# Efficient separate quantification of state preparation errors and
measurement errors on quantum computers and their mitigation

Hongye Yu(余泓烨)  C. N. Yang Institute for Theoretical Physics, State University
of New York at Stony Brook, Stony Brook, NY 11794-3840, USA  Department of
Physics and Astronomy, State University of New York at Stony Brook, Stony
Brook, NY 11794-3800, USA  Tzu-Chieh Wei  C. N. Yang Institute for Theoretical
Physics, State University of New York at Stony Brook, Stony Brook, NY
11794-3840, USA  Department of Physics and Astronomy, State University of New
York at Stony Brook, Stony Brook, NY 11794-3800, USA

###### Abstract

Current noisy quantum computers have multiple types of errors, which can occur
in the state preparation, measurement/readout, and gate operation, as well as
intrinsic decoherence and relaxation. Partly motivated by the booming of
intermediate-scale quantum processors, measurement and gate errors have been
recently extensively studied, and several methods of mitigating them have been
proposed and formulated in software packages (e.g., in IBM Qiskit). Despite
this, the state preparation error and the procedure to quantify it have not
yet been standardized, as state preparation and measurement errors are usually
considered not directly separable. Inspired by a recent work of Laflamme, Lin,
and Mor [Phys. Rev. A 106, 012439 (2022)], we propose a simple and resource-
efficient approach to quantify separately the state preparation and readout
error rates. With these two errors separately quantified, we also propose
methods to mitigate them separately, especially mitigating state preparation
errors with linear (with the number of qubits) complexity. As a result of the
separate mitigation, we show that the fidelity of the outcome can be improved
by an order of magnitude compared to the standard measurement error mitigation
scheme. We also show that the quantification and mitigation scheme is
resilient against gate noise and can be immediately applied to current noisy
quantum computers. To demonstrate this, we present results from cloud
experiments on IBM’s superconducting quantum computers. The results indicate
that the state preparation error rate is also an important metric for qubit
metrology that can be efficiently obtained.

††preprint: APS/123-QED

##  I Introduction

There has been dramatic growth in multi-qubit quantum processors; some possess
qubits well over one hundred. Nevertheless, they are characterized as noisy-
intermediate quantum (NISQ) devices [1] as it is currently not yet feasible to
implement quantum error correction on these devices to reduce logical error
rates and prolong coherent computation. Hence, a short-term goal is to develop
error mitigation schemes to enhance the performance of NISQ devices. Errors
and noise arise in different stages of quantum computation: state preparation,
gate operation, and readout/measurement. In the past few years, substantial
efforts have been spent on gate error mitigation [2, 3, 4, 5, 6, 7], as gate
operation is the bulk of quantum computation.

However, measurement errors also affect the computation outcome. To mitigate
the measurement error, several methods have been proposed [8, 9, 10, 11, 12]
to mitigate the final readout statistics, and they have improved the
estimation of physical observables from quantum computers. State preparation
is perhaps the most omitted one among the three error types. At present, there
does not seem to be an efficient and practical approach to even quantifying
state preparation and readout errors separately. Usually, state preparation
error is combined with the measurement error as state preparation and
measurement (SPAM) error. Moreover, current readout mitigation schemes on
superconducting qubits, such as in the Qiskit software [13], still mixed with
the state preparation error, although its rate is typically lower than that of
the readout error.

The goal of this work is two-fold. First, we propose a simple scheme that
allows separate quantification of state preparation and readout errors. This
issue was recently discussed by Laflamme, Lin, and Mor [14] employing
algorithmic cooling [15, 16, 17]. Our approach extends theirs but dramatically
simplifies the procedure. Both schemes rely on coupling the original qubit to
other qubits. Ours requires only one additional qubit to characterize the
errors in state preparation and readout separately. Secondly, with separate
quantification, the readout mitigation matrix, in principle, is free of the
state preparation error and can be directly applied to mitigate the readout
error, just like the current standard practice. Moreover, we propose a novel
approach to mitigate the state preparation error. Even though we focus our
attention and implementation on superconducting qubits, we envision that our
method can be applied to other systems as well.

We also remark we do assume that single-qubit gates, in particular, the X
gate, are of high fidelity and that the zero-noise extrapolation (ZNE)
techniques can mitigate the CNOT gate error [2, 3, 4, 5, 6, 7] if needed.
Another approach to characterize gates and SPAM errors with very few
assumptions is the gate set tomography [18]; however, it requires many gate
sequences.

The remainder of the paper is organized as follows. In Sec. II, we present a
simple scheme using an ancilla to characterize the target qubit’s state
preparation and readout errors separately. In Sec. III, we discuss how the
characterized state preparation and readout errors can be mitigated. In Sec.
IV, we compare our proposed mitigation scheme with the standard readout
mitigation (which does not characterize the state preparation error) and
illustrate the difference in mitigated outcomes for simple circuits. In Sec.
V, we present results from experiments performed on cloud IBM quantum
computers to demonstrate our theoretical proposal. We make concluding remarks
in Sec. VI. Furthermore, in Appendix A, we show that the algorithmic cooling
idea of Laflamme, Lin, and Mor can also lead to the separate characterization
of state preparation and readout errors despite the fact that the equations to
be solved are nonlinear and more complicated. In Appendix B, we present the
zero-noise extrapolation of CNOT errors when characterizing the state
preparation and readout errors.

##  II Separate characterization of state preparation and readout errors

A single-qubit readout can be characterized by a two-element POVM
$\\{M_{0},M_{1}\\}$, so that a one-qubit state $\rho$ will be measured to be
‘0’ with probability ${\rm Tr}(\rho M_{0})$ and ‘1’ with probability ${\rm
Tr}(\rho M_{1})$; see, e.g., Ref. [8], which also demonstrated that on
superconducting qubits of IBM’s devices, the readout errors are dominantly
classical flips, i.e., the POVM elements are well approximated by the
following forms,

$M_{0}=\begin{pmatrix}1-\delta_{M}^{0}&0\\\ 0&\delta_{M}^{1}\end{pmatrix},\ \
M_{1}=\begin{pmatrix}\delta_{M}^{0}&0\\\ 0&1-\delta_{M}^{1}\end{pmatrix}.$ (1)

We note that the POVM for the general single-qubit readout can be more general
than the above form, but it can be enforced to the above form by twirling with
a random Pauli Z operator (with 50% probability) before the measurement [8,
14].

In Ref. [14], a measurement-based algorithmic cooling was developed to reduce
the state preparation error, where the error ruins the preparation of a
perfect $|0\rangle$, i.e.,

$(1-\delta_{\rm SP})|0\rangle\langle 0|+\delta_{\rm SP}|1\rangle\langle 1|,$
(2)

and the cooling was achieved by coupling to ancillary qubits via CNOTs. There,
symmetric readout errors were considered, i.e.,
$\delta_{M}^{0}=\delta_{M}^{1}=:\delta_{\rm M}$ and a procedure employing
multiple ancillary qubits was proposed to separately characterize a target
qubit’s $\delta_{\rm SP}$ and $\delta_{\rm M}$ (in the limit of many
ancillas), which comprise the total SPAM error $\delta_{\rm SPAM}=\delta_{\rm
SP}+\delta_{\rm M}-2\delta_{\rm SP}\delta_{\rm M}$, giving the measurable
probability of reading out ‘1’.

In this section, we propose a simplified procedure using one ancilla to
extract separately the state preparation error $\delta_{\rm SP}$ and readout
errors $\delta_{M}^{0}$ and $\delta_{M}^{1}$. First, we note that a coherent
state preparation error, e.g., $\sqrt{1-\delta_{\rm
SP}}|0\rangle+\sqrt{\delta_{\rm SP}}e^{i\phi}|1\rangle$, can be turned into
the above incoherent state preparation error (2) by twirling with a Pauli Z
operator randomly with 50% probability [14]. (We note that the coherent error
can be probed using quantum state tomography once the readout errors have been
characterized and mitigated; see Sec. V.2.) We will not assume the symmetric
readout errors but actively employ the twirling to ensure the form of
incoherent state preparation error. The parameter $\delta_{M}^{0}$ is the
probability that the ideal ‘0’ state will be read as ‘1’, and $\delta_{M}^{1}$
is the probability that the ideal ‘1’ will be read as ‘0’. Similar values are
being reported on the device properties of all IBM quantum computers, except
that the reported values contain the state preparation error. Can the true
readout errors be obtained experimentally and separately from the state
preparation error? Indeed, Ref. [14] describes an approach by algorithmic
cooling to reduce the state preparation error, and in the limit that it
becomes infinitesimally small, only the readout errors remain. However, the
reduction to zero state preparation error is not practical as CNOT gates
(which were assumed to be error-free in Ref. [14]) that are required to
perform the measurement-based algorithmic cooling also introduce errors in
practice. Our procedure below solves all these problems and can practically
improve the device property characterization, separating the state preparation
and readout errors.

###  II.1 Access $\delta_{sp}$ via an ancillary qubit

$\Qcircuit@C=.7em@R=.7em@!{\lstick{\ket{0}_{\text{a}}}&\targ\meter\rstick{\tilde{\delta}_{\rm
SPAM}^{a,0}}\qw\\\
\lstick{\ket{0}_{\text{t}}}\ctrl{-1}\qw\meter\rstick{\delta_{\rm
SPAM}^{t,0}}\qw}$

$\Qcircuit@C=.7em@R=.7em@!{\lstick{\ket{0}_{\text{a}}}&\qw\targ\meter\rstick{\tilde{\delta}_{\rm
SPAM}^{a,1}}\qw\\\
\lstick{\ket{0}_{\text{t}}}\gate{X}\ctrl{-1}\qw\meter\rstick{\delta_{\rm
SPAM}^{t,1}}\qw}$

Figure 1: Circuits for measuring $\delta_{\rm SP}$ with initial state prepared
to be: (top) $\ket{00}$ and (bottom) $\ket{01}$. Here we assume the X gate is
noiseless and can be considered as a part of state preparation.

Here we present an efficient way to estimate the state preparation error
$\delta_{\rm SP}^{t}$ for the target qubit $q_{t}$, which only requires one
additional qubit $q_{a}$, as shown in the Fig. 1. The SPAM error of both
qubits can be easily measured. To measure the probability $\delta_{\rm
SPAM}^{0}$ that a qubit is prepared with $\ket{0}$ but measured in $\ket{1}$,
one only needs to measure the initialized qubit directly; to measure the
probability $\delta_{\rm SPAM}^{1}$ that the qubit is prepared with $\ket{1}$
but measured in $\ket{0}$, one needs to first initialize the qubit in
$\ket{0}$, apply an X gate, and then measure it. The measured error
$\delta_{\rm SPAM}$ is a combination of both state preparation error
$\delta_{\rm SP}$ and measurement error $\delta_{M}$’s, and, without assuming
symmetric readout errors, we have

$\displaystyle\delta_{\rm
SPAM}^{0}=(1-\delta_{M}^{0}-\delta_{M}^{1})\delta_{\rm SP}+\delta_{M}^{1},$
(3) $\displaystyle\delta_{\rm
SPAM}^{1}=(1-\delta_{M}^{0}-\delta_{M}^{1})\delta_{\rm SP}+\delta_{M}^{1}.$
(4)

After adding a noiseless CNOT gate (Fig. 1), the SPAM error of $q_{t}$ does
not change, whereas the ancilla qubit readout error $\tilde{\delta}_{\rm
SPAM}^{a,0}$ becomes (note: the derivation is straightforward but may appear
tedious)

$\displaystyle\tilde{\delta}_{\rm SPAM}^{a,0}=(1-\delta_{\rm
SPAM}^{a,0}-\delta_{\rm SPAM}^{a,1})\delta_{\rm SP}^{t}+\delta_{\rm
SPAM}^{a,0},$ (5) $\displaystyle\tilde{\delta}_{\rm SPAM}^{a,1}=(1-\delta_{\rm
SPAM}^{a,0}-\delta_{\rm SPAM}^{a,1})\delta_{\rm SP}^{t}+\delta_{\rm
SPAM}^{a,1}.$ (6)

By using either of these two equations and the measured values of $\delta_{\rm
SPAM}^{a,0/1}$, one can solve the $\delta_{\rm SP}^{t}$. Combined with the
measured SPAM errors from $q_{t}$, one can further obtain the value of
measurement errors $\delta_{M}^{0/1}$ for qubit $q_{t}$.

We note that solving $\delta_{M}$’s and $\delta_{\rm SP}$ involves only linear
equations. One can also extend the algorithmic cooling of Laflamme, Lin, and
Mor in the two-qubit setting to solve for (the symmetric) $\delta_{M}$ and
$\delta_{\rm SP}$, but this will involve non-linear equations; see Appendix A.

##  III Complete mitigation for measurement error and state-preparation error

Measurement mitigation is an essential tool to extract information from noisy
quantum devices. The measurement error with only bit flips can be
characterized by the error assignment matrix. In the single-qubit case, the
matrix is simply

$\displaystyle A_{M}=\begin{pmatrix}1-\delta_{M}^{0}&\delta_{M}^{1}\\\
\delta_{M}^{0}&1-\delta_{M}^{1}\end{pmatrix}.$ (7)

Note that we assume there are only bit-flip errors, or else X and Y errors
have been removed by the Z randomization procedure mentioned above. The effect
of the measurement error on the outcome probability distribution
$\hat{\bm{P}}$ from ideal measurement is then

$\bm{P}_{\rm noisy}=A_{M}\hat{\bm{P}},$ (8)

where $A_{M}$ can be the general assignment matrix characterizing $n$-qubit
measurement error. Suppose we know the exact form of $A_{M}$; we can exactly
mitigate the measurement error by

$\hat{\bm{P}}=A_{M}^{-1}\bm{P}_{\rm noisy}.$ (9)

However, due to the state preparation error, the matrix $A_{M}$ is usually
hard to obtain exactly. In practice, one usually lumps together the state
preparation and readout errors as the SPAM error $\delta_{\rm SPAM}$. The
corresponding assignment matrix $A_{\rm SPAM}$ is obtained by naive
calibration circuits, such as in Qiskit [13], where $2^{n}$ computational-
basis states are prepared and then measured. Subsequently, the inverse of the
$A_{\rm SPAM}$, instead of $A_{M}$, is applied for the mitigation in practice

$\tilde{\bm{P}}=A_{\rm SPAM}^{-1}\bm{P}_{\rm noisy}=A_{\rm
SPAM}^{-1}A_{M}\hat{\bm{P}}.$ (10)

Since $A_{\rm SPAM}^{-1}A_{M}$ is generally not identity, the mitigated result
$\tilde{\bm{P}}$ differs from the ideal distribution $\hat{\bm{P}}$. Suppose
the assignment matrix for the state-preparation error on the initial state is
$A_{\rm SP}$. In single-qubit case, it is

$\displaystyle A_{\rm SP}=\begin{pmatrix}1-\delta_{\rm SP}&\delta_{\rm SP}\\\
\delta_{\rm SP}&1-\delta_{\rm SP}\end{pmatrix}.$ (11)

What calibration circuits measure is effectively the successive application of
$A_{\rm SP}$ and $A_{M}$ on the initial state, which gives

$A_{\rm SPAM}=A_{M}A_{\rm SP}.$ (12)

Thus, the result mitigated by $A_{\rm SPAM}$ is

$\tilde{\bm{P}}=A_{\rm SP}^{-1}\hat{\bm{P}}$ (13)

Note that the distribution $\hat{\bm{P}}$ contains the state preparation
error, which takes place at the beginning of the circuit and, in general,
cannot be mitigated by applying the inverse at the end of the circuit. Thus,
the mitigation by $A_{\rm SPAM}^{-1}$ usually over-mitigates the outcome and
can give rise to some unphysical probability distributions (in addition to
those caused by statistical fluctuations), such as negative probabilities.
Nevertheless, the negative probability problem can be dealt with by using the
nearest physical probability distribution instead [19], after applying the
mitigation matrix $A_{\rm SPAM}^{-1}$.

While fully characterizing those error matrices requires exponentially many
circuits, if we assume all the errors are local, those error assignment
matrices can be decomposed into a direct product of all single-qubit
measurement errors $\\{A^{(i)}\\}$

$\displaystyle A_{M}=A_{M}^{(1)}\otimes A_{M}^{(2)}\cdots A_{M}^{(n)},$ (14)
$\displaystyle A_{\rm SP}=A_{\rm SP}^{(1)}\otimes A_{\rm SP}^{(2)}\cdots
A_{\rm SP}^{(n)},$ (15) $\displaystyle A_{\rm SPAM}=A_{\rm SPAM}^{(1)}\otimes
A_{\rm SPAM}^{(2)}\cdots A_{\rm SPAM}^{(n)},$ (16)

where

$\displaystyle A_{\rm SPAM}^{(i)}=\begin{pmatrix}1-\delta_{\rm
SPAM}^{i,0}&\delta_{\rm SPAM}^{i,1}\\\ \delta_{\rm SPAM}^{i,0}&1-\delta_{\rm
SPAM}^{i,1}\end{pmatrix}.$ (17)

The above conditions approximately hold and match many current superconducting
qubit devices [20], where the uncorrelated errors dominate, and correlated
errors among qubits are usually small. It is easy to check that $A_{\rm
SPAM}^{(i)}=A_{M}^{(i)}A_{\rm SP}^{(i)}$. Thus, one can measure the
$\delta_{\rm SPAM}$ locally and then combine them to get the full $A_{\rm
SPAM}$ matrix, which only takes $O(n)$ number of circuits.

Nevertheless, the state preparation error usually does not commute with the
circuit gates, which, in principle, cannot be automatically combined with the
measurement errors at the end of the circuit. Thus, we need a new technique to
mitigate the state preparation error and the measurement error separately.

###  III.1 Measurement error mitigation

In this section, we present a similar mitigation scheme for the measurement
error as Eq. 9. Here, we assume that all the errors are uncorrelated and Eqs.
(14), (15), and (16) hold. We can firstly measure the SPAM error $\delta_{\rm
SPAM}$ for each qubit, then use the methods in Sec. II.1 to obtain the state
preparation errors $\delta_{\rm SP}$’s and measurement errors $\delta_{M}$’s.
After getting the single-qubit measurement errors $\delta_{M}^{i,0}$ and
$\delta_{M}^{i,1}$ for each qubit, we construct the full measurement error
assignment matrix $A_{M}$ according to Eq. (7) and Eq. (14). The measurement
error is mitigated by applying the inverse of $A_{M}$

$\hat{\bm{P}}=A_{M}^{-1}\bm{P}_{\rm noisy}.$ (18)

Ideally, after such a readout error mitigation process, the remaining errors
come from state preparation and gate operations.

###  III.2 State preparation error mitigation

The state preparation error affects the outcome at the beginning of the
circuit, so the output probability distribution $\bm{P}$ is a combination of
all possible initial assignments with the corresponding probability

$\bm{P}(\delta_{\rm SP}^{(1)},\delta_{\rm
SP}^{(2)}...)=\sum_{\\{s_{i}\\}\in\\{0,1\\}^{n}}\left(\prod_{j=1}^{n}q_{\rm
SP}^{(j)}(s_{i},0)\right)\hat{\bm{P}}_{s_{1},s_{2}...s_{n}},$ (19)

where $\hat{\bm{P}}_{s_{1},s_{2}...s_{n}}$ is the $\delta_{\rm SP}$-free
output distribution of the quantum circuit with the same gates and
measurements but different initial qubit assignments
$\ket{s_{1}s_{2}...s_{n}}$’s, and

$\displaystyle q_{\rm SP}^{(j)}(s,s^{\prime})=\begin{cases}1-\delta_{\rm
SP}^{(j)},&s=s^{\prime}\\\ \delta_{\rm SP}^{(j)},&s\neq s^{\prime}\end{cases}$
(20)

and $\delta_{\rm SP}^{(j)}$ is the state preparation error of the $j$-th
qubit. Note that the desired probability distribution

$\bm{P}(0,0...)=\hat{\bm{P}}_{0,0...}$ (21)

can, in principle, be obtained by solving $2^{n}$ linear equations with
regards to $2^{n}$ variables $\hat{\bm{P}}_{s_{1},s_{2}...s_{n}}$

$\bm{P}^{s_{1}^{\prime},s_{2}^{\prime},...s_{n}^{\prime}}=\sum_{\\{s_{i}\\}\in\\{0,1\\}^{n}}\left(\prod_{j=1}^{n}q_{\rm
SP}^{(j)}(s_{i},s_{i}^{\prime})\right)\hat{\bm{P}}_{s_{1},s_{2}...s_{n}},$
(22)

where $\bm{P}^{s_{1}^{\prime},s_{2}^{\prime},...s_{n}^{\prime}}$ is the noisy
experimental output distribution with the same gates and measurement but
different erroneous initialization
$\ket{s_{1}^{\prime},s_{2}^{\prime},...s_{n}^{\prime}}$. For a small number of
qubits, one can solve for $\hat{\bm{P}}$ exactly.

(a) | (b)
---|---
|
Figure 2: Simulation results for comparison between two mitigation schemes for
the (a) 2-qubit circuit (Fig. 3) and (b) 4-qubit circuit (Fig. 4). We set
$\delta_{\rm SP}=0.05$, $\delta_{M}^{0}=0.04$, and $\delta_{M}^{1}=0.06$ for
all qubits. The circuits for calculating $\delta_{\rm SP}$ are repeated
$4\times 10^{6}$ times, and the target circuits are repeated $6.4\times
10^{5}$ times.

However, for a large number of qubits, exactly solving $\hat{\bm{P}}_{0,0...}$
requires exponentially many circuits and is not practical in experiments.
Here, we present an approximate approach that takes a linear number of
circuits to achieve $O(\delta_{\rm SP}^{2})$ accuracy. To do this, we first
expand the distribution $\bm{P}$ near the $\delta_{\rm SP}^{(j)}$,

$\displaystyle\bm{P}(x_{1},x_{2}...)$ $\displaystyle=\bm{P}(\delta_{\rm
SP}^{(1)},\delta_{\rm SP}^{(2)},...)+$ (23)
$\displaystyle\sum_{i=1}^{n}\frac{\partial\bm{P}(\delta_{\rm
SP}^{(1)},...)}{\partial x_{i}}(x_{i}-\delta_{\rm SP}^{(j)})+...$

If we assume the $\delta_{\rm SP}^{(i)}$’s are small and approximate the above
equation to the order of $O(\delta_{\rm SP})$, we can neglect higher order
terms and get the estimation of $\bm{P}(0,0...)$

$\bm{P}(0,0...)=\bm{P}(\delta_{\rm SP}^{(1)},\delta_{\rm
SP}^{(2)}...)-\sum_{i=1}^{n}\frac{\partial\bm{P}(\delta_{\rm
SP}^{(1)},...)}{\partial x_{i}}\delta_{\rm SP}^{(i)},$ (24)

where $\bm{P}(\delta_{\rm SP}^{(1)},\delta_{\rm SP}^{(2)}...)$ is the
experimental result, and the derivatives $\frac{\partial\bm{P}(\delta_{\rm
SP}^{(1)},...)}{\partial x_{i}}$ can be obtained by one additional
distribution $\bm{P}_{i}$ from the same circuit except with one additional
initial X gate on the $i$-th qubit. This can be seen by rewriting Eq. (19)
into

$\bm{P}(x_{1},\delta_{\rm
SP}^{(2)},...)=(1-x_{1})\tilde{\bm{P}}_{0,X}+x_{1}\tilde{\bm{P}}_{1,X},$ (25)

where

$\displaystyle\tilde{\bm{P}}_{0,X}$ $\displaystyle=$
$\displaystyle\sum_{\\{s_{2},s_{3}...\\}\in\\{0,1\\}^{n-1}}\left(\prod_{j=2}^{n}q_{\rm
SP}^{(j)}(s_{i})\right)\tilde{\bm{P}}_{0,s_{2}...s_{n}},$
$\displaystyle\tilde{\bm{P}}_{1,X}$ $\displaystyle=$
$\displaystyle\sum_{\\{s_{2},s_{3}...\\}\in\\{0,1\\}^{n-1}}\left(\prod_{j=2}^{n}q_{\rm
SP}^{(j)}(s_{i})\right)\tilde{\bm{P}}_{1,s_{2}...s_{n}}.$

The original distribution $\bm{P}_{0}$ and the additional distribution
$\bm{P}_{1}$ (the distribution from the same circuit with additional initial X
gate on the first qubit) can be viewed as the result of substituting
$x_{1}=\delta_{\rm SP}^{(1)}$ and $x_{1}=1-\delta_{\rm SP}^{(1)}$,
respectively, in Eq. (25),

$\displaystyle\bm{P}_{0}$ $\displaystyle=$ $\displaystyle\bm{P}(\delta_{\rm
SP}^{(1)},...)=(1-\delta_{\rm SP}^{(1)})\tilde{\bm{P}}_{0,X}+\delta_{\rm
SP}^{(1)}\tilde{\bm{P}}_{1,X},$ $\displaystyle\bm{P}_{1}$ $\displaystyle=$
$\displaystyle\bm{P}(1-\delta_{\rm SP}^{(1)},...)=\delta_{\rm
SP}^{(1)}\tilde{\bm{P}}_{0,X}+(1-\delta_{\rm SP}^{(1)})\tilde{\bm{P}}_{1,X}.$

Thus, we obtain the derivative of $x_{1}$

$\displaystyle\frac{\partial\bm{P}(\delta_{\rm SP}^{(1)},...)}{\partial
x_{1}}$ $\displaystyle=$
$\displaystyle\tilde{\bm{P}}_{1,X}-\tilde{\bm{P}}_{0,X}$ (26) $\displaystyle=$
$\displaystyle\frac{\bm{P}_{1}-\bm{P}_{0}}{1-2\delta_{\rm SP}^{(1)}}.$

The other derivatives w.r.t. $x_{i}$’s can be obtained in a similar way.
Substituting them into the Eq. (24), we arrive at the equation for the first-
order state preparation error mitigation

$\hat{\bm{P}}=\bm{P}_{0}+\sum_{i=1}^{n}\frac{\delta_{\rm
SP}^{(i)}}{1-2\delta_{\rm SP}^{(i)}}(\bm{P}_{0}-\bm{P}_{i}).$ (27)

Note that the state preparation error mitigation (27) is linear among
different distributions $\bm{P}$’s and that measurement mitigation (18) is a
linear transformation within each distribution $\bm{P}$, so swapping the order
of measurement error mitigation and state preparation error mitigation here
does not change the final results. We remark that one can achieve $r$-th-order
accuracy mitigation at the cost of $O(n^{r})$ additional circuits with a
similar procedure, where $n$ is the qubit number, and we do not explicitly
write down the derivation.

##  IV comparison between two mitigation schemes

In current standard practice on superconducting qubits, readout mitigation is
carried out first by measuring the mitigation assignment matrix $A_{\rm SPAM}$
obtained by measuring the distribution in the computational basis when states
are prepared in all different computational states. Then, the inverse of the
assignment matrix is applied to the actual measurement distribution. However,
the preparation of computational states incurs errors, and these are not
separately characterized and are thus lumped together in the mitigation
matrix. We call this the standard readout mitigation (SRM) and refer to ours
as state preparation and readout mitigation (SPRM).

$\Qcircuit@C=.7em@R=.7em@!{\lstick{\ket{0}_{\text{1}}}&\gate{R_{y}(\theta)}\ctrl{1}\qw\meter\\\
\lstick{\ket{0}_{\text{2}}}\gate{R_{y}(\theta)}\targ\meter}$ Figure 3: The
2-qubit benchmark circuit for comparing two mitigation schemes SRM and SPRM.
$\Qcircuit@C=.7em@R=.7em@!{\lstick{\ket{0}_{\text{1}}}&\gate{R_{y}(\theta)}\qw\targ\meter\\\
\lstick{\ket{0}_{\text{2}}}\gate{R_{y}(\theta)}\ctrl{1}\qw\ctrl{-1}\qw\meter\\\
\lstick{\ket{0}_{\text{3}}}\gate{R_{y}(\theta)}\targ\ctrl{1}\qw\meter\\\
\lstick{\ket{0}_{\text{4}}}\gate{R_{y}(\theta)}\qw\targ\meter\\\ }$ Figure 4:
The 4-qubit benchmark circuit for comparing two mitigation schemes SRM and
SPRM.

###  IV.1 Summary of the two mitigation schemes

target qubit(ancilla qubit) $q_{0}(q_{1})$ $q_{1}(q_{2})$ $q_{2}(q_{1})$ $q_{3}(q_{5})$ $q_{4}(q_{5})$ $q_{5}(q_{6})$ $q_{6}(q_{5})$ $\delta_{\rm SPAM}^{0}$ 0.0108(4) 0.0134(5) 0.0086(4) 0.0087(4) 0.0120(4) 0.0162(5) 0.0086(4) $\delta_{\rm SPAM}^{1}$ 0.0514(9) 0.0394(8) 0.085(1) 0.0380(8) 0.0457(8) 0.102(1) 0.0319(7) $\delta_{M}^{0}$ 0.0005(8) 0.0037(8) 0.0018(9) 0.0020(8) 0.0039(8) 0.010(1) 0.0021(7) $\delta_{M}^{1}$ 0.0411(8) 0.0297(8) 0.0780(9) 0.0312(8) 0.0376(8) 0.096(1) 0.0253(7) $\delta_{\rm SP}$ 0.011(1) 0.0101(4) 0.0074(3) 0.0070(5) 0.0085(7) 0.0069(3) 0.0067(4) |
---|---
Table 1:  The state preparation error and measurement error for all the qubits
in ibm_nairobi have been characterized using methods described in the text.
The qubit connection of ibm_nairobi is shown on the right. All of the circuits
are repeated $6\times 10^{5}$ times.

First, we summarize the procedure of standard readout mitigation (SRM) as
follows:

> 1\. Execute the quantum circuit $U$ on qubits $\\{q_{i}\\}$ and get the
> noisy raw probability distribution $P_{\rm raw}$;
> 2\. Run the standard SPAM error calibration circuits on qubits $\\{q_{i}\\}$
> and get the SPAM error assignment matrix $A_{\rm SPAM}$;
> 3\. Apply the inverse of $A_{\rm SPAM}$ to the raw distribution and get the
> mitigated quasi-probability distribution $\tilde{P}=A_{\rm SPAM}^{-1}P_{\rm
> raw}$;
> 4\. Find the nearest probability distribution $\hat{P}$ from $\tilde{P}$ in
> case of the existence of negative probabilities in $\tilde{P}$. Then, the
> output $\hat{P}$ is the desired mitigated probability distribution.

Next, we summarize our proposed procedure of state preparation and readout
mitigation (SPRM) as follows:

> 1\. Execute the quantum circuit $U$ on qubits $\\{q_{i}\\}$ and get the
> noisy raw probability distribution $P_{\rm raw}$;
> 2\. Run the calibration circuits in sec. II.1 on each qubit pair and
> separately get the state preparation error $\\{\delta_{\rm SP}^{(i)}\\}$ and
> readout error assignment matrix $A_{M}$;
> 3\. Adding an initial X gate on $q_{i}$ and then apply circuit $U$ for each
> qubit to get the calibration distribution $\\{P_{i}\\}$;
> 4\. Apply the inverse of $A_{M}$ to the raw distribution $P_{\rm raw}$ and
> calibration distribution $\\{P_{i}\\}$s, and get the mitigated quasi-
> probability distribution $\tilde{P}_{M}=A_{M}^{-1}P_{\rm raw}$ and
> $\\{\tilde{P}_{i}=A_{M}^{-1}P_{i}\\}$;
> 5\. Use the Eq. (27) to get the further mitigated quasi-probability
> distribution $\tilde{P}_{\rm M,SP}$ from $\tilde{P}_{M}$,
> $\\{\tilde{P}_{i}\\}$, and $\\{\delta_{\rm SP}^{(i)}\\}$;
> 4\. Find the nearest probability distribution $\hat{P}$ from $\tilde{P}_{\rm
> M,SP}$ in case of the existence of negative probabilities in $\tilde{P}_{\rm
> M,SP}$. Then, the output $\hat{P}$ is the desired mitigated probability
> distribution.

###  IV.2 Results from simulator

Here, we compare mitigating the state preparation error and measurement error
separately and together for a simple circuit shown in Fig. 3. (A four-qubit
example circuit that will be tested later is shown in Fig. 4.) The $R_{y}$
gate is defined as $R_{y}(\theta)=\exp(-i\theta Y/2)$ and the ideal output
state is

$\displaystyle\ket{\hat{\psi}}$ $\displaystyle=$
$\displaystyle\cos^{2}\frac{\theta}{2}\ket{00}+\sin^{2}\frac{\theta}{2}\ket{01}$
(28)
$\displaystyle+\sin\frac{\theta}{2}\cos\frac{\theta}{2}\ket{10}+\sin\frac{\theta}{2}\cos\frac{\theta}{2}\ket{11}.$

Since the measurement can only give probability distributions instead of wave
functions, we define the overlap (or fidelity) between two probability
distributions [21]

$F(\bm{P},\bm{P}^{\prime})=\left(\sum_{i=1}^{2^{n}}\sqrt{p_{i}p_{i}^{\prime}}\right)^{2}\,,$
(29)

where $p_{i}$ is the probability in the distribution $\bm{P}$ for the state
$\ket{i}$. The above definition is related to their Heilinger distance [22]
$H(\bm{P},\bm{P^{\prime}})$ via
$\sqrt{F(\bm{P},\bm{P^{\prime}})}=1-H^{2}(\bm{P},\bm{P^{\prime}})$ between the
two distributions. The fidelity of an output distribution $\bm{P}_{\theta}$ is
then defined as the overlap with the ideal probability distribution
$\hat{\bm{P}}$

$f(\theta)=F(\bm{P}_{\theta},\hat{\bm{P}}),$ (30)

where
$\hat{\bm{P}}=(\cos^{4}\frac{\theta}{2},\sin^{4}\frac{\theta}{2},\frac{1}{4}\sin^{2}\theta,\frac{1}{4}\sin^{2}\theta)$.
We use a simulator with a state preparation error $\delta_{\rm SP}=0.05$ and
$\delta_{M}^{0}=0.04,\delta_{M}^{1}=0.06$ for both qubits.

The theoretical calculation is straightforward but gives rise to messy
expressions for both mitigation schemes, except for $\theta=0$, at which we
have

$\displaystyle f_{\rm SRM}(0)=1-\frac{\delta_{\rm SP}^{(1)}}{2},$ (31)
$\displaystyle f_{\rm SPRM}(0)=1-\frac{4}{3}\delta_{\rm SP}^{(1)}\delta_{\rm
SP}^{(2)},$ (32)

and, at $\theta=\pi/2$, we have

$\displaystyle f_{\rm SRM}(\frac{\pi}{2})=f_{\rm SPRM}(\frac{\pi}{2})=1.$ (33)

Therefore, we expect that, near $\theta=0$, the two mitigation schemes achieve
fidelity with the ideal distribution at different orders, with the SPRM scheme
fairly better. The simulation results for both the two-qubit and four-qubit
circuits are shown in Fig. 2, where SPRM has better fidelity and less
fluctuation in a large proportion of the parameter region in both cases.

##  V experimental demonstration on superconducting quantum computers

Figure 5: Experimental results, expected exact results, and mitigated results
(using both SRM and SPRM) for the 4-qubit circuit, shown in Fig. 4, with
$\theta=\pi/5$. The outcome labels on the horizontal axis are in the order
$[q_{4},q_{3},q_{2},q_{1}]$. The results are obtained from ibmq_mumbai (whose
machine layout is shown above the histogram), with the initial layout of our
experiment being the four qubits [21,18,15,12] (highlighted in green). The
circuits for calculating $\\{\delta_{\rm SP}\\}$ are repeated $6\times 10^{5}$
times, and the target circuits are repeated $10^{5}$ times.

In this section, we experimentally implement the $\delta_{\rm SP}$
characterizing circuits and obtain the estimated values of $\delta_{\rm SP}$
on superconducting quantum computers of IBM’s cloud platform. Then, we perform
quantum state tomography (with readout mitigation) to confirm that the
$\delta_{\rm SP}$ values we obtain match the tomography results. Finally, we
compare the two mitigation schemes on a 4-qubit circuit and demonstrate that
the SPRM obtains better probability fidelity than the SRM.

###  V.1 Characterization of $\delta_{\rm SP}$ with noisy gates

We implement the circuit shown in Fig. 1 on IBM’s superconducting quantum
computers, where the gate noise becomes the major error source in the
circuits. Ideally, the SPAM error $\delta^{t}_{\rm SPAM}$ measured in the
circuit of Fig. 1 should be exactly the same as $\delta^{t,\rm raw}_{\rm
SPAM}$ that is measured directly without the CNOT gate. However, adding the
CNOT gate can bring additional gate errors, and the measured SPAM error rate
$\delta^{t,\rm noisy}_{\rm SPAM}$ can be larger than $\delta^{t,\rm raw}_{\rm
SPAM}$.

Suppose the noise of the CNOT can be described by a noise channel $\Lambda$,
which we assume is a direct product of two Pauli-like single-qubit channels

$\Lambda(\rho)=\sum_{a\in\\{I,X,Y,Z\\}^{2}}p_{a}P_{a}(\rho)P_{a}.$ (34)

By adding a random Pauli Z operator with 50% probability for each qubit before
the CNOT gate, we ensure that the state-preparation error produces effectively
$\rho=(1-\delta_{\rm SP})|0\rangle\langle 0|+\delta_{\rm SP}|1\rangle\langle
1|$, on which the Pauli Z channel has no effect but other Pauli channels act
effectively like the bit-flip error, whose error assignment matrix can be
combined with the measurement error

$A_{\rm M}^{\rm noisy}=A_{M}A_{\rm CNOT}.$ (35)

The SPAM error directly measured from circuits in Fig. 1 becomes

$A_{\rm SPAM}^{\rm noisy}=A_{M}^{\rm noisy}A_{\rm SP}.$ (36)

If we assume the noise effect of CNOT gates acts symmetrically on control and
target qubits, the $A_{M}^{\rm noisy}$ matrix will not change when swapping
the control qubit and target qubit of the CNOT gate. Thus, one can use the
SPAM error with gate noise to calculate $\delta_{\rm SP}$

$\displaystyle\\!\\!\\!\\!\tilde{\delta}_{\rm SPAM}^{a,0,\rm
noisy}=(1-\delta_{\rm SPAM}^{a,0,\rm noisy}-\delta_{\rm SPAM}^{a,1,\rm
noisy})\delta_{\rm SP}^{t}+\delta_{\rm SPAM}^{a,0,\rm noisy},$ (37)
$\displaystyle\\!\\!\\!\\!\\!\\!\tilde{\delta}_{\rm SPAM}^{a,1,\rm
noisy}=(1-\delta_{\rm SPAM}^{a,0,\rm noisy}-\delta_{\rm SPAM}^{a,1,\rm
noisy})\delta_{\rm SP}^{t}+\delta_{\rm SPAM}^{a,1,\rm noisy},$ (38)

where $\delta_{\rm SPAM}^{a,\rm noisy}$ is obtained by setting $q_{a}$ as the
control qubit of CNOT and $\tilde{\delta}_{\rm SPAM}^{a,\rm noisy}$ is
obtained by setting $q_{t}$ as the control qubit. Then we use the SPAM error
without the gate noise to calculate $\delta_{M}$

$\displaystyle\delta_{\rm SPAM}^{0,\rm
raw}=(1-\delta_{M}^{0}-\delta_{M}^{1})\delta_{\rm SP}+\delta_{M}^{0},$ (39)
$\displaystyle\delta_{\rm SPAM}^{1,\rm
raw}=(1-\delta_{M}^{0}-\delta_{M}^{1})\delta_{\rm SP}+\delta_{M}^{1}.$ (40)

We experimentally demonstrate the above method on the qubit $q_{12}$ of the
backend ibmq_mumbai. With two different ancilla qubits $q_{10}$ and $q_{15}$,
we obtain the state preparation error $\delta_{\rm SP}^{(12)}=0.042(2)$ (using
$q_{10}$ as the ancilla) and $\delta_{\rm SP}^{(12)}=0.041(3)$ (using $q_{15}$
as the ancilla), respectively. We also perform zero-noise extrapolation for
the CNOT gate for the same qubit and obtain similar results (see Appendix B).

We use the above method to characterize all of the qubits in the ibm_nairobi,
and the results are shown in Table 1. We find that the state preparation
error, if not separated from the readout error, is lumped in the SPAM error
that is used in the current standard approach for readout mitigation.

###  V.2 One-qubit state tomography

To fully characterize the state-preparation noise channel before the
measurement, we also perform state tomography for $\ket{0}$ state and
$\ket{1}$ state after obtaining the $\delta_{\rm SP}$ and $\delta_{M}$’s. We
perform the experiment on $q_{12}$ of ibmq_mumbai and get $\delta_{\rm
SP}=0.018(1)$. After the measurement mitigation with regard to $\delta_{M}$’s,
the tomography results for preparing $\ket{0}$ and $\ket{1}$ are

$\displaystyle\rho_{0}=\begin{pmatrix}0.983(2)&-0.003(2)-0.016(2)i\\\
-0.003(2)+0.016(2)i&0.017(2)\end{pmatrix},$ (41)
$\displaystyle\rho_{1}=\begin{pmatrix}0.018(2)&0.002(2)+0.019(2)i\\\
0.002(2)-0.019(2)i&0.982(2)\end{pmatrix},$ (42)

where the diagonal part is consistent with the $\delta_{\rm SP}$ obtained from
our quantification scheme.

###  V.3 Experimental comparison between two mitigation schemes

To demonstrate the direct application to circuit runs on current NISQ devices,
we use the circuit shown in Fig. 4 to benchmark both mitigation schemes. We
choose $\theta=\pi/5$. We run the circuit on the ibmq_mumbai with the initial
qubit layout [21,18,15,12]. Using the technique shown in Sec. V.1, we obtained
the state preparation errors $\\{\delta_{\rm
SP}\\}=\\{0.007(3),0.007(2),0.006(1),0.029(1)\\}$. Then, we perform the SRM
and SPRM, respectively. The raw and mitigated probability results are shown in
Fig. 5. The SRM gives 99.6(4)% probability fidelity while the SPRM gives
99.9(2)% probability fidelity. The noiseless simulation indicates a 0.4%
improvement in fidelity, close to the experimental results.

##  VI Conclusion

We have proposed a resource-efficient approach to separately quantify the
state preparation and readout error rates. With these two errors separately
quantified, we have also described efficient methods to mitigate them,
especially mitigating state preparation errors with linear complexity. In
addition to theoretical analysis, we have additionally presented results from
cloud experiments on IBM’s superconducting quantum computers, and the results
well match our theoretical predictions.

In this work, we have assumed high fidelity of X gates, which is usually the
case in real quantum devices. We also proposed an efficient way to obtain
$\delta_{\rm SP}$ in the presence of CNOT gate noise, which, in the
experiment, gives consistent results compared to those obtained by the ZNE.
Thus, we believe our methods can be easily extended beyond superconducting
quantum computers to other physical systems.

The experimental results show practical applications of our quantification and
mitigation scheme for $\delta_{\rm SP}$ on current noisy quantum computers.
Although the improvement of fidelity is limited due to the relatively small
$\delta_{\rm SP}$ compared to the $\delta_{M}$ on IBM’s superconducting
quantum computer, the improvement would be much evident when the $\delta_{M}$
is reduced to similar scale as $\delta_{\rm SP}$ in the future. We advocate
that in reporting the properties of quantum bits, the state preparation error
rate should also be provided as an important metric (at least the portion in
the incoherent mixture of $|1\rangle\langle 1|$ when preparing $|0\rangle$;
but, in principle, quantum state tomography can be used for further
characterization).

###### Acknowledgements.

We thank Yusheng Zhao for many useful discussions. This work was partly
supported by the National Science Foundation under Grant No. PHY 2310614, in
particular, on the part of the state preparation error’s characterization and
mitigation, and by the U.S. Department of Energy, Office of Science, National
Quantum Information Science Research Centers, Co-design Center for Quantum
Advantage (C2QA) under Contract No. DE-SC0012704, in particular, on the part
of the algorithmic cooling. This research also used resources from the Oak
Ridge Leadership Computing Facility, which is a DOE Office of Science User
Facility supported under Contract DE-AC05-00OR22725, and the Brookhaven
National Laboratory operated IBM-Q Hub. The results presented in this work do
not reflect the views of IBM and its employees.

##  Appendix A Extension of the Laflamme-Lin-Mor algorithmic-cooling scheme
for separate quantification of state preparation and readout errors

(a) | (b)
---|---
|
(c) | (d)
|
Figure 6:  Zero-noise extrapolation for $\delta_{\rm SP}$ characterization
circuit with $q_{12},q_{10}$ on ibmq_mumbai. Each circuit is repeated 64000
times. (a) Fidelity of $q_{10}$ with initial state $\ket{00}$ for circuit 1
with $q_{12}$ as target qubit. The fitted result gives $1-\tilde{\delta}_{\rm
SPAM}^{10,0,\rm miti}=0.938(1)$. (b) Fidelity of $q_{10}$ with initial state
$\ket{01}$ for circuit 1 with $q_{12}$ as target qubit. The fitted result
gives $1-\tilde{\delta}_{\rm SPAM}^{10,1,\rm miti}=0.937(1)$. (c) Fidelity of
$q_{10}$ with initial state $\ket{00}$ for circuit 1 with $q_{10}$ as target
qubit. The fitted result gives $1-\delta_{\rm SPAM}^{10,0,\rm miti}=0.973(1)$.
(d) Fidelity of $q_{10}$ with initial state $\ket{01}$ for circuit 1 with
$q_{10}$ as target qubit. The fitted result gives $1-\delta_{\rm
SPAM}^{10,0,\rm miti}=0.980(1)$.

In this Appendix, we assume for simplicity that the readout error is symmetric
(i.e., $\delta_{\rm M}^{0}=\delta_{\rm M}^{1}$) or else we can use the
symmetrized readout protocol. We will use scripts to label the two qubits 1
and 2. First, we use 1 as the control and 2 as the target and go through the
procedure of post-selecting ‘0’ for qubit 2. This reduces the state
preparation error of qubit 1 to $\tilde{\delta}_{\rm SP,1}=\delta_{\rm
SP,1}f_{12}$, where (see Ref. [14])

$f_{12}\equiv\frac{2(\delta_{\rm SP,1}+\delta_{\rm M,1}-2\delta_{\rm
SP,1}\delta_{\rm M,1})}{1+(1-2\delta_{\rm SP,1})(1-2\delta_{\rm
M,1})(1-2\delta_{\rm SP,2})},$ (43)

and the subscripts are used to denote which qubits. The total measured SPAM
error (after post-selecting qubit 2) is $\tilde{\delta}_{\rm
SPAM,1}=\tilde{\delta}_{\rm SP,1}+\delta_{\rm M,1}$.

Next, we use 2 as the control and 1 as the target and go through the procedure
of post-selecting ‘0’ for qubit 1. This reduces the state preparation error of
qubit 2 to $\tilde{\delta}_{\rm SP,2}=\delta_{\rm SP,2}f_{21}$, where

$f_{21}\equiv\frac{2(\delta_{\rm SP,2}+\delta_{\rm M,2}-2\delta_{\rm
SP,2}\delta_{M,2})}{1+(1-2\delta_{\rm SP,2})(1-2\delta_{\rm
M,2})(1-2\delta_{\rm SP,1})}.$ (44)

The total measured SPAM error (after post-selecting qubit 2) is
$\tilde{\delta}_{\rm SPAM,2}=\tilde{\delta}_{\rm SP,2}+\delta_{\rm M,2}$.
Together with two other measurable quantities, $\delta_{\rm
SPAM,i}=\delta_{\rm SP,i}+\delta_{\rm M,i}-2\delta_{\rm SP,i}\delta_{M,i}$
(for $i=1,2$), we have a total of four equations for four unknowns, with which
one can solve in principle. We have performed numerical tests and confirmed
that the equations can be solved.

For example, if we measure the four SPAM errors (using the symmetrized readout
protocol to average the two readout errors) and obtain the measured results:

$\displaystyle\\{\delta_{\rm SPAM,1},\delta_{\rm SPAM,2},\tilde{\delta}_{\rm
SPAM,1},\tilde{\delta}_{\rm SPAM,2}\\}$
$\displaystyle=\\{0.22923,0.277084,0.222717,0.275828\\}.$ (45)

Then, solving the above equations leads to a single solution:

$\displaystyle\\{\delta_{\rm SP,1},\delta_{\rm M,1},{\delta}_{\rm
SP,2},{\delta}_{\rm M,2}\\}$
$\displaystyle=\\{0.0242911,0.215404,0.0172848,0.269102\\}.$ (46)

With additional characterization of the asymmetry in the readout, we can
obtain a complete and separate characterization of SPAM errors.

We note that the algorithmic cooling approach here requires solving coupled
nonlinear equations, whereas the method presented in the main text requires
solving only linear equations.

##  Appendix B Comparison for calculating $\delta_{\rm SP}$ with and without
CNOT gate noise extrapolation

In this appendix, we compare the estimation of $\delta_{\rm SP}$ from noisy-
CNOT results (37) and mitigated-CNOT results. To mitigate CNOT gate noise, we
use the zero-noise extrapolation technique [5], where CNOT gates are repeated
$m=1,3,5,...$ times to get different noise-scale results. After that, we use
linear extrapolation to extrapolate them to zero-noise ($m=0$) point and get
the mitigated data. The extrapolating results for calculating $\delta_{\rm
SP}^{(12)}$ via ancilla qubit $q_{10}$ of ibmq_mumbai are shown in Fig. 6.
From the mitigated results, we get the $\delta_{\rm SP}^{(12)}=0.041(4)$,
which is very close to the results from unmitigated data $\delta_{\rm
SP}^{(12)}=0.042(2)$ (shown in Sec. V.1).

## References

  * Preskill [2018] J. Preskill, Quantum Computing in the NISQ era and beyond, Quantum 2, 79 (2018).
  * Temme  _et al._ [2017] K. Temme, S. Bravyi, and J. M. Gambetta, Error mitigation for short-depth quantum circuits, Physical review letters 119, 180509 (2017).
  * Endo  _et al._ [2018] S. Endo, S. C. Benjamin, and Y. Li, Practical quantum error mitigation for near-future applications, Physical Review X 8, 031027 (2018).
  * Dumitrescu  _et al._ [2018] E. F. Dumitrescu, A. J. McCaskey, G. Hagen, G. R. Jansen, T. D. Morris, T. Papenbrock, R. C. Pooser, D. J. Dean, and P. Lougovski, Cloud quantum computing of an atomic nucleus, Physical review letters 120, 210501 (2018).
  * Kandala  _et al._ [2019] A. Kandala, K. Temme, A. D. Córcoles, A. Mezzacapo, J. M. Chow, and J. M. Gambetta, Error mitigation extends the computational reach of a noisy quantum processor, Nature 567, 491 (2019).
  * Giurgica-Tiron  _et al._ [2020] T. Giurgica-Tiron, Y. Hindy, R. LaRose, A. Mari, and W. J. Zeng, Digital zero noise extrapolation for quantum error mitigation, in  _2020 IEEE International Conference on Quantum Computing and Engineering (QCE)_ (IEEE, 2020) pp. 306–316.
  * Kim  _et al._ [2023] Y. Kim, C. J. Wood, T. J. Yoder, S. T. Merkel, J. M. Gambetta, K. Temme, and A. Kandala, Scalable error mitigation for noisy quantum circuits produces competitive expectation values, Nature Physics , 1 (2023).
  * Chen  _et al._ [2019] Y. Chen, M. Farahzad, S. Yoo, and T.-C. Wei, Detector tomography on ibm quantum computers and mitigation of an imperfect measurement, Physical Review A 100, 052315 (2019).
  * Maciejewski  _et al._ [2020] F. B. Maciejewski, Z. Zimborás, and M. Oszmaniec, Mitigation of readout noise in near-term quantum devices by classical post-processing based on detector tomography, Quantum 4, 257 (2020).
  * Geller and Sun [2021] M. R. Geller and M. Sun, Toward efficient correction of multiqubit measurement errors: Pair correlation method, Quantum Science and Technology 6, 025009 (2021).
  * Bravyi  _et al._ [2021] S. Bravyi, S. Sheldon, A. Kandala, D. C. Mckay, and J. M. Gambetta, Mitigating measurement errors in multiqubit experiments, Physical Review A 103, 042605 (2021).
  * Van Den Berg  _et al._ [2022] E. Van Den Berg, Z. K. Minev, and K. Temme, Model-free readout-error mitigation for quantum expectation values, Physical Review A 105, 032620 (2022).
  * Qiskit contributors [2023] Qiskit contributors, Qiskit: An open-source framework for quantum computing (2023).
  * Laflamme  _et al._ [2022] R. Laflamme, J. Lin, and T. Mor, Algorithmic cooling for resolving state preparation and measurement errors in quantum computing, Phys. Rev. A 106, 012439 (2022).
  * Baugh  _et al._ [2005] J. Baugh, O. Moussa, C. A. Ryan, A. Nayak, and R. Laflamme, Experimental implementation of heat-bath algorithmic cooling using solid-state nuclear magnetic resonance, Nature 438, 470 (2005).
  * Ryan  _et al._ [2008] C. Ryan, O. Moussa, J. Baugh, and R. Laflamme, Spin based heat engine: demonstration of multiple rounds of algorithmic cooling, Physical review letters 100, 140501 (2008).
  * Schulman  _et al._ [2005] L. J. Schulman, T. Mor, and Y. Weinstein, Physical limits of heat-bath algorithmic cooling, Physical review letters 94, 120501 (2005).
  * Blume-Kohout  _et al._ [2017] R. Blume-Kohout, J. K. Gamble, E. Nielsen, K. Rudinger, J. Mizrahi, K. Fortier, and P. Maunz, Demonstration of qubit operations below a rigorous fault tolerance threshold with gate set tomography, Nature communications 8, 14485 (2017).
  * Smolin  _et al._ [2012] J. A. Smolin, J. M. Gambetta, and G. Smith, Efficient method for computing the maximum-likelihood quantum state from measurements with additive gaussian noise, Physical review letters 108, 070502 (2012).
  * Nation  _et al._ [2021] P. D. Nation, H. Kang, N. Sundaresan, and J. M. Gambetta, Scalable mitigation of measurement errors on quantum computers, PRX Quantum 2, 040326 (2021).
  * Nielsen and Chuang [2010] M. A. Nielsen and I. L. Chuang,  _Quantum computation and quantum information_ (Cambridge university press, 2010).
  * Hellinger [1909] E. Hellinger, Neue begründung der theorie quadratischer formen von unendlichvielen veränderlichen., Journal für die reine und angewandte Mathematik 1909, 210 (1909).