††thanks: These authors contributed equally.††thanks: These authors
contributed equally.

# Pauli Check Extrapolation for Quantum Error Mitigation

Quinn Langfitt1 Ji Liu1 Benchen Huang2 Alvin Gonzales1 Kaitlin N. Smith3 Nikos
Hardavellas3 Zain H. Saleem1 1Mathematics and Computer Science Division,
Argonne National Laboratory, Lemont, IL, USA  2Department of Chemistry,
University of Chicago, Chicago, IL, USA  3Department of Computer Science,
Northwestern University, Evanston, IL, USA

###### Abstract

Pauli Check Sandwiching (PCS) is an error mitigation scheme that uses pairs of
parity checks to detect errors in the payload circuit. While increasing the
number of check pairs improves error detection, it also introduces additional
noise to the circuit and exponentially increases the required sampling size.
To address these limitations, we propose a novel error mitigation scheme,
Pauli Check Extrapolation (PCE), which integrates PCS with an extrapolation
technique similar to Zero-Noise Extrapolation (ZNE). However, instead of
extrapolating to the ‘zero-noise’ limit, as is done in ZNE, PCE extrapolates
to the ‘maximum check’ limit—the number of check pairs theoretically required
to achieve unit fidelity. In this study, we focus on applying a linear model
for extrapolation and also derive a more general exponential ansatz based on
the Markovian error model. We demonstrate the effectiveness of PCE by using it
to mitigate errors in the shadow estimation protocol, particularly for states
prepared by the variational quantum eigensolver (VQE). Our results show that
this method can achieve higher fidelities than the state-of-the-art Robust
Shadow (RS) estimation scheme, while significantly reducing the number of
required samples by eliminating the need for a calibration procedure. We
validate these findings on both fully-connected topologies and simulated IBM
hardware backends.

††preprint: APS/123-QED

##  I Introduction

Quantum error mitigation is essential for the practical application of noisy
intermediate-scale quantum (NISQ) devices in various fields, including machine
learning [1], combinatorial optimization [2], quantum simulation [3], and
cryptography [4]. Several error mitigation techniques have been developed to
improve the performance of NISQ devices, such as zero-noise extrapolation
(ZNE) [5, 6, 7], probabilistic error cancellation [8], Pauli Check Sandwiching
(PCS) [9, 10], symmetry verification [11, 12, 13, 14], and simulated quantum
error mitigation [15].

Zero-Noise Extrapolation (ZNE) is an error mitigation technique that reduces
noise in quantum circuits by artificially amplifying the noise intensity in
the circuit. This can be achieved by inserting logical identity operations
into the circuit which increase the noise without altering the intended
computations. ZNE then measures the expectation values of the circuit at
varying noise intensities and uses these measurements to extrapolate back to a
zero-noise limit. By applying a regression model to these data points, ZNE
predicts the output of the circuit as if it were operating in an ideal, noise-
free environment.

Pauli Check Sandwiching (PCS) mitigates errors in the computational circuit by
imposing constraints on the circuit and post-selecting on the outputs that
violate those constraints. Pauli gates are applied at both ends of the circuit
with an ancilla qubit appended for each pair of checks. Since the checks are
applied in a layering fashion on the original circuit, each pair of checks is
typically referred to as a ‘layer.’ Each sample is then followed by a
measurement of the ancilla qubit(s), where only the results corresponding to
zero(s) on the ancila(s) are kept, therefore acting as a post-selection
scheme. In the limit that the checks are ideal, Proposition 2 in [10] states
that there exist $2n$ number of PCS checks (possibly employing checks not in
the Pauli group), where $n$ is the number of compute qubits, such that the
postselected state is noiseless. The checks consist of left and right pairs
and we can always employ weight-one Pauli checks for one of the checks for
each pair.

In this work, we introduce Pauli Check Extrapolation (PCE), which integrates
an extrapolation technique with PCS to extend our expectation value
calculations to the ‘maximal check’ limit. Employing an approach similar to
ZNE, we develop a model— linear or exponential—and fit it to the data points
corresponding to the expectation values for each check using least squares
fitting. We then extrapolate the expectation value to the maximum limit of
checks. Like ZNE, this simulates the effect of adding additional layers of
checks to estimate the expectation value at the maximum check limit. PCS
allows us to reduce noise, whereas traditional ZNE requires the increase in
noise levels, which may not be suitable for highly noisy circuits.
Additionally, extrapolation methods are typically more accurate when the
collected data points are near the target domain value. Thus, a very important
advantage of our scheme over ZNE is that we do not need to sample circuits
with a significantly larger gate count than the original circuit we are
mitigating. Our approach also avoids the noise typically associated with the
physical implementation of additional check layers and eliminates the
exponentially increasing sampling overhead with each added layer.
Additionally, for non-Clifford circuits where the number of applicable Pauli
checks is limited, we can extrapolate to checks that cannot be physically
performed, thus bypassing this limitation.

We apply PCE to improve the accuracy of the shadow estimation scheme for the
variational quantum eigensolver (VQE) [16]. We compare our results with that
of the state-of-the-art Robust Shadow (RS) estimation scheme [17]. Our
simulations employ circuits with depolarizing noise affecting each single-
qubit and two-qubit gate, including the checks. Under fully-connected qubit
topologies, we find that the extrapolated PCS technique significantly improves
fidelity over the RS estimation for a $4$-qubit H2 VQE circuit when the entire
circuit (including state preparation and shadow portions) is protected. We
also demonstrate comparable or even improved fidelities for the $6$-qubit H2O
VQE circuit under homogeneous and inhomogeneous noise distributions,
respectively. Furthermore, we find that PCE technique can offer comparable,
and sometimes improved, performance on fake IBM hardware backends, depending
on the device connectivity.

In addition to improved performance, we note that the PCE technique does not
require a calibration procedure, unlike RS estimation. By eliminating this
calibration step, PCE offers the additional benefit of reducing the total
required sampling size. This advantage is significant, even if the fidelity is
slightly lower for some instances.

##  II Background

###  II.1 Pauli Check Sandwiching

PCS sandwiches the target computational circuit, denoted by $U$, between pairs
of controlled Pauli gates. Mathematically, these Pauli gates are chosen from
the Pauli group $\mathcal{P}_{n}=\\{I,X,Y,Z\\}^{\otimes n}\times\\{\pm 1,\pm
i\\}$. The controlled Pauli gates, $L,R\in\mathcal{P}_{n}$, are selected such
that

$LUR=U.$ (1)

To detect errors, an ancilla qubit is appended to the system for each layer of
Pauli checks. The detection of an error is indicated by a measurement outcome
of $1$ on the ancilla qubit(s). Conversely, a measurement outcome of $0$ on
all ancilla qubits signifies no detected errors, and thus, the results of the
computation are kept. Through this process of post-selection, PCS effectively
mitigates the impact of errors in the computational circuit. An example of a
circuit with one layer of Pauli checks is shown in Fig. 1.

Figure 1: Pauli Check Sandwiching circuit with a single layer of checks. Here
$U$ is the compute circuit.

We can introduce additional layers to our circuit, where each pair of Pauli
gates $L_{n}$, $R_{n}$, where $n$ is the layer number, satisfy the condition
in Eq. 1. This will theoretically allow the circuit to detect more errors and
therefore improve the state fidelity. Furthermore, as stated in Proposition
$2$ of [10], in the theoretical limit of noiseless checks, there exists a set
of checks (which may include non-Pauli checks for general circuits) that
ensures the post-selected state is noiseless. An example of a circuit with
multiple layers is shown in Fig. 2.

Figure 2:  Pauli Check Sandwiching circuit with multiple layers. Here $U$ is
the compute circuit.

###  II.2 Classical Shadow Tomography

The goal of shadow tomography [18] is to predict the expectation value of $M$
physical observables for a certain quantum state (simultaneously) from
relatively few measurements. It works by i). performing random measurements
over the quantum state and ii). estimating all the expectation values through
post-processing measurement outcomes. Shadow tomography is an approximate
theory, and it requires a number of samples:

$N=O\left(\log(M)\max_{i}\left\|O_{i}\right\|^{2}_{\text{shadow}}/\epsilon^{2}\right),$

to keep errors for the estimation of each observable within $\epsilon$. What’s
essential about this protocol is that the number of samples required only
scales logarithmically with $M$, making it promising for practical quantum
simulation applications [19, 20, 21]. In the above equation,
$\left\|O_{i}\right\|_{\text{shadow}}$ denotes the shadow norm of the $i$-th
observable, which depends on the set of randomized measurements we use. For
our experiments, we use the global Clifford group $\text{Cl}(2^{n})$, which
has a shadow norm of
$\left\|O_{i}\right\|_{\text{shadow}}<3\operatorname{tr}(O^{2})$.

To apply PCS to classical shadow tomography, we can sandwich the global
Clifford at the end of the circuit. For any given Pauli check $P_{2}$, there
will always exist a corresponding Pauli check $P_{1}$ for a circuit with only
Clifford gates. This configuration is theoretically guaranteed to protect that
portion of the circuit up to unit fidelity if the checks are noiseless
(Theorem 1 in [10]). Alternatively, we can include both the state preparation
circuit and the random global Clifford. An advantage of extrapolation is that,
when including the state preparation circuit—potentially a non-Clifford
circuit—we can extrapolate to checks that cannot be physically applied. This
is important because the number of pairs $L$ and $R$ that satisfy Eq. 1
decreases with the number of non-Clifford gates. Moreover, for larger quantum
circuits, we can theoretically guarantee finding pairs of Pauli checks for
portions of the circuit by focusing on protecting blocks of Clifford circuits.
We present results for both configurations in Section IV.

###  II.3 Robust Shadow Estimation

RS estimation [17, 22] is a protocol designed for mitigating errors in the
shadow circuit, originally for global/local Clifford group
$\text{Cl}(2^{n}),\text{Cl}_{2}^{\otimes n}$, and recently generalized to the
fermionic case [23, 24]. This protocol could in principle recover the noise-
free expectation value if i). the state preparation is perfect, and ii). the
shadow circuit noise is gate-independent, time-stationary, and Markovian
(GTM).

The scheme operates by first characterizing the noisy channel of classical
shadow estimation, which is ultimately used to compensate for the effect of
noise during post-processing. This characterization is accomplished through a
calibration procedure involving sampling over shadow circuits. This
calibration contributes to a significant sampling overhead compared to PCS,
which does not require such a step. Theorem $7$ in Ref. [17] states that when
sampling from the global Clifford group, the number of total required samples
is

$R=136\ln(2\delta^{-1})\frac{(1+\varepsilon^{2})(1+\frac{1}{d})^{2}}{\varepsilon^{2}(F_{Z}-\frac{1}{d})^{2}},$

where $\varepsilon$ sets the threshold for calibrating the noise channel with
a success probability of at least $1-\delta$, $d$ represents the dimension of
the problem, and $F_{Z}$ is called the $Z$-basis fidelity of the noise channel
which can be approximated as unity when the noise is small [17]. Although we
note that the sampling complexity is (approximately) independent of system
size when the noise is small, in practice this calibration step can introduce
a fairly large sampling overhead. For example, if we set $\varepsilon=1\%$,
this increases the number of samples by a factor of $1/\varepsilon^{2}=10,000$
to the already existing $136$ pre-factor. These additional calibration samples
can be challenging to implement on real hardware.

##  III Methodology

###  III.1 Pauli Check Extrapolation

The PCS scheme faces the following challenges:

  1. 1.

On real hardware, each additional layer of checks introduces more noise into
the circuit.

  2. 2.

The post-selection rate decreases exponentially with the increase in the
number of layers.

  3. 3.

For non-Clifford circuits, the number of Pauli checks we can apply is limited.

To overcome these limitations, we propose an extrapolation technique that
simulates the effect of maximizing the number of PCS layers without physically
implementing them. This maximum layer count is tailored to the system size and
the specific observables required; for instance, only incorporating a
controlled Pauli Z gate for each qubit when measurements are solely in the
Z-basis. Extrapolation effectively bypasses the additional noise and sampling
overhead that would result from actually implementing additional layers.
Additionally, for non-Clifford circuits where the number of applicable Pauli
checks is limited, extrapolation allows us to bypass this limitation, provided
we can at least find a few checks.

The extrapolation can be done as follows: First, we implement the first $m$
layers of Pauli checks on the quantum circuit and measure the corresponding
expectation values ${E_{i}}_{n=1}^{m}$. The collected expectation values form
a dataset ${(n,E_{n})}$, where $n$ represents the number of check layers and
$E_{n}$ is its corresponding measured expectation value.

Next, we assume an extrapolation model $E(n)$ to fit the collected data. For
example, the model could be a linear model with the form $E(n)=\alpha+\beta
n$, where $n$ is the number of check layers, and $\alpha$ and $\beta$ are the
regression coefficients. We apply linear regression to the dataset
${(n,E_{n})}$ to determine the optimal parameters $\alpha$ and $\beta$ by
minimizing the sum of squared differences between the observed expectation
values $E_{n}$ and the predicted values $\alpha+\beta n$. Different choices of
the extrapolation model can produce different results, and while we primarily
use a linear model, other models such as polynomial or exponential can also be
considered. For instance, an exponential model can be expressed as
$E(m)=ab^{m}+c$, where $a$, $b$, and $c$ are fitting parameters.

Once the model is fitted, we use it to extrapolate the expectation value to
the theoretical maximum number of check layers $n_{\text{max}}$. The
extrapolated expectation value $E(n_{\text{max}})=\alpha+\beta n_{\text{max}}$
provides an estimate as if the circuit had $n_{\text{max}}$ check layers that
would theoretically achieving unit fidelity. Figure 3 visually depicts the
extrapolation process.

[width=0.45]linear_model_fit

Figure 3: Schematic of a linear model fitted to the expectation values for
each physically implemented check layer (blue circle) and extrapolating to the
fourth check (red diamond).

###  III.2 Expectation Value Ansatz

We can use the Markovian error model described in Ref. [25] to derive an
ansatz for the expectation value. Let $\pi_{1}$, $\pi_{2}$, and $\pi_{3}$
denote the states of detected error, undetected error, and no error,
respectively. Let $\epsilon$ denote the probability of an error on the data
qubits. Initially (without checks), the state of the quantum computer is
described by the vector

$\displaystyle\vec{\pi}=\begin{pmatrix}\pi^{(0)}_{1}\\\ \pi^{(0)}_{2}\\\
\pi^{(0)}_{3}\end{pmatrix}=\begin{pmatrix}0\\\ \epsilon\\\
1-\epsilon\end{pmatrix}$ (2)

and the transition matrix for each check is given by the upper triangular
matrix

$\displaystyle T=\begin{pmatrix}1&\frac{1}{2}&t_{d}\\\ 0&\frac{1}{2}&t_{u}\\\
0&0&t_{ok}\end{pmatrix},$ (3)

where $t_{d}$, $t_{u}$, $t_{ok}$ are the probabilities of the check
introducing a detectable error, an undetectable error, and no error,
respectively [25]. The probability of a logical error is given by

$\displaystyle P(\text{logical error})=\frac{\pi_{2}}{\pi_{2}+\pi_{3}}.$ (4)

Since we are interested in the logical error, we can ignore the detected error
state and focus on the block matrix

$\displaystyle T^{\prime}=\begin{pmatrix}\frac{1}{2}&t_{u}\\\
0&t_{ok}\end{pmatrix}.$ (5)

Let the checks be near perfect, i.e., $t_{d}=0$, $t_{u}=0$, and $t_{ok}=1$.
The logical error rate after $m$ checks for this scenario is

$\displaystyle P(\text{logical
error})=\frac{\left(\frac{1}{2}\right)^{m}\epsilon}{1-\epsilon+\left(\frac{1}{2}\right)^{m}\epsilon}=\frac{\epsilon}{2^{m}(1-\epsilon)+\epsilon},$
(6)

which is exponential in $m$. Therefore, the expectation value as a function of
the number of checks $m$ should also be exponential. In this paper, we use the
form

$\displaystyle E(m)=ab^{m}+c,$ (7)

where $a$, $b$, and $c$ are scalars determined from fitting. This result is
intuitive because the set of undetectable commuting Pauli strings
exponentially decreases (by a factor of $\frac{1}{2})$ with each check.

##  IV Results

###  IV.1 Experimental Setup

In the following set of experiments, we focus on the application of error
mitigating the expectation value of tensor products of Pauli operators derived
from classical shadow tomography for VQE circuits. We focus on two examples: a
$4$-qubit circuit for modeling the H2 molecule and a $6$-qubit circuit for
H2O. The circuit for the H2 molecule is shown in Fig. 4. For classical shadow
tomography, a random global Clifford circuit unitary is appended to the VQE
circuit, followed by $Z$-basis measurements.

Figure 4: State preparation circuit for the H2 molecule using $4$ qubits. The
variational parameters in the circuit is optimized from a prior noiseless VQE
calculation.

A widely used type of ansatz for solving the eigenvalue problem of molecular
and solid-state systems is the unitary coupled-cluster (UCC) [26] circuit,
which is inspired from coupled-cluster theory [27]. UCC ansatz is
systematically improvable but suffers from long circuit depth, which makes it
hard for near-term quantum hardware. A workaround is the qubit coupled-cluster
(QCC) [28] ansatz, which reduces the circuit depth significantly by a pre-
screening procedure. For the circuits used in this work, we adopted the QCC
circuits with variational parameters optimized from a prior noiseless VQE
calculation.

In our experiments, we compare four types of scenarios when executing our VQE
circuits with classical shadow tomography: the ideal noiseless circuit, the
noisy circuit without any error mitigation, the noisy circuit with robust
shadow estimation applied, and the noisy circuit with various number of check
layers implemented.

For a given state preparation circuit, we generate $10000$ shadow circuits by
appending random global Cliffords to the VQE circuit. We then collect a total
of $100$ computational basis measurement samples from each circuit. We post-
process the measurement data by estimating observables across various number
of shadow circuits $N$, which include $100$, $400$, $1000$, $4000$, and
$10000$ shadow circuits. For each shadow circuit number, we sample $N$ shadow
circuits from the total pool. These $N$ shadow circuits are then partitioned
into $20$ equally sized sets, each containing $N/20$ circuits. To reduce the
effect of outliers, we compute the mean observable estimate for each set and
then take the median of these means for each number of shadow circuits. This
process follows the procedure outlined in Algorithm 1 from [18].

In this implementation of PCS within classical shadow tomography, the
measurement is conducted exclusively in the $Z$-basis. Therefore, errors on
the phase do not impact the output, limiting the utility of checks beyond the
number of qubits in the compute circuit. Accordingly, we use the first $3$
check expectation values to extrapolate to a maximum of $4$ checks for the H2
circuit and use the first $4$ checks to extrapolate to $6$ checks for the H2O
circuit.

###  IV.2 Fully connected topology

We begin by examining a fully connected topology where each gate, including
single-qubit and two-qubit gates, is subjected to depolarizing noise. The
error rates are denoted as $p_{1}$ for single-qubit gates and $p_{2}$ for two-
qubit gates. We allow the depolarizing noise to affect all gates in the
circuit, including the checks $L_{n}$ and $R_{n}$.

As mentioned in Section II, in the context of a Clifford circuit, it is always
possible to identify a Pauli check $L_{n}$ corresponding to any given $R_{n}$
such that the condition in Eq. 1 is satisfied. Figure 5 illustrates the
scenario where the global Clifford at the end of the circuit in the classical
shadow protocol is protected, denoted as ‘check4’ and ‘check4 (extrap)’. A
distinct advantage of PCS over robust shadow estimation is its capability to
extend protection to the state preparation portion of the circuit, in addition
to the global Clifford. This extended protection is evident in Figure 5, where
PCS outperforms robust shadow estimation by error mitigating the entire
circuit. We can also see in Figure 5 that the extrapolated check provides
similar performance to its actual check implementation, suggesting that the
extrapolation technique accurately predicts the associated expectation values
for the maximal check layer implementation. We noticed similar accuracy across
all the scenarios we tested, including the IBM fake backends we test in
Section IV.3.

Figure 5:  4-qubit H2 circuit with full connectivity under a depolarizing
error channel ($p_{1}=0.002$, $p_{2}=0.02$). Results compare protecting only
the Clifford portion (Check) versus the entire circuit (PrepCheck). ‘Check 4
(extrap)’ and ‘PrepCheck 4 (extrap)’ show extrapolated expectation values from
the first three check implementations.

Extending our analysis to the $6$-qubit H2O circuit, we find that PCS, when
extrapolated to the maximum of six layers, performs comparably to robust
shadow estimation as we scale up the circuit size. This observation is
supported by the results shown in Figure 6. Moreover, RS estimation generally
assumes a uniform noise distribution, (assumption A1 in [17]), which is often
not the case in real hardware where error rates can vary between qubits, even
for the same quantum gate [29]. However, PCS does not make any such
assumptions on the error model of the circuit.

Figure 6(b) shows the results when we apply a random Gaussian distribution of
error rates across the qubits, with mean error rates of $p_{2}=0.02$ and
$p_{1}=0.002$ and standard deviations of $0.005$ and $0.0005$, respectively.
The advantage of PCS over RS in mitigating circuits with inhomogeneous noise
distribution is evident, as PCS shows lower error rates in the presence of
unevenly distributed noise for extrapolated check 6 compared to robust shadow
estimation. This improvement suggests that PCS may perform better under more
realistic error rates that reflect real-world quantum computing environments.

(a)

(b)

Figure 6: Fully connected 6-qubit H2O circuit under different error
conditions. (a) has a homogeneous depolarising error rate across all qubits of
$p_{1}=0.002$ and $p_{2}=0.02$ while (b) has varying depolarizing error with
two-qubit error rate $p_{2}=0.02\pm 0.005$, and mean single-qubit error rate
$p_{1}=0.002\pm 0.0005$.

###  IV.3 IBM device topology

In this subsection, we analyze PCS performance using realistic noise models
and circuit topologies that mimic IBM quantum devices, specifically using mock
backends provided by Qiskit. Here we focus on only the case where PCS protects
the entire $4$-qubit H2 shadow circuit (both state preparation and Clifford
portions) and compare the results to RS.

Figure 7 shows the performance of PCS on two IBM mock backends: Cairo and
Melbourne. We allow the error rates provided by the backends to affect the
entire circuit, including the controlled Pauli checks. By protecting both the
state preparation and the Clifford portions of the circuit, PCS, when
extrapolated to four layers, shows performance comparable to that of robust
shadow estimation. Additionally, the error rate curves across the number of
check layers follow a similar trend, even when extrapolating the final fourth
check. This trend suggests that the extrapolation model is well suited even
for noise models which are not exclusively depolarizing.

We have also tested various other IBM mock backends and observed a similar
trend: PCS can perform comparably to RS estimation, and in some cases, offer
slightly improved fidelities, but this improvement was only evident when the
entire circuit was protected. Our analysis suggests that the performance of
PCS depends more on the topology of the circuit than on the specific types of
noise. This difference in performance is noticeable when comparing the results
from Figure 5 to Figure 7. In Figure 5, the difference between PCS PrepCheck4
and RS is significantly greater than in Figure 7, where PCS PrepCheck4 and RS
perform almost equally.

(a)

(b)

Figure 7: 4-qubit H2 circuit with realistic device connectivity and noise
models. Both panels demonstrate the protection of the state preparation and
Clifford portions of the circuit, referred to as ‘PrepCheck’: (a) Cairo device
and (b) Melbourne device.

##  V Conclusions and Future Directions

In this paper, we introduced a novel error mitigation scheme that extends upon
PCS by extrapolating the checks from the physically implemented checks. This
approach helps us to achieve similar fidelities associated with the maximum
number of checks while avoiding the practical issues such as the additional
noise and the exponetially decreasing postselection rate that arise from
physically adding more checks. We applied our scheme to mitigate errors in the
classical shadow scheme and demonstrated that it can offer equal or superior
performance compared to the current state-of-the-art classical shadow error
mitigation scheme. Additionally, a key advantage of our scheme over the RS
method is that it does not require a calibration step, which significantly
reduces the overall number of samples needed for implementation.

One future direction is to evaluate how PCE performs when protecting the
Clifford portion and a partial amount of the state preparation circuit. This
approach is particularly relevant when the number of non-Clifford gates is
significantly large, as is often the case with increasing system sizes in VQE
circuits, making it impractical to find Pauli checks to protect the entire
state preparation circuit. Our results show that optimal performance is
achieved when the entire circuit, rather than just the Clifford circuit, is
sandwiched. Protecting both the state preparation and Clifford circuits is a
another advantage of the PCS scheme over RS estimation. However, a key
consideration is the trade-off between performance and the time required to
search for Pauli checks when including non-Clifford portions of the circuit.

Additionally, extending PCE to larger quantum circuits is another important
future direction. As system sizes grow and circuits become more complex,
particularly those with a higher number of non-Clifford gates, it becomes
increasingly important to evaluate the scalability of PCE. This includes
investigating the efficiency and practicality of finding suitable Pauli checks
for large-scale circuits and exploring the performance trade-offs involved.

For this paper, we utilized a linear ansatz to extrapolate expectation values
from our check implementations. Additionally, we tested a more general
exponential ansatz, $E(m)=a\left(b\right)^{m}+c$, where $a$, $b$, and $c$ are
scalars determined through fitting. This model was derived in Section III.2
using the Markovian model described in Ref. [25]. In our tests, the
exponential model produced results similar to the linear model, with the
fitted parameter $b$ often being very close to $1$, effectively resulting in a
linear fit. It is possible that the exponential model may perform better with
a larger number of checks or under different noise models. Additionally, other
models used in ZNE, such as various polynomial ansatzes, could be tested to
improve performance.

Finally, another future direction is to test PCE on real quantum devices with
fully connected architectures. The effectiveness of PCE appears to be
significantly influenced by the device’s connectivity, likely due to the
additional noise overhead introduced by the checks themselves, as the noise
overhead imposed on the compute circuit $U$ would likely be detected by the
Pauli checks and thus mitigated. Testing on fully or near-fully connected
hardware will provide further insights into the practical performance and
robustness of PCE in mitigating errors in real-world quantum computing
environments.

##  VI Acknowledgements

QL, JL, BH, ZHS, and AG acknowledge support by the Q-NEXT Center. NH and KNS
acknowledge support from NSF award CCF-2119069 and Northwestern University’s
McCormick School of Engineering.

## References

  * Maria Schuld and Petruccione [2015] I. S. Maria Schuld and F. Petruccione, Contemporary Physics 56, 172 (2015), https://doi.org/10.1080/00107514.2014.964942 .
  * Farhi  _et al._ [2014] E. Farhi, J. Goldstone, and S. Gutmann, “A quantum approximate optimization algorithm,” (2014), arXiv:1411.4028 [quant-ph] .
  * Li and Benjamin [2017] Y. Li and S. C. Benjamin, Phys. Rev. X 7, 021050 (2017).
  * Buhrman  _et al._ [2014] H. Buhrman, N. Chandran, S. Fehr, R. Gelles, V. Goyal, R. Ostrovsky, and C. Schaffner, SIAM Journal on Computing 43, 150–178 (2014).
  * Temme  _et al._ [2017] K. Temme, S. Bravyi, and J. M. Gambetta, Phys. Rev. Lett. 119, 180509 (2017).
  * Kim  _et al._ [2023] Y. Kim, C. J. Wood, T. J. Yoder, S. T. Merkel, J. M. Gambetta, K. Temme, and A. Kandala, Nature Physics 19, 752–759 (2023).
  * Kandala  _et al._ [2019] A. Kandala, K. Temme, A. D. Córcoles, A. Mezzacapo, J. M. Chow, and J. M. Gambetta, Nature 567, 491–495 (2019).
  * Endo  _et al._ [2018] S. Endo, S. C. Benjamin, and Y. Li, Physical Review X 8 (2018), 10.1103/physrevx.8.031027.
  * Debroy and Brown [2020] D. M. Debroy and K. R. Brown, Phys. Rev. A 102, 052409 (2020).
  * Gonzales  _et al._ [2023] A. Gonzales, R. Shaydulin, Z. H. Saleem, and M. Suchara, Scientific Reports 13 (2023), 10.1038/s41598-023-28109-x.
  * Bonet-Monroig  _et al._ [2018] X. Bonet-Monroig, R. Sagastizabal, M. Singh, and T. E. O’Brien, Physical Review A 98 (2018), 10.1103/physreva.98.062339.
  * McArdle  _et al._ [2019] S. McArdle, X. Yuan, and S. Benjamin, Physical Review Letters 122 (2019), 10.1103/physrevlett.122.180501.
  * Shaydulin and Galda [2021] R. Shaydulin and A. Galda, in  _2021 IEEE International Conference on Quantum Computing and Engineering (QCE)_ (IEEE, 2021).
  * Cai [2021] Z. Cai, Quantum 5, 548 (2021).
  * Liu  _et al._ [2022] J. Liu, A. Gonzales, and Z. H. Saleem, “Classical simulators as quantum error mitigators via circuit cutting,” (2022), arXiv:2212.07335 [quant-ph] .
  * Peruzzo  _et al._ [2014] A. Peruzzo, J. McClean, P. Shadbolt, M.-H. Yung, X.-Q. Zhou, P. J. Love, A. Aspuru-Guzik, and J. L. O’brien, Nature Communications 5, 4213 (2014).
  * Chen  _et al._ [2021] S. Chen, W. Yu, P. Zeng, and S. T. Flammia, PRX Quantum 2 (2021), 10.1103/prxquantum.2.030348.
  * Huang  _et al._ [2020] H.-Y. Huang, R. Kueng, and J. Preskill, Nature Physics 16, 1050–1057 (2020).
  * Huggins  _et al._ [2022] W. J. Huggins, B. A. O’Gorman, N. C. Rubin, D. R. Reichman, R. Babbush, and J. Lee, Nature 603, 416 (2022).
  * O’Brien  _et al._ [2022] T. E. O’Brien, M. Streif, N. C. Rubin, R. Santagati, Y. Su, W. J. Huggins, J. J. Goings, N. Moll, E. Kyoseva, M. Degroote, _et al._ , Physical Review Research 4, 043210 (2022).
  * Avdic and Mazziotti [2024] I. Avdic and D. A. Mazziotti, Physical Review Letters 132, 220802 (2024).
  * Koh and Grewal [2022] D. E. Koh and S. Grewal, Quantum 6, 776 (2022).
  * Zhao and Miyake [2024] A. Zhao and A. Miyake, npj Quantum Information 10, 57 (2024).
  * Wu and Koh [2024] B. Wu and D. E. Koh, npj Quantum Information 10, 39 (2024).
  * van den Berg  _et al._ [2023] E. van den Berg, S. Bravyi, J. M. Gambetta, P. Jurcevic, D. Maslov, and K. Temme, Phys. Rev. Res. 5, 033193 (2023).
  * Romero  _et al._ [2018] J. Romero, R. Babbush, J. R. McClean, C. Hempel, P. J. Love, and A. Aspuru-Guzik, Quantum Science and Technology 4, 014008 (2018).
  * Bartlett and Musiał [2007] R. J. Bartlett and M. Musiał, Reviews of Modern Physics 79, 291 (2007).
  * Ryabinkin  _et al._ [2018] I. G. Ryabinkin, T.-C. Yen, S. N. Genin, and A. F. Izmaylov, Journal of Chemical Theory and Computation 14, 6317 (2018).
  * Aseguinolaza  _et al._ [2024] U. Aseguinolaza, N. Sobrino, G. Sobrino, J. Jornet-Somoza, and J. Borge, Quantum Information Processing 23 (2024), 10.1007/s11128-024-04384-z.

The submitted manuscript has been created by UChicago Argonne, LLC, Operator
of Argonne National Laboratory (“Argonne”). Argonne, a U.S. Department of
Energy Office of Science laboratory, is operated under Contract No. DE-
AC02-06CH11357. The U.S. Government retains for itself, and others acting on
its behalf, a paid-up nonexclusive, irrevocable worldwide license in said
article to reproduce, prepare derivative works, distribute copies to the
public, and perform publicly and display publicly, by or on behalf of the
Government. The Department of Energy will provide public access to these
results of federally sponsored research in accordance with the DOE Public
Access Plan. http://energy.gov/downloads/doe-public-access-plan.


# Utilization of Reconfigurable Intelligent Surfaces with Context Information:
Use Cases

Łukasz Kułacz

###### Abstract

In terms of complex radio environments especially in dense urban areas, a very
interesting topic is considered - the utilization of reconfigurable
intelligent surfaces. Basically, based on simple controls of the angle of
reflection of the signal from the surface, it is possible to achieve different
effects in a radio communication system. Maximizing or minimizing the received
power at specific locations near the reflecting surface is the most important
effect. Thanks to this, it is possible to: receive a signal in a place where
it was not possible, detect spectrum occupancy in a place where the sensor
could not make a correct detection, or minimize interference in a specific
receiver. In this paper, all three concepts are presented, and, using a simple
ray tracing simulation, the potential profit in each scenario is shown. In
addition, a scenario was analyzed in which several of the aforementioned
situations are combined.

##  1 Introduction

In recent times, wireless communication development is based, in many cases,
on introducing an ever greater level of intelligence to the network. By the
aforementioned intelligence, we mean dynamic spectrum access systems, which
often require access to a large amount of contextual information about the
environment and network users. This information is necessary to make optimal
decisions - based on many parameters (not necessarily directly related to the
network itself) and their historical values. Historical data allows, among
other things, to capture trends in the behavior of network users and thus the
ability to predict upcoming events. The urban environment has always been
highly demanding for designers of wireless systems, where the transmitted
signal itself is subjected to many propagation phenomena, i.e., reflections or
dispersion. The huge number and variety of buildings and objects affecting the
propagation of the signal collectively create a highly complex radio channel.
Additionally, in an urban environment, there is a huge number of users in a
relatively small area. Combining these two facts is a significant challenge in
the operation of wireless systems. It is worth noting that, in general, signal
reflections are unavoidable. While they hinder the analysis of signal
propagation, they are generally a beneficial phenomenon - because the signal
can reach the receiver that is not in the direct view of the transmitter. In
this context, an exciting approach is the recently popular and considered
topic of Reconfigurable Intelligent Surfaces (RIS) [1]. At the core of this
solution, RIS allows you to influence the process of radio wave reflection on
this surface. This means that it is possible to modify the basic principle
where the angle of incidence of the wave is equal to the angle of reflection
of the wave. This creates great possibilities for creating and controlling the
propagation environment. On the one hand, it makes the possibility of
directing the radio signal to previously unattainable places (due to the
geometry of the objects). On the other hand, the possibility of removing the
radio signal in a specific place (minimizing interference) by directing it in
a completely different direction [2]. Only these two concepts seem to have
great potential for use in intelligent networks. They can be used to detect
the signal or improve the quality of the received signal (by increasing the
received power if it is a desirable signal or reducing the received power if
it is an unwanted signal). Signal detection could be possible for a sensor
that is somehow hidden from the transmitter in the sense of direct visibility
and reflected signals with a sufficiently high power necessary for the
operation of signal detection algorithms. However, the indicated solutions
require the development of new algorithms whose task is to control the
described RIS [3].

##  2 Scenario Description

In this work, we consider a simple scenario of an elongated room with a wall
between the signal transmitter and three receivers, which prevents direct
visibility. In the depths of the room, however, there is a surface that will
simulate RIS.

Figure 1: Analyzed area

All devices are at the same height, and the effects of the ceiling and floor
have been ignored - simplifying the analyzed case to two-dimensional space.
The projection of the investigated area is presented in Fig. 1. The described
room was chosen on purpose as a demonstration of the potential possibilities
of using RIS. This decision is dictated by the desire to highlight the concept
of using RIS, and the ideas themselves can be used in more complex analyzes,
where, however, it would be much more challenging to isolate the impact of
RIS. The work described in this paper compares three simulation scenarios:

  1. 1.

Scenario 1 assumes no use of RIS, so a fragment of the wall was placed instead
- this scenario is a reference for the other scenarios.

  2. 2.

Scenario 2 assumes a simplified RIS operation that works independently of
external factors, i.e., periodically and alternately changes the angle of the
reflected signal - which simulates the use of a mirror rotated from left to
right and back. This scenario has been proposed to demonstrate the potential
profits from using even such a simple control mechanism.

  3. 3.

Finally, Scenario 3 is in which network users have access to information about
the current RIS setting and indirectly influence the RIS settings. The last
scenario is an example of using context information (the current RIS setting)
to achieve the current goal (e.g., minimizing interference).

Three use cases of RIS were analyzed (utilizing all three scenarios) in the
paper: improved spectrum occupancy detection by the sensor, improvement in the
received signal level for two receivers, and reduction of interference in the
receiver. During the simulation, the operation of RIS was simulated as a
change in the rotation of the selected surface by an angle from -20 to 20
degrees with a step of 5 degrees.

##  3 Simulation Results

Fig. 2 presents the spatial distribution of the received power in the analyzed
area for the given transmitting parameters and uses a propagation model based
on ray tracing.

Figure 2: Received power without RIS influence Figure 3: Received power in
receivers versus RIS setting

Preliminary analysis of the indicated distribution shows that part of the area
is characterized by less signal attenuation (warmer colors, receiver C). In
comparison, the remaining part of the area has greater signal attenuation
(darker colors, receivers A and B - remaining ”in the shadow” of the signal).
The received power in this situation results directly from the perpendicular
arrangement of the wall marked as ”RIS” in Fig. 1 \- however, here plays a
role of a wall. It can be seen that receivers A and B would have a problem
with receiving and detecting the analyzed signal. However, they would be
deprived of excessive interference. The situation is exactly the opposite for
receiver C. Nevertheless, it should be remembered that in the opposite
situation, when receivers A and B consider the analyzed signal undesirable and
receiver C a desirable signal, influencing the current propagation would not
be beneficial. Fig. 3 shows the received power of all three analyzed receivers
depending on all considered RIS settings. It shows all possibilities of
received power for all receivers, and it will be used depending on the current
role of a particular receiver.

###  3.1 First Use Case: Spectrum occupancy detection

For the first use case, i.e., improving the spectrum occupancy detection by
receiver B, preliminary simulations showed that in the first scenario, correct
detection is challenging; in the second scenario, correct detection is
possible for 31.25% of the time, and in the third scenario, for 79.17% of time
there is a possibility of correct detection. Received power in the second
scenario (depending on RIS angle) is presented in Fig. 3 (see ”RX A”). The
last scenario highly depends on the algorithm used (and its purpose). Still,
even just information about the current RIS setting allows the sensor to
assume - that the occupancy decision is updated only at a particular time.
This approach is not ideal, and its effectiveness depends very much on the
transmission characteristics (frequency, length of transmission, etc.). The
solution, which assumes the ability to control the RIS even in a very simple
scenario, enables, for example, periodic ”combing” of all available settings
and then maintaining one (or several alternating) settings for a certain time.
In the considered system with one sensor, it is possible to correctly set the
RIS for the entire observation time after a short time, achieving almost 100%
of the possible detection time. Fig. 4 presents the received power map with
the best RIS setting for receiver A, which is treated like a spectrum
detection sensor in this use case.

Figure 4: Received power map for RIS setting (angle = 20)

###  3.2 Second Use Case: Received signal level improvement

For the second use case, i.e., improving the received signal level for
receivers B and C, the preliminary results of the simulation show that in the
first scenario, receiver B cannot receive the signal correctly, while receiver
C can receive the signal all the time. In the second scenario, 35.29% of the
time, receiver B receives a signal with sufficient power level, while receiver
C receives 41.18% of the time. In this case, we see the profit from the
possibility of servicing the receiver, which was in the ”shadow” of the signal
and could not receive the signal without changing its position. In the third
scenario, however, it was assumed that for an initial time, RIS checks all
available settings and then, based on reports from receivers, alternately uses
the best-reported settings. As a result, receiver B could receive the signal
45.1% of the time and receiver C 47.06% of the time. At this point, the
possibility of considering traffic steering algorithms to adjust the duration
of settings for individual receivers is worth mentioning. For example, on Fig.
5 received power map with the best RIS setting for receiver B was shown.

Figure 5: Received power map for RIS setting (angle = -15)

###  3.3 Third Use Case: Interference reduction

For the third use case, i.e., interference reduction in receiver C,
preliminary simulation results show that in the first scenario, 100% of the
time, the receiver observes significant interference. In the second scenario,
the interference exceeds the threshold 35.29% of the time. It is worth
mentioning that the average observed interference level decreases by about
3.715 dB, the median by about 7.972 dB, and the 10th percentile by about
24.762 dB. The 90th percentile remains the same. The knowledge of the RIS
setting could at least provide information about the occurrence of potential
interference - which can, for example, be used in the transmission planning
process. In the third scenario, we again assume a specified period of checking
all RIS settings and then, based on the receiver C report, setting the
configuration, causing the minimum level of interference. Thanks to this, it
was possible to reduce the interference time above the threshold from 100% to
about 19.61%.

Figure 6: Received power map for RIS setting (angle = -20)

This time, of course, again is hugely dependent on the given configuration and
the presence of other devices. Fig. 6 presents the received power map with the
best RIS setting for receiver C, which treats the analyzed signal as
interference. It should be remembered that the simulations carried out assumed
constant in-time transmission schema during the whole experiment. In the case
of only temporary signal transmissions for all three use cases, this would be
an additional challenge for the algorithm that controls the operation of the
RIS.

###  3.4 All Use Cases combined

In addition, an experiment was carried out combining all three analyzed use
cases, i.e., receiver A is a spectrum occupancy detection sensor, receiver B
is a receiver of the analyzed signal (desired signal), and receiver C is not a
receiver of the analyzed signal (unwanted signal). Recall that in the
reference scenario, the objectives of none of the receivers are met. Then, RIS
operating according to the following scheme was used: initially, all possible
RIS configurations are checked, and then all the best settings reported by the
receivers are used alternately. As a result of preliminary simulations,
receiver A can successfully detect the signal about 43.4% of the time;
receiver B can successfully receive the desired signal about 45.28% of the
time, and receiver C is free of significant interference about 79.25% of the
time. In addition, the average interference power observed in receiver C
decreases by about 8.58 dB. Received power in the time domain for each
receiver is shown in Fig. 7.

Figure 7: Received power in all receivers in time domain

##  4 Conclusions

The simulations presented in the paper show that using RIS, even with the most
straightforward control mechanism, can benefit each of the analyzed use cases.
However, much better results can be obtained by adding a simple algorithm to
the RIS control mechanism and providing interested receivers with information
about the current RIS setting. This gives a very flexible solution that should
be carefully considered. In particular, the nature of the environment, the
nature of the signal source or sources, potential signal receivers, their
positions, and the purpose of use should be considered. Of course, the
presented concepts, on the one hand, create the possibility of directing the
signal to previously inaccessible places. However, this situation can be both
positive and negative because there is a risk of introducing interference in a
place where it has not occurred so far. This problem is exacerbated,
especially concerning devices unaware of the existence and presence of RIS.
Nevertheless, the ideas presented in this paper seem promising regarding
potential gains for specific applications.

## Acknowledgements

The presented work has been funded by the National Science Centre in Poland
within the project (no. 2021/43/B/ST7/01365) of the OPUS programme.

## References

  * [1] Y. Liu et al., ”Reconfigurable Intelligent Surfaces: Principles and Opportunities,” in IEEE Communications Surveys & Tutorials, vol. 23, no. 3, pp. 1546-1577, third quarter 2021, doi: 10.1109/COMST.2021.3077737.
  * [2] M. Di Renzo, F. H. Danufane and S. Tretyakov, ”Communication Models for Reconfigurable Intelligent Surfaces: From Surface Electromagnetics to Wireless Networks Optimization,” in Proceedings of the IEEE, vol. 110, no. 9, pp. 1164-1209, Sept. 2022, doi: 10.1109/JPROC.2022.3195536.
  * [3] C. Liaskos, S. Nie, A. Tsioliaridou, A. Pitsillides, S. Ioannidis and I. Akyildiz, ”A New Wireless Communication Paradigm through Software-Controlled Metasurfaces,” in IEEE Communications Magazine, vol. 56, no. 9, pp. 162-169, Sept. 2018, doi: 10.1109/MCOM.2018.1700659.
  * [4] M. Hoyhtya et al., ”Spectrum Occupancy Measurements: A Survey and Use of Interference Maps,” in IEEE Communications Surveys & Tutorials, vol. 18, no. 4, pp. 2386-2414, fourth quarter 2016.