respectively. On the other hand, the  NPE associated with the least parametrized ansätz, UiCCSDn-A-pp,  is found to be 0.11  $mE_{h}$. The NPE and the parameters count associated with all the ansätze have been plotted in Fig. <ref>.

In order to assess the efficiency of the newly developed ansätze, it is important to
elucidate on the circuit complexity, which is quantified by the number of variational
parameters and the quantum gate count. We should note that the NPE corresponding to
UiCCSDn-A-pp is in the order of 0.11 $mE_{h}$ at the expense of an additional 32 parameters
over the conventional UCCSD (with a NPE of 4.74 $mE_{h}$). Also it is worth mentioning at
this point that the conventional UCCSDT leads to a NPE of 2.169 $mE_{h}$ at the expense of
188 parameters (which is of the order $\sim\mathcal{O}(n_o^3n_v^3)$), whereas the UiCCSDn-A-diag (with lesser number of parameters than UCCSDT)
and the UiCCSDn-A-op (with same number of parameters as UCCSDT) exhibit 0.048 $mE_{h}$
and 0.0209 $mE_{h}$ NPE respectively. Looking at the number of parameters (and hence, the  gate count), we can infer that the newly developed ansätze strikes a right balance between the two aspects i.e., implementation cost and the accuracy.

§.§ Potential energy curve for $LiH$ dissociation

We have evaluated the VQE energies employing different variants of UiCCSDn ansätze
for a few selected geometries of LiH, as listed in Table <ref>. According to our
findings, the conventional UCCSD approach performs pretty well around the equilibrium geometry, but it
starts to deteriorate for the strongly correlated regions when the $Li-H$ bond is stretched.
However, the ground state energies evaluated through our ansätze uniformly show a
sub micro-hartree accuracy for both weakly and strongly correlated regions irrespective of
the UiCCSDn variant.

§ CONCLUSION AND FUTURE OUTLOOK

In this paper, we have developed the unitary version of the dual
exponential coupled cluster methodology (UiCCSDn) and have adapted it for use in a quantum computing framework.
We have presented theoretical arguments for the non-viability of implementing the iCCSDn methodology or for that matter, its unitary version on a classical device without approximation in the rank and number of
terms in the similarity transformed many-body Hamiltonian. The implementation of UiCCSDn
in a quantum device can bypass these drawbacks entirely. The dual exponential structure,
by construction, allows the scattering operators to act on entangled states to
induce high-rank excitations. We have introduced different variants
by judiciously including a selection of the scattering operators while keeping the cluster
operators in full. Apart from the parent UiCCSDn, all the variants have a gate count
and circuit depth in the same order as the conventional UCCSD without significantly
compromising the expressibility of the anstze. We have shown with a prototypical
molecular application that all the different variants show high accuracy compared to
classically computed FCI over the entire molecular surface, but with a significantly
less implementation cost than allied theories.

It would be interesting to implement the UiCCSDn in the ADAPT-VQE framework to further
reduce the parameter count. Evaluation of the parameters in PQE framework would be another
avenue which would keep us engaged in coming years.

§ ACKNOWLEDGEMENTS

RM thanks Science and Engineering Research Board (SERB),
Government of India for providing financial support. DH thanks
SERB and IRCC, IIT Bombay for research fellowship.
Some of the preliminary calculations were carried out on National Supercomputing Mission's (NSM) computing resource, `PARAM Siddhi-AI', at C-DAC Pune, which is implemented by C-DAC and supported by the Ministry of Electronics and Information Technology (MeitY) and Department of Science and Technology (DST), Government of India.

§ DATA AVAILABILITY

The data generated in this study is available upon reasonable request to the corresponding author.

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CERN-TH-2019-039

# The Holographic Fishchain

Nikolay Gromov<EMAIL_ADDRESS>Mathematics Department, King’s
College London, The Strand, London WC2R 2LS, UK  St.Petersburg INP, Gatchina,
188 300, St.Petersburg, Russia  Amit Sever<EMAIL_ADDRESS>School of
Physics and Astronomy, Tel Aviv University, Ramat Aviv 69978, Israel  CERN,
Theoretical Physics Department, 1211 Geneva 23, Switzerland

###### Abstract

We present the first-principle derivation of a weak-strong duality between the
fishnet theory in four dimensions and a discretized string-like model living
in five dimensions. At strong coupling, the dual description becomes classical
and we demonstrate explicitly the classical integrability of the model. We
test our results by reproducing the strong coupling limit of the $4$-point
correlator computed before non-perturbatively from the conformal partial wave
expansion. Due to the extreme simplicity of our model, it could provide an
ideal playground for holography with no super-symmetry. Furthermore, since the
fishnet model and ${\cal N}=4$ SYM theory are continuously linked our
consideration could shed light on the derivation of AdS/CFT for the latter.

###### pacs:

Valid PACS appear here

##  I Introduction

In recent years the ideas of holography Maldacena:1997re ; Gubser:1998bc ;
Witten:1998qj  conquered almost all corners of theoretical physics. The idea
that some (or any?) strongly coupled quantum system with many degrees of
freedom should have an alternative dual description in terms of the
gravity/string theory in a higher dimensional spacetime is becoming more and
more popular. Despite this enormous attention the holographic principle has
received in the last two decades, we are still lacking the first principle
derivation of it. There are, however, numerous and extremely non-trivial tests
of the duality. Due to its strong-weak character, it is very hard to produce
these tests. In some special models such as ${\cal N}=4$ SYM theory, tools
such as super-symmetric localization, or integrability, provide ways to
compute observables for arbitrary coupling strengths and compare with the
holographic predictions.

In this paper we will provide the first principle derivation of a holographic
dual of the fishnet model Gurdogan:2015csr

${\cal L}_{4d}=N\,{\rm
tr}\left(|\partial\phi_{1}|^{2}+|\partial\phi_{2}|^{2}+(4\pi)^{2}\xi^{2}\phi_{1}^{\dagger}\phi_{2}^{\dagger}\phi_{1}\phi_{2}\right)\,,$
(1)

in the planar expansion where $N$ is taken large while $\xi^{2}$ is held fixed
but arbitrary 111Here we have suppressed double trace interactions which are
not relevant non-perturbatively Fokken:2013aea ; Gromov:2018hut .. Here,
$\phi_{1,2}$ are two $N\times N$ complex scalar fields. The model can be
obtained from ${\cal N}=4$ SYM theory in a double scaling limit and was shown
to be conformal and integrable in the planar limit Fokken:2013aea ;
Gurdogan:2015csr ; Gromov:2017cja ; Grabner:2017pgm . This breaks the
$\mathfrak{su}^{*}(4|4)$ superconformal symmetry down to
$\mathfrak{u}(1)_{1}\times\mathfrak{u}(1)_{2}\times\mathfrak{so}(1,5)$ living
us with no super-symmetry at all. Yet, being a solvable interacting CFT in
four dimensions, this model attracted a lot of attention. In particular, one
can compute the spectrum of anomalous dimensions Gromov:2017cja  as well as
some structure constants and correlation functions Gromov:2018hut  at any
$\xi$. In all of these cases, there were indications of the existence of the
holographic dual – the scaling dimensions $\Delta$ generically scale as $\xi$
and the $4$-point correlation functions behave as $e^{-\xi A(z,\bar{z})}$
Gromov:2018hut . At the same time, these indications were somehow puzzling as
the dual description of ${\cal N}=4$ SYM become weakly coupled at infinitely
large $\lambda$, whereas the fishnet model obtained in the opposite
$\lambda=R^{2}_{\rm AdS}/l^{2}_{s}\to 0$ limit Gurdogan:2015csr . Furthermore,
the corresponding deformation is known to produce tachionic instability in the
string background Pomoni:2008de . As a result, it is not clear how to link the
holographic string description of ${\cal N}=4$ SYM to that of the fishnets or
even if that is possible at all.

Indeed, the dual description presented here is not in terms of a smooth
string. Instead, we found a chain of $J$ particles or string-bits with nearest
neighbor interactions. More precisely the dual model action functional $S_{\rm
dual}=\xi\int dt\,\sum_{i}L_{i}$, is given in terms of the Lagrangian-density

$L_{i}=-\frac{\dot{X}_{i}^{2}}{2}-\prod_{k=1}^{J}\left(-X_{k}.X_{k+1}\right)^{-\frac{1}{J}}-\eta_{i}(X_{i}^{2}+R^{2})+R^{2}\;.$
(2)

Here, $X_{i}(t)\in{\mathbb{R}}^{1,5}$ with $-++\dots$ signature and
$\eta_{i}(t)$, $R^{2}(t)$ are Lagrange multipliers. The world-sheet
coordinates $X(t)$ are further subjected to Virasoro-type constraints
described below in (11) and (13). Note in particular that the square root of
the ’t Hooft coupling $\xi$, stands in front of the action and plays the role
of $1/\hbar$.

The field $R(t)$ looks like an AdS radius in string units. It satisfies a
dynamical evolution equation and will be set to zero consistently. The $X_{i}$
coordinates are not projective and hence, the action $S_{\rm dual}$ describe a
discretized string propagating on the five dimensional lightcone of
${\mathbb{R}}^{1,5}$, subject to the Virasoro constraints. The fifth dimension
naturally emerges when making the symmetries manifest. It is encoded in a non-
trivial way in all the original $J$ four-dimensional degrees of freedom and is
related to an emergent local scale invariance.

Below we give the derivation of this result and also show how one can
reproduce the classical limit of the anomalous dimensions and also the 4-point
functions.

##  II Derivation of the dual action

Figure 1: Typical Feynman diagram in the $\mathfrak{u}(1)$ sector of the model
has the wheel, or fishnet Nielsen:1970bc , structure which can be resummed.
This structure also leads to integrability Zamolodchikov:1980mb .

One of the main features of the fishnet theory is the simple structure of its
Feynman diagrams. In this paper we consider the ${\mathfrak{u}}(1)$ sector of
the model, where the ${\mathfrak{u}}(1)_{1}$ charge is $J$ and the
${\mathfrak{u}}(1)_{2}$ charge is set to zero. It consists of all operators of
the type ${\rm
tr}\;(\partial^{m}\phi_{1}^{J}(\phi_{2}\phi_{2}^{\dagger})^{n},\dots)$,
containing any number of derivatives, $J$-scalar fields $\phi_{1}$ and any
neutral combination of $\phi_{2}$ and $\phi_{2}^{\dagger}$.

The Feynman diagrams which contribute to the correlation functions of these
operators and their conjugates are of iterative fishnet type, after all
$\phi_{2}$’s annihilate with $\phi_{2}^{\dagger}$, (see Fig.1). It is possible
to resum, at least formally, infinitely many Feynman graphs by introducing the
“graph-building” operator $\widehat{B}$, defined by its integral kernel
Gurdogan:2015csr

$B(\\{\vec{y}_{i}\\}_{i=1}^{J},\\{\vec{x}_{j}\\}_{j=1}^{J})=\prod_{i=1}^{J}\frac{\xi^{2}/\pi^{2}}{(\vec{y}_{i}-\vec{y}_{i+1})^{2}(\vec{x}_{i}-\vec{y}_{i})^{2}}\;.$
(3)

Applying this operator once, we add one wheel to the graph on Fig.1, thus the
sum of all wheels inside the graph forms a geometric series

${\rm all\;wheels}=\frac{1}{1-\widehat{B}}\;.$ (4)

We see that the zeros of the denominator play a special role. By diagonalizing
$\widehat{B}$, one finds that the eigenfunctions are parameterized by the
continuous parameter $\Delta$, conjugated to the dilatation operator. The sum
over the complete set of eigenfunctions will involve the integration over
$\Delta$, which then can be computed by residues giving distinct meaning to
those values of $\Delta$ where $\widehat{B}=1$. Namely, those poles can be
identified as the anomalous dimensions of the local operators. This procedure
was exemplified in detail in Gromov:2018hut . The output of this discussion is
that we need to solve $(\widehat{B}-1)\Psi=0$, or, equivalently, acting on
both sides with $\prod_{i}\Box_{i}$, to cancel factors
$1/(4\pi^{2}(x_{i}-y_{i})^{2})$, we find

$H\circ\Psi(\\{x_{i}\\})=0\
,\quad{H}=\prod_{i=1}^{J}\vec{p}_{i}^{2}-\prod_{i=1}^{J}\frac{4\xi^{2}}{(\vec{x}_{i}-\vec{x}_{i+1})^{2}}$
(5)

where $\vec{p}_{i}\equiv-i\vec{\partial}_{x_{i}}$. Under the operator-state
correspondence, the wave function $\Psi$ is dual to a local operator. The key
step in our derivation is to interpret (5) as the constraint appearing in a
system with time reparametrization symmetry $t\to f(t)$, where $t$ is
conjugate to $H$. To see this gauge symmetry manifestly, we write the
Lagrangian corresponding to the Hamiltonian $H$ in (5). After solving for
$\vec{p}_{i}$ in term of $\dot{\vec{x}}_{i}=\frac{\partial
H}{\partial\vec{p}_{i}}$ we arrive at

${L}=\frac{2J-1}{2^{\frac{2J}{2J-1}}}\left(\frac{1}{\gamma}\prod_{i=1}^{J}\vec{\dot{x}}_{i}^{2}\right)^{\frac{1}{2J-1}}+\gamma\prod_{i=1}^{J}\frac{4\xi^{2}}{(\vec{x}_{i}-\vec{x}_{i+1})^{2}}$
(6)

where $\gamma$ transforms under reparametrization $t\to f(t)$ as
$\gamma\to\gamma/f^{\prime}$ and (5) is the constraint that corresponds to
fixing $\gamma=1$. Instead of fixing the gauge $\gamma=1$ it is more
beneficial to eliminate the auxiliary field $\gamma$, by setting it to its
extremum to obtain

$S=\xi\int
L\,dt=2J\xi\int\left(\prod_{i=1}^{J}\frac{\dot{\vec{x}}_{i}^{2}}{(\vec{x}_{i}-\vec{x}_{i+1})^{2}}\right)^{\frac{1}{2J}}dt\;.$
(7)

One may draw analogies between (6), (7) and the Polyakov, Nambu-Goto actions
respectively. In this analogy, the initial equation (5) corresponds to the
Virasoro constraint. There is a number of significant observations one can
make about (7). First, we see that the coupling $\xi$ is playing the role of
$1/\hbar$ in the quasiclassical analysis in accordance with the previous
observations Gromov:2017cja ; Gromov:2018hut . In particular, that explains
the scaling $\Delta\sim\xi$ observed numerically in Gromov:2017cja . We note,
however, that our starting point (3) contained $\xi^{2J}$, implying that all
roots $e^{\pi in/J}\xi,\;n\in{\mathbb{Z}}$ should be considered. Different
$n$’s correspond to different branches in the spectrum as we demonstrate in
Sec. III. In addition to the time reparametrization symmetry the action $S$ in
the form (7) is also invariant under global conformal transformations, which
is of course a highly expected property for a CFT dual. In the next section,
we will make the conformal symmetry manifest by uplifting the action into a 6D
embedding space.

Finally, we comment about an interpretation of the action (7). One may think
of it as describing $J$ string bits, Bergman:1995wh ; Dorey:2008zy . Each bit
corresponds to a spike of a holographic string that propagates at the AdS
boundary, see figure 2. The dynamics of the string segment between the spikes
results in the nearest neighbors interaction of the model.

### Embedding space formulation.

As it is well known the conformal group in 4D coincides with the group of
rotations in ${\mathbb{R}}^{1,5}$. Under this identification, the flat space
coordinate $x^{\mu=1,\dots,4}$ is mapped to the projective lightcone of
${\mathbb{R}}^{1,5}$, parametrized by $X_{i}^{M},\;M=-1,0,1,\dots,4$, as

Figure 2: Spiky string in AdS.

$x_{i}^{\mu}=X_{i}^{\mu}/X_{i}^{+}\ ,\quad X_{i}^{2}=0\ ,\quad
X_{i}^{+}=X_{i}^{0}+X_{i}^{-1}\;.$ (8)

After this change of variables, the action (7) becomes

$L=2J\left(\prod_{i=1}^{J}\frac{\dot{X}_{i}\cdot\dot{X}_{i}}{-2X_{i}\cdot
X_{i+1}}\right)^{\frac{1}{2J}}\;.$ (9)

By uplifting to a projective space, we have introduced a new local gauge
symmetry $X_{i}\to g_{i}(t)X_{i}$.

We now introduce auxiliary fields to disentangle the action as

$L=-\sum_{i}\left[\frac{\dot{X}_{i}^{2}}{2\alpha_{i}}+\eta_{i}X_{i}^{2}+\gamma\prod_{k}\left(-X_{k}.X_{k+1}\right)^{-\frac{1}{J}}\right]\;,$
(10)

where the last term is independent of $i$. In order to get back to (9) one
should extremize in $\alpha_{i}$ and $\gamma$, assuming
$\prod\alpha_{i}=\gamma^{J}$. We also introduced the remaining constraint
$X_{i}^{2}=0$ with the Lagrange multipliers $\eta_{i}$. The symmetries of the
action are 1) manifest conformal symmetry; 2) time-dependent re-scaling
symmetry $X_{i}\to g_{i}(t)X_{i},\;\alpha_{i}\to
g_{i}^{2}(t)\alpha_{i},\;\gamma\to\gamma\prod_{i}g_{i}^{2/J}(t)$ and
$\eta_{i}\to g^{-2}_{i}(t)\eta_{i}$; 3) time reparameterization symmetry $t\to
f(t),\;\eta_{i}\to\eta_{i}/f^{\prime},\;\gamma\to\gamma/f^{\prime}$; 4)
translation along the chain $X_{i}\to X_{i+1}$. To fix the gauge symmetries we
can set $\alpha_{i}=\gamma=1$, leading to the constraints

$\dot{X}_{k}^{2}=2\prod_{i}\left(-X_{i}.X_{i+1}\right)^{-\frac{1}{J}}\equiv{\cal
L}\;,\;\;k=1,\dots,J\;,$ (11)

which is very reminiscent of the Virasoro constraints in the conformal gauge,
telling us that the energy density is zero along the string. Finally, we
notice that there is still one remaining gauge symmetry left $t\to
f(t),\;X_{i}\to X_{i}/\sqrt{f^{\prime}},\;\eta_{i}\to\eta_{i}/{f^{\prime}}$,
which we fix by further imposing $\sum\eta_{i}=J$, with the Lagrange
multiplier $R^{2}$, leading to (2). This action together with (11) is our main
result 222alternative gauge choice ${\cal L}=1$ could be convenient too.

### Equations of motion.

The variation of (2) with respect to $X_{i}$ gives

$\ddot{X}_{i}=2\eta_{i}X_{i}-{{\cal L}\over 2}\left({X_{i+1}\over
X_{i+1}.X_{i}}+{X_{i-1}\over X_{i}.X_{i-1}}\right)\;.$ (12)

By contracting (12) with $X_{i}$ and using that $X_{i}^{2}=0$ we arrive back
at (11). Contracting (12) with $\dot{X}_{i}$ however, leads to the secondary
constraint that is analogous of the second Virasoro constraint, imposing that

${\dot{X}_{i}.X_{i+1}\over X_{i}.X_{i+1}}+{\dot{X}_{i}.X_{i-1}\over
X_{i}.X_{i-1}}=-\partial_{t}\log{{\cal L}}$ (13)

does not depend on the site index $i$, 333The relative sign may look strange.
However, in the continuum limit the analogue of the r.h.s. has the effect of
correcting it longpaper ..

Finally, $\eta_{i}$ can be extracted from the derivative of (13). Instead, we
eliminate $\eta_{i}$ by introducing the $SO(1,5)$ charge density
$q_{i}^{MN}=2\dot{X}_{i}^{[M}X_{i}^{N]}$. The equation of motion (12) can be
equivalently written as

$\dot{q}_{i}=\frac{{\cal L}}{2}({j}_{i+1}-{j}_{i})\
,\quad{j}_{i}^{MN}=2{X_{i-1}^{[M}X_{i}^{N]}\over X_{i-1}.X_{i}}$ (14)

where $j_{i}$ can be interpreted an $SO(1,5)$ current density. The $SO(1,5)$
charge is given by $Q^{MN}\equiv\xi{\cal Q}^{MN}=\xi\sum_{i}{q_{i}^{MN}}$. We
can always assume that ${\cal Q}^{MN}$ is block diagonal, with non-zero
elements ${\cal Q}^{-1,0}=i{\cal D}$, ${\cal Q}^{1,2}={\cal S}_{1}$ and ${\cal
Q}^{3,4}={\cal S}_{2}$, where ${\cal D}=\Delta/\xi$ and ${\cal
S}_{a}=S_{a}/\xi$ are the appropriate notations for the large $\xi$ classical
limit.

After introducing $R^{2}$, we are no longer constrained to the lightcone and
one could be worried about the consistency of the initial condition
$X_{i}^{2}=-R^{2}=0$. By contracting (12) with $X_{i}$ we get
$2\partial_{t}^{2}R^{2}=\eta_{i}R^{2}$. Since $\sum_{i}\eta_{i}=J$ we obtain
$2\partial_{t}^{2}R^{2}=R^{2}$, meaning that once we set $R=0$ at some moment
of time it will stay so forever.

### Integrability.

Our fishchain model at $R=0$ is dual to the integrable fishnet model and
hence, it is expected to be integrable too. Similarly to the Toda chain, we
find a pair of spacelike and timelike connections, dependant on the spectral
parameter $u$, ${\mathbb{L}}_{i}(u)$ and ${\mathbb{V}}_{i}(u)$, that satisfy
the zero curvature condition book

$\dot{\mathbb{L}}_{i}={\mathbb{V}}_{i+1}.{\mathbb{L}}_{i}-{\mathbb{L}}_{i}.{\mathbb{V}}_{i}\;.$
(15)

This condition ensures that each coefficient of the polynomial
${\mathbb{T}}(u)={\rm tr}\;\Omega(u)$ where
$\Omega\equiv{\mathbb{L}}_{J}\dots\mathbb{L}_{2}.{\mathbb{L}}_{1}$ gives an
integral of motion, constant in time on equations of motions. In the irrep
$\bf 6$ of $SO(1,5)$ these matrices are

${\mathbb{L}}^{\bf 6}_{i}=u^{2}+uq_{i}+\frac{q_{i}^{2}}{2}\
,\qquad{\mathbb{V}}^{\bf 6}_{i}={j_{i}\over u}\frac{{\cal L}}{2}\;.$ (16)

To derive the discrete zero curvature condition (15) we use (14) and the
identity $\left(q_{i}^{2}\right)^{MN}=-{\cal L}X_{i}^{M}X_{i}^{N}$, which
implies, using (12), that $\partial_{t}q_{i}^{2}={\cal
L}(j_{i+1}q_{i}-q_{i}j_{i})$ and $j_{i+1}q_{i}^{2}-q_{i}^{2}j_{i}=0$.
Interestingly, the constraint (11) results in the relation ${\mathbb{T}}^{\bf
6}(0)=(-1)^{J}$. Similarly to (16), the spacelike and timelike connections in
the irrep 4 take the form ${\mathbb{L}}^{\bf
4}_{k}=u-\frac{i}{2}q_{k}^{MN}\Sigma_{MN}$ and ${\mathbb{V}}^{\bf
4}_{k}=-\frac{i{\cal L}}{4u}j_{k}^{MN}\Sigma_{MN}$, where $\Sigma_{MN}$ are
the 6D $\sigma$-matrices. ${\mathbb{L}}^{\bf 6}$ can be constructed from
${\mathbb{L}}^{\bf 4}_{k}$ by projecting ${\mathbb{L}}^{\bf
4}_{k}\otimes{\mathbb{L}}^{\bf 4}_{k}$ on the 6.

The key objects in integrability are the $4$ quasi-momenta $p_{a}$ which are
defined as $\det(\Omega^{\bf 4}(u)-u^{J}e^{ip_{a}(u)})=0$. Their large $u$
asymptotic is determined by the global charges $p_{a}\simeq\frac{\pm\Delta\pm
S_{1}\pm S_{2}}{2\xi u}$. At the origin $p_{a}$’s have a logarithmic
singularity $\pm iJ\log u$. In addition, $p_{a}(u)$ has square-root
singularities, coming from the diagonalization procedure. Together they form
an algebraic curve, whose genus depends on the number of degrees of freedom.
We expect the number of cuts to be equal to the number of independent cross-
ratios for $2J$ points (i.e. $2$ for $J=2$ and $8J-15$ for $J>2$). Each
a-cycle on the curve corresponds to an action variable
$I_{a}\equiv\frac{\xi}{\pi i}\oint_{a}p(u)\,du$. The action variables are
expected to become integers in the Bohr-Sommerfeld quantization procedure. We
postpone more detailed investigation of the algebraic curve and separation of
variables in this model for the future longpaper .

##  III Explicit example

First we consider the simplest case where $J=2$. This case was studied in
detail at the quantum level, in particular the spectrum is know exactly
Grabner:2017pgm

$\Delta_{t=2/4}=2+\sqrt{(S_{1}+1)^{2}+1\mp 2\sqrt{(S_{1}+1)^{2}+4\xi^{4}}}$
(17)

where $\pm$ correspond to the twist $t=2$ and twist $t=4$ branches in the
spectrum. The $4$-point function was computed in Gromov:2018hut  as an
infinite sum. In the classical limit, it was shown to sit on saddle points
with classical dimension and spins that are related to the two conformal cross
ratios as

$\displaystyle S_{\rm
cl}^{2}=\pm\frac{4\xi^{2}\theta^{2}}{\theta^{2}+\rho^{2}}\quad\text{and}\quad\Delta_{\rm
cl}^{2}=S_{\rm cl}^{2}\mp 4\xi^{2}\;,$ (18)

where the second relation follows from (17). Here, $\rho$ and $\theta$
parametrize the two conformal cross ratios
$u=\frac{4}{(\cos\theta-\cosh\rho)^{2}}$ and
$v=\frac{(\cos\theta+\cosh\rho)^{2}}{(\cos\theta-\cosh\rho)^{2}}$.
Furthermore, the 4-point correlation function itself takes the form $e^{-\xi
A_{\rm cl}}$ ($e^{i\xi A_{\rm cl}}$) for $t=2$ ($t=4$), with $A_{\rm
cl}=2i\sqrt{\theta^{2}+\rho^{2}}$. Next, we try to reproduce this data from
our classical dual description.

For $J=2$ we have two 6D null-vectors, $X_{1}(t)$ and $X_{2}(t)$. Using global
symmetries we can always go to the centre of mass frame and set the last two
components to zero $X_{1,2}=\frac{r}{\sqrt{2}}\left(\cosh s,\sinh
s,\pm\cos\phi,\mp\sin\phi,0,0\right)$. In this parametrization the coordinates
$s$ and $\phi$ are conjugate to conserved charges, ${\cal D}=ir^{2}\dot{s}$
and ${\cal S}_{1}=r^{2}\dot{\phi}$. The constraint (11) gives $\pm
4=r^{4}(\dot{s}^{2}+\dot{\phi}^{2})={\cal D}^{2}+{\cal S}_{1}^{2}$, where the
$\pm$ sign comes from the different choice of the branch of the root in the
r.h.s. of (11). It perfectly reproduces the spectrum (17), (18) in the
classical limit where $S_{1},\Delta\sim\xi\to\infty$, with different twists
$t=2,4$ corresponding to different branches of the interaction term. The
classical action, with the constraint taken into account, becomes $S=\mp
4\xi\int\frac{dt}{r^{2}}$. We see that it is beneficial to define the proper
time $d\tau=\frac{dt}{r^{2}(t)}$, in terms of which

$S=\mp 4\xi\,\tau\ ,\quad s=-i{\cal D}\,\tau\ ,\quad\phi={\cal
S}_{1}\,\tau\;.$ (19)

Next, computing the cross-ratios between $X_{1,2}(\tau=0)$ and
$X_{1,2}(\tau=T)$ we find in our parameterization $\theta={\cal S}_{1}T$ and
$\rho=i{\cal D}\,T$. Next, solving for ${\cal S}_{1}$ and $T$, with the
constraint ${\cal S}_{1}^{2}-{\cal D}^{2}=\pm 4$, we find
$T^{2}=\pm\frac{1}{4}(\theta^{2}+\rho^{2})$ and ${\cal
S}_{1}^{2}=\pm\frac{4\theta^{2}}{\theta^{2}+\rho^{2}},$ leading via (19) to
$e^{iS}=e^{-2i\xi\sqrt{\theta^{2}+\rho^{2}}}$ and
$e^{iS}=e^{-2\xi\sqrt{\theta^{2}+\rho^{2}}}$ for $t=2$ and $t=4$
correspondingly, in perfect agreement with Gromov:2018hut .

Figure 3: Chain of $J$ particles on the light-cone of ${\mathbb{R}}^{1,5}$.

##  IV Discussion and speculation

There are two fundamentally important properties of the fishchain model (2).
First, the square root of the ’t Hooft coupling constant, $\xi$, stands in
front of the action, playing the role of $1/\hbar$. It emerged naturally from
our interpretation of the graph building operator. Second, the fishchain
propagates in five-dimensional target space, see Fig. 3. The fifth dimension
has emerged from the principle of realizing all symmetries in a manifestly
covariant way. It may be thought of as a concrete realization of the
holographic map and the original prediction of ’t Hooft tHooft:1973alw .

Going away from the model (1), one may add back the rest of the fields of
${\cal N}=4$ SYM in a controlled expansion around the fishnet limit
Bykov:2012sc  and incorporate their effect on the dual fishchain. Such
expansion may open the path for a rigorous proof of AdS/CFT. One way in which
this path may materialize is the following. The radius $R^{2}$ came about as a
Lagrange multiplier, associated with a global rescaling gauge symmetry. It is
consistently set to zero in the relevant classical solutions discussed above.
If the correction away from the fishnet limit will stabilize it at some fixed
$R^{2}>0$ then the discretized string will propagate in AdS5 instead of the
lightcone of ${\mathbb{R}}^{1,5}$. Moreover, for fixed $R^{2}>0$, the model
has a smooth large $J$ continuum limit where
$X_{i}.(X_{i+1}-X_{i})\simeq{1\over 2}\epsilon^{2}X^{\prime\prime}$. As a
result, a new local time reparametrization symmetry emerges. Using similar
manipulations to the ones in section II one arrives at a string action in AdS5
that is subject to the standard local Virasoro constraint, including a
constant contribution from the extension of the string on the sphere longpaper
. In the $J\to\infty$ limit we may also be able to find a connection with the
proposal of Basso:2018agi .

There are many future directions to pursue, some of them we list below.
${\mathrm{I}}$) One would like to extend the fishchain model away from the
${\mathfrak{u}}(1)$ sector by incorporating extra $\phi_{2}$ fields.
$\mathrm{II}$) One may incorporate quasi-classical corrections.
$\mathrm{III}$) Systematic $1/N$ expansion should lead to fishchain
interaction vertices. $\mathrm{IV}$) The open fishchain version of the model,
dual to Wilson lines in the ladder limit Correa:2012nk , can be obtained from
the derivation above by adding two more sites, replacing
$\vec{p}^{2}_{1,J+2}\to\vec{p}^{2}_{1,J+2}+m^{2}$ and taking the large mass
limit. $\mathrm{V}$) We expect the fishchain to exhibit T-duality.
$\mathrm{VI}$) The simplicity of the classical model could help with the
separation of variables approach Cavaglia:2018lxi ; twistingpaper  to the
correlation functions in ${\cal N}=4$ SYM.

Finally, analogous fishnet diagrams also exist in 2,3 and 6 dimensions
Zamolodchikov:1980mb  and one may try to derive their duals. In particular,
one may consider the large twist limit of the ABJM model Aharony:2008ug ;
Caetano:2016ydc , or more general fishnets, which could also include fermions
Kazakov:2018gcy .

Acknowledgements: We thank D. Anninos, B. Basso, A. Cavaglià, G. Korchemsky,
F. Levkovich-Maslyuk, I. Kostov, S. Lukyanov, F. Smirnov, K. Zarembo, B.
Vicedo, S. Zhiboedov and especially to V. Kazakov for invaluable discussions.
N.G. acknowledges the hospitality of the theory group at CERN during the main
phase of this work. N.G. was supported by the STFC grant (ST/P000258/1). A.S.
was supported by the I-CORE Program of the Planning and Budgeting Committee,
The Israel Science Foundation (1937/12) and by the Israel Science Foundation
(grant number 968/15).

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