# Dual Map Framework for Noise Characterization of Quantum Computers

James Sud<EMAIL_ADDRESS>Quantum Artificial Intelligence Laboratory (QuAIL),
NASA Ames Research Center, Moffett Field, CA, 94035, USA  USRA Research
Institute for Advanced Computer Science (RIACS), Mountain View, CA, 94043, USA
Jeffrey Marshall  Quantum Artificial Intelligence Laboratory (QuAIL), NASA
Ames Research Center, Moffett Field, CA, 94035, USA  USRA Research Institute
for Advanced Computer Science (RIACS), Mountain View, CA, 94043, USA  Zhihui
Wang  Quantum Artificial Intelligence Laboratory (QuAIL), NASA Ames Research
Center, Moffett Field, CA, 94035, USA  USRA Research Institute for Advanced
Computer Science (RIACS), Mountain View, CA, 94043, USA  Eleanor Rieffel
Quantum Artificial Intelligence Laboratory (QuAIL), NASA Ames Research Center,
Moffett Field, CA, 94035, USA  Filip A. Wudarski<EMAIL_ADDRESS>Quantum
Artificial Intelligence Laboratory (QuAIL), NASA Ames Research Center, Moffett
Field, CA, 94035, USA  USRA Research Institute for Advanced Computer Science
(RIACS), Mountain View, CA, 94043, USA

(August 27, 2024)

###### Abstract

In order to understand the capabilities and limitations of quantum computers,
it is necessary to develop methods that efficiently characterize and benchmark
error channels present on these devices. In this paper, we present a method
that faithfully reconstructs a marginal (local) approximation of the effective
noise (MATEN) channel, that acts as a single layer at the end of the circuit.
We first introduce a dual map framework that allows us to analytically derive
expectation values of observables with respect to noisy circuits. These
findings are supported by numerical simulations of the quantum approximate
optimization algorithm (QAOA) that also justify the MATEN, even in the
presence of non-local errors that occur during a circuit. Finally, we
demonstrate the performance of the method on Rigetti’s Aspen-9 quantum
computer for QAOA circuits up to six qubits, successfully predicting the
observed measurements on a majority of the qubits.

##  I Introduction

Appropriate and accurate error characterization and benchmarking is vital for
many aspects of quantum computation. Understanding dominant forms of error
allows for improvements on quantum hardware, bringing these devices closer to
the fault-tolerant regime, and possibly allowing for the tailoring of error
correcting codes to specific error channels Piveteau  _et al._ (2021). On the
algorithms side, error characterization opens the possibility for error-aware
algorithm design and error mitigation strategies, improving the performance of
algorithms on hardware Temme  _et al._ (2017); Li and Benjamin (2017). A
plethora of protocols have been designed for understanding error. These can be
divided into benchmarking protocols, which aim to return numerical values that
capture the rate of errors in a process (usually defined as an average
fidelity Nielsen (2002); Wudarski  _et al._ (2020)), and characterization
protocols, which aim to return information about both the level and form of
the error channels themselves. Benchmarking protocols include randomized
benchmarking Emerson  _et al._ (2005); Magesan  _et al._ (2011) (along with
extensions such as Claes  _et al._ (2021)), cycle benchmarking Erhard  _et
al._ (2019), and direct fidelity estimation Flammia and Liu (2011).
Characterization protocols include quantum process tomography Chuang and
Nielsen (1997), gate set tomography Greenbaum (2015), Hamiltonian estimation
Schirmer  _et al._ (2004), and robust phase estimation Kimmel  _et al._
(2015), as well as state preparation and measurement (SPAM) error
characterization methods such as Sun and Geller (2020); Lin  _et al._ (2021);
Werninghaus  _et al._ (2021). So far, benchmarking and characterization
methods have suffered substantial shortcomings - either returning limited
information (e.g. average fidelity for RB) or restricted to small systems due
to exponential scaling (tomographic methods). In this work, we develop a
characterization scheme that efficiently returns information about process
matrix of the marginal noise channel acting on a single qubit. The method
combines ease and efficiency of benchmarking techniques with substantially
richer information content. Additionally, the introduced protocol operates
without additional compilation overhead, as opposed to RB approaches, which
require twirling subroutine to cast the noisy channel into a convenient form
of a Pauli channel.

Quantum noise, which can lead to computational errors, are inevitable
companions of quantum evolution. In order to properly describe physically
admissible errors, one has to employ the framework of completely positive and
trace preserving (CPTP) maps, which are referred to as error channels. These
channels can be represented in numerous ways Milz  _et al._ (2017); Breuer
_et al._ (2002); Bengtsson and Życzkowski (2017). For our purposes, the most
natural representation is of the following form

$\mathcal{E}[\rho]=\sum_{k,l=0}^{d^{2}-1}\chi_{k,l}P_{k}\rho P_{l}^{\dagger},$
(1)

where $P_{k}$ are operators, $d=2^{N}$ is dimensionality of the Hilbert space
for $N$ qubits and $\chi$ is referred to as a process matrix. Setting $P_{k}$
to orthonormal basis elements (e.g. Pauli matrices), one can determine all
elements of $\chi$ matrix via quantum process tomography Chuang and Nielsen
(1997). In order to represent a valid quantum channel, Eq. (1) has to be CPTP,
which happens when $\chi\geq 0$ (CP condition) and

$\sum_{k,l=0}^{n^{2}-1}\chi_{k,l}P_{l}^{\dagger}P_{k}=\mathbbm{1},\quad\quad\mathrm{(TP\
condition)}.$ (2)

On noisy intermediate-scale quantum (NISQ) devices, levels of noise are too
high for error correction and fault tolerance to occur. Thus, error
mitigation, and error-aware algorithm co-design strategies are needed to
maximize the performance of algorithms run on these devices. In order to
determine these optimal mitigation and co-design strategies, it is imperative
to characterize and understand error. A popular and well studied algorithm for
NISQ devices is the Quantum Approximate Optimization Algorithm or Quantum
Alternating Operator Ansatz (QAOA) Farhi  _et al._ (2014); Hogg (2000);
Hadfield  _et al._ (2019), which aims to find approximately optimal solutions
to optimization problems. In this work, we study the application of our
characterization method for QAOA run on combinatorial optimization problems.
The characterization and effect of local noise in QAOA circuits has been
previously studied Marshall  _et al._ (2020); Xue  _et al._ (2021); Wang  _et
al._ (2021); Streif  _et al._ (2021), however here we provide analytical
treatment for popular classes of error channels, as well as for a generic
single-qubit noise channel.

Given this understanding of prior work on error characterization and
benchmarking, as well as the introduction of error maps and QAOA, we lay out
the rest of our paper as follows. In Sec. II, we introduce the framework of
the dual map that is essential for analytical derivation of the expectation
values of arbitrary observables in the noisy setup. In Sec. III, we use these
results to reverse-engineer a method to introduce a marginal approximation to
the effective noise (MATEN), the core procedure of this contribution, that
allows to estimate local contribution to the error channels. In Sec. IV, we
discuss limitations of the MATEN protocol for spatially correlated noise.
Finally, in V we demonstrate the efficacy of the method in characterizing
error on classical simulations, as well as on the Aspen-9 quantum computer
device from Rigetti Computing.

##  II Dual Map Framework

Given a quantum state $\rho$, an error channel $\mathcal{E}$ (in the form of
Eq. (1)), and an Hermitian operator $O$, the noisy expectation value of $O$
with respect to $\rho$ is typically evaluated as
$\mathrm{Tr}[O\mathcal{E}(\rho)]$. In this work, however, we consider the dual
action of $\mathcal{E}$, and compute the same expectation as

$\langle
O\rangle=\mathrm{Tr}[\mathcal{E}^{\\#}(O)\rho]=\mathrm{Tr}[O^{\prime}\rho]$
(3)

where $\mathcal{E}^{\\#}$ is the dual channel of $\mathcal{E}$, defined as

$\mathcal{E}^{\\#}[O]=\sum_{k,l=0}^{n^{2}-1}\chi_{k,l}P_{l}^{\dagger}OP_{k},$
(4)

which can be derived from the cyclic property of trace.

We can then study properties of the modified operator
$O^{\prime}=\mathcal{E}^{\\#}(O)$, which means that noise affects only the
observable and not the state $\rho$. Therefore, the expectation values with
respect to the ideal quantum state $\rho$ (e.g. output of a quantum circuit),
can potentially benefit from the local structure of noise of the observables.
In particular, if $\rho$ is a pure state (i.e. $\rho^{2}=\rho$), we can avoid
costly simulations of the density matrix, and focus on unitary simulations and
local measurements of a state vector.

Finally note that it is always possible to decompose a quantum map $\Lambda$
associated with a noisy quantum circuit as a composition
$\Lambda=\mathcal{E}\circ\mathcal{U}$, where $\mathcal{U}$ is the circuit’s
(ideal) unitary map, and $\mathcal{E}$ a noisy channel (e.g with the trivial
example $\mathcal{E}=\Lambda\circ\mathcal{U}^{{\dagger}}$). Therefore, instead
of characterizing the total error map $\Lambda$ (which includes contribution
from the ideal unitary), we focus on determining $\mathcal{E}$, which one can
perceive as an effective noise channel for the circuit (in general dependent
on $\mathcal{U}$, e.g. rotational angles in QAOA). This mathematical trick,
allows us to “move” effects of noise to the very last layer of the quantum
circuit (see Fig. 1), and exploit the dual map framework. The main advantage
of using this formalism, is that many observables of interest (e.g.
combinatorial or molecular Hamiltonians) can be expressed as a combination of
$k$-local terms, and, as explained above, the simulation of which can be
significantly more efficient in the dual map framework. We demonstrate this
idea in the subsequent examples.

Figure 1: Operational framework for the noisy circuit characterization that is
described by a quantum map $\Lambda(\vec{\theta})$, where $\vec{\theta}$ is a
collection of circuit parameters. In a) noise is “moved” to the last layer
(according to
$\mathcal{E}=\Lambda(\vec{\theta})\circ\mathcal{U}^{\dagger}(\vec{\theta})$,
and in general it is of non-local character. The non-locality is neglected in
b) by approximating circuit with only local quantum channels $\mathcal{E}_{i}$
acting on $i$-th qubit. In c) we depict the marginal approximation for a
single-layered QAOA.

###  II.1 Example: Noisy Single-Layered QAOA

In this section we analyze the example of QAOA circuits, which are constructed
by interleaving layers of parameterized unitaries of a mixing Hamiltonian $B$
and a phasing Hamiltonian $H$, as so

$\ket{\Psi_{\mathrm{QAOA}}(\vec{\gamma},\vec{\beta})}=e^{-i\beta_{p}B}e^{-i\gamma_{p}H}\cdots
e^{-i\beta_{1}B}e^{-i\gamma_{1}H}\ket{\psi_{\mathrm{0}}},$ (5)

where $p$ is the number of layers in the circuit, $(\vec{\gamma},\vec{\beta})$
represent length $p$ parameter vectors, and $\ket{\psi_{\mathrm{0}}}$
corresponds to an initial state. Given this form, one can then choose
$(\vec{\gamma},\vec{\beta})$ such that the expectation value of $H$ is
optimized (minimized or maximized) when the circuit is applied to a suitably
chosen initial state. Strategies for optimizing $(\vec{\gamma},\vec{\beta})$
Streif and Leib (2020); Zhou  _et al._ (2020); Shaydulin  _et al._ (2019);
Brandao  _et al._ (2018) as well as choosing optimal starting states and
mixing Hamiltonians Hadfield  _et al._ (2019); Sack and Serbyn (2021); Wang
_et al._ (2020) have been intensively analyzed. In the original formulation,
and most applications of QAOA, the cost function is classical, ensuring that
its corresponding Hamiltonian consists of Pauli terms only containing the
Pauli $Z$ operators. For this section, we restrict to QAOA applied to the
well-studied form of quadratic unconstrained binary optimization (QUBO), with
cost functions given by a Hamiltonian of the form

$H=H_{1}+H_{2}=\sum_{i}h_{i}Z_{i}+\sum_{ij}J_{ij}Z_{i}Z_{j}.$ (6)

Many popular combinatorial optimization problems can be cast into QUBO form
Glover  _et al._ (2019). In order to understand the effects of various noise
channels on specific problems under certain noise assumptions then, it
suffices to compute the action of the dual channel on Pauli terms with limited
locality, and analyze how the modified $H^{\prime}$ Hamiltonians relate to the
original cost Hamiltonians.

For parameterized circuits, such as QAOA, in order to perform mathematical
analysis, we assume $\mathcal{E}$ is independent of the parameters
$(\vec{\gamma},\vec{\beta})$, and is a product of local channels such that we
can write

$\rho(\vec{\gamma},\vec{\beta})=\mathcal{E}\mathcal{U}(\vec{\gamma},\vec{\beta})\left[|\psi_{0}\rangle\langle\psi_{0}|\right].$
(7)

This assumption for QAOA circuits is visualized in Fig. 1 c), with
$\mathcal{E}=\bigotimes_{i=1}^{N}\mathcal{E}_{i}$ that is equivalent to MATEN
(thoroughly described in the next section). A similar noise structure was
considered in Marshall  _et al._ (2020); Xue  _et al._ (2021).

In the following subsections, we analyze the effects of the dual map of
various common error channels on Hamiltonians of form Eq. (6) wit one layer.
For these examples, we assume the error channel is identical on each qubit in
order to simplify the equations, but this assumption can straightforwardly be
relaxed.

####  II.1.1 Single Qubit Depolarizing Channel

We first define a single qubit depolarizing channel, parameterized by a
depolarization rate $p$, as

$\mathcal{E}_{p}^{i}(\rho)=\frac{1+3p}{4}\rho+\frac{1-p}{4}\sum_{k=1}^{3}\sigma_{k}^{i}\rho\sigma_{k}^{i},$
(8)

with $\sigma_{1},\sigma_{2},\sigma_{3}$ corresponding to Pauli X,Y, and Z
respectively, and the indices $i$ corresponding to the qubit that Pauli
operators act upon. Note that each Pauli matrix is an eigenmatrix of the
depolarizing channel with eigenvalue $p$, i.e.
$\mathcal{E}_{p}(\sigma_{k})=p\sigma_{k}$, and this map is self-dual
($\mathcal{E}=\mathcal{E}^{\\#}$), so we also have
$\mathcal{E}^{\\#}_{p}(\sigma_{k})=p\sigma_{k}$.

For an $N$ qubit system we have noise channel acting on each qubit, i.e.
$\mathcal{E}_{p}=\mathcal{E}_{p}^{1}\otimes\mathcal{E}_{p}^{2}\otimes\ldots\otimes\mathcal{E}_{p}^{n}$,
where $\mathcal{E}_{p}^{i}$ corresponds to a one-local channel on qubit $i$.
Therefore, if $\mathcal{E}_{p}$ acts on a $k$-local term in Hamiltonian, and
we assume that $p$ is constant on all qubits, $\mathcal{E}_{p}$ effectively
multiplies this term by $p^{k}$. Specifically, we have

$\displaystyle\mathcal{E}^{\\#}_{p}(Z_{i})=pZ_{i},$ (9)
$\displaystyle\mathcal{E}^{\\#}_{p}(Z_{i}Z_{j})=p^{2}Z_{i}Z_{j}.$ (10)

This allows us to easily identify the action of local depolarizing noise on
QAOA for depth one with the noise channel applied at the end of the circuit,
by moving to the dual picture

$\displaystyle H^{\prime}=\mathcal{E}_{p}^{\\#}(H)=pH_{1}+p^{2}H_{2}.$ (11)

Thus, for single qubit depolarizing channels, the effect on QAOA cost
operators is simply that one-qubit terms are rescaled by $p$, two-qubit terms
are rescaled by $p^{2}$, and $k$-qubit terms by $p^{k}$ (although $k>2$ are
not considered for QUBO problems). For a strictly 2-local problem such as
MaxCut, this would mean that the cost is simply rescaled by $p^{2}$. For
optimization purposes, this simple rescaling means that the optimal parameter
settings stay unchanged.

####  II.1.2 Amplitude Damping

Another common error channel is amplitude damping, given by the following map

$\mathcal{E}{\rho}=A_{1}\rho A_{1}^{\dagger}+A_{2}\rho A_{2}^{\dagger},$ (12)

where $A_{1}$, $A_{2}$ are Kraus operators parameterized by a damping rate
$\gamma$ and given by

$A_{1}=\left(\begin{array}[]{cc}1&0\\\
0&\sqrt{1-\gamma}\end{array}\right)\quad,\quad
A_{2}=\left(\begin{array}[]{cc}0&\sqrt{\gamma}\\\ 0&0\end{array}\right).$ (13)

The action of the dual of this error channel on single and two qubit Pauli Z
operators are as follows

$\displaystyle\mathcal{E}_{\gamma}^{\\#}(Z_{i})=(1-\gamma)Z_{i}+\gamma I,$
(14)
$\displaystyle\mathcal{E}_{\gamma}^{\\#}(Z_{i}Z_{j})=(1-\gamma)^{2}Z_{i}Z_{j}+\gamma(1-\gamma)(Z_{i}+Z_{j})+\gamma^{2}I.$
(15)

We can see then that the 1-local terms are simply scaled and shifted. For the
2-local terms, we get not only a scale and a shift from the first and third
terms, respectively, but also an extra contribution of 1-local terms from the
middle term. We can write out the action of this channel on the general
Hamiltonian given in Eq. (6)

$\displaystyle
H^{\prime}=(1-\gamma)H_{1}+\gamma\sum_{i}h_{i}+(1-\gamma)^{2}H_{2}$
$\displaystyle+\gamma^{2}\sum_{i<j}J_{ij}+\gamma(1-\gamma)\sum_{i<j}J_{ij}(Z_{i}+Z_{j}).$
(16)

Now the only term that is neither a scale nor a constant shift is the last
term. We first note that if $h_{i}=0$ for all $i$ and we start in a
$\mathbb{Z}_{2}$ symmetric state, the resultant QAOA state is $\mathbb{Z}_{2}$
symmetric, thus all single qubit $Z$ terms go to zero Shaydulin  _et al._
(2021), so this added 1-local term has no effect on the observed cost function
value.

Another case where this term has a nice solution can be seen as follows: We
can rewrite $\sum_{i<j}J_{ij}(Z_{i}+Z_{j})$ as $\sum_{i}Z_{i}(\sum_{j\neq
i}J_{ij})$. Next, if $(\sum_{j\neq i}J_{ij})=ah_{i}$ for all $i$ and for some
constant $a$, then this term gives us $aH_{1}$, meaning that $H_{1}$ is
rescaled by $1-\gamma+a$ instead. This occurs in some cases enumerated below

  1. 1.

if all $h$’s and $J$’s are constant (all equal $h$, $J$, respectively):

$\sum_{i}Z_{i}(\sum_{j\neq i}J_{ij})=\frac{J(N-1)}{h}H_{1},$ (17)

  2. 2.

$d$-regular graph, all $h$’s constant, all nonzero $J$’s are constant (all
equal $h$, $J$, respectively):

$\sum_{i}Z_{i}(\sum_{j\neq i}J_{ij})=\frac{Jd}{h}H_{1},$ (18)

  3. 3.

max-$k$-colorable-subgraph Wang  _et al._ (2020) ($h_{i}=d_{i}$ where $d_{i}$
is degree of vertex $i$, $J_{ij}=-1$ if edge $(i,j)$ exists in the graph):

$\sum_{i}Z_{i}(\sum_{j\neq i}J_{ij})=-\sum_{i}d_{i}Z_{i}=-H_{1}.$ (19)

Notably, if $a=-1$ as in case 3), the Hamiltonian reduces to

$H^{\prime}=(1-\gamma)^{2}H+\gamma\sum_{i}h_{i}+\gamma^{2}\sum_{i<j}J_{ij},$
(20)

where we see that the entire Hamiltonian is simply scaled and shifted.

For an analysis of the effects of other common error channels on QAOA
operators, such as Pauli Channels, T1/T2 error, and overrotations, etc),
please see Appendix A.

##  III Single Qubit Noise Characterization

So far we have shown how certain local noise channels affect 1-local and
2-local observables, given complete knowledge of the noise. In this section,
however, we demonstrate the opposite direction, showing how to exploit the
dual map framework to find a marginal approximation to the effective noise
(MATEN), which is defined as follows

######  Definition 1 (MATEN).

For a unitary quantum circuit $\mathcal{U}$ acting on $N$-qubits, and its
noisy realization $\Lambda_{U}$, we call
$\mathcal{E}=\Lambda_{U}\circ\mathcal{U}^{{\dagger}}$ an effective noise
channel, that acts as the final CPTP circuit layer. Additionally we define a
marginal approximation to the effective noise (MATEN) as

$\tilde{\mathcal{E}}=\bigotimes_{k=1}^{N}\mathrm{Tr}_{\bar{k}}\left(\mathcal{E}\right)=\bigotimes_{k=1}^{N}\mathcal{E}_{k},$
(21)

where $\mathrm{Tr}_{\bar{k}}(\mathcal{E})=\mathcal{E}_{k}$ traces out all
subsystem except $k$-th (see Fig. 1 b)).

In order to determine a MATEN, we express a noisy map in terms of so-called
process matrix (or $\chi$ matrix), that can be in principle measured directly
in a set of experiments via quantum process tomography Chuang and Nielsen
(1997). The map takes the form (for a single qubit),

$\mathcal{E}_{\chi}(\rho)=\sum_{k,l=0}^{3}\chi_{kl}\sigma_{k}\rho\sigma_{l},$
(22)

where $\chi_{kl}$ are elements of the $\chi$ matrix, which in general can be
expressed as

$\chi=\left(\begin{array}[]{cccc}p_{0}&t_{0,1}+iv_{0,1}&t_{0,2}+iv_{0,2}&t_{0,3}+iv_{0,3}\\\
t_{0,1}-iv_{0,1}&p_{1}&t_{1,2}-it_{0,3}&t_{1,3}+it_{0,2}\\\
t_{0,2}-iv_{0,2}&t_{1,2}+it_{0,3}&p_{2}&t_{2,3}-it_{0,1}\\\
t_{0,3}-iv_{0,3}&t_{1,3}-it_{0,2}&t_{2,3}+it_{0,1}&p_{3}\\\
\end{array}\right),$ (23)

with $t_{kl}$ and $v_{kl}$ representing real and imaginary parts of
$\chi_{kl}$ elements, respectively. This form, combined with conditions
$\chi\geq 0$ and $\sum_{k=0}^{3}p_{k}=1$, guarantees that the map is
completely positive and trace preserving, and is the most general for the
qubit systems. Note, that in total we have 12 free parameters, and
diagonalizing the $\chi$ matrix will lead to a Kraus form (note that the Kraus
form is not unique). Given $\chi$ we can then evaluate the effect of the dual
map on the Pauli observables

$\displaystyle\tilde{I}$ $\displaystyle=I,$ (24) $\displaystyle\tilde{X}$
$\displaystyle=(p_{0}+p_{1}-p_{2}-p_{3})X+4t_{01}I$
$\displaystyle+2((t_{12}-v_{03})Y+(t_{13}+v_{02}))Z,$ (25)
$\displaystyle\tilde{Y}$ $\displaystyle=(p_{0}+p_{2}-p_{1}-p_{3})Y+4t_{02}I$
$\displaystyle+2((t_{23}-v_{01})Z+(t_{12}+v_{03}))X,$ (26)
$\displaystyle\tilde{Z}$ $\displaystyle=(p_{0}+p_{3}-p_{1}-p_{2})Z+4t_{03}I$
$\displaystyle+2((t_{13}-v_{02})X+(t_{23}+v_{01}))Y,$ (27)

where $\tilde{I}$,$\tilde{X}$,$\tilde{Y}$,$\tilde{Z}$ represent the noisy
transformations of the Pauli operators, and $\tilde{I}=I$ due to the property
that the dual of trace preserving maps are unital (i.e.
$\mathcal{E}^{\\#}(I)=I$). We can rewrite the coefficients in each equations
as $P_{AB}$, forming a vector of coefficients $\vec{P}$, so for example
$\tilde{X}=P_{XI}I+P_{XX}X+P_{XY}Y+P_{XZ}Z$. We can also write a simple matrix
$A$ that relates coefficients $P_{AB}$ to the $\chi$ matrix elements as in
$\vec{P}=A\vec{\chi}$, where $\vec{\chi}$ is a 12-dimensional vector having
all independent $\chi$ matrix elements (i.e. $p_{k},t_{kl}$, and $v_{kl}$).

Given Eqs. (24)-(27) we can then perform the following procedure for a
parameterized circuit of interest111Extension to parameter-free circuit is
straightforward, and requires only altering some gates, e.g. $X\to Z$.
However, this procedure would disturb the investigated algorithm, and could
serve only as a characterization protocol.:

  1. 1.

Choose a set $\mathbb{S}$ of parameters to the circuit. E.g. for level-1 QAOA
this corresponds to choosing $|\mathbb{S}|$ different ($\gamma_{1}$,
$\beta_{1}$) pairs.

  2. 2.

Implement the circuit on a quantum device, take many measurements in the $X$,
$Y$, and $Z$ bases to approximate $\braket{\tilde{X}}$, $\braket{\tilde{Y}}$,
and $\braket{\tilde{Z}}$ for each parameter setting in $\mathbb{S}$ on each
qubit. Since $\braket{\tilde{I}}$ is trivial, the measurement is not needed.

  3. 3.

On a classical simulator or via analytic derivation, determine the ideal
values of the $\braket{X}$, $\braket{Y}$, and $\braket{Z}$ for each parameter
setting in $\mathbb{S}$ for each qubit. $\braket{I}$ is trivial to calculate.

  4. 4.

Using the ideal and noisy values of all four Pauli observables, determine the
coefficients $\vec{P}$ via linear regression on Eqs. (24)-(27) for each qubit.

  5. 5.

Given the coefficients $\vec{P}$, along with the matrix $A$ relating $\vec{P}$
to $\vec{\chi}$ matrix elements, perform $\vec{\chi}_{pred}=A^{-1}\vec{P}$ for
each qubit, where $\vec{\chi}_{pred}$ are the elements of the predicted $\chi$
matrix.

The above protocol is visualized in the chart in Fig. 2.

Figure 2: Protocol for characterizing local approximation to effective noise
in the case of a single-layered QAOA algorithm. 1) run multiple times a
circuit with different parameters from the set $\mathbb{S}$, next 2) measure
all qubit registers in $X,Y$, and $Z$ bases. Use output bitstrings to infer
(on a classical computer) noisy expectation values
$\langle\tilde{X}\rangle,\langle\tilde{Y}\rangle$, and
$\langle\tilde{Z}\rangle$. For the same set of parameters, 3) simulate ideal
circuits on a classical computer to obtain expectation values. Based on the
ideal expectation values 4) construct vector $\vec{P}$, and then 5) determine
$\vec{\chi}_{pred}$. Perform this procedure for each qubit in order to
reconstruct the MATEN that approximates effective noise.

We note two limitations with the presented procedure. First is that of step 3,
in general, it may be prohibitive to determine the ideal values of single
qubit expectation values in simulation. However, for shallow circuits, one can
use reverse light cone arguments to calculate local operator expectation
values in time and memory growing exponentially with circuit depth, rather
than circuit size Peng  _et al._ (2020). Additionally, if the noise channels
_mildly_ depend on circuits 222Here by mildly we mean, that noise is static,
and parameter (e.g. angle) independent to the leading order., one could
perform this characterization process on a few sets of qubits individually;
this approach would work especially for shallow circuits. Finally, state of
the art classical simulators can handle circuits with relatively large depth
and qubit number, depending on simulation methods and computational resources.

Second, for problems with $\mathbb{Z}_{2}$ symmetry, the ideal values of
$\braket{Y}$ and $\braket{Z}$ vanish, so it may be impossible to fully
determine $\vec{P}$, and it remains an open question if we can reliably
determine nonzero elements of $\vec{P}$. If this is the case, one can derive
similar equations as Eqs. (24)-(27), but for two-qubit operators, although
this becomes much more complicated. For our analysis, we restrict to problems
that lack $\mathbb{Z}_{2}$ symmetry. For a problem such as MaxCut, this can be
achieved by simply adding single qubit $Z$ terms to the Hamiltonian.
Presumably, these single qubit $Z$ gates do not introduce a significant amount
of noise (on Rigetti devices, they are indeed implemented in software), so the
$\chi_{pred}$ matrix should remain close to that of the original circuit.
Thus, the characterized channels for these modified problems should match very
well those of the original problems. One can also break this symmetry by
starting in a different initial state. For QAOA problems the initial state is
usually $\ket{+}^{\otimes N}$, which is $\mathbb{Z}_{2}$ itself, but changing
$\ket{+}$ to a different non-$\mathbb{Z}_{2}$ symmetric state would break that
symmetry.

##  IV Local vs Non-Local Channels

One of the major challenges in current technology is understanding spatial
correlations in noise. Whether or not noise is confined locally to a single
qubit, or can be correlated across neighboring (or even distant) qubits (such
as in crosstalk Ash-Saki  _et al._ (2020); Sarovar  _et al._ (2020))
determines the efficacy of error mitigation techniques, and quantum error
correction (where errors are typically assumed to be independent). Here we aim
to find out, how well one can approximate non-local noise channels with the
MATEN approach. Our strategy is as follows: i) first we derive a lower bound
for the worst case scenario, ii) then we numerically compute accuracy of the
method for random non-local channels, iii) finally we repeat numerical
analysis from ii), but for random Pauli channels and analyze some scaling
properties.

Since single-qubit $\chi$ matrix (in Pauli basis) is a positive operator of
trace one, we can treat it as a 4-dimensional quantum state (with some extra
constraints imposed by the structure of $\chi$). This enables us to
incorporate results from the theory of quantum entanglement for the analysis
of non-local channels. In particular, all the marginal states for maximally
entangled states are maximally mixed states, i.e. they are proportional to the
identity matrix. Therefore, the marginal approximation (MA), which on the
level of $\chi$ matrix is translated to

$\chi=\ket{\Psi}\bra{\Psi}\to\chi_{\mathrm{MA}}=\bigotimes_{k=1}^{N}\left(\frac{1}{4}\mathbbm{1}\right)=\frac{1}{4^{N}}\mathbbm{1}_{4^{N}},$
(28)

also yields the maximally mixed state in the full $4^{N}$ dimensional space
(where $N$ is the number of considered qubits), which corresponds to the fully
depolarizing channel. Above we denote the non-local process matrix $\chi$ as
the maximally entangled state, that is defined as a projector onto
$\ket{\Psi}=\frac{1}{2}\sum_{i=0}^{3}\ket{ii\ldots i}$, with $N$ 4D subsystems
(each corresponding to a qubit), note that this is a GHZ state Greenberger
_et al._ (2007). We conjecture that the effective channel with the maximally
entangled $\chi$ matrix is the worst case scenario for the proposed MATEN
protocol. Since the MATEN approach neglects all non-trivial correlations
between different subsystems, and maximally entangled states exhibit the
strongest correlations among quantum objects resulting in minimal knowledge of
the subsystem’s structure (maximally mixed state), the protocol yields the
minimum fidelity value between the marginal approximation (MA) and the full
$\chi$. However, this conjecture requires more rigorous treatment, which we
leave as an open problem. Note that maximally entangled $\chi$ is a completely
valid choice, since $\chi\geq 0$ and the map associated with it is trace
preserving.

Having established that the maximally entangled $\chi$ matrix is the limiting
case for the protocol, now we determine the accuracy of this approximation.
For this purpose we incorporate the fidelity of quantum states as a useful
figure of merit. We compute it for the non-local $\chi$ matrix and its MA.
Since, the MA gives a trivial state, one can easily compute the fidelity
Życzkowski and Sommers (2005); Nielsen (2002)

$F\left(\chi,\chi_{MA}\right)=\frac{1}{4^{N}}\mathrm{Tr}(\sqrt{\chi})^{2}=\frac{1}{4^{N}},$
(29)

where we used the fact that $\chi$ is a projector (i.e. $\sqrt{\chi}=\chi$,
and $\mathrm{Tr}(\chi)=1$). As mentioned before, this result represents the
worst case scenario, and is unlikely to happen in real experiments (especially
if one is interested in low depth circuits), where hardware building blocks
operate on fairly high gate fidelity (95-99%, with lower fidelities for multi-
qubit gates, and higher for single-qubit ones). Therefore, we can escape this
unfavorable scaling by restricting to channels that are close to perfect
(noiseless) case, i.e. to the identity channel ($\chi_{00}=1$ and all other
elements equal to zero). This also implies that non-local effects are
comparably small to the leading order, which is predominantly determined by
the $\chi_{00}$ element. Similar restrictions are commonly considered in
benchmarking literature (see for example Flammia and Wallman (2020)), since
they represent noise regimes that are more relevant for the current hardware
technology and help tailor error correcting schemes. In order to properly
address this issue, we incorporate numerical methods to find out how well the
MA can represent the true non-local noise process. Here, we use random
sampling of full $\chi$ matrices and random samples of Pauli channels (i.e.
$\chi$ matrices with a random probability vector on the diagonal and all other
elements equal to zero). For the case of the full random $\chi$ processes, we
explore systems composed of $N=2,3,4$ qubits, while for Pauli channels we
additionally look at $N=5$. The results are displayed in Fig. 3, where we took
10,000 samples of random channels (generated with QuTiP Johansson  _et al._
(2012)), and computed all marginals of the multi-qubit $\chi$ matrix (i.e.
tracing out all but one qubit) and compared fidelity between a tensor product
of the marginals (essentially what we call the MA) and the non-local one.

Figure 3: Comparison between random non-local $\chi$ matrix and its MA in
terms of fidelity. Left plot depicts case of full random channels, while right
plot restricts the analysis to random Pauli channels. Additionally, we provide
a numerical lower bound (LB) on Pauli channels with two and three qubits
(right plot).

For random Pauli channels, we additionally numerically minimize the fidelity
between the $4^{N}$ dimensional probability vector representing the Pauli
channel, and its MA 333The marginal approximation for Pauli channels is also
done on the level of matrices, and not on the probability vectors.. In order
to guarantee a genuine probability distribution over our parameters (without
having to impose any constraint) we use the modified Hurwitz parametrization
for the probability vector Hurwitz (1897); Zyczkowski and Sommers (2001)

$\displaystyle\chi=$
$\displaystyle\mathrm{diag}[\cos^{2}(\theta_{4^{N}-1}),\cos^{2}(\theta_{4^{N}-2})\sin^{2}(\theta_{4^{N}-1}),\ldots,$
$\displaystyle\sin^{2}(\theta_{1})\sin^{2}(\theta_{2}),\cdots\sin^{2}(\theta_{4^{N}-1})]$
. (30)

We employ Sequential Least Squares Programming (SLSQP) Kraft  _et al._ (1988)
optimization routine to find the lower bound. Surprisingly, two and three
qubit channels display similar lower bounds (in particular for high fidelity
channels, i.e. $\chi_{00}$ close to one). The key observation is that for
channels with reasonably large $\chi_{00}$ (corresponding to the identity
channel), which is directly related to the gate/circuit fidelity, the MA can
provide results with acceptable accuracy. Therefore, the MATEN protocol,
identifies a MA that can estimate the leading order of the effective noise
channel.

##  V Results

In this section we present the success of the method presented in III for
noisy simulations.

###  V.1 Classical Simulation

For classical simulations, we test our characterization method against a
variety of noise sources. Noiseless and noisy classical simulations are
performed via pure state and density matrix simulations with HybridQ, an open-
source hybrid quantum simulator Mandrà  _et al._ (2021). In some cases, we
additionally generate and apply error channels via QuTiP Johansson  _et al._
(2012) an open-source toolbox that allows for classical simulation of open
quantum systems. With this capability of finding ideal and noisy states and
operators, we can easily compute metrics needed to evaluate our method. For
all of these experiments, we test the characterization method on parameterized
QAOA circuits for QUBO problems.

####  V.1.1 Purely Local Noise

First we test the efficacy of the characterization method laid out in Sec. III
for predicting $\chi$ matrices that we manually apply at the end of noiseless
classical simulation. To do this, we pick a $\chi_{in}$ matrix by iteratively
selecting elements uniformly randomly from the interval $[0,1]$ for the
elements $\vec{p}$, and $[-1,1]$ for $\vec{t}$ and $\vec{v}$ in Eq. (23), and
checking if the resultant map is physical (i.e. $\chi\geq 0$) until we
succeed. We further choose the same $\chi_{in}$ matrix on each qubit, although
this is relaxed in the next section. We additionally choose random QUBO
problems by randomly drawing $J$ and $h$ from a uniform distribution in range
$[0,1]$. In this experiment, we should expect that for some reasonable number
of parameter settings (size of $\mathbb{S}$) and for a sufficient number of
shots (measurements), we should be able exactly recover the input $\chi_{in}$
matrix to arbitrary precision, as the noise is taken to fit perfectly within
the MATEN approximation. We quantify the accuracy of determining $\chi_{in}$
by taking the L2 distance between the elements of $\chi_{in}$ and
$\chi_{pred}$, the process matrix our method predicts. The results for various
values of shot number and number of regression angles are plotted in Fig. 4.
These plots are generated using statevector simulations for perfect evaluation
of observables.

Figure 4: Average L2 distance between the true and predicted $\chi$ for
classical simulations for randomly chosen two qubit QUBO instances with
weights in the range $[0,1]$, as a function of the number of angles
$|\mathbb{S}|$ used in the regression and the number of shots used in the
estimation of expectation values. Solid lines depict the average over 100
runs, shading depicts one standard deviation above and below the average. For
large number of shots and angles, the distance is below $10^{-6}$.

Indeed, we see that for a typical case, increasing the number of angles and
the number of shots used in regression allows for more accurate determination
of $\chi_{in}$. We further see from this figure that the L2 distance shrinks
with added number of shots, by roughly a factor of $10$ when the number of
shots increases by a factor of $10$. We later numerically see this roughly
polynomial scaling with the number of shots for various values of
$|\mathbb{S}|$. For instance, with $|\mathbb{S}|=16$ we find the L2 distance
goes as $numshots^{-1.39}$. We note that the L2 distance between randomly
chosen $\chi$ matrices was numerically found to be $.800\pm.125$, but we see
fidelities much higher than this value for sufficiently large $|\mathbb{S}|$
and number of shots.

####  V.1.2 Non-Local Noise At End of Circuit

In order to test the resiliency of the noise characterization procedure, we
must test the method against noise models that a MATEN is not suited to
perfectly capture. For the first of these models, we choose a constant error
channel that exists only at the end of a quantum circuit, but is not a simple
tensor product of single qubit channels. In order to apply this combination of
local and non-local noise, then, we apply an error map of the form

$\mathcal{E}=(1-c)\mathcal{E}_{1}^{(n)}+c\mathcal{E}_{n}^{(n)},$ (31)

where we have a combination of purely local channel $\mathcal{E}_{1}^{(n)}$
and nonlocal channel $\mathcal{E}_{n}^{(n)}$ weighted by a correlation factor
$c\in[0,1]$. In this section we allow the $\chi_{in}$ matrices to vary for
each qubit (in $\mathcal{E}_{1}^{(n)}$). Since due to the addition of extra
noise ($\mathcal{E}_{n}^{(n)}$) we no longer expect that
$\chi_{pred}\approx\chi_{in}$, we no longer report the fidelity between the
two. Instead, we use
$(1/3)*r(\braket{X},\braket{\tilde{X}})+r(\braket{Y},\braket{\tilde{Y}})+r(\braket{Z},\braket{\tilde{Z}})$,
the average Pearson correlation coefficient $r(x,y)$ between the measured and
predicted expectation values of $\braket{X}$, $\braket{Y}$, and $\braket{Z}$,
as these tell us how well our noise model predicts simple observables of
interest on the quantum device. However, it is possible to induce overfitting,
especially when the number of considered parameter settings ($|\mathbb{S}|$)
is small. Thus we additionally look at correlations for an additional “testing
set” of parameter settings. For our experiments at around $|\mathbb{S}|=50$
however, these correlations very closely matched that of the training set, so
we only present correlations of the testing set for the following cases. In
addition to correlation, we also use Choi fidelity, defined as
$F(\Phi_{1},\Phi_{2})$, the state fidelity between Choi matrices
$\Phi_{1},\;\Phi_{2}$, representing respectively the entire $n$-qubit maps
generated from the chosen error channels, and the predicted MATEN from our
method. We present the results from the characterization of this noise model
in Fig. 5. For these experiments, we fix our problem Hamiltonian to a fully-
connected QUBO instance with all $J=1$ and all $h=0$. Evolution and
expectation values are evaluated using density matrix simulation.

Figure 5: Average testing correlations and full state fidelities between the
actual noise model and predicted noise model for two (left) and four (right)
qubit fully connected QUBO problems with all $J=1$ and all $h=0$, as a
function of the weight of the applied non-local channel, or $c$ in Eq. (31).
Solid lines depict the average over 100 runs, shading depicts one standard
deviation above and below the average.

From these simulations we see that, as expected, when $c=0$ and there is only
local noise, the model works extremely well. However, as more non-local noise
is added into the system, the ability to accurately predict the expectation
values of Pauli observables begins to falter. At $c=1$, we typically see a
sharp downturn of correlations, as at this point there we are not injecting
any purely local noise to our system, thus weakening the accuracy of the
MATEN.

####  V.1.3 Sampling Noise

In addition to the above tests, we also experimented with adding in sampling
noise to our noisy simulations. To accomplish this, we choose random Gaussian
perturbations with mean $0$ and standard deviation of $1/\sqrt{numshots}$ to
add to all expectation value measurements, simulating the effect of sampling
error on the evaluation of expectation values. Given this form of noise, we
repeated analysis from above, running QAOA with cost Hamiltonian given by Eq.
(6) with two qubits and all $h=0$, $J=1$. We varied the number of shots on the
x-axis, and the results of this setup are shown in Fig. 6.

Figure 6: Average testing correlations and Choi fidelities between the actual
noise model and predicted noise model for two qubit QUBO problem with $J=1$
and both $h=0$, as a function of the number of shots used to estimate
expectation values. Solid lines depict the average over 100 runs, shading
depicts one standard deviation above and below the average.

From the simulations we can see that sampling noise diminishes the ability of
the method to accurately fit the noisy measurements to ideal measurements, as
well as predict the value of noisy measurements. The stochastic noise in
causes expectation values to fluctuate between measurements, thus essentially
introducing a non-constant noise model. This may cause poor performance as our
method depends on having the same error channels for all angles and
measurement bases.

Additionally, we note that poor performance may arise if errors are angle-
dependent, leading to an error model that is non-constant between different
angles in a similar manner to sampling noise. Errors can be extremely angle-
dependent on quantum computers, especially for parameterized two qubit gates
such as Abrams  _et al._ (2020), so this feature could be an important
limitation in the success of the method in the near term.

####  V.1.4 Larger Systems: Phasing and Mixing Rotation Error

Finally, in order to scale our simulations to larger system sizes, we
performed our characterization routine on $10$-qubit instances. For these
runs, selecting and applying randomly generated $10$-qubit Kraus maps becomes
numerically prohibitive, so we switch to a simpler and more realistic noise
model. For these experiments, we assume that there is some stochastic error,
or deviation in the parameters for both the phasing and mixing operators. In
particular, the QAOA angles are assumed to be normally distributed about the
desired mean value, with a non-zero standard deviation that defines the total
amount of noise. For a given phasing gate $e^{-i\gamma Z_{i}Z_{j}}$, this
noise is introduced through the Kraus operators

$A_{1}=\sqrt{1-\omega}I,\ A_{2}=\sqrt{\omega}Z_{i}Z_{j},$ (32)

and for a mixing gate $e^{-i\beta X_{i}}$ we apply

$A_{1}=\sqrt{1-\omega}I,\ A_{2}=\sqrt{\omega}X_{i}.$ (33)

Here $\omega$ defines the amount of noise (related to the standard deviation
in the angles’ values). A derivation of these noise models is shown in
Appendix Sec A.3. This model applies the two-qubit dephasing noise layer (Eq.
(32)) on each pair qubits the phase gates act, after the dephasing unitaries
and directly before the mixing layer. After the mixing layer the one qubit $X$
noise is applied on each qubit (Eq. (33)).

Under this noise model we can test our characterization method on larger
systems, and test against the assumption that all noise is applied at the end
of the circuit. We display the results for the method on $10$-qubit ring and
fully-connected QUBO problems in Fig. 7.

Figure 7: Average testing correlations and Choi fidelities between the actual
noise model and predicted noise model for 10-qubit ring (right) and fully-
connected (right) QUBO problems with all $J=1$ and all $h=0$, as a function of
the deviation $\omega$ of both the phasing and mixing operators.

For these plots, no matter the value of $\omega$, we saw that we were able to
perfectly reproduce $1$-qubit correlations, so we chose to add in the average
of all $2$-qubit correlations as well. Additionally, we report the average
fidelity between the actual $10$-qubit noisy density matrix and the predicted
density matrix using the characterized noise model. From these results we find
that the fidelity drops rapidly, especially for the fully connected case.
Crucially, however, the $1$ and $2$-qubit correlations remain very high, even
as the $\omega$ grows. We note that on the fully connected plot, the fidelity
rises after $\omega\approx.02$. This is likely explained by the fact that
$\omega=0.5$ corresponds to the maximally dephasing channel, which our model
can capture well. Thus we expect to see the fidelity drop initially as
$\omega$ grows, then rise back to $1$ when $\omega=.5$, and then follow a
symmetric pattern once $\omega>.5$. From these results, however, our main
takeaway is that even in the presence of noise which is not local and not
strictly at the end of the circuit, the method finds a suitable MATEN
approximation that is able to replicate single-qubit expectation values
perfectly and two-qubit expectation values very well, even as we scale to
large system sizes.

##  VI Characterization of Rigetti’s Aspen-9 Device

In this section we apply the error characterization method from Sec. III to
the Aspen-9 Quantum Processing Unit (QPU) from Rigetti Computing rig . We run
the characterization procedure for QAOA circuits with phase separation given
by Hamiltonians of the form in Eq. (6), with all $h_{i}=1$ (to break
$\mathbb{Z}_{2}$ symmetry) and $J_{ij}=\delta_{i+1,j}$ (forming a line
topology), and implemented using a single CPHASE($\gamma$) gate, and with
mixing via the standard X-mixer. These experiments were run at $N=2$ and $N=6$
with $|\mathbb{S}|=100$, where $N$ is the number of qubits and $|\mathbb{S}|$
is the number of different parameter settings used. For these experiments, we
run under three cases.

  1. a)

(1q only) Remove all two-qubit (CPHASE) gates (equivalent to setting all
$J_{ij}$ to $0$). The intention of this is to make sure that our method works
when only single qubit gates are present, removing main sources of crosstalk
and non-local noise, which could distort the results.

  2. b)

(2q idle) Add back in two qubit (CPHASE) gates, but set the angles ($\gamma$)
of all two-qubit gates to $0$ (again equivalent to setting all $J_{ij}$ to
$0$). This ideally implements the same circuit as the previous case, but two-
qubit gates are physically implemented in the circuit.

  3. c)

(2q active) Lift the restriction of setting two-qubit gate angles to $0$, thus
performing the method completely as intended.

For these experiments, much like Sec. V.1.2, we present statistics on the
correlations between predicted and observed Pauli expectation values. These
are shown for both the two and six qubit cases in table 1.

qubit | 1q only | 2q idle | 2q active
---|---|---|---
34 | 0.9964 | 0.5154 | 0.9622
35 | 0.9982 | 0.7758 | 0.9704
Table 1: Correlations for the method performed on QAOA circuits for qubits 34
and 35 on Rigetti Aspen-9 device. The low values for the 2q idle case are
likely explained by day-to-day changes in qubit calibration and error sources
on the device.

For the two-qubit experiments, we see that the method is able to predict
expectation values of all Pauli observables with a high correlation to the
experimental values. We note that there is low fidelity for the “2q idle”
case. This is likely explained by the fact that this case was run a few days
after the other experiments, as this experiment idea was conceived after
running the “1q only” and “2q active” cases. Due to day-to-day changes in
calibration, qubits 34 and 35 may have experienced calibration issues on the
day of running. Unfortunately, the Aspen-9 device was de-commissioned before a
re-run of the experiment began. This faulty qubit can result in elevated
angle-dependent, nonlocal, or inter-circuit noise, which have the potential to
reduce the accuracy of a MATEN.

The six-qubit experiments are presented in Table 2.

qubit | 1q only | 2q idle | 2q active
---|---|---|---
30 | 0.9963 | 0.9254 | 0.5477
31 | 0.9901 | 0.9041 | 0.6692
32 | 0.9923 | 0.6584 | 0.5363
33 | 0.9948 | 0.0480 | 0.0449
34 | 0.9936 | 0.8674 | 0.7855
35 | 0.9985 | 0.9908 | 0.9801
Table 2: Correlations for the method performed on QAOA circuits for qubits
30-35 on Rigetti Aspen-9 device.

Here, we see that all metrics remain high for the “1q only” case, but for the
“2q idle” case for qubits 32 and 33 we see a significant drop in regression
score and average correlation. In the “2q active” case, we see a further
decline in the correlations of qubits 30, 31, 32, and 34. For this case, which
matches most closely the type of experiments we would like to characterize,
our method gives an average expectation value correlation of $.69\pm.30$.
These values are far from the ideal values of 1, but the positive correlation
values suggest that the method approximately captures the dominant error
channels present on the QPU. The wide variability in performance on various
qubits suggest that certain qubits may have more angle-dependent noise, or may
have larger sources of crosstalk, as analyzed in Sec. V.1. In particular,
qubits 32 and 33 experience a sharp decline in correlations in both the “2q
idle/active” cases, with the correlations of qubit 33 plummeting to roughly
$.05$. The correlations on qubit 33 of roughly $.05$ are additionally much
lower than we see even on the right side of Fig. 5 or anywhere in Fig. 7. This
indicates that the errors introduced by two-qubit gates are in a sense worse
than both of these cases. We suspect this may be due to the fact that the
added two qubit gate, even with all angles set to zero, may introduce some
significant crosstalk between the two qubits that is far from the intended
phasing operation, which the MATEN is not equipped to accurately handle. In
the simulations we perform, artificially added errors come in the form of
randomly chosen Kraus maps or overrotations, but the error maps on a quantum
device may be of a specific, more detrimental for. Additionally, even with a
two-qubit gate with angle set to $0$, it can be the case that a different
unitary is applied from shot-to-shot, approaching the case of Sec. V.1.3,
which is the only source of noise we found to reduce correlations to such a
low number. Thus we suspect that this error or shot-dependent noise may play a
role in the extremely low correlations, as we would not expect to be able to
accurately characterize any noise procedure that is changing over time.

##  VII Discussion

In this paper we introduce the dual map framework for computing the effects of
error maps on expectation values evaluated on a quantum computer. We then
presented a method to compute a marginal approximation to the effective noise
(MATEN) of a parameterized quantum circuit, that is efficient in terms of
number of measurements needed to perform on a quantum computer and is simple
to implement. We demonstrate that the method effectively computes a MATEN for
local noise at the end of a circuit, and demonstrate that it can be effective
even in the presence of nonlocal and inter-circuit noise, especially when the
noise is only weakly correlated. We finally show that the method is effective
in computing a MATEN on a few qubits of Rigetti’s Aspen-9 quantum computer.
Lower values in extracted correlations of expectation values can be inform us
that the system exhibits a fair amount of angle dependant (gate) noise, as
well as errors that are absent in the theoretical model, e.g. readout or
leakage to the non-computational subspace. The latter can be modelled in a
similar fashion as qubits under our scheme, with the difference that the
$\chi$ process matrix now needs to represent a qudit process. This, however,
introduces an extra layer of complexity, which we leave for future analysis.

The error characterization method can additionally be used as a proxy for the
fidelity of a gate, layer, or entire circuit, as the values of the computed
$\chi$ matrices for each qubit (specifically $\chi_{00}$) quantify the
difference between the ideal and noisy evolution. Furthermore, the returned
$\chi$ can inform dominant sources of error, which can in turn point to
particularly effective strategies from error mitigation, leading to
algorithmic improvements on NISQ devices. Once dominant sources of error are
determined, we leave these error-specific mitigation approaches as open
problems for the reader.

The dual map framework introduced can be used to understand which error
channels can be specifically detrimental for a circuit. For instance, with
QAOA, we show that depolarizing noise simply flattens the energy landscape,
thus it does not affect the location of optimal parameters for the algorithm.
However, error sources such as amplitude damping may introduce non-trivial
behavior. The characterization procedure we introduce can be used to
characterize error in NISQ devices, especially for shallow circuits in which
the effective noise channels are expected to be non-correlated. Overall, the
dual map picture for error channels provides a simple and elegant method for
researching the interplay between quantum error and algorithms in the future,
and our characterization approach can significantly aid hardware-aware
algorithm design on today’s devices.

##  VIII Acknowledgements

This material is based upon work supported by the U.S. Department of Energy,
Office of Science, National Quantum Information Science Research Centers,
Superconducting Quantum Materials and Systems Center (SQMS) under contract
number DE-AC02-07CH11359 through NASA-DOE interagency agreement SAA2-403602.
All authors appreciate support from the NASA Ames Research Center. JS, JM, ZW,
FW are thankful for support from NASA Academic Mission Services, Contract No.
NNA16BD14C.

## References

  * Piveteau  _et al._ (2021) Christophe Piveteau, David Sutter, Sergey Bravyi, Jay M. Gambetta, and Kristan Temme, “Error mitigation for universal gates on encoded qubits,” Phys. Rev. Lett. 127, 200505 (2021).
  * Temme  _et al._ (2017) Kristan Temme, Sergey Bravyi, and Jay M. Gambetta, “Error mitigation for short-depth quantum circuits,” Phys. Rev. Lett. 119, 180509 (2017).
  * Li and Benjamin (2017) Ying Li and Simon C. Benjamin, “Efficient Variational Quantum Simulator Incorporating Active Error Minimization,” Phys. Rev. X 7, 021050 (2017).
  * Nielsen (2002) Michael A Nielsen, “A simple formula for the average gate fidelity of a quantum dynamical operation,” Physics Letters A 303, 249–252 (2002).
  * Wudarski  _et al._ (2020) Filip Wudarski, Jeffrey Marshall, Andre Petukhov, and Eleanor Rieffel, “Augmented fidelities for single-qubit gates,” Phys. Rev. A 102, 052612 (2020).
  * Emerson  _et al._ (2005) Joseph Emerson, Robert Alicki, and Karol \\.Zyczkowski, “Scalable noise estimation with random unitary operators,”  7, S347–S352 (2005), publisher: IOP Publishing.
  * Magesan  _et al._ (2011) Easwar Magesan, J. M. Gambetta, and Joseph Emerson, “Scalable and Robust Randomized Benchmarking of Quantum Processes,” Physical Review Letters 106, 180504 (2011), publisher: American Physical Society.
  * Claes  _et al._ (2021) Jahan Claes, Eleanor Rieffel, and Zhihui Wang, “Character Randomized Benchmarking for Non-Multiplicity-Free Groups With Applications to Subspace, Leakage, and Matchgate Randomized Benchmarking,” PRX Quantum 2, 010351 (2021), publisher: American Physical Society.
  * Erhard  _et al._ (2019) Alexander Erhard, Joel J. Wallman, Lukas Postler, Michael Meth, Roman Stricker, Esteban A. Martinez, Philipp Schindler, Thomas Monz, Joseph Emerson, and Rainer Blatt, “Characterizing large-scale quantum computers via cycle benchmarking,” Nature Communications 10, 5347 (2019).
  * Flammia and Liu (2011) Steven T. Flammia and Yi-Kai Liu, “Direct Fidelity Estimation from Few Pauli Measurements,” Physical Review Letters 106, 230501 (2011), publisher: American Physical Society.
  * Chuang and Nielsen (1997) Isaac L. Chuang and M. A. Nielsen, “Prescription for experimental determination of the dynamics of a quantum black box,” Journal of Modern Optics 44, 2455–2467 (1997).
  * Greenbaum (2015) Daniel Greenbaum, “Introduction to Quantum Gate Set Tomography,” arXiv:1509.02921 [quant-ph] (2015), arXiv: 1509.02921.
  * Schirmer  _et al._ (2004) S. G. Schirmer, A. Kolli, and D. K. L. Oi, “Experimental Hamiltonian identification for controlled two-level systems,” Physical Review A 69, 050306 (2004), publisher: American Physical Society.
  * Kimmel  _et al._ (2015) Shelby Kimmel, Guang Hao Low, and Theodore J. Yoder, “Robust calibration of a universal single-qubit gate set via robust phase estimation,” Physical Review A 92, 062315 (2015), publisher: American Physical Society.
  * Sun and Geller (2020) Mingyu Sun and Michael R. Geller, “Efficient characterization of correlated SPAM errors,” arXiv:1810.10523 [quant-ph] (2020), arXiv: 1810.10523.
  * Lin  _et al._ (2021) Junan Lin, Joel J. Wallman, Ian Hincks, and Raymond Laflamme, “Independent state and measurement characterization for quantum computers,” Physical Review Research 3, 033285 (2021), publisher: American Physical Society.
  * Werninghaus  _et al._ (2021) M. Werninghaus, D.J. Egger, and S. Filipp, “High-Speed Calibration and Characterization of Superconducting Quantum Processors without Qubit Reset,” PRX Quantum 2, 020324 (2021), publisher: American Physical Society.
  * Milz  _et al._ (2017) Simon Milz, Felix A. Pollock, and Kavan Modi, “An introduction to operational quantum dynamics,” Open Systems & Information Dynamics 24, 1740016 (2017).
  * Breuer  _et al._ (2002) Heinz-Peter Breuer, Francesco Petruccione, _et al._ ,  _The theory of open quantum systems_ (Oxford University Press on Demand, 2002).
  * Bengtsson and Życzkowski (2017) Ingemar Bengtsson and Karol Życzkowski,  _Geometry of quantum states: an introduction to quantum entanglement_ (Cambridge university press, 2017).
  * Farhi  _et al._ (2014) Edward Farhi, Jeffrey Goldstone, and Sam Gutmann, “A Quantum Approximate Optimization Algorithm,” arXiv:1411.4028 [quant-ph] (2014), arXiv: 1411.4028.
  * Hogg (2000) Tad Hogg, “Quantum search heuristics,” Physical Review A 61, 052311 (2000), publisher: American Physical Society.
  * Hadfield  _et al._ (2019) Stuart Hadfield, Zhihui Wang, Bryan O’Gorman, Eleanor G. Rieffel, Davide Venturelli, and Rupak Biswas, “From the Quantum Approximate Optimization Algorithm to a Quantum Alternating Operator Ansatz,” Algorithms 12, 34 (2019), number: 2 Publisher: Multidisciplinary Digital Publishing Institute.
  * Marshall  _et al._ (2020) Jeffrey Marshall, Filip Wudarski, Stuart Hadfield, and Tad Hogg, “Characterizing local noise in QAOA circuits,”  1, 025208 (2020), publisher: IOP Publishing.
  * Xue  _et al._ (2021) Cheng Xue, Zhao-Yun Chen, Yu-Chun Wu, and Guo-Ping Guo, “Effects of Quantum Noise on Quantum Approximate Optimization Algorithm,”  38, 030302 (2021), publisher: IOP Publishing.
  * Wang  _et al._ (2021) Samson Wang, Enrico Fontana, M. Cerezo, Kunal Sharma, Akira Sone, Lukasz Cincio, and Patrick J. Coles, “Noise-induced barren plateaus in variational quantum algorithms,” (2021), arXiv:2007.14384 [quant-ph] .
  * Streif  _et al._ (2021) Michael Streif, Martin Leib, Filip Wudarski, Eleanor Rieffel, and Zhihui Wang, “Quantum algorithms with local particle-number conservation: Noise effects and error correction,” Physical Review A 103 (2021), 10.1103/physreva.103.042412.
  * Streif and Leib (2020) Michael Streif and Martin Leib, “Training the quantum approximate optimization algorithm without access to a quantum processing unit,”  5, 034008 (2020), publisher: IOP Publishing.
  * Zhou  _et al._ (2020) Leo Zhou, Sheng-Tao Wang, Soonwon Choi, Hannes Pichler, and Mikhail D. Lukin, “Quantum Approximate Optimization Algorithm: Performance, Mechanism, and Implementation on Near-Term Devices,” Physical Review X 10, 021067 (2020), publisher: American Physical Society.
  * Shaydulin  _et al._ (2019) Ruslan Shaydulin, Ilya Safro, and Jeffrey Larson, “Multistart Methods for Quantum Approximate optimization,” in  _2019 IEEE High Performance Extreme Computing Conference (HPEC)_ (2019) pp. 1–8, iSSN: 2643-1971.
  * Brandao  _et al._ (2018) Fernando G. S. L. Brandao, Michael Broughton, Edward Farhi, Sam Gutmann, and Hartmut Neven, “For Fixed Control Parameters the Quantum Approximate Optimization Algorithm’s Objective Function Value Concentrates for Typical Instances,” arXiv:1812.04170 [quant-ph] (2018), arXiv: 1812.04170.
  * Sack and Serbyn (2021) Stefan H. Sack and Maksym Serbyn, “Quantum annealing initialization of the quantum approximate optimization algorithm,” Quantum 5, 491 (2021), publisher: Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften.
  * Wang  _et al._ (2020) Zhihui Wang, Nicholas C. Rubin, Jason M. Dominy, and Eleanor G. Rieffel, “$XY$ mixers: Analytical and numerical results for the quantum alternating operator ansatz,” Physical Review A 101, 012320 (2020), publisher: American Physical Society.
  * Glover  _et al._ (2019) Fred Glover, Gary Kochenberger, and Yu Du, “A Tutorial on Formulating and Using QUBO Models,” arXiv:1811.11538 [quant-ph] (2019), arXiv: 1811.11538.
  * Shaydulin  _et al._ (2021) Ruslan Shaydulin, Stuart Hadfield, Tad Hogg, and Ilya Safro, “Classical symmetries and the Quantum Approximate Optimization Algorithm,” Quantum Information Processing 20, 359 (2021), arXiv: 2012.04713.
  * Note (1) Extension to parameter-free circuit is straightforward, and requires only altering some gates, e.g. $X\to Z$. However, this procedure would disturb the investigated algorithm, and could serve only as a characterization protocol.
  * Peng  _et al._ (2020) Tianyi Peng, Aram W. Harrow, Maris Ozols, and Xiaodi Wu, “Simulating Large Quantum Circuits on a Small Quantum Computer,” Physical Review Letters 125, 150504 (2020), publisher: American Physical Society.
  * Note (2) Here by mildly we mean, that noise is static, and parameter (e.g. angle) independent to the leading order.
  * Ash-Saki  _et al._ (2020) Abdullah Ash-Saki, Mahabubul Alam, and Swaroop Ghosh, “Experimental Characterization, Modeling, and Analysis of Crosstalk in a Quantum Computer,” IEEE Transactions on Quantum Engineering 1, 1–6 (2020), conference Name: IEEE Transactions on Quantum Engineering.
  * Sarovar  _et al._ (2020) Mohan Sarovar, Timothy Proctor, Kenneth Rudinger, Kevin Young, Erik Nielsen, and Robin Blume-Kohout, “Detecting crosstalk errors in quantum information processors,” Quantum 4, 321 (2020), publisher: Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften.
  * Greenberger  _et al._ (2007) Daniel M. Greenberger, Michael A. Horne, and Anton Zeilinger, “Going beyond bell’s theorem,” (2007), arXiv:0712.0921 [quant-ph] .
  * Życzkowski and Sommers (2005) Karol Życzkowski and Hans-Jürgen Sommers, “Average fidelity between random quantum states,” Physical Review A 71 (2005), 10.1103/physreva.71.032313.
  * Flammia and Wallman (2020) Steven T. Flammia and Joel J. Wallman, “Efficient estimation of pauli channels,” ACM Transactions on Quantum Computing 1, 1–32 (2020).
  * Johansson  _et al._ (2012) J. R. Johansson, P. D. Nation, and Franco Nori, “QuTiP: An open-source Python framework for the dynamics of open quantum systems,” Computer Physics Communications 183, 1760–1772 (2012), arXiv: 1110.0573.
  * Note (3) The marginal approximation for Pauli channels is also done on the level of matrices, and not on the probability vectors.
  * Hurwitz (1897) A. Hurwitz, “über die erzeugung der invarianten durch integration,” Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse 1897, 71–2 (1897).
  * Zyczkowski and Sommers (2001) Karol Zyczkowski and Hans-Jürgen Sommers, “Induced measures in the space of mixed quantum states,” Journal of Physics A: Mathematical and General 34, 7111–7125 (2001).
  * Kraft  _et al._ (1988) Dieter Kraft _et al._ , “A software package for sequential quadratic programming,” (1988).
  * Mandrà  _et al._ (2021) Salvatore Mandrà, Jeffrey Marshall, Eleanor G. Rieffel, and Rupak Biswas, “HybridQ: A Hybrid Simulator for Quantum Circuits,” (2021), arXiv:2111.06868 [quant-ph] .
  * Abrams  _et al._ (2020) Deanna M. Abrams, Nicolas Didier, Blake R. Johnson, Marcus P. da Silva, and Colm A. Ryan, “Implementation of XY entangling gates with a single calibrated pulse,” Nature Electronics 3, 744–750 (2020).
  * (51) “Rigetti quantum computing,” Https://www.rigetti.com/.

##  Appendix A More error maps

In this section we derive the effects of various other error channels on the
pauli $Z$ and $ZZ$ terms found in QUBO problems, in the same style as section
II.1. We first present the following proof to aid in the analysis:

######  Theorem 1.

For $\mathbb{Z}_{2}$ symmetric states, the expectation value of a Pauli string
$\mathcal{P}$ is $0$ if the number of Pauli Z’s plus Pauli Y’s in
$\mathcal{P}$ is odd.

###### Proof.

We assume an operator $\mathcal{O}$ on $N$ qubits that is of the form
$\prod_{i=1}^{N}\sigma_{p_{i}}$ where $p_{i}$ represents the Pauli that acts
on qubit $i$: either X,Y,Z, or I. We can then define $S_{X},S_{Y},S_{Z}$ to be
the set of qubits in which our operator is X,Y,and Z. We then look at the
expectation value of the most general operator of this form

$\bra{\Psi}\mathcal{O}\ket{\Psi}=\bra{\Psi}\prod_{i\in S_{Y}}Z_{i}\prod_{j\in
S_{Y}}Y_{j}\prod_{k\in S_{X}}X_{k}\ket{\Psi}.$ (34)

Then noting that Y = -iZX and write

$-i\bra{\Psi}\prod_{i\in S_{Z}}Z_{i}\prod_{j\in S_{Y}}Z_{j}X_{j}\prod_{k\in
S_{X}}X_{k}\ket{\Psi}=-i\bra{\Psi}\prod_{i\in S_{Z}\cup S_{Y}}Z_{i}\prod_{k\in
S_{X}\cup S_{Y}}X_{k}\ket{\Psi}.$ (35)

We can then expand out $\ket{\Psi}$ in terms of bitstrings $l$

$-i\bra{\Psi}\prod_{i\in S_{Z}\cup S_{Y}}Z_{i}\prod_{k\in S_{X}\cup
S_{Y}}X_{k}\sum_{l}c_{l}(\ket{l}+\ket{\overline{l}}).$ (36)

We can then define $\ket{l^{*}}=\prod_{k\in S_{X}\cup S_{Y}}X_{k}\ket{l}$ and
write

$-i\bra{\Psi}\prod_{i\in S_{Z}\cup
S_{Y}}Z_{i}\sum_{l}c_{l}(\ket{l^{*}}+\ket{\overline{l^{*}}})$ (37)
$-i\sum_{l}c_{l}(\bra{\Psi}\prod_{i\in S_{Z}\cup
S_{Y}}Z_{i}\ket{l^{*}}+(\bra{\Psi}\prod_{i\in S_{Z}\cup
S_{Y}}Z_{i}\ket{\overline{l^{*}}})$ (38)

. Then we can note following two properties. First, since $\Phi$ is
$\mathbb{Z}_{2}$ symmetric,
$\braket{\Phi}{l^{*}}=\braket{\Phi}{\overline{l^{*}}}$ by definition. Also,
for general state $\ket{l}$, we note that
$Z_{i}\ket{\overline{l}}=\ket{\overline{l}}\bra{l}(-Z_{i})\ket{l}$ for $i$ on
any qubit. Using the second property we have

$-i\sum_{l}c_{l}(\bra{\Psi}\prod_{i\in S_{Z}\cup
S_{Y}}Z_{i}\ket{l^{*}}+(\bra{\Psi}\prod_{i\in S_{Z}\cup
S_{Y}}(-Z_{i})\ket{\overline{l^{*}}}\bra{l^{*}}(-Z)\ket{l^{*}}).$ (39)

where $\ket{\overline{x}}$ represent the inverse of $\ket{x}$, obtained by
flipping all qubits in the state. Rewriting and then using the first property
we see

$\displaystyle-i\sum_{l}c_{l}(\bra{\Psi}\prod_{i\in S_{Z}\cup
S_{Y}}Z_{i}\ket{l^{*}}+(-1)^{S_{Y}+S_{Z}}\bra{\Psi}\prod_{i\in S_{Z}\cup
S_{Y}}\ket{\overline{l^{*}}}\bra{l^{*}}(-Z_{i})\ket{l^{*}})$ (40)
$\displaystyle=-i\sum_{l}c_{l}(\bra{\Psi}\prod_{i\in S_{Z}\cup
S_{Y}}Z_{i}\ket{l^{*}}+(-1)^{|S_{Y}|+|S_{Z}|}\bra{\Psi}\prod_{i\in S_{Z}\cup
S_{Y}}\ket{\overline{l^{*}}}\bra{l^{*}}Z_{i}\ket{l^{*}})$ (41)
$\displaystyle=-i\sum_{l}c_{l}(\bra{\Psi}\prod_{i\in S_{Z}\cup
S_{Y}}Z_{i}\ket{l^{*}}+(-1)^{|S_{Y}|+|S_{Z}|}\bra{\Psi}\prod_{i\in S_{Z}\cup
S_{Y}}\ket{\overline{l^{*}}}\bra{l^{*}}Z_{i}\ket{l^{*}})$ (42)
$\displaystyle=-i\sum_{l}c_{l}(\bra{\Psi}\prod_{i\in S_{Z}\cup
S_{Y}}Z_{i}\ket{l^{*}}+(-1)^{|S_{Y}|+|S_{Z}|}\bra{\Psi}\prod_{i\in S_{Z}\cup
S_{Y}}Z_{i}\ket{l^{*}})$ (43)
$\displaystyle=-i\sum_{l}c_{l}(1+(-1)^{|S_{Y}|+|S_{Z}|})\bra{\Psi}\prod_{i\in
S_{Z}\cup S_{Y}}Z_{i}\ket{l^{*}}.$ (44)

Where we use the first property again in the second to last step. Now we can
clearly see that if there if $|S_{Y}|+|S_{Z}|$ is odd this inner product will
vanish for all $l$ ∎

We will reference this theorem in the following analyses

###  A.1 Generic Single Qubit Channel

The action of the dual of a generic single qubit channel defined by $\chi$ in
1 on $Z$ and $ZZ$ is as follows

$\displaystyle\mathcal{E}^{\\#}(Z)=p_{Z}Z+p_{Y}Y+p_{X}X+p_{I}I,$ (45)
$\displaystyle\mathcal{E}^{\\#}(ZZ)=p_{Z}^{2}ZZ+p_{Y}^{2}YY+p_{X}^{2}XX+p_{I}^{2}I+$
$\displaystyle+P_{Z}P_{Y}(ZY+YZ)+P_{Z}P_{X}(ZX+XZ)$
$\displaystyle+P_{Z}P_{I}(ZI+IZ)+P_{Y}P_{X}(YX+XY)$
$\displaystyle+P_{Y}P_{I}(YI+IY)+P_{X}P_{I}(XI+IX).$ (46)

where $p_{Z}=p_{0}+p_{3}-p_{1}-p_{2}$, $p_{Y}=2(t_{23}+v_{02})$,
$p_{X}=2(t_{13}-v_{02})$, and $p_{I}=4t_{03}$.

Thus, the action on QAOA Hamiltonians of form given in Eq. (6) is:

$\displaystyle H^{\prime}=$
$\displaystyle\sum_{i}h_{i}(p_{Z}Z_{i}+p_{Y}Y_{i}+p_{X}X_{i}+P_{I}I)$ (47)
$\displaystyle+\sum_{i<j}J_{ij}(p_{Z}^{2}Z_{i}Z_{j}+p_{Y}^{2}Y_{i}Y_{j}+p_{X}^{2}X_{i}X_{j}+p_{I}^{2}I)$
$\displaystyle+\sum_{i,j}J_{i,j}(P_{Z}P_{Y}Z_{i}Y_{j}+P_{Z}P_{X}Z_{i}X_{j}+P_{Z}P_{I}Z_{i}I$
$\displaystyle+P_{Y}P_{X}Y_{i}X_{j}+P_{Y}P_{I}Y_{i}I+P_{X}P_{I}X_{i}I).$ (48)

From here various assumptions can be made. If all h’s=0, which is the case for
MaxCut and strictly 2-local QUBO problems we can eliminate all terms with odd
number of $Z$+$Y$ terms from Thm.1 in the appendix. We could also assume that
all but $p_{Z}$ are small, meaning that the noise channel is relatively close
to the identity, which is a condition that would be satisfied on quantum
hardware with low levels of noise. This would allow us to eliminate all terms
quadratic in $p_{X}$,$p_{Y}$,and $p_{Z}$. If we make these two assumptions we
reduce to

$H^{\prime}=p_{Z}^{2}H+p_{Z}p_{Y}\sum_{i,j}J_{ij}Z_{i}Y_{j}.$ (49)

This corresponds to a simple rescaling of the Hamiltonian, plus an additional,
nontrivial term, which is examined in the appendix Sec. A.2.

###  A.2 Constant Mixing Overrotations

Take a very simple model, where the phase is applied correctly, but the mixer
applied as $H_{M}=\sum_{i}\tilde{\beta}_{i}X_{i}$ where
$\beta_{i}=\beta+\delta\beta_{i}$, i.e. a small over/under rotation in the x
direction. This means the mixing unitary is of the form $U_{M}^{\prime}U_{M}$
where $U_{M}=e^{-i\beta\sum_{i}X_{i}}$ and
$U_{M}^{\prime}=e^{-i\sum_{i}\delta\beta_{i}X_{i}}$.

This locally rotates the Hamiltonian,
$H^{\prime}=U_{M}^{\prime{\dagger}}HU_{M}^{\prime}$. Now use $e^{-i\theta
X}Ze^{i\theta X}=\cos(2\theta)Z-\sin(2\theta)Y$.

Assuming the standard form $H=\sum_{i<j}J_{ij}Z_{i}Z_{j}$,

$\begin{split}&H^{\prime}=\sum_{i<j}J_{ij}\cos(2\delta\beta_{i})\cos(2\delta\beta_{j})Z_{i}Z_{j}+\sum_{i<j}J_{ij}\cos(2\delta\beta_{i})\sin(2\delta\beta_{j})Z_{i}Y_{j}\\\
&+\sum_{i<j}J_{ij}\sin(2\delta\beta_{i}\cos(2\delta\beta_{j})Y_{i}Z_{j}+\sum_{i<j}J_{ij}\sin(2\delta\beta_{i}\sin(2\delta\beta_{j})Y_{i}Y_{j}.\end{split}$
(50)

Let’s average the fluctuations $\langle\cos 2\delta\beta_{i}\cos
2\delta\beta_{j}\rangle=\cos^{2}(2\delta\beta)$ etc.

This gives the noise averaged Hamiltonian

$H^{\prime}=\cos^{2}(2\delta\beta)H+\sin(4\delta\beta)\sum_{i,j}\frac{J_{ij}}{2}Z_{i}Y_{j}+\sin^{2}(2\delta\beta)\sum_{i<j}J_{ij}Y_{i}Y_{j}.$
(51)

where in the middle sum, we now sum over all $i$ and all $j$.

We see that first, the spectrum is flattened by a factor of
$\cos^{2}(2\delta\beta)$, but second, the terms in $Y$ modify it in a non-
trivial way. Let us look at a perturbation to order $\delta\beta$:

$H^{\prime}=H+2\delta\beta\sum_{i,j}J_{ij}Z_{i}Y_{j}+O(\delta\beta^{2}).$ (52)

The first order correction to any energy level is

$\sum_{i,j}J_{ij}\langle E|Z_{i}Y_{j}|E\rangle=0,$ (53)

using that $|E\rangle$ is just some $z$ bit string.

This suggests we must go to second order perturbation, looking at terms of the
form

$\langle E_{1}|Y_{j}|E_{2}\rangle,$ (54)

where $E_{2}$ is a single bit-flip from $E_{1}$. Since the magnitude of the
change depends on the energy difference $E_{2}-E_{1}$, the correction depends
strongly on the spectrum of the original problem.

Since the eigenstates of classical Hamiltonians are computational basis
states, just like in the Hamming weight Hamiltonian, the shifted eigenstates
are as before. We can then again calculate:

$\displaystyle\braket{H^{\prime}}=\sum_{n}E_{n}\bigl{\lvert}\bra{m}(\frac{\ket{n}-i\epsilon\sum_{k}X_{k}\ket{n}}{\sqrt{1+N\epsilon^{2}}})\bigr{\rvert}^{2}$
(55)
$\displaystyle=\frac{1}{1+N\epsilon^{2}}(E_{m}+\epsilon^{2}\sum_{n}E_{n}\lvert\bra{m}\sum_{k}X_{k}\ket{n})\rvert^{2}.$
(56)

We now note that the inner product is 1 if and only if $n$ is a bitflip away
from $m$ and if $k$ is the index of the bit that is flipped. We can also
calculate the energy difference between $E_{m}$ and $E_{n}$ in this case. Here
we compute w.l.o.g the case where let $k$ be the vertex of highest index.

$E_{m}-E_{n}=\Bigl{(}\sum_{\begin{subarray}{c}i<j\\\ j\neq
k\end{subarray}}J_{ij}\braket{m}{Z_{i}Z_{j}}{m}+\sum_{i\neq
k}J_{ik}\braket{m}{Z_{i}Z_{k}}{m}\Bigr{)}-\\\
\Bigl{(}\sum_{\begin{subarray}{c}i<j\\\ j\neq
k\end{subarray}}\braket{m}{X_{k}Z_{i}Z_{j}X_{k}}{m}+\sum_{i\neq
k}J_{ik}\braket{m}{X_{k}Z_{i}Z_{k}X_{k}}{m}\Bigr{)}.$ (57)

Then we use the fact that $X_{k}Z_{k}X_{k}=-Z_{k}$ and that $X_{k}$ commutes
through $Z_{i}Z_{j}$. We can then cancel and add terms, giving us

$2\sum_{i\neq k}J_{ik}\braket{m}{Z_{i}Z_{k}}{m}$ (58)

Then we can easily rearranged to see that $E_{n}$ = $E_{m}-2\sum_{i\neq
k}J_{ik}\braket{m}{Z_{i}Z_{k}}{m}$. We can then plug this expression back in
for $E_{n}$ in Eq. (55):

$\displaystyle\frac{1}{1+N\epsilon^{2}}\Bigl{(}E_{m}+\epsilon^{2}\bigl{(}\sum_{k}E_{m}-2\sum_{i\neq
k}J_{ik}\braket{Z_{i}Z_{k}}\bigr{)}\Bigr{)}$ (59)
$\displaystyle=\frac{1}{1+N\epsilon^{2}}\bigl{(}E_{m}+\epsilon^{2}NE_{m}-4\sum_{i<k}J_{ik}\braket{Z_{i}Z_{k}}\bigr{)}$
(60)
$\displaystyle=E_{m}\bigl{(}\frac{1+(N-4)\epsilon^{2}}{1+N\epsilon^{2}}\bigr{)}$
(61) $\displaystyle\approx
E_{m}(1-4\epsilon^{2}+4N\epsilon^{4}+\mathcal{O}(\epsilon^{6})).$ (62)

So for this case overrotations also just scales the old eigenvalues

###  A.3 Non-constant Mixing Overrotations

We may consider that instead of having perfect angles, they demonstrate small
stochastic fluctuations. We will exploit model based on von Mises distribution
of angles (i.e. normal distribution on a circle) and we’re looking for the
maps

$\mathcal{E}_{U}^{\alpha}(\rho)=\int_{0}^{2\pi}\frac{e^{\kappa\cos(\varepsilon})}{2\pi
I_{0}(\kappa)}U(\alpha+\varepsilon)\rho
U^{\dagger}(\alpha+\varepsilon)d\varepsilon,$ (63)

where $1/\kappa$ is variance, $I_{0}(\kappa)$ is modified Bessel function, and
$U(\alpha+\varepsilon)$ is our mixer of phase operator set to angle $\alpha$
with fluctuation $\varepsilon$.

The integral for mixer leads to the following map

$\mathcal{E}_{M}^{\beta}(\rho)=\frac{1}{2}\Big{(}1+\frac{I_{2}(\kappa)}{I_{0}(\kappa)}\cos(2\beta)\Big{)}\rho+\frac{1}{2}\Big{(}1-\frac{I_{2}(\kappa)}{I_{0}(\kappa)}\cos(2\beta)\Big{)}X\rho
X-\frac{iI_{2}(\kappa)}{I_{0}(\kappa)}\frac{\sin(2\beta)}{2}[X,\rho],$ (64)

while for a single gate of phase (assuming that we have cost Hamiltonian
$H=\sum J_{ij}Z_{i}Z_{j}$), the single $ZZ(J_{ij}\gamma)$ gate (for
$J_{ij}=\\{+1,-1\\}$) is given by

$\mathcal{E}_{P}^{\gamma,J}(\rho)=\frac{1}{2}\Big{(}1-\frac{I_{2}(\kappa)}{I_{0}(\kappa)}\Big{)}A_{1}(\gamma,J)\rho
A_{1}^{\dagger}(\gamma,J)+\frac{1}{2}\Big{(}1+\frac{I_{2}(\kappa)}{I_{0}(\kappa)}\Big{)}A_{2}(\gamma,J)\rho
A_{2}^{\dagger}(\gamma,J),$ (65)

where

$A_{1}(\gamma,J)=\mathrm{Diag}\big{[}(1,-e^{2i\gamma J},-e^{2i\gamma
J},1)\big{]}\quad,\quad A_{2}(\gamma,J)=\mathrm{Diag}\big{[}(1,e^{2i\gamma
J},e^{2i\gamma J},1)\big{]}.$ (66)

The mixer error map can also be looked at as a composition of a perfect
rotation by $\beta$, then an application of $\mathcal{E}_{M}^{\beta}(\rho)$
with $\beta=0$. In this case we get a self-dual channel of form

$\mathcal{E}_{M}^{\beta}(\rho)=\frac{1}{2}\Big{(}1+\frac{I_{2}(\kappa)}{I_{0}(\kappa)}\cos(2\beta)\Big{)}\rho+\frac{1}{2}\Big{(}1-\frac{I_{2}(\kappa)}{I_{0}(\kappa)}\cos(2\beta)\Big{)}X\rho
X.$ (67)

This maps

$Z\rightarrow\frac{I_{2}(\kappa)}{I_{0}(\kappa)}Z.$ (68)

And the analysis follows local depolarizing noise. We note that in the limit
of high variance, $\kappa\rightarrow 0$, and
$\frac{I_{2}(\kappa)}{I_{0}(\kappa)}\rightarrow 0$, so we have complete
disorder in $Z$ ($Z\rightarrow 0)$. In the limit of 0 variance,
$\kappa\rightarrow\infty$, and $\frac{I_{2}(\kappa)}{I_{0}(\kappa)}\rightarrow
1$, so there is no effect of the channel ($Z\rightarrow Z)$.

###  A.4 Pauli Channel

Interesting class of noisy channel is so-called Pauli channels, that have form

$\mathcal{E}(\rho)=\sum_{k=0}^{3}p_{k}\sigma_{k}\rho\sigma_{k},$ (69)

where $\sigma_{0}=\mathbbm{1}$, and for $k=1,2,3$ we get Pauli X, Y, and Z,
respectively. $p_{k}\geq 0$ and $\sum_{k}p_{k}=1$. One may interpret that a
given noisy channel (e.g. bit-flip corresponding to $X$, happens with a
respective probability). One can set $p_{1}=p_{2}=p_{3}=(1-p)/4$ and
$p_{0}=(1+3p)/4$ and get depolarizing channel as in (8).

The action of the dual of a Pauli channel on Z is:

$\mathcal{E}^{\\#}(Z)=(p_{0}+p_{3}-(p_{1}+p_{2}))*Z,$ (70)

This then reduces to depolarizing noise where we set
$p=p_{0}+p_{3}-(p_{1}+p_{2})$. The results are similar for other Pauli
matrices, with the difference that for Pauli $k$ matrix (1=x, 2=y, 3=z), we
will have $p_{0}+p_{k}-(p_{i}+p_{j})$, where i,j are the coefficients standing
in front of remaining matrices.

###  A.5 Phase damping

Phase damping has also two Krauses, with $A_{1}$ same as for the amplitude
damping case analyzed in Sec II.1.2 and $A_{2}$ given in

$A_{2}=\left(\begin{array}[]{cc}0&0\\\ 0&\sqrt{\gamma}\end{array}\right).$
(71)

In this case, the error maps Z to itself with no scaling or shifting, so the
error channel has no effect on classical Hamiltonians.