# Optimality conditions for spatial search with multiple marked vertices

Mathieu Roget  Hachem Kadri  Giuseppe Di Molfetta  giuseppe.dimolfetta@lis-
lab.fr Aix Marseille Univ, Université de Toulon, CNRS, LIS, Marseille, France

###### Abstract

We contribute to fulfil the long-lasting gap in the understanding of the
spatial search with multiple marked vertices. The theoretical framework is
that of discrete-time quantum walks (QW), i.e. local unitary matrices that
drive the evolution of a single particle on the lattice. QW based search
algorithms are well understood when they have to tackle the fundamental
problem of finding only one marked element in a $d-$dimensional grid and it
has been proven they provide a quadratic advantage over classical searching
protocols. However, once we consider to search more than one element, the
behaviour of the algorithm may be affected by the spatial configuration of the
marked elements and even the quantum advantage is no longer guaranteed. Here
our main contribution is threefold : (i) we provide sufficient conditions for
optimality for a multi-items QWSearch algorithm ; (ii) we provide analytical
evidences that almost, but not all spatial configurations with multiple marked
elements are optimal; and (iii) we numerically show that the computational
advantage with respect to the classical counterpart is not always certain and
it does depend on the proportion of searched elements over the total number of
grid points.

One of the main applications of quantum computing is algorithmic. Considered
to be still beyond the reach of today’s quantum computers, it has had a major
impact in several fields, from cryptography [1] and quantum machine learning
[2] to quantum simulation [3]. The first quantum algorithms were formulated in
the early 1990s [4, 5], and since then researchers have continued to create
new ones over the past 30 years 111For a comprehensive catalog of quantum
algorithms visit https://quantumalgorithmzoo.org, trying to optimise computing
time and quantum resources. However, compared to the thousands of non-quantum
algorithms, the number of quantum algorithms is still modest. This is
essentially due to the difficulty of proving the advantage that each of them
has over its classical counterpart. An example is given by one of the most
studied problem in computer science : the quantum search in an unstructured
set of $N$ elements. The first algorithm aiming to solve this problem appeared
in L. K. Grover [7]’s work. The basic ideas was to introduce a quantum oracle,
which recognizes a solution to a search problem when it sees one. The Grover
algorithm could solve this problem in $O\left(\sqrt{N}\right)$ time, i.e.
quadratically faster than what a classical computer needs to complete the same
task, and it soon seemed to be extraordinary advantageous to speed-up many
classical algorithms that use search heuristic [8, 9, 10]. In fact the Grover
search algorithm can be applied to any decision problems whose solutions can
be checked efficiently [11], with a clear polynomial speed-up. Such quantum
advantage has been proven for several generalizations, for instance when the
target elements are multiple [12]. However, the situation may be dramatically
different for spatial searching, as in [13], where quadratic speed-up is known
to be possible only for some specific case. Spatial search may come in
different forms, in continuous time and discrete time. The first example of
spatial search algorithm in continuous time has been introduced by Childs and
Goldstone [14] in the quantum walk framework, where the searching method
involves now a Hamiltonian defined over an arbitrary graph, which has to be
able to solve the searching problem. In this context, it has been proven that
only for some certain graphs, such as a complete graph or the hypercube, the
hitting time shows a quadratic speed-up with respect to the classical
counterpart. This long-standing problem has been recently addressed by
Chakraborty, Novo and Roland [15], who obtained the necessary and sufficient
conditions for the Childs and Goldstone algorithm to be optimal for any graph
that meets certain general spectral properties. Another open problem that has
remained poorly understood until now is the spatial search of multiple target
items, both in continuous and discrete time. While the hitting time in the
case we search only one target item is in line with the one recovered by the
Grover algorithm, when the items are multiple, their spatial configuration can
affect significantly the performance (aka the time complexity) of the
algorithm. The intuitive reason behind it is that the marked vertices
interfere among them, and the interplay between constructive and destructive
interference may determine very different scenario. Such scenario has been
remarked first by Aaronson and Ambainis [16] and recently observed by Bezerra
et al [17], but never been studied and fully understood. Yet traditionally, it
has been assumed that when the number of elements searched is low, the spatial
search algorithm is optimal. In this paper, we will prove this assumption
false. In fact, we will prove that even when considering only two searched
elements, there is always a set of spatial configurations for which the search
algorithm is sub-optimal. Fortunately, this need not worry because this number
is small compared to the number of optimal configurations, for all practical
purposes. More specifically, we will show that almost all configurations with
two marked elements are optimal (complexity in $O(\sqrt{N}\text{polylog}(N)$)
and we will able to precisely upper bound the number of non-optimal
configurations (complexity in $O(N)$). As a byproduct of this rigorous result,
we will provide a set of sufficient conditions on the searched elements’
relative position on the grid to ensure the optimality of the algorithm.
Finally, we discuss the results and we provide strong numerical evidences that
such non-optimality issue is not only present when the number of marked
elements is two. More in general, the quantum advantage will be shown to be
not always guaranteed and also strictly depending on the ratio of marked
elements $M/N$, with $M$ the number of the searched elements on the grid.

#### A 2-dimensionnal, discrete time Quantum Walk Search (QWSearch) algorithm

A discrete time quantum walk on a grid is the quantum analog of a
2-dimensional random walk. The quantum walker lives in a composite Hilbert
space : the coin state space, encoding the walker direction and the position
state space. The physical space here is a grid of size
$\sqrt{N}\times\sqrt{N}$. A generic state of the walker reads as follows:

$\ket{\psi}=\sum_{v\in\\{0,1\\}}\sum_{x=0}^{\sqrt{N-1}}\sum_{y=0}^{\sqrt{N-1}}\alpha_{v,x,y}\ket{v,x,y},$
(1)

where the coin state space is spanned by the $z$-basis
$\\{\ket{0},\ket{1}\\}$. The walker evolves driven by the usual split-step
operator :

$U=\Sigma_{y}(C_{y}\otimes\mathbb{I}_{N})\Sigma_{x}(C_{x}\otimes\mathbb{I}_{N}),$
(2)

where $C_{y}$ and $C_{x}$ are two non commutative $U(2)$ operators and the
$\Sigma_{i}$, are coin state dependent shift operator along direction $i=x,y$,
defined as follows :

$\displaystyle\Sigma_{i}\ket{v}\ket{i}$
$\displaystyle=\ket{v}\ket{i-(-1)^{v}}.$

The search algorithm based on the aforementioned quantum walk scheme
(QWSearch), for a set of marked vertices $|m\rangle\in\mathcal{M}$ is
implemented as follows: (i) Initialize the quantum walker $\psi(0)$ to the
equal superposition over all states, by applying $n$ single bit Hadamard
operation to the ground state $\ket{0}$. (ii) Given a coin oracle
$R=\mathbb{1}-2\sum_{m\in\mathcal{M}}\ket{d,m}\bra{d,m}$, where $\ket{d}$ is
the diagonal state in the coin state space, apply the perturbed evolution
operator $U^{\prime}=UR$, for $t_{opt}$ steps, the hitting time. (iii) Measure
the state in the $\ket{v,x,y}$ basis. The success probability of the walker
$p(t)$ after $t$ time steps is given by :

$p(t)=\sum_{m\in\mathcal{M}}|\bra{d,m}{U^{\prime}}^{t}\ket{\psi(0)}|^{2}=\\\
\sum_{j=0,1}|\sum_{\theta}e^{i\theta
t}\braket{d,m_{j}}{\theta}\braket{\theta}{\psi(0)}|^{2}.$ (3)

where the operator $U^{\prime}$ has been diagonalised on its basis. The main
aim here is to find the hitting time $t_{opt}$ which maximises the above
probability. Assuming that the search algorithm converges in finite time, one
shall expect that the eigenspace of $U^{\prime}$ be approximately spanned by
$\\{\ket{\lambda_{+}},\ket{\lambda_{-}}\\}$, with $e^{i\lambda_{+}}$ and
$e^{i\lambda_{-}}$ the two closest eigenvalues to unity :

$\lambda_{+}=\min_{e^{i\theta}\in Sp(U^{\prime}),\;\theta>0}\theta\;\text{ and
}\;\lambda_{-}=\max_{e^{i\theta}\in Sp(U^{\prime}),\;\theta<0}\theta.$ (4)

Thus, let us cast all the negligible contributions as $\epsilon_{m}$ :

$p(t)=\sum_{j=0,1}|\beta_{+,j}e^{i\lambda_{+}t}+\beta_{-,j}e^{i\lambda_{-}t}+\epsilon_{m}|^{2}.$
(5)

To analytically calculate the probability of success, we must therefore obtain
an explicit expression of the coefficients
$\beta_{i,j}=\braket{d,m_{j}}{\lambda_{i}}\braket{\lambda_{i}}{\psi(0)}$. To
begin with, let us consider the eigenmodes of the QW operator $\ket{\psi_{\pm
k,l}}$, with eigenvalues $e^{i\phi_{\pm k,l}}$. We can point out that
$\braket{\psi_{\pm k,l}}{\lambda_{i}}$, can be expressed in terms of
$\braket{d,m_{j}}{\lambda_{i}}$. In fact :

$\braket{\psi_{\pm k,l}}{U^{\prime}}{\lambda_{i}}=e^{i\phi_{\pm
k,l}}\braket{\psi_{\pm k,l}}{\lambda_{i}}\\\ -2e^{i\phi_{\pm
k,l}}\sum_{j=0,1}\braket{\psi_{\pm
k,l}}{d,m_{j}}\braket{d,m_{j}}{\lambda_{i}}$ (6)

which leads to

$\braket{\psi_{\pm k,l}}{\lambda_{i}}=\\\
\frac{2}{1-e^{i(\lambda_{i}-\phi_{\pm k,l})}}\sum_{j=0,1}\braket{\psi_{\pm
k,l}}{d,m_{j}}\braket{d,m_{j}}{\lambda_{i}}.$ (7)

Moreover, since $\lambda_{i}$ is close to zero, the pre-factor of the sum can
be approximated as follows :

$\frac{2}{1-e^{i(\lambda_{i}-\phi_{\pm k,l})}}=1+ib_{\pm k,l}^{\lambda},$ (8)

where

$b_{\pm,k,l}^{\lambda}=\left\\{\begin{matrix}\frac{2}{\lambda_{i}}+O(\lambda_{i})&\text{if
}\phi_{\pm,k,l}=0\\\
\frac{-1}{1-\cos\phi_{\pm,k,l}}\left(\lambda_{i}+\sin\phi_{\pm,k,l}\right)+O(\lambda_{i}^{2})&\text{otherwise}.\\\
\end{matrix}\right.$ (9)

From Eq. 7 and using the completeness relation, we recover the characteristic
equation :

$\Lambda^{\lambda}\braket{d,m_{j}}{\lambda_{i}}=0,$ (10)

where the symmetric square matrix $\Lambda$ has elements

$\Lambda_{j,j^{\prime}}^{\lambda}=\sum_{a=\pm,k,l}b_{a,k,l}^{\lambda}\braket{d,m_{j}\mid\psi_{a,k,l}}\braket{\psi_{a,k,l}\mid
d,m_{j^{\prime}}}.$ (11)

Leaving a detailed calculation to the supplementary material, it can be shown
that there exists two distinct cases :

#### Case i

For $x+y$ odd, then

$\displaystyle
N\Lambda_{j,j^{\prime}}^{\lambda}\sim\left\\{\begin{matrix}\frac{4}{\lambda}-\lambda\frac{N\ln{N}}{\pi}&\text{if
}j=j^{\prime}\\\ 0&\text{otherwise},\\\ \end{matrix}\right.$ (12)

#### Case ii

For $x+y$ even :

$\displaystyle
N\Lambda_{j,j^{\prime}}^{\lambda}\sim\left\\{\begin{matrix}\frac{4}{\lambda}-\lambda\frac{N\ln{N}}{\pi}&\text{if
}j=j^{\prime}\\\
\frac{4}{\lambda}-\mathcal{I}-\lambda\mathcal{M}&\text{otherwise},\\\
\end{matrix}\right.$ (13)

where

$\displaystyle\mathcal{I}$
$\displaystyle=\frac{-1}{2}\sum_{k,l=0}^{\sqrt{N}-1}\cos\left(2\pi\frac{kx+ly}{\sqrt{N}}\right)\frac{\sin\frac{2\pi
k}{\sqrt{N}}\sin\frac{2\pi l}{\sqrt{N}}}{1-\cos\frac{2\pi
k}{\sqrt{N}}\cos\frac{2\pi l}{\sqrt{N}}}.$ $\displaystyle\hskip
2.84544pt\mathcal{M}$
$\displaystyle=\sum_{k,l=0}^{\sqrt{N}-1}\frac{\cos\left(2\pi\frac{kx+ly}{\sqrt{N}}\right)}{1-\cos\frac{2\pi
k}{\sqrt{N}}\cos\frac{2\pi l}{\sqrt{N}}}.$ (14)

Now, being the columns of $\Lambda$ linearly dependent, its determinant is
zero. Thus, using Eqs. 12,13, we can compute the roots of the equation
$\det\Lambda=0$ respectively for :

Case i :

$\lambda_{\pm}\sim\pm 2\sqrt{\frac{\pi}{N\ln(N)}},$ (15)

and for Case ii :

$\lambda_{\pm}\sim\frac{-\mathcal{I}\pm\sqrt{\mathcal{I}^{2}+32(\frac{N\ln{N}}{\pi}+\mathcal{M})}}{2(\frac{N\ln{N}}{\pi}+\mathcal{M})}.$
(16)

Having calculated lambdas allows us to explicitly get all the coefficients
$\beta_{i,j}$ in the Eq. 5. Let us study Case i and Case ii separately. In the
former, the $\lambda_{i}$ are equal up to a sign, thus the Eq. 5 reduces to :

$p(t)\sim 4|\beta_{+,j}|^{2}\sin^{2}(\lambda t+\text{cte})+O(\epsilon_{m})$
(17)

where $\lambda=|\lambda_{\pm}|$ and $4|\beta_{+,j}|^{2}\sim\frac{\pi}{8\ln N}$
and the hitting time $t_{opt}\sim\frac{\sqrt{\pi N\ln N}}{4}$. Thus the
overall complexity time is $O(\sqrt{N\ln N})$ for a success probability
scaling as $O(\ln^{-1}N)$, in line with previous QW search for only one
element on the grid. We know that this bound is unlikely to be improved, given
the strong arguments given by [18, 19, 20]. This is not unexpected because the
QW operator acts independently onto the sub-lattice
$\mathcal{L}_{e}=\\{(i,j)\mid i+j\text{ even}\\}$, with even vertices and the
one with odd vertices $\mathcal{L}_{o}=\\{(i,j)\mid i+j\text{ odd}\\}$, as we
can see in Fig. 1.a. This is due to the bipartite nature of a QW in discrete
time makes. So that, if the two searched elements are located respectively on
white cells and black cells of a chessboard, the QWsearch is running two
parallel and independent searching.

1(a) Optimal configuration. The searched elements are respectively on the odd
and even partition of the grid. The two elements are searched independently.
The algorithm is always optimal for this configuration.

1(b) Optimal configuration. Here the optimality of the algorithm is ensured by
the condition $\mathcal{I}\ll\mathcal{M}+\frac{N\ln{N}}{\pi}$, leading to a
success probability $O(1/\ln N)$

1(c) Non-optimal configuration. For this configuration the condition
$\text{min}(x,n-x)\text{min}(y,n-x)\geq n$ does not hold, leading to a success
probability $O(1/N)$

Figure 1: Three examples of spatial configuration for which the QWSearch
algorithms behaves differently.

Now let us consider Case ii. The coefficients
$\beta_{i,j}~{}=~{}\braket{d,m_{j}}{\lambda_{i}}\braket{\lambda_{i}}{\psi(0)}$
in the Eq. 5 read :

$|\beta_{i,j}|^{2}\sim\frac{N}{\lambda_{i}^{2}(\frac{N\ln
N}{\pi}+\mathcal{M})^{2}}.$

Notice that there are cases where we can manifestly recover optimality. One
example is given in Fig. 1.b. Indeed, in the limit

$\mathcal{I}^{2}\ll\mathcal{M}+\frac{N\ln{N}}{\pi}$ (18)

the
$\lambda_{\pm}\sim\pm\frac{2\sqrt{2}}{\sqrt{\mathcal{M}+\frac{N\ln{N}}{\pi}}},$
and again the success probability reduces to Eq. 17 with
$\lambda=|\lambda_{\pm}|$ and $|\beta_{+,j}|^{2}\sim\frac{N}{8(\frac{N\ln
N}{\pi}+\mathcal{M})}$. This results lead to the very same complexity we have
in Case i. It is easy to convince oneself that condition expressed by the Eq.
18 is satisfied in a finite number of cases. In fact, after have computed the
sums in the Eq. Case ii and simplified for large $n=\sqrt{N}$,
$\mathcal{I}=O\left(\frac{n^{2}}{\text{min}(x,n-x)\text{min}(y,n-x)}\right)$
and $0\leq\mathcal{M}\leq N\log N$. Then the sufficient condition for a given
spatial configuration to be optimal is $\text{min}(x,n-x)\text{min}(y,n-x)\geq
n.$ But how many of them do not meet the above condition ? It is possible to
prove that they are always bounded by $O(n\ln{n})=O(\sqrt{N}\ln{\sqrt{N}})$.
Although finite, this number is statistically negligible for large values of
$n$. In fact, if one tosses a random configuration, then the probability to
find a non-optimal configuration, such as one in Fig. 1.c, is given by
$O(\sqrt{N}\ln{\sqrt{N}}/N)\xrightarrow{N\rightarrow\infty}0$.

Extending the results obtained in this letter to more than two elements is
mathematically prohibitive, although feasible in theory. In any case, however,
it is possible to show that the number of searched elements, at constant grid
size, dramatically affects the advantage over classical search algorithms.
Let’s note $M$ the number of marked vertices and $\tau=M/N$. In order to
search $M$ vertices, we may define the following procedure: (i) Shuffle the
grid to get a random configuration; (ii) Apply the searching algorithm for
$t_{opt}=\lfloor\sqrt{\pi N\ln N}/4\rfloor$ steps; (iii) Measure the final
state over the computational basis of the walker. The Fig. 2 shows the average
success probability in function of $\tau$ for several grid sizes. Quite
remarkably the success probability does not depend on the grid size.
Furthermore, for $\tau$ sufficiently large, the quantum advantage is
completely lost and the success probability coincides with the classical one.

Figure 2: Average probability of success of QW Search assuming uniform
distribution of the marked elements configuration in function of $\tau=M/N$.
Figure 3: Critical number of marked elements $M_{c}$ in function of the grid
size $N$.

To best characterize at what point the advantage is lost, in Fig. 3 we compare
the success probability of our quantum algorithm with respect to a classical
algorithm. For the same number of query, we consider a stochastic search
algorithm with success probability :
$p_{cl}=1-\frac{\binom{N-t_{opt}}{M}}{\binom{N}{M}}\sim 1-(1-\tau)^{t_{opt}}$.
Using a numerical solver (here a binary search), Fig. 3 shows the critical
number of marked vertices $M_{c}$ upon which the quantum advantage is lost
(i.e. the classical algorithm has a higher probability of success).

#### Discussion

In this letter, we have provided definitive proof that the spatial search
algorithm, based on a quantum discrete-time walker, is not always optimal. We
have proved the existence of an upper bound on the number of sub-optimal
configurations, which remains small for large grid sizes. The proof was
obtained in the special case of only two elements searched on the rectangular
grid. We have also shown that there is strong numerical evidence that the
advantage of the quantum algorithm depends not only on the spatial
configuration but also on the ratio of the number of elements searched over
the grid size. Such evidence is crucial in order to be able to use the quantum
search algorithm correctly, being sure of a quantum advantage, albeit a
polynomial one. The sufficiency conditions for optimality are thus relevant
for a wide range of applications where the searching algorithm is used as
subroutine : from simulation to optimisation, from machine learning to
distributed algorithmic.

#### Acknowledgements

This work is supported by grant DisQC ANR-22-CE47-0002-01 from the French
National Research Agency and the Amidex fondation.

#### Author Contribution Statement

The authors confirm contribution to the paper as follows: study conception and
design : GDM and MR; data collection: MR; analysis and interpretation of
results: GDM, MR, HK; draft manuscript preparation : GDM and MR. All authors
reviewed the results and approved the final version of the manuscript.

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##  Appendix A Supplementary material

In this section, we give further detail on the computations ranging from
Equation 10 to the coefficients $\beta_{i,j}$.

Before that, we provide the explicit expressions for the eigenvalues and
eigenvectors of $U$ :

$\displaystyle\ket{\psi_{a,k,l}}$ $\displaystyle=$
$\displaystyle\ket{v_{a,k,l}}\otimes\frac{1}{\sqrt{N}}\sum_{j,j^{\prime}}\exp\left(2i\pi\frac{kj+lj^{\prime}}{\sqrt{N}}\right)\ket{j,j^{\prime}}$
(19) $\displaystyle\phi_{a,k,l}$ $\displaystyle=$ $\displaystyle
a\arccos\left(\cos\frac{2\pi k}{\sqrt{N}}\cos\frac{2\pi l}{\sqrt{N}}\right)$
(20) $\displaystyle|\braket{d\mid v_{a,k,l}}|^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}-a\frac{1}{4}\frac{\sin\frac{2\pi
k}{\sqrt{N}}\sin\frac{2\pi
l}{\sqrt{N}}}{\sin\phi_{+,k,l}}\mathbb{1}_{\phi_{a,k,l}\neq 0}$ (21)

###  A.1 Computation of $\Lambda^{\lambda}$

We recall that

$\Lambda_{j,j^{\prime}}^{\lambda}=\sum_{a=\pm,k,l}b_{a,k,l}^{\lambda}\braket{d,m_{j}\mid\psi_{a,k,l}}\braket{\psi_{a,k,l}\mid
d,m_{j^{\prime}}}.$ (22)

In order to simplify the coefficients of $\Lambda^{\lambda}$, we use the
explicit form of $\psi_{a,k,l}$ to show that

$\displaystyle\braket{d,m_{j}\mid\psi_{a,k,l}}\braket{\psi_{a,k,l}\mid
d,m_{j^{\prime}}}$ $\displaystyle=\braket{d\mid
v_{a,k,l}}\braket{m_{j}\mid\psi_{a,k,l}}\braket{v_{a,k,l}\mid
d}\braket{\psi_{a,k,l}\mid m_{j^{\prime}}}$
$\displaystyle=\frac{|\braket{d\mid
v_{a,k,l}}|^{2}}{N}\exp\left(2i\pi\frac{k(x_{j}-x_{j^{\prime}})+l(y_{j}-y_{j^{\prime}})}{\sqrt{N}}\right).$

then,

$\braket{d,m_{j}\mid\psi_{a,k,l}}\braket{\psi_{a,k,l}\mid
d,m_{j^{\prime}}}=\left\\{\begin{matrix}\frac{|\braket{d\mid
v_{a,k,l}}|^{2}}{N}&\text{if }j=j^{\prime}\\\ \frac{|\braket{d\mid
v_{a,k,l}}|^{2}}{N}e^{\pm 2i\pi\frac{kx+ly}{\sqrt{N}}}&\text{otherwise}.\\\
\end{matrix}\right.$ (23)

The main idea now is to use Eq. 9 to cut the coefficients
$\Lambda^{\lambda}_{j,j^{\prime}}$ into three cases. Let us start for the
coefficients $\Lambda^{\lambda}_{j,j}$ :

$\displaystyle N\Lambda^{\lambda}_{j,j}$
$\displaystyle=N\sum_{a=\pm,k,l}b_{a,k,l}^{\lambda}\braket{d,m_{j}\mid\psi_{a,k,l}}\braket{\psi_{a,k,l}\mid
d,m_{j}}$ according to Equation 10
$\displaystyle=\sum_{a=\pm,k,l}b_{a,k,l}^{\lambda}|\braket{d\mid
v_{a,k,l}}|^{2}$ using Equation 23
$\displaystyle\sim\sum_{\phi_{a,k,l}=0}\frac{2}{\lambda}|\braket{d\mid
v_{a,k,l}}|^{2}-\sum_{\phi_{a,k,l}\neq
0}\frac{(\sin\phi_{a,k,l})+\lambda}{1-\cos\phi_{a,k,l}}|\braket{d\mid
v_{a,k,l}}|^{2}$ using Equation 9
$\displaystyle=\frac{1}{\lambda}\sum_{\phi_{a,k,l}=0}1+\frac{1}{2}\sum_{\phi_{+,k,l}\neq
0}\frac{\sin\frac{2\pi k}{\sqrt{N}}\sin\frac{2\pi
l}{\sqrt{N}}}{1-\cos\frac{2\pi k}{\sqrt{N}}\cos\frac{2\pi
l}{\sqrt{N}}}-\lambda\sum_{\phi_{+,k,l}\neq 0}\frac{1}{1-\cos\frac{2\pi
k}{\sqrt{N}}\cos\frac{2\pi l}{\sqrt{N}}}$ $\displaystyle\text{using Equation
\ref{eq:U_explicit}}.$

First, it is straightforward to prove that there are four possible cases for
$\phi_{a,k,l}=0$, thus $\sum_{\phi_{a,k,l}=0}1=4$. Furthermore, one can show
that the second sum $\displaystyle\frac{1}{2}\sum_{\phi_{+,k,l}\neq
0}\frac{\sin\frac{2\pi k}{\sqrt{N}}\sin\frac{2\pi
l}{\sqrt{N}}}{1-\cos\frac{2\pi k}{\sqrt{N}}\cos\frac{2\pi
l}{\sqrt{N}}}=\frac{1}{2}\sum_{\phi_{k,l}\neq 0}S_{k,l}$ is vanishing, using
the symmetry $S_{k,l}~{}=~{}-S_{n-k,l}$. Moreover the last sum can be
approximated using integral bounding, resulting in
$\displaystyle\sum_{\phi_{+,k,l}\neq 0}\frac{1}{1-\cos\frac{2\pi
k}{\sqrt{N}}\cos\frac{2\pi l}{\sqrt{N}}}\sim\frac{1}{\pi}N\ln N$. And finally,

$\Lambda^{\lambda}_{j,j}\sim\frac{4}{\lambda}-\lambda\frac{N\ln N}{\pi}.$ (24)

Let us now compute the other half of the $\Lambda^{\lambda}$’s coefficients.
By using a similar procedure :

$N\Lambda^{\lambda}_{j,j}\sim\frac{1}{\lambda}\sum_{\phi_{a,k,l}=0}e^{\pm
2i\pi\frac{kx+ly}{\sqrt{N}}}+\frac{1}{2}\sum_{\phi_{+,k,l}\neq
0}\frac{\sin\frac{2\pi k}{\sqrt{N}}\sin\frac{2\pi
l}{\sqrt{N}}}{1-\cos\frac{2\pi k}{\sqrt{N}}\cos\frac{2\pi l}{\sqrt{N}}}e^{\pm
2i\pi\frac{kx+ly}{\sqrt{N}}}-\lambda\sum_{\phi_{+,k,l}\neq
0}\frac{1}{1-\cos\frac{2\pi k}{\sqrt{N}}\cos\frac{2\pi l}{\sqrt{N}}}e^{\pm
2i\pi\frac{kx+ly}{\sqrt{N}}}.$

Now, we can study separately case i and case ii.

#### Case i: $x+y$ odd

All three above addends are identically vanishing by symmetry. In particular,
for each addend $S_{k+\frac{n}{2},l+\frac{n}{2}}=(-1)^{x+y}S_{k,l}$.
Consequently,

$\Lambda^{\lambda}_{j,j^{\prime}}=0,$ (25)

with $j\neq j^{\prime}$ and $x+y$ odd.

#### Case ii: $x+y$ even

The first sum $\sum_{\phi_{a,k,l}=0}e^{\pm 2i\pi\frac{kx+ly}{\sqrt{N}}}=4$.
The other two sums are harder to compute. We can, however, retrieve the
expressions of $\mathcal{I}$ and $\mathcal{M}$ by using the symmetry
$S_{n-k,n-l}=\overline{S_{k,l}}$. This symmetry implies that the sum is real.
We can thus discard the imaginary part. In conclusion

$N\Lambda^{\lambda}_{j,j}\sim\frac{4}{\lambda}-\mathcal{I}-\lambda\mathcal{M},$

where

$\mathcal{I}=\frac{-1}{2}\sum_{\phi_{+,k,l}\neq 0}\frac{\sin\frac{2\pi
k}{\sqrt{N}}\sin\frac{2\pi l}{\sqrt{N}}}{1-\cos\frac{2\pi
k}{\sqrt{N}}\cos\frac{2\pi
l}{\sqrt{N}}}\cos\left(2\pi\frac{kx+ly}{\sqrt{N}}\right)\qquad\text{and}\qquad\mathcal{M}=\sum_{\phi_{+,k,l}\neq
0}\frac{1}{1-\cos\frac{2\pi k}{\sqrt{N}}\cos\frac{2\pi
l}{\sqrt{N}}}\cos\left(2\pi\frac{kx+ly}{\sqrt{N}}\right).$

###  A.2 Computation of $\lambda_{i}$ and $|\beta_{i,j}|^{2}$

In this section we give further details about how we obtained each
$\lambda_{i}$. We assume that $x+y\text{ even
}\Rightarrow\mathcal{I}^{2}\ll\mathcal{M}+\frac{N\ln{N}}{\pi}$. We start by
using Eqs. 12 and 13 combined with $\det\Lambda^{\lambda}=0$ (which can be
deduced from Eq. 10) to get $\lambda_{\pm}$. Once done, we note that
$|\braket{d,m_{0}\mid\lambda_{\pm}}^{2}=|\braket{d,m_{1}\mid\lambda_{\pm}}^{2}$
because of the symmetry of $\Lambda^{\lambda}$. Combining the latter with the
Eq. 7, we can deduce that

$|\braket{\lambda_{a}}{\psi(0)}|^{2}=\left|1+ib^{\lambda_{a}}_{+,0,0}\right|^{2}\left|\sum_{j=0,1}\braket{\psi_{\pm
k,l}}{d,m_{j}}\braket{d,m_{j}}{\lambda_{a}}\right|^{2}\sim\frac{8|\braket{d,m\mid\lambda}|^{2}}{N\lambda^{2}}.$
(26)

where $\ket{\psi(0)}=\ket{\psi_{+,0,0}}$. One can similarly use Eq. 7 in
$\sum_{\pm,k,l}|\braket{\psi_{\pm,k,l}\mid\lambda}|^{2}=1$ to show that

$|\braket{d,m\mid\lambda}|^{2}\sim\frac{1}{4}\frac{N}{\frac{N\ln
N}{\pi}+\mathcal{M}\mathbb{1}_{\\{x+y\text{ even}\\}}}.$ (27)

Combining the above result into the expression of
$|\braket{\lambda_{a}}{\psi(0)}|^{2}$, we get

$|\braket{\lambda_{a}}{\psi(0)}|^{2}\sim\frac{2+2.\mathbb{1}_{x+y\text{
even}}}{\lambda^{2}\left(\frac{N\ln
N}{\pi}+\mathcal{M}\mathbb{1}_{\\{x+y\text{ even}\\}}\right)}.$ (28)

Again, let us consider case i and case ii separately.

#### Case i: $x+y$ odd

$\det\Lambda^{\lambda}=0\Leftrightarrow\frac{4}{\lambda}-\lambda\frac{N\ln
N}{\pi}\sim 0\Leftrightarrow\lambda^{2}\sim 4\frac{\pi}{N\ln
N}\Leftrightarrow\lambda_{\pm}\sim\pm 2\sqrt{\frac{\pi}{N\ln N}}.$ (29)

We recognise the Eq. 15. We can now solve explicitly Eqs. 27 and 28 for Case i
:

$|\braket{d,m\mid\lambda_{\pm}}|^{2}\sim\frac{\pi}{4\ln
N}\qquad\text{and}\qquad|\braket{\lambda_{\pm}}{\psi(0)}|^{2}\sim\frac{1}{2}.$
(30)

Notice that the latter expression implies that $\epsilon_{m}\to 0$. We recover
the right expression $4|\beta_{+,j}|^{2}\sim\frac{\pi}{8\ln N}$.

#### Case ii: $x+y$ even

$\displaystyle\det\Lambda^{\lambda}=0$
$\displaystyle\Leftrightarrow\left(\Lambda^{\lambda}_{0,0}\right)^{2}-\left(\Lambda^{\lambda}_{1,0}\right)^{2}=0$
because $\Lambda^{\lambda}$ is symmetric
$\displaystyle\Leftrightarrow\left(\Lambda^{\lambda}_{0,0}-\Lambda^{\lambda}_{1,0}\right)\left(\Lambda^{\lambda}_{0,0}+\Lambda^{\lambda}_{1,0}\right)=0$
$\displaystyle\Leftrightarrow\left[\mathcal{I}-\lambda\left(\frac{N\ln
N}{\pi}-\mathcal{M}\right)\right]\left[\frac{8}{\lambda}-\mathcal{I}-\lambda\left(\frac{N\ln
N}{\pi}+\mathcal{M}\right)\right]=0$ using Equation 13
$\displaystyle\Leftrightarrow\lambda_{\pm}\in\left\\{\frac{-\mathcal{I}+\sqrt{\mathcal{I}^{2}+32(\frac{N\ln{N}}{\pi}+\mathcal{M})}}{2(\frac{N\ln{N}}{\pi}+\mathcal{M})},\frac{-\mathcal{I}-\sqrt{\mathcal{I}^{2}+32(\frac{N\ln{N}}{\pi}+\mathcal{M})}}{2(\frac{N\ln{N}}{\pi}+\mathcal{M})},\frac{\mathcal{I}}{\frac{N\ln{N}}{\pi}-\mathcal{M}}\right\\}.$

Using the hypothesis $\mathcal{I}^{2}\ll\mathcal{M}+\frac{N\ln{N}}{\pi}$ and
the fact that $0\leq\mathcal{M}\leq\frac{N\ln N}{\pi}$, it holds :

$\lambda_{\pm}\sim\frac{-\mathcal{I}-\pm\sqrt{\mathcal{I}^{2}+32(\frac{N\ln{N}}{\pi}+\mathcal{M})}}{2(\frac{N\ln{N}}{\pi}+\mathcal{M})}\sim\pm\frac{2\sqrt{2}}{\sqrt{\frac{N\ln{N}}{\pi}+\mathcal{M}}}=O\left(\frac{1}{\sqrt{N\ln
N}}\right).$

We recognise Eq. 16.Again we solve the Eqs. 27 and 28 for Case ii and note
that

$|\braket{d,m\mid\lambda_{\pm}}|^{2}\sim\frac{1}{4}\frac{N}{\frac{N\ln
N}{\pi}+\mathcal{M}}\qquad\text{and}\qquad|\braket{\lambda_{\pm}}{\psi(0)}|^{2}\sim\frac{1}{2}.$
(31)

As for case i, the latter implies $\epsilon_{m}\to 0$, meaning that all other
eigenvalues of $U^{\prime}$ can be neglected (including the last solution of
$\det\Lambda^{\lambda}=0$). We get finally
$4|\beta_{+,j}|^{2}\sim\frac{1}{2}\frac{N}{\frac{N\ln N}{\pi}+\mathcal{M}}$
and using $M\leq N\ln N$, we can deduce that the success probability
$p_{succ}~{}|\beta_{+,j}|^{2}=O(\ln^{-1}N)$.


# Gottesman-Kitaev-Preskill encoding in continuous modal variables of single
photons

Éloi Descamps1 Arne Keller1,2 Pérola Milman1 corresponding author:
<EMAIL_ADDRESS>1Laboratoire Matériaux et Phénomènes Quantiques,
Université Paris Diderot, CNRS UMR 7162, 75013, Paris, France  2Department de
Physique, Université Paris-Saclay, 91405 Orsay Cedex, France

###### Abstract

GKP states, introduced by Gottesman, Kitaev, and Preskill, are continuous
variable logical qubits that can be corrected for errors caused by phase space
displacements. Their experimental realization is challenging, in particular
using propagating fields, where quantum information is encoded in the
quadratures of the electromagnetic field. However, travelling photons are
essential in many applications of GKP codes involving the long-distance
transmission of quantum information. We introduce a new method for encoding
GKP states in propagating fields using single photons, each occupying a
distinct auxiliary mode given by the propagation direction. The GKP states are
defined as highly correlated states described by collective continuous modes,
as time and frequency. We analyze how the error detection and correction
protocol scales with the total photon number and the spectral width. We show
that the obtained code can be corrected for displacements in time-frequency
phase space - which correspond to dephasing, or rotations, in the quadrature
phase space - and to photon losses. Most importantly, we show that generating
two-photon GKP states is relatively simple, and that such states are currently
produced and manipulated in several photonic platforms where frequency and
time-bin biphoton entangled states can be engineered.

Derived from classical error correction protocols, quantum error correction
plays a central role in quantum information theory. The counterintuitive
features of quantum mechanics are inherently fragile and necessitate error
correction to enable the manifestation of the quantum advantage of protocols
over their classical counterparts. Originally devised for qubits and finite-
dimensional discrete systems [1, 2], quantum error-correcting codes rely on
creating redundant states composed of multiple physical qubits to define
logical qubits. By measuring well chosen observables that do not affect the
state of the logical qubit - the stabilizers - one can detect and correct
errors affecting the physical qubits. Entanglement is usually an important
ingredient in error correction, as exemplified by the states that withstand
physical qubit flips and dephasing [3, 4].

When dealing with continuous variables such as position and momentum, it is
also possible to encode quantum information into states that can be corrected
for errors [5, 6, 7, 8]. One of the most successful model for error correction
in continuous variables is known as the GKP code [9] (see, for instance, [10]
for a review), named after its creators: Gottesman, Kitaev, and Preskill. GKP
states are a direct extension of the discrete variable codes into the
continuous domain [11, 12] and are correctable for errors modeled as
displacements in phase space. In the realm of quantum optics, one usually
thinks of encoding GKP states using two orthogonal quadratures of the
electromagnetic field, since the associated observables obey the same
commutation relation as position and momentum. In such a system, phase space
displacements can be the consequence of unwanted interference with a parasite
classical field or model photon losses [13, 14]. However, a main difficulty
consists of the experimental production of such highly non-classical states,
which involves, for instance, the prior (non-deterministic) production of
Schrödinger cat-like states (or, in practice, kittens) that are made to
interfere [15]. Several proposals exist [16, 17, 18, 19, 20, 21], as well as a
first experimental realization [22]. Of course, one can also encode GKP states
using other bosonic systems, as superconducting circuits [23] or the motional
states of trapped ions [24, 25]. However, building a robust code adapted to
propagating fields is clearly of major importance if one wills to transmit
quantum information, in particular, to several independent users [26].

In the present Letter we propose a new way to encode GKP states in quantum
optics using an original approach to continuous variables. We use continuous
collective variables of single photons occupying distinct auxiliary modes, as
the propagation direction [27], to define redundant states. Our model can
apply to different single photon continuous modes, as time and frequency (that
we discuss in detail), the transverse position and momentum [28], the
propagation direction [29, 30] and also to the collective modes of massive
particles, as the normal modes of trapped ions [31, 32]. Using the established
formalism, we theoretically investigate how GKP codewords can be defined in a
system comprising $n$ individual photons, elucidating how some known
properties of the encoded states can be retrieved, as for instance the scaling
of the error rate with the number of photons, their possibility to recover
from photon losses and the effects of imperfect preparation and measurement.
Notably, we show that in the time-frequency (TF) encoding scheme, these
properties have a fundamentally different physical origin compared to
quadrature-based (QB) encoding. Finally, we demonstrate that the generation
and manipulation of TF GKP states in quantum optics is already a reality in
laboratories. This is particularly evident in experimental setups where
entangled photon pairs with a correlated comb-like temporal or spatial
structure are observed, as exemplified in references [33, 34, 35, 36, 37, 38,
39, 40, 26, 41, 42, 43, 44, 45]. Consequently, these experimental platforms
and the associated quantum protocols can immediately benefit from our
findings.

We start by defining as ${\cal S}_{n}$ the subspace consisting of $n$ single
photons occupying each an auxiliary mode, as the propagation direction.
Photons are characterized by a collective spectral function that also depends
on the auxiliary mode. The auxiliary modes can be seen as external degrees of
freedom and frequency as internal degrees of freedom [46]. Pure states are
written as

$\ket{\psi}=\int
d\omega_{1}...d\omega_{n}f(\omega_{1},...,\omega_{n})\ket{\omega_{1},...,\omega_{n}},$
(1)

where $f$ is a normalized function (the spectral amplitude) that determines
the properties of state (1), as entanglement and its mode decomposition [47,
48], and $\hat{a}_{i}^{\dagger}(\omega_{i})\ket{0}=\ket{\omega_{i}}$. In
${\cal S}_{n}$, errors are represented in the basis of time and frequency
displacements, and they can affect the photons locally, i.e., they act
independently on the photons of each auxiliary mode, a situation similar to
the one affecting a collection of physical qubits. Such displacements are
described by operators acting on a given mode $j\in\\{1,...,n\\}$ as
$\hat{D}_{\hat{\omega}_{j}}(\delta_{t_{j}})=e^{-i\hat{\omega}_{j}\delta_{t_{j}}}$
and
$\hat{D}_{\hat{t}_{j}}(\delta_{{\omega}_{j}})=e^{-i\hat{t}_{j}\delta_{{\omega}_{j}}}$,
where $\hat{\omega}_{i}=\int
d\omega\omega\hat{a}_{i}^{\dagger}(\omega)\hat{a}_{i}(\omega)$ and
$\hat{t}_{i}=\int dtt\tilde{\hat{a}}_{i}^{\dagger}(t)\tilde{\hat{a}}_{i}(t)$,
with $\tilde{\hat{a}}_{i}(t)=\frac{1}{\sqrt{2\pi}}\int d\omega e^{i\omega
t}\hat{a}(\omega)$, where $\hat{a}_{i}^{\dagger}(\eta)$ creates one photon at
frequency $\eta$ at the $i$-th auxiliary mode and
$[\hat{\omega}_{k},\hat{t}_{j}]=i\delta_{k,j}\int
d\omega\hat{a}_{k}^{\dagger}(\omega)\hat{a}_{k}(\omega)=\hat{n}_{k}i\delta_{k,j}$
($\hat{n}_{k}=\mathbb{1}$ on ${\cal S}_{n}$) [49, 27]. Now we show that
entangled states of $n$ photons in the TF continuous variables sharing the
same properties of GKP states can be corrected for this type of errors. Such
states can be written in the general separable form in collective variables:

$\ket{\overline{k}}=\int
d\Omega_{1}...d\Omega_{n}F_{k}(\Omega_{1})\Pi_{i>1}^{n}G_{i}(\Omega_{i})\ket{\omega_{1},...,\omega_{n}},$
(2)

where $\Omega_{j}=\sum_{i}^{n}\alpha_{i,j}\omega_{i}$ are collective
variables, $\alpha_{i,j}$ is an invertible matrix with
$\alpha_{i,j}\in\\{-1/\sqrt{n},1/\sqrt{n}\\}$ and $k\in\\{0,1\\}$. We consider
for simplicity and without loss of generality that
$\alpha_{i,1}=1/\sqrt{n}~{}\forall~{}i$, and that $n=2^{m}$, $m\in\mathbb{Z}$.
The matrix $\alpha$ is unitary, hence
$\omega_{i}=\sum_{j}\alpha_{j,i}\Omega_{j}$.

An ideal $n$ photon GKP state $\ket{\overline{k}}$ in ${\cal S}_{n}$ can be
defined from (2) using
$F_{k}(\Omega_{1})=\sum_{s=-\infty}^{s=\infty}\delta(\Omega_{1}-(2s+k)\Omega_{o})$,
where $\delta$ is the Dirac delta function and $\Omega_{o}$ is an arbitrary
(constant) frequency. Hence, the logical qubits
$\ket{\overline{0}(\overline{1})}$ are non-physical states formed by an
infinity of peaks localized at frequencies which are integer multiples of
$\Omega_{o}$, and $2\Omega_{o}$ is the peak interspacing in each logical qubit
(notice that the choice of $\alpha_{i,j}$ means that we have supposed that all
the photons are equally spaced [50]). States $\ket{\overline{k}}$ are defined
uniquely using the collective variable $\Omega_{1}$. The functions $G_{i}$ are
arbitrary and their role, not crucial for the code working principles, will be
discussed later in this manuscript. Thus, all the relevant information for
error diagnosis and correction is contained only in variable $\Omega_{1}$, and
we disregard the information contained in $\Omega_{i>1}$. This type of
situation is current in quantum optics, and different physical properties, as
group and phase velocity for multi-modal fields, are associated to different
collective variables. We can mention for instance the Hong-Ou-Mandel (HOM)
experiment [51] and its generalization to many photons [52, 53]. In the two-
photon configuration, the variable
$\Omega_{1}=(\omega_{1}-\omega_{2})/\sqrt{2}$ is directly measured (group
velocity), while the information in variable
$\Omega_{2}=(\omega_{1}+\omega_{2})/\sqrt{2}$ (phase velocity) is totally
disregarded [54]. However, by combining different interferometric techniques
[55], one can access different collective variables measuring not only
frequency but other continuous modes, as the transverse position and momentum
[28, 39].

TF GKP states differ from GKP states defined by optical combs in single mode
states using spectral engineering of classical (coherent) states or single
photons [33, 56]: the TF GKP states are entangled states of $n$ photons
displaying quantum features that will prove to be essential to determine the
scaling of the error tolerance with the number of photons $n$. In addition, TF
GKP are intrinsically multimode states relying on the particle-mode non-
separability, contrary to the QB GKP states - that can be defined in single
modes -, and differently from [57] where modes are used to define a QB multi-
dimensional GKP state.

An alternative and intuitive way to express $\ket{\overline{k}}$ is setting
$G_{i}(\Omega_{i})=\delta(\Omega_{i})~{}\forall~{}i$ in (2) [58]:

$\ket{\overline{k}}=\sum_{s=-\infty}^{\infty}\ket{(2s+k)\frac{\Omega_{o}}{\sqrt{n}}}_{1}...\ket{(2s+k)\frac{\Omega_{o}}{\sqrt{n}}}_{n},$
(3)

Analogously to the QB GKP states, we can identify non-Hermitian operators that
act in $\ket{\overline{k}}$ as Pauli matrices [33]. One way to see this is
using displacements in the collective TF variables,
$\hat{D}_{\hat{T}_{1}}(\Delta_{\omega})=e^{-i\hat{T}_{1}\Delta_{\omega}}$ and
$\hat{D}_{\hat{\Omega}_{1}}(\Delta_{t})=e^{-i\hat{\Omega}_{1}\Delta_{t}}$,
where ${\Delta_{t(\omega)}}\in\mathbb{R}$,
$\hat{\Omega}_{1}=\sum_{i}^{n}\hat{\omega}_{i}/\sqrt{n}$ and
$\hat{T}_{1}=\sum_{i}^{n}\hat{t}_{i}/\sqrt{n}$, with
$[\hat{\Omega}_{1},\hat{T}_{1}]=\mathbb{1}i$ in ${\cal S}_{n}$ [49].
Collective operators can also be associated to variables $\Omega_{i>1}$. By an
appropriate choice of ${\Delta_{t(\omega)}}$, we can define the Pauli-like
operators in the TF GKP subspace as $\hat{X}=e^{-i\Omega_{o}\hat{T}_{1}}$ and
$\hat{Z}=e^{-i\hat{\Omega}_{1}T_{o}}$, with $T_{o}=\pi/\Omega_{o}$ and
$\hat{Y}=i\hat{Z}\hat{X}$ so $\ket{\overline{1}}=\hat{X}\ket{\overline{0}}$.
As we will see in the following, this construction of the logical operators
$\hat{X}$, $\hat{Y}$ and $\hat{Z}$ is not unique.

Figure 1: Left: A schematic example of a TF GKP state with peak spacing
$2\Omega_{o}$ in the collective variable $\Omega_{1}$ and with a finite
spectral width, where $\Omega_{\perp}$ is a variable orthogonal to
$\Omega_{1}$. The figure is a projection in the two-dimensional space formed
by $\Omega_{1}$ and $\Omega_{\perp}$. Right: The effect of a displacement
$\sqrt{n}\Omega_{o}$ in the local variable $\omega_{j}$. The TF GKP state is
indeed displaced in variable $\Omega_{1}$ thanks to a local displacement with
the proper choice of amplitude. The displacement in the orthogonal direction
$\Omega_{\perp}$ is not relevant since we ignore the information it contains.

The formal construction of the TF GKP code is identical to the QB one, so all
the properties of the latter can be retrieved here, but now associated to
different continuous variables: states (3) enable correcting for collective
time and frequency errors corresponding to TF displacements such that
$|\Delta_{\omega}|<\Omega_{o}/2$ and $|\Delta_{t}|<\pi/(2\Omega_{o})$,
delimiting a TF phase space area of correctable errors satisfying
$4|\Delta_{\omega}||\Delta_{t}|<\pi$. Physically, displacement errors can
correspond to imperfect aligning of an interferometer and the effects of a
non-linear device as an optical fiber.

Restricting to collective errors is missing our main goal, which is creating
states which are robust against local displacements in the TF variables of
each photon. Physically, this corresponds to a situation where each photon
occupying a different propagation mode is distributed throught different
channels and frequency and time displacements occur independently in each mode
(and consequently each photon). Such TF displacements are rotations in the
quadrature phase space, a type of error against which the usual QB GKP are not
very efficient [10] and require using rotation symmetric states in the
quadrature space [59]. As for states (3), they are robust against global and
local rotations by construction. We now study the effect of local noise in the
TF GKP states (2). We have:

$\displaystyle\hat{D}_{\hat{\omega}_{j}}(\delta_{t_{j}})\ket{\overline{k}}=$
(4) $\displaystyle\int
d\Omega_{1}...d\Omega_{n}e^{i\frac{\Omega_{1}\delta_{t_{j}}}{\sqrt{n}}}F_{k}(\Omega_{1})\prod\limits_{k>1}^{n}\tilde{G}_{k}(\Omega_{k})\ket{\omega_{1},...,\omega_{n}},$

where
$\tilde{G}_{k}(\Omega_{k})=e^{i\frac{\alpha_{j,k}\Omega_{k}\delta_{t_{j}}}{\sqrt{n}}}G_{k}(\Omega_{k})$.
We can also see that

$\displaystyle\hat{D}_{\hat{t}_{j}}(\delta_{\omega_{j}})\ket{\overline{k}}=$
(5) $\displaystyle\int
d\Omega_{1}...d\Omega_{n}F_{k}\left(\Omega_{1}-\frac{\delta_{\omega_{j}}}{\sqrt{n}}\right)\prod\limits_{k>1}^{n}G_{k}(\Omega_{k}-\bar{\Omega}_{k}^{\prime})\ket{\omega_{1},...,\omega_{n}},$

where $\bar{\Omega}_{k}=\alpha_{j,k}\delta_{\omega_{j}}$. Eqs. (4) and (5)
lead us to an important result: the codewords (3) protect against shifts in
local variables $\omega_{j}$ in a way that scales with $\sqrt{n}$ with the
number of photons $n$. Some examples of correctable errors are then: a single
photon in mode $j$ that is displaced by
$|\delta_{t_{j}}|<\sqrt{n}\pi/(2\Omega_{o})$; up to $\approx\sqrt{n}$ photons
in different modes $j$ that are each displaced by
$|\delta_{t_{j}}|<\pi/(2\Omega_{o})$; $n$ photons that are each equally
displaced by $|\delta_{t_{j}}|<\pi/(2\Omega_{o}\sqrt{n})$. Thus, if one focus
on local errors, the codewords (3) can protect for them provided that they lie
in a phase space area of size
$4|\sum_{j}^{n}\delta_{\omega_{j}}||\sum_{j}^{n}\delta_{t_{j}}|<n\pi$ [60]. We
can see this as a re-scaling of the TF phase space, since the overall
protection corresponds to the one one would obtain using single photon GKP
states formed by peaks that are distant of $2\sqrt{n}\Omega_{o}$ and
$\sqrt{n}/(2\Omega_{o})$ in time and frequency variables, respectively.
Interestingly, contrary to what one would observe in a classical time
frequency Fourier relation where the dilatation of the frequency space is
accompanied by the shrinking of the time space and vice-versa, the observed
effective phase space dilatation is a consequence of encoding information in
collective variables. Furthermore, this re-scaling is the equivalent of the
photon number dependency of the probability error rate for QB GKP encoding, as
we’ll see later.

The definition of the $\hat{X}$ and $\hat{Z}$ operators is not unique. We can
for instance define $\hat{X}_{j}=e^{i\hat{\omega}_{j}T_{o}\sqrt{n}}$ and check
by computing $\hat{X}_{j}\ket{\bar{k}}$ that $\hat{X}_{j}$ acts in variables
$\Omega_{1}$ in the same way as $\hat{X}$ does (see Fig. 1 and [61]). Using
this, we show how to detect the loss of one photon in mode $j^{\prime}$
(unknown) and adapt to its effects by measuring time and frequency
displacements only. If a photon is lost, the TF GKP state becomes
$\ket{\bar{k}}_{-1}=\int d\omega\hat{a}_{j}(\omega)\ket{\bar{k}}=\hat{\cal
E}_{j}\ket{\bar{k}}$ (we considered that the photon loss rate in independent
of the frequency [62]). Defining
$\hat{S}_{j}=e^{-i\eta_{j}\Omega_{o}\hat{t}_{j}}\hat{X}_{j}^{2}e^{i\eta_{j}\Omega_{o}\hat{t}_{j}}=e^{i2(\hat{\omega}_{j}-\hat{n}_{j}\eta_{j}\Omega_{o})T_{o}\sqrt{n}}$,
$\eta_{j}\in\mathbb{R}$, and only considering the information contained in
variable $\Omega_{1}$, we have that $\hat{S}_{j}\hat{S}_{j+1}$ stabilizes
$\ket{\overline{k}}$ in ${\cal S}_{n}$ if $(\eta_{j}+\eta_{j+1})\sqrt{n}=m$,
$m\in\mathbb{Z}$, for all $j$, and it stabilizes $\hat{\cal
E}_{j^{\prime}}\ket{\bar{k}}=\ket{\bar{k}}_{-1}$ if $j^{\prime}\neq j,j+1$. In
addition, $\hat{S}_{j}\hat{S}_{j+1}\hat{\cal
E}_{j}\ket{\overline{k}}=\hat{\cal
E}_{j}\hat{S}_{j+1}\ket{\overline{k}}=e^{-2\eta_{j}\pi\sqrt{n}}\hat{\cal
E}_{j}\ket{\overline{k}}$, so by judiciously choosing $\eta_{i}$ we can detect
a photon loss and the mode from which it was lost (among $j$ and $j+1$ in this
example). Notice that we can also define $\hat{S}=\Pi_{j}\hat{S}_{j}$, which
is also a stabilizer of $\ket{\overline{k}}$. In this case, as detailed in
[61], using that $\hat{S}\hat{\cal
E}_{j}\ket{\overline{k}}=e^{-i\eta_{j}\pi\sqrt{n}}\hat{\cal
E}_{j}\ket{\overline{k}}$ we can detect in a single shot that a photon has
been lost and from which mode, by judiciously choosing $\eta_{j}$’s and if
$\sum_{j}^{n}\eta_{j}\sqrt{n}=m$. Rather than enabling the re-insertion of the
lost photon, the stabilizer measurement provides essential information about
the mode that lost a photon, permitting to adapt the operations and
measurements to a $n-1$ photon configuration. For instance, if a photon is
lost, the collective effects in displacements are smaller in the resulting
state, and in order to have
$e^{2i\hat{\Omega}_{1}T_{o}^{\prime}}\ket{\overline{k}}_{-1}=\ket{\overline{k}}_{-1}$
we must use $T_{o}^{\prime}=T_{o}(n)/(n-1)$.

States (3) are not physical. We’ll follow again the steps of [9] and define
their normalizable version, $\ket{\tilde{0}(\tilde{1})}$, using
$\tilde{F}_{k}(\Omega_{1})=\sum_{s=-\infty}^{s=\infty}e^{-\kappa^{2}(2s+k)\Omega_{o}^{2}}e^{-\frac{(\Omega_{1}-(2s+k)\Omega_{o})^{2}}{\Delta^{2}}}$,
$k\in\\{0,1\\}$. Now, each peak of the TF GKP code has a Gaussian spectrum of
width $\Delta$ (small compared to $\Omega_{o}$) in variable $\Omega_{1}$ and
the comb of peaks distribution is modulated by a Gaussian envelope of width
$\kappa^{-1}$ with $\kappa$ small compared to $T_{o}/\pi$ [9]. We’ll consider
for simplicity and with no lack of generality that $\Delta=\kappa$. A finite
width provides an intrinsic error probability to states
$\ket{\tilde{0}(\tilde{1})}$, which can be seen as perfect states (3) that
have been subjected to a distribution of displacements (errors). Consequently,
an error probability ${\cal E}(\Delta/\Omega_{o})$ is associated to the error
correction protocol through the definition of non-perfectly orthogonal states
as codewords. Before discussing the effect of such imperfection, we’ll model
the physical origin of a finite spectral width by supposing that each photon
is independently displaced and the displacements amplitudes distribution is a
Gaussian of width $\Delta_{j}$. Each photon $j$ in (3) is then described by
state

$\displaystyle\ket{\widetilde{(2s+k)\frac{\Omega_{o}}{\sqrt{n}}}}_{j}=$ (6)
$\displaystyle\int d\omega
e^{-\Delta_{j}^{2}(2s+k)\left(\frac{\Omega_{o}}{\sqrt{n}}\right)^{2}}e^{-\frac{\left(\omega-\frac{(2s+k)\Omega_{o}}{\sqrt{n}}\right)^{2}}{\Delta_{j}^{2}}}\ket{\omega}_{j}.$

The local variables $\omega_{i}$ behave as independent random variables, and
we suppose without loss of generality that $\Delta_{j}=\Delta$. Each $n$
photon peak has now the form
$\Pi_{j}^{n}\ket{\widetilde{(2s+k)\frac{\Omega_{o}}{\sqrt{n}}}}_{j}$, and from
(6) we see that the width of the total state’s temporal envelope is
$\Delta^{-1}$. As for the width of each peak, it’s easy to show by changing
$\omega_{j}$ to the collective variables $\Omega_{i}$ that it is also
described by a Gaussian spectral distribution of width $\Delta$ in all
variables $\Omega_{i}$, leading to $\tilde{F}_{k}(\Omega_{1})$ shown above.
This means that, differently from the QB GKP encoding, the spectral width does
not depend on the average photon number, and the probability of mistaking
$\ket{\tilde{0}}$ and $\ket{\tilde{1}}$ is also given by ${\cal
E}(\Delta/\Omega_{o})=\frac{\Delta}{\pi\Omega_{o}}e^{\frac{-\pi\Omega_{o}^{2}}{4\Delta^{2}}}$,
as in [9]. We should recall however that in the present encoding the effective
displacement on the collective variable $\Omega_{1}$ decreases with
$\sqrt{n}$, increasing the effective distance between peaks. This is
equivalent to modifying the region of potential overlap between the two
supposedly orthogonal QB GKP qubit states [63].

We now discuss the role of a finite frequency width in the collective
variables $\Omega_{j>1}$. For simplicity, we’ll consider $\Omega_{\perp}$ to
be one of these variables, and ignore all the others, effectively considering
a state with spectral amplitude $F_{k}(\Omega_{1})G(\Omega_{\perp})$. Since
many photon pair sources are based on non-linear collective effects, the width
of the spectral distributions $F_{k}$ and $G$ in (2) can be independent and
related to different physical constraints, as for instance energy conservation
or the phase matching condition, as is the case of photons generated by
spontaneous parametric down-conversion (SPDC). If states (2) are perfectly
prepared, the spectral function is indeed separable in variables $\Omega_{1}$
and $\Omega_{\perp}$, and if all the measurements are performed on variable
$\Omega_{1}$, the spectral width or the particular shape of $G$ have no
importance. However, state preparation may be imperfect, which we model here
as a state that can still be separable but in the rotated variables
$\Omega_{1}^{\prime}=\cos\theta\Omega_{1}+\sin\theta\Omega_{\perp}$ and
$\Omega_{\perp}^{\prime}=\cos\theta\Omega_{\perp}-\sin\theta\Omega_{1}$. This
model also applies to the situation of imperfect measurements, where variable
$\Omega_{1}^{\prime}$ is measured instead of $\Omega_{1}$. In these cases, as
shown in Fig. 2 (details in [61]) the peaks’ width in the measured variable is
broadened by an additive factor $\sigma\sin\theta$ and the peak spacing is re-
scaled, becoming $2\Omega_{o}\cos\theta$. This effective width and peak-
spacing can be seen as errors. In order not to significantly affect the code,
the cumulated change in peak spacing
$2\Omega_{o}(1-\cos\theta)(1/(2\Omega_{o}\Delta)$ \- where
$(1/(2\Omega_{o}\Delta)$ is of the order of the number of peaks - should lie
within the peak’s width $\Delta$, leading to
$(1-\cos\theta)\lesssim\Delta^{2}\ll\Omega_{o}T_{o}/\pi=1$. If this condition
is not observed, we can adapt the code to the modified interspacing
$2\Omega_{o}\cos\theta$ (where $\theta$ is known), to which one should
associate an error probability ${\cal E}(\sigma\tan\theta/\Omega_{o})$ in
frequency and ${\cal E}(\tan\theta/(\sigma T_{o}))$ in time [61]. These
adapted errors probabilities lead to $\tan\theta\ll{\rm
min}\\{\pi\sigma/(2\Omega_{o}),\Omega_{o}/(2\sigma)\\}$. Interestingly,
choosing $\sigma\ll\Omega_{o}$ is not the most resilient strategy for this
type of imperfection.

We can compare our results to other encodings based on GKP-like states within
the continuous modes of single photons. In the TF domain, for instance, it is
possible to define GKP qubits using a frequency comb spectral distribution in
single photons. In such scenario, one photon corresponds to one qubit, and the
spectral function of each photon is used to define a two-level-like system
that exhibits local robustness against TF displacements [33, 64]. This is
analogous to employing the polarization modes of photons [65], with the
difference that the comb-like spectral function of each photon offers an
additional protection against local displacements. Nevertheless, this is a
single-mode classical-like effect, independent of the number of photons
involved. In fact, in this single-photon comb encoding, if the photon’s peaks
are separated by $2\Omega_{o}/\sqrt{n}$ (as in (3)), the protection against
displacement errors is limited to amplitudes
$\delta_{\omega_{j}}\sim\Omega_{o}/(2\sqrt{n})$ per photon. Of course, it’s
possible to entangle these photons to enhance protection against local
(single-photon) qubit flipping and dephasing. However, the so constructed
codes operate similarly to discrete ones [1, 2, 3, 4], and correction for
qubit flip and dephasing requires using complex entangled states of at least
five photons. In contrast, the encoding proposed here demonstrates enhanced
protection starting from $n=2$, which can be verified using HOM experiments.

Figure 2: Schematic view of the effect of imperfect state preparation or
measurement in the rotated variables $\Omega_{1}^{\prime}$ and
$\Omega_{\perp}^{\prime}$. The TF GKP state in variable $\Omega_{1}$ is
represented by the red lines, and they are close to ideal in this variable
(close to zero spectral width). Due to imperfect preparation or measurement,
each peak’s width is broadened by a factor $\sin\theta\sigma$, where $\sigma$
is the spectral width in the collective variables $\Omega_{\perp}$ and the
peak interspacing becomes $2\Omega_{o}\cos\theta$.

Based on our analysis, we can reinterpret the results of [33] as the
production of a two-photon GKP state. While it remains a GKP state on a small
scale, the existing techniques for producing QB GKP states involve the
manipulation of Schrödinger kittens with a mean photon number of the order of
one. Therefore, TF GKP states hold significant promise: techniques to directly
generate larger entangled states [66, 67, 68, 69] or to combine photons in
pairs using non-linear entangling devices [70] are being developed, as well as
non-destructive single-photon detectors [71, 72] and frequency gates [73, 74,
75].

In conclusion, we have introduced and conducted an extensive study on a novel
quantum optical encoding method for GKP states. This encoding can be
implemented in small scale in many laboratories using current technology.
Using the presently available TF GKP states, typically generated in photon
pair sources with a frequency comb structure through broadband SPDC, we can
already envision applications in various domains, as quantum communications
[26, 76], quantum computation [77], and quantum metrology [78]. Moreover, we
can broaden the scope of potential applications by applying the proposed GKP
encoding to other continuous degrees of freedom of mode entangled photons,
such as their transverse position and momentum [39, 28], the propagation
direction [29, 30] and even to individual electrons in distinguishable modes
with an underlying bosonic structure [79]. An interesting perspective involves
adapting recent theoretical and experimental advancements related to QB GKP
states in quantum information to the modal domain [80, 81, 82, 83, 84, 16, 85,
86]. Finally, the tools we provide for defining quantum continuous variables
using collective variables of single photons can also be extended to other
error-correcting codes, such as cat codes [87, 88, 89, 90, 91]. However, this
remains a subject for future investigation.

## Acknowledgements

We acknowledge funding from the Plan France 2030 through the project
ANR-22-PETQ-0006 and N. Fabre, F. Baboux and S. Ducci for fruitful
discussions.

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