††thanks: These two authors contributed equally to this work.††thanks: These
two authors contributed equally to this work.

# Deterministic Benchmarking of Quantum Gates

Vinay Tripathi  Department of Physics & Astronomy, University of Southern
California, Los Angeles, California 90089, USA  Center for Quantum Information
Science and Technology, University of Southern California, Los Angeles,
California 90089, USA  Daria Kowsari  Department of Physics & Astronomy,
University of Southern California, Los Angeles, California 90089, USA  Center
for Quantum Information Science and Technology, University of Southern
California, Los Angeles, California 90089, USA  Kumar Saurav  Center for
Quantum Information Science and Technology, University of Southern California,
Los Angeles, California 90089, USA  Ming Hsieh Department of Electrical &
Computer Engineering, University of Southern California, Los Angeles,
California 90089, USA
Haimeng Zhang  Center for Quantum Information Science and Technology,
University of Southern California, Los Angeles, California 90089, USA  Ming
Hsieh Department of Electrical & Computer Engineering, University of Southern
California, Los Angeles, California 90089, USA  Eli M. Levenson-Falk
<EMAIL_ADDRESS>Department of Physics & Astronomy, University of Southern
California, Los Angeles, California 90089, USA  Center for Quantum Information
Science and Technology, University of Southern California, Los Angeles,
California 90089, USA  Ming Hsieh Department of Electrical & Computer
Engineering, University of Southern California, Los Angeles, California 90089,
USA  Daniel A. Lidar  Department of Physics & Astronomy, University of
Southern California, Los Angeles, California 90089, USA  Center for Quantum
Information Science and Technology, University of Southern California, Los
Angeles, California 90089, USA  Ming Hsieh Department of Electrical & Computer
Engineering, University of Southern California, Los Angeles, California 90089,
USA  Department of Chemistry, University of Southern California, Los Angeles,
California 90089, USA

###### Abstract

We introduce _deterministic benchmarking_ (DB), a protocol designed to
identify the interplay of coherent and incoherent errors overlooked by
randomized benchmarking (RB) and related benchmarking methods. DB provides a
set of four parameters that characterize both incoherent and coherent errors
in the single-qubit gate set. Furthermore, DB reveals asymmetries in gate
performance induced by strong relaxation errors ($T_{1}$). We experimentally
demonstrate DB using a superconducting transmon qubit and support these
results with a simple analytical model and master equation simulations. Our
findings uncover critical errors missed by conventional RB and point to
strategies to mitigate these errors.

Fault-tolerant quantum error correction is essential for the ultimate
scalability of quantum computing, but requires high-fidelity operations that
exceed an accuracy threshold [1, 2, 3, 4, 5]. Coherent errors that arise from
systematic deviations in quantum gate operations pose a significant challenge
in this context [6, 7]. The reason is that, while stochastic errors add up as
probabilities, coherent errors add up as amplitudes, generally leading to a
quadratically smaller accuracy threshold [8]. Traditional error
characterization methods such as Randomized Benchmarking (RB) [9, 10, 11],
which deliberately randomize noise channels, do not capture coherent error
accumulation unless appropriately modified [12]. This can result in an
underestimation of the gate performance needed to achieve accuracy thresholds,
critical for realizing fault-tolerant quantum computing [13, 14]. Various
techniques exist that can mitigate the accumulation of coherent errors by
converting them into stochastic noise, but only at the expense of significant
additional cost in the number of required circuits [15, 16, 17, 18]. It can be
advantageous to, instead, characterize coherent gate errors [19], and
subsequently eliminate or at least suppress such errors at the control level,
rather than placing the burden for doing so on fault-tolerant quantum error
correction using large codes [20].

To address these characterization and control challenges, here we introduce a
method we call _deterministic benchmarking_ (DB) and validate its utility
through experiments conducted on a superconducting transmon qubit [21]. Unlike
RB, which averages over many and lengthy random Clifford gate sequences to
approximate all errors as a depolarizing channel and lacks the sensitivity
needed to detect small changes in decoherence times and qubit frequencies [22,
23], DB is highly sensitive to such changes. As illustrated in Fig. 1(a,b), DB
utilizes a carefully chosen small set of two-gate sequences that specifically
target the real noise affecting single-qubit gates. The DB protocol is
significantly more efficient than RB or other well-established methods such as
process tomography [24] or gate set tomography [25]. Indeed, we show below
that DB requires only four sets of state-fidelity measurements to accurately
estimate all incoherent and coherent errors relevant to single-qubit gate
operations. Additionally, DB can also be used to characterize the bath
temperature, which cannot be obtained using other benchmarking techniques.
Moreover, we show that DB also enables exploration of how coherent and
incoherent errors interact to cause biased noise, whose impact on fault
tolerance has been thoroughly studied [26, 27].

Figure 1: Schematic representations of the gate sequences for (a) randomized
benchmarking (RB) and (b) deterministic benchmarking (DB), respectively. RB
uses $N$ random Clifford gates $C_{i}$, followed by a single recovery gate
$C_{r}=C_{N}^{\dagger}\cdots C_{1}^{\dagger}$. DB prepares an initial state
via $U|{0}\rangle$, applies $n$ repetitions of a fixed, deterministic two-
pulse sequence $P_{1}P_{2}$, and unprepares the initial state via
$U^{\dagger}$.

The gate sequences used in DB resemble the simplest dynamical decoupling (DD)
pulse sequences introduced decades ago in nuclear magnetic resonance [28, 29,
30]. But, while DD is nowadays typically used to _suppress_ incoherent and
coherent noise during _idling_ —whether between gates [31, 32, 33, 34, 35, 36]
or as a result of crosstalk [37, 38, 39, 40, 41, 42, 43], the goal of DB is
instead to _characterize_ the noise arising during _gate operation_.

Qubit model.—We model the transmon qubit as a driven two-level system and
consider it in the drive frame under the rotating wave approximation (e.g.,
Ref. [44]). Let $R_{\alpha}(\theta)\equiv\exp[-i(\theta/2)\sigma_{\alpha}]$,
where $\\{\sigma_{\alpha}\\}$ is the set of Pauli matrices. The Hamiltonian
that generates the single-qubit gates $R_{x}(\theta)$ and $R_{y}(\theta)$,
including both phase and rotation errors, is given by:

$H_{\alpha}=\left(\varepsilon+\varepsilon_{\mathrm{err}}\right)\frac{\sigma_{\alpha}}{2}+\Delta_{\mathrm{err}}\frac{\sigma_{z}}{2}\
,\quad\alpha\in\\{x,y\\}.$ (1)

Ideally, $\varepsilon_{\mathrm{err}}=\Delta_{\mathrm{err}}=0$. In reality,
both are present and give rise to rotation and phase errors

$\delta\theta\equiv\varepsilon_{\mathrm{err}}t_{g}\
,\quad\delta\phi\equiv\frac{\Delta_{\mathrm{err}}}{\bar{\varepsilon}}$ (2)

respectively, with $t_{g}$ denoting the gate duration and $\bar{\varepsilon}$
the effective pulse amplitude. We expect to see pulse-shape dependence in the
phase error $\delta\phi$ through $\bar{\varepsilon}$ which can be computed
numerically. For instance, for a square pulse $R_{\alpha}(\pi)$ gate ,
$\bar{\varepsilon}t_{g}={\varepsilon}t_{g}=\pi$. We assume square-shaped
pulses for our analytical theory and cosine-shaped pulses for our numerical
calculations using the Lindblad master equation (cosine pulses are used in
experimental measurements).

Eq. 1 is an effective model, where $\Delta_{\mathrm{err}}$ captures a phase
error resulting from both detuning—a discrepancy between the drive and qubit
frequencies—and the ac-Stark shift induced by driving the higher levels of the
transmon off-resonantly during the gate [45]. The two-level model is
justifiable given that leakage to higher levels of the transmon during our
gates is very small [46].

Deterministic Benchmarking.—Any completely positive trace-preserving (CPTP)
map acting on a $d$-dimensional Hilbert space can be characterized using no
more than $d^{4}-1$ parameters, i.e., $15$ for a single qubit. RB reduces this
to a single effective parameter. DB makes a less drastic reduction to four
parameters: the relaxation time $T_{1}$, the coherence time $T_{2}$ (which
contains information on pure dephasing $T_{\phi}$), the rotation error
$\delta\theta$, and the phase error $\delta\phi$. Assuming a Markovian noise
model, we show below that these parameters can be straightforwardly extracted
and, at the same time, provide just the right amount of information about the
errors affecting single-qubit gates to enable such gates to be accurately
calibrated. We denote $X\equiv R_{x}(\pi)$, $\overline{X}\equiv R_{x}(-\pi)$,
and similarly for $Y$ and $\overline{Y}$. To define DB, we need two-gate
sequences of the form $P_{1}P_{2}$. In particular, we consider
$P_{1}P_{2}\in\\{XX,X\overline{X},YY,Y\overline{Y},\overline{Y}Y\\}$. Note
that in the absence of any errors, i.e., when
$\varepsilon_{\mathrm{err}}=\Delta_{\mathrm{err}}=0$ and without coupling to a
bath, $P_{1}P_{2}=I$ (identity operator) for this set. In reality,
$P_{1}P_{2}\neq I$ and we consider the corresponding fidelity
$\tilde{F}(n)=|\langle{\psi}|(P_{1}P_{2})^{n}|{\psi}\rangle|^{2}$ where
$|{\psi}\rangle$ is the initial state and $n\in\mathbb{Z}^{+}$ is the number
of repetitions of the $P_{1}P_{2}$ sequence; hence the fidelity is defined
stroboscopically. We use a tilde to denote the experimentally measured
fidelity.

The core DB protocol consists of a set of four “learning” experiments designed
to extract $\\{T_{1},T_{2},\delta\theta,\delta\phi\\}$, followed by tests
involving new experimental data. In each case, the notation
$\\{P_{1}P_{2};|{\psi}\rangle\\}$ means a circuit comprising preparation of
the state $|{\psi}\rangle$ from the $|{0}\rangle$ state, varying number of
repetitions of the $P_{1}P_{2}$ sequence, unpreparation of $|{\psi}\rangle$,
and finally a measurement in the $\sigma^{z}$ eigenbasis. Finally, the
empirical fidelity $\tilde{F}$ is found for each experiment as the ratio of
$|{0}\rangle$ outcomes to the total number of experimental shots ($800$ in our
case). We find that in all these experiments—learning as well as testing—the
fidelity can be accurately described using a simple analytical expression:

$F(t_{n})=\frac{1}{2}(1+a)+\frac{1}{2}(1-a)e^{-t_{n}/T_{D}}\cos(2\omega
t_{n}),$ (3)

where $t_{n}=2nt_{g}$ represents the total evolution time. The parameter
$T_{D}$ denotes a characteristic timescale associated with incoherent noise
processes, while $\omega$ is a frequency associated with coherent errors. The
parameter $a$ quantifies the deviation from the infinite temperature limit,
where $F(\infty)=\frac{1}{2}$. Eq. 3 effectively captures the interplay
between coherent and incoherent noise sources. It is derived from a Markovian
master equation model for some of our sequences in [46]. Finally, we note that
Eq. 3 can be further generalized if we replace $\frac{1}{2}(1-a)\mapsto A$ and
$\frac{1}{2}(1+a)\mapsto B$, which adds an extra parameter. If we set
$\omega=0$ (no coherent errors), this reduces to an RB fidelity-like
expression $Ap^{N}+B$ (the average gate infidelity is $r_{\rm
RB}=(1-p)(d-1)/d$, where $d$ is the Hilbert space dimension). RB’s ability to
quantify state preparation and measurement (SPAM) errors as $1-(A+B)$ is thus
captured by this generalization of DB; we do not consider this here as our
focus is the determination of $\\{T_{1},T_{2},\delta\theta,\delta\phi\\}$.

Protocol.—We now detail the DB protocol, which comprises the following
experiments or steps:

  1. 1.

$\\{\text{free};|{1}\rangle\\}$: This is a standard $T_{1}$ measurement, where
we do not apply any pulses (free evolution). Eq. 3 reduces to
$\frac{1}{2}(1+a)+\frac{1}{2}(1-a)e^{-t/T_{1}}$ in this case which we fit to
determine $T_{1}$. This yields the relaxation component of the Lindbladian.

  2. 2.

$\\{XX;|{+}\rangle\\}$: Fitting this experiment with Eq. 3 yields
$T_{D}=T_{2}$, the coherence time from which we deduce the pure dephasing time
$T_{\phi}=2T_{1}T_{2}/(2T_{1}-T_{2})$. This is analogous to an echo-Ramsey
measurement [47]. This yields the dephasing component of the Lindbladian.

  3. 3.

$\\{YY;|{+}\rangle\\}$: This yields fidelity oscillations which we fit to find
the rotation error $\delta\theta$ where $\omega\approx\delta\theta/(2t_{g})$
in Eq. 3.

  4. 4.

$\\{X\overline{X};|{+}\rangle\\}$: Similarly, this yields fidelity
oscillations which we fit to find the phase error $\delta\phi$ where
$\omega=\delta\phi/t_{g}$ in Eq. 3. The decay times $T_{D}$ obtained here and
in step 3 are a combination of the incoherent and coherent noise and hence
different from $T_{2}$ obtained in step $2$. We now have all four parameters,
which we use to numerically calculate predictions for the test experiments.

  5. 5.

Test 1: $\\{Y\overline{Y};|{+}\rangle\\}$ and
$\\{\overline{Y}Y;|{+}\rangle\\}$: This highlights the interplay of asymmetric
relaxation noise and gate operations and can also be used to characterize the
qubit temperature [46].

  6. 6.

Test 2: Various other experiments, e.g., DD. In this work, we report tests
using the UR6 sequence [48].

Next, we demonstrate that this protocol enables a detailed assessment of both
coherent and incoherent errors affecting quantum gate operations.

Figure 2: Empirical ($\tilde{F}$, symbols), analytical fit ($F$, solid
curves), and Lindblad-based ($\bar{F}$, dashed curves) fidelities for the
initial states $|{1}\rangle$ under free evolution and $|{+}\rangle$ under
various gate sequences $P_{1}P_{2}$. The analytical fit yields $T_{1}=23.36\pm
0.40~{}\mu$s, $T_{2}=44.13\pm 2.49~{}\mu$s, $\delta\theta=0.398\pm
0.004^{\circ}$, and $\delta\phi=0.426\pm 0.004^{\circ}$.

Experimental results.—We conduct all our experiments using a superconducting
transmon qubit dispersively coupled to its readout resonator [46]. We
implement $R_{y}(\theta)$ gates by adjusting the phase of the microwave pulses
used for $R_{x}(\theta)$ gates, thus eliminating the need for additional
calibrations. All single-qubit gates are calibrated using the Derivative
Removal of Adiabatic Gates (DRAG) technique [49] to minimize leakage and phase
errors induced by higher levels of the transmon. To emulate drifts in
experimental parameters, we first perform measurements on an outdated set of
gate calibration parameters. The results are shown in Fig. 2.

We observe that $\tilde{F}_{YY}$ oscillates, indicating rotation errors
$\delta\theta$, as explained below. Similarly, $\tilde{F}_{X\overline{X}}$
oscillates due to phase errors $\delta\phi$. The $XX$ sequence is insensitive
to coherent errors and, instead, $\tilde{F}_{XX}$ exhibits an exponential
decay characterizing the $T_{2}$ time during time-dependent gate operations.
The $T_{D}$ from $XX$ is chosen as the effective $T_{2}$ because, subject to
this sequence, the state always remains $|{+}\rangle$. Also shown in Fig. 2
are results from the $\overline{Y}Y$ and $Y\overline{Y}$ sequences. We discuss
the reason for the pronounced difference they exhibit below.

We use Eq. 3 to fit (solid curves) each of the four experimental fidelities
(symbols) in Fig. 2 prescribed by steps 1-4 of the DB protocol and find close
agreement. We contrast the resulting
$\\{T_{1},T_{2},\delta\theta,\delta\phi\\}$ values with the single metric
$r_{\rm RB}$ derived from the RB protocol. The RB protocol requires averaging
over multiple Clifford gate sequences ($30$ in our case, each consisting of
$700$ Clifford gates). This process yields an average gate infidelity of
$0.262\pm 0.004\%$ for the single-qubit Clifford gate set [46], a far less
detailed characterization that misses the coherent oscillations and the
distinction between $T_{1}$ and $T_{2}$.

As we show next, all observed features can be understood using either a
simple, analytically solvable closed-system model or a numerical model based
on a Lindblad master equation with time-dependent gates. We next provide the
analytical fidelities in the closed system case, which exhibit oscillations
for certain sequences. We derive the fidelity expressions in the open system
setting, justifying Eq. 3, in Ref. [46].

Fidelity with coherent errors.—Assuming the system is closed, Eq. 1 can be
used to analytically compute the fidelities of the gate sequences comprising
our DB protocol. The closed system fidelity $F^{\text{clsd}}_{P_{1}P_{2}}(n)$
then reflects only the coherent errors contained within $P_{1}$ and $P_{2}$.
For $YY$ and $XX$ and the initial state $|{+}\rangle$, we find:

$\displaystyle F^{\text{clsd}}_{YY}(n)$
$\displaystyle=\cos^{2}(n\theta_{\mathrm{err}})$ (4a) $\displaystyle
F^{\text{clsd}}_{XX}(n)$
$\displaystyle=1-\left(\frac{\pi\delta\phi\sin(n\theta_{\mathrm{err}})}{\theta_{\mathrm{err}}}\right)^{2},$
(4b)

where $\theta_{\mathrm{err}}\equiv\sqrt{(\varepsilon
t_{g}+\delta\theta)^{2}+(\pi\delta\phi)^{2}}$. Expanding to the leading order
in $\delta\phi$ and $\delta\theta$, with $\varepsilon t_{g}=\pi$,
$F^{\text{clsd}}_{YY}(n)$ simplifies to $\cos^{2}(n\delta\theta)$, indicating
that its dependence is dominated by rotation errors $\delta\theta$. Regarding
$F^{\text{clsd}}_{XX}(n)$, in the small error limit, the second term in Eq. 4b
is suppressed, and $F^{\text{clsd}}_{XX}(n)\approx 1$. Consequently, there is
no dependence on either the rotation or the phase error. Finally:

$\displaystyle F^{\text{clsd}}_{X\overline{X}}(n)$
$\displaystyle=\cos^{2}\left(n\varphi_{\mathrm{err}}\right)$ (5a)
$\displaystyle\varphi_{\mathrm{err}}$
$\displaystyle\equiv\tan^{-1}\left(2\pi\frac{\delta\phi}{\theta_{\mathrm{err}}}\sin\left(\frac{\theta_{\mathrm{err}}}{2}\right)\frac{\sqrt{1-\Lambda/2}}{1-\Lambda}\right)$
(5b) $\displaystyle\Lambda$
$\displaystyle\equiv\left(\pi\frac{\delta\phi}{\theta_{\mathrm{err}}}\right)^{2}(1-\cos{\theta_{\mathrm{err}}})$
(5c)

Expanding to leading order again, with $\varepsilon t_{g}=\pi$, we have
$F^{\text{clsd}}_{X\overline{X}}(n)\approx\cos^{2}\left[2n\delta\phi\left(1-\frac{1}{\pi}\delta\theta\right)\right]$,
showing that this fidelity is dominated by phase errors.

These results justify our assignment of the oscillations observed in the $YY$
and $X\overline{X}$ experiments shown in Fig. 2 to rotation and phase errors,
respectively.

Open system model.—The cosine dependence in Eq. 3 is well motivated by the
closed system (coherent error) fidelities given above, corresponding to
$T_{D}=\infty$ and $a=0$. The choice of the two incoherent parameters in the
DB protocol, specifically $T_{1}$ (associated with free evolution) and $T_{2}$
(associated with the $XX$ sequence), warrants further investigation through
numerical simulation to better understand their contributions to the
parameters $T_{D}$ and $a$ in Eq. 3. We use a simple Lindblad master equation
to capture the effects of non-unitary evolution, with a Lindbladian of the
form
$\mathcal{L}=-i[H,\cdot]+\sum_{\alpha\in\\{1,\phi\\}}\gamma_{\alpha}(L_{\alpha}\cdot
L_{\alpha}^{\dagger}-\frac{1}{2}\\{L_{\alpha}^{\dagger}L_{\alpha},\cdot\\})$.
The Lindblad operators are $L_{1}=\sigma^{-}=|{0}\rangle\\!\langle 1|$ and
$L_{\phi}=\sigma_{z}/\sqrt{2}$ with rates $\gamma_{1}=1/T_{1}$ (relaxation)
and $\gamma_{\phi}=1/T_{\phi}$ (dephasing).

Using the fitted parameters $\\{T_{1},T_{2},\delta\theta,\delta\phi\\}$ from
Steps 1-4 of the DB protocol, we numerically solve the Lindblad equation and
find good agreement with the experimental results, as shown by the dashed
lines in Fig. 2. Moreover, knowledge of these parameters allows us to proceed
to step 5 of the protocol and predict, as a test, the outcomes of the
$Y\overline{Y}$ and $\overline{Y}Y$ experiments. As shown in Fig. 2, there is
again good agreement between the simulations and the experimental results.
This validates the choice of $\\{T_{1},T_{2},\delta\theta,\delta\phi\\}$ as an
appropriate set of parameters. In contrast, $r_{\rm RB}$ is clearly
insufficient for predicting the outcomes of the $Y\overline{Y}$ and
$\overline{Y}Y$ experiments.

Figure 3: Experimental fidelities $\tilde{F}^{|{+}\rangle}$ showing the very
different sensitivity of DB and RB to coherent errors. (a)
$\delta\theta=0^{\circ}$, $\delta\phi=0^{\circ}$, (b)
$\delta\theta=0^{\circ}$, $\delta\phi=0.893\pm 0.002^{\circ}$, (c)
$\delta\theta=0.932\pm 0.007^{\circ}$, $\delta\phi=0^{\circ}$, (d)
$\delta\theta=0.995\pm 0.009^{\circ}$, $\delta\phi=0.90\pm 0.002^{\circ}$.
These angles are obtained from fits using Eq. 3. Dashed curves are the
numerically computed fidelities $\bar{F}^{|{+}\rangle}$ from the open system
Lindblad model using the DB parameters above along with the corresponding
$T_{1}$ and $T_{2}$. The UR$6$ sequence suppresses even large coherent errors.

Sensitivity to coherent errors.—Having confirmed that DB effectively
identifies various errors, we next assess the sensitivity of both DB and RB to
coherent errors. Initially, we calibrate the gates to minimize both
$\delta\phi$ and $\delta\theta$ [46], and immediately follow this calibration
by a series of DD experiments, as depicted in Fig. 3(a). These experiments
focus on the $X\overline{X}$ and $YY$ sequences, which are sensitive to phase
and rotation errors. As a second test (Step 6 in the DB protocol), we include
a Universally Robust (UR$n$) DD sequence, specifically designed to suppress
coherent errors [48]. We also conduct RB on these gates. We then deliberately
introduce coherent errors with ($\delta\theta$, $\delta\phi$) values given in
Fig. 3. The phase error $\delta\phi$, is introduced by mistuning the DRAG
weighting parameter $\alpha$, which effectively varies the detuning error term
$\Delta_{\mathrm{err}}$ in our model. The increase in rotation error
$\delta\theta$ from Fig. 3(c) (in the absence of $\delta\phi$) to (d) (in the
presence of $\delta\phi$) can be attributed to the second order dependence of
$\theta_{\mathrm{err}}$ on $\delta\phi$, given by
$\frac{1}{2}(\pi\delta\phi)^{2}/(\pi+\delta\theta)$.

As anticipated, immediately following gate calibration using the parameters
extracted from the DB protocol, neither $X\overline{X}$ nor $YY$ causes
oscillations, as shown in Fig. 3(a), and the coherent error is negligible.
However, after deliberately introducing phase errors, significant oscillations
appear in $\tilde{F}_{X\overline{X}}$, as depicted in Fig. 3(b), while
$\tilde{F}_{YY}$ simply decays exponentially without exhibiting any
oscillations. In contrast, introducing an over-rotation error in Fig. 3(c)
reveals clear oscillations in $\tilde{F}_{YY}$ but none in
$\tilde{F}_{X\overline{X}}$. When both errors are introduced, Fig. 3(d)
displays clear oscillations for both sequences. The UR6 sequence never
exhibits oscillations, confirming that coherent errors are their source. The
Lindblad model with parameters obtained via the DB protocol (dashed curves) is
again in good agreement with the experimental fidelities, further confirming
the utility of DB. In contrast, the error rate derived from RB, $r_{\rm RB}$,
is unchanged when a phase error ($\delta\phi\approx 0.90^{\circ}$) is
introduced and increases only slightly ($\approx 0.02\%$) when an over-
rotation error ($\delta\theta\approx 0.93^{\circ}$). We see that small
coherent errors can cause large, rapid oscillations in sequence fidelity even
when the RB fidelity is almost unchanged. This illustrates RB’s limitations in
detecting specific types of coherent errors.

Interplay of $T_{1}$ asymmetry and gates.—An intriguing observation arising
from Fig. 2 is the strong asymmetry in the decay of the $Y\overline{Y}$ and
$\overline{Y}Y$ fidelity curves. Transmon qubits operate at frequencies in the
$3-6$ GHz range and temperatures around $10$ mK ($\sim 0.2$ GHz). In this
regime, the thermal energy is negligible, and the ground state $|{0}\rangle$
is approximately the thermal state of the transmon qubit, i.e., excitation
from $|{0}\rangle$ to $|{1}\rangle$ is negligible and relaxation dominates.
Tracing the Bloch sphere trajectories corresponding to different sequences
reveals that $Y\overline{Y}$ confines the state to the southern (excited
state) hemisphere, while $\overline{Y}Y$ limits it to the northern (ground
state) hemisphere. Therefore, the former is more susceptible to relaxation
($T_{1}$) errors, which explains the faster decay and lower saturation
fidelity of $Y\overline{Y}$ compared to $\overline{Y}Y$. This asymmetry is
also well captured by the time-dependent Lindblad model, as shown by the
dashed curves in Fig. 2.

To demonstrate this effect, we performed additional experiments in the same
outdated gate calibration cycle used in Fig. 3. We applied the $X\overline{X}$
sequence to $|{\pm i}\rangle$ (eigenstates of $\sigma_{y}$), as well as to
$|{\psi_{\frac{\pi}{4},\frac{3\pi}{4}}}\rangle$ and
$|{\psi_{\frac{3\pi}{4},-\frac{3\pi}{4}}}\rangle$, where
$|{\psi_{\theta,\phi}}\rangle=\cos(\theta/2)|{0}\rangle+e^{i\phi}\sin(\theta/2)|{1}\rangle$.
We observe a clear asymmetry between these pairs of orthogonal states under
$X\overline{X}$, as shown in Fig. 4(a). We further confirm this asymmetry
using $Y\overline{Y}$ applied to $|{\pm}\rangle$ (eigenstates of $\sigma_{x}$)
and
$\\{|{\psi_{\frac{\pi}{4},\frac{3\pi}{4}}}\rangle,|{\psi_{\frac{3\pi}{4},-\frac{3\pi}{4}}}\rangle\\}$,
finding the same asymmetric decay patterns both experimentally and numerically
in Fig. 4(b). These results illustrate the dependence of relaxation during
time-dependent gates on the initial state and its trajectory under different
pulse sequences. Numerical modeling of this asymmetry enables qubit
temperature calibration [46].

Figure 4: Asymmetry in the decay pattern of DB sequences due to the interplay
of gates and $T_{1}$. Fidelity decay for (a) $X\overline{X}$ and (b)
$Y\overline{Y}$ applied to two pairs of orthogonal initial states. Dashed
curves represent the Lindblad master equation simulation results with
parameters obtained by fitting Eq. 3.

Conclusions and outlook.—We have developed a deterministic benchmarking (DB)
protocol and demonstrated it using a superconducting transmon qubit. Unlike
traditional randomized benchmarking schemes that rely on random Clifford
gates, DB employs a small and fixed set of simple two-pulse sequences. The DB
protocol quantifies both incoherent and coherent gate errors that elude RB and
does so without averaging over random circuits, thus yielding more information
than RB at a reduced cost. Moreover, DB highlights noise bias, which might
benefit the development of compilation strategies [50] provided circuits can
be compiled so as to minimize relaxation by steering quantum trajectories away
from excited states. The benefits could extend to resource estimation for
fault-tolerant quantum algorithms [51, 52] and biased noise error correction
[27].

The implementation of the DB protocol is straightforward and adaptable beyond
superconducting qubits. We anticipate that DB will contribute to the
benchmarking and enhancement of quantum gates across various platforms. A
natural open problem is to extend DB to two-qubit gates and identify the
corresponding minimal extended parameter set that describes incoherent and
coherent errors.

Acknowledgement— We thank Namit Anand, Nic Ezzell and Alexandru Paler for
useful discussions. Devices were fabricated and provided by the
Superconducting Qubits at Lincoln Laboratory (SQUILL) Foundry at MIT Lincoln
Laboratory, with funding from the Laboratory for Physical Sciences (LPS) Qubit
Collaboratory. This research was supported by the National Science Foundation,
the Quantum Leap Big Idea under Grant No. OMA-1936388, the Army Research
Office MURI Grant No. W911NF-22-S-0007, and the Intelligence Advanced Research
Projects Activity (IARPA) under Cooperative Agreement No. W911NF-23-2-0216.

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Supplemental Material

##  Appendix A Fidelities of the $YY$ and $X\overline{X}$ sequences

We assume a Markovian environment which we model via the Lindblad master
equation with time-dependent driving:

$\displaystyle\dot{\rho}$
$\displaystyle=-i\left[H(t),\rho\right]+\mathcal{L}_{D}(\rho)$ (6a)
$\displaystyle\mathcal{L}_{D}(\rho)$
$\displaystyle=\sum_{\alpha}\gamma_{\alpha}(L_{\alpha}\rho
L_{\alpha}^{\dagger}-\frac{1}{2}\\{L_{\alpha}^{\dagger}L_{\alpha},\rho\\}).$
(6b)

We recognize that this is a phenomenological model and that rigorously derived
master equations for time-dependent driving have a different structure, where
the Lindblad operators $L_{\alpha}$ become time-dependent [53, 54, 55, 56, 57,
58, 59]. However, this simplified model is analytically solvable and suffices
for our purposes.

###  A.1 Background

For analysis purposes, we work in the Bloch vector picture, which substitutes
$\rho=\frac{1}{2}(I+v_{x}\sigma_{x}+v_{y}\sigma_{y}+v_{z}\sigma_{z})$,
$H=\frac{1}{2}(h_{x}\sigma_{x}+h_{y}\sigma_{y}+h_{z}\sigma_{z})$, where
$\mathbf{v}=(v_{x},v_{y},v_{z})\in\mathbb{R}^{3},\mathbf{h}=(h_{x},h_{y},h_{z})\in\mathbb{R}^{3}$.
The dissipator is replaced by $(R,\mathbf{c})$ with
$R_{ij}\equiv\frac{1}{2}\mathrm{Tr}(\sigma_{i}\mathcal{L}_{D}(\sigma_{j}))$
and $c_{i}\equiv\frac{1}{2}\mathrm{Tr}(\sigma_{i}\mathcal{L}_{D}(I))$. The
Lindblad master equation then is
$\dot{\mathbf{v}}=\mathbf{h}\times\mathbf{v}+R\mathbf{v}+\mathbf{c}$ (see,
e.g., Ref. [60] for derivations of all these results). The corresponding
fidelity with the $|{+}\rangle$ state is given by

$F(t)=\frac{1}{2}(1+v_{x}(t)).$ (7)

We model our system as a qubit with an energy gap of $\omega_{01}$ coupled to
a thermal environment at inverse temperature $\beta$ along with dephasing
noise. The combined noise model translates to
$R=\text{diag}(-\gamma_{2},-\gamma_{2},-\gamma_{1})$ and
$\mathbf{c}=(0,0,\eta\,\gamma_{1})$ where

$\gamma_{2}=\frac{1}{2}\gamma_{1}+\gamma_{\phi},$ (8)

and

$\eta=\frac{1-\exp(-\beta\omega_{01})}{1+\exp(-\beta\omega_{01})}\in[0,1].$
(9)

###  A.2 $YY$ sequence

For a square $YY$ pulse,
$\mathbf{h}(t)=(0,\varepsilon_{\mathrm{tot}},\Delta_{\mathrm{err}})$ where
$\varepsilon_{\mathrm{tot}}=\varepsilon+\varepsilon_{\mathrm{err}}$. Then the
Bloch vector equation becomes $\dot{\mathbf{v}}=G\mathbf{v}+\mathbf{c}$, where

$G=\begin{pmatrix}-\gamma_{2}&-\Delta_{\mathrm{err}}&\varepsilon_{\mathrm{tot}}\\\
\Delta_{\mathrm{err}}&-\gamma_{2}&0\\\
-\varepsilon_{\mathrm{tot}}&0&-\gamma_{1}\end{pmatrix}.$ (10)

We first solve the case where $\Delta_{\mathrm{err}}=0$ ($\delta\phi=0$).
Define

$\delta G=\begin{pmatrix}0&-\Delta_{\mathrm{err}}&0\\\
\Delta_{\mathrm{err}}&0&0\\\ 0&0&0\end{pmatrix}.$ (11)

####  A.2.1 Exact solution for $\Delta_{\mathrm{err}}=0$

The Bloch vector equation for $\Delta_{\mathrm{err}}=0$ therefore becomes

$\dot{\mathbf{v}}=(G-\delta G)\mathbf{v}+\mathbf{c}.$ (12)

In this equation, $v_{y}$ is decoupled from the other two components and to
solve for $v_{x}(t)$ and $v_{z}(t)$, the equation requires exponentiation of a
$2\times 2$ matrix. Given that $\mathbf{v}(0)=(1,0,0)$, the relevant solutions
for the Bloch vector components take the following form:

$\displaystyle v_{y}(t)$ $\displaystyle=0$ (13a) $\displaystyle v_{x}(t)$
$\displaystyle=v_{\infty}+(1-v_{\infty})e^{-\gamma_{*}t}\left[\cos{(\omega_{*}t)}\vphantom{\frac{\gamma}{\omega}}\right.$
(13b)
$\displaystyle\qquad\left.+\left(\frac{(\gamma_{1}-\gamma_{2})-(\gamma_{1}+\gamma_{2})(1-v_{\infty})}{2\omega_{*}(1-v_{\infty})}\right)\sin{(\omega_{*}t)}\right]$

where

$\displaystyle v_{\infty}$
$\displaystyle=\eta\frac{\varepsilon_{\mathrm{tot}}\gamma_{1}}{\varepsilon_{\mathrm{tot}}^{2}+\gamma_{1}\gamma_{2}}$
(14a) $\displaystyle\gamma_{*}$
$\displaystyle=\frac{1}{2}(\gamma_{1}+\gamma_{2})$ (14b)
$\displaystyle\omega_{*}$
$\displaystyle=\sqrt{\varepsilon_{\mathrm{tot}}^{2}-\frac{1}{4}(\gamma_{1}-\gamma_{2})^{2}}.$
(14c)

Along with Eq. 7, Eq. 13b is close to, but not quite Eq. 3.

####  A.2.2 Perturbation theory for $\Delta_{\mathrm{err}}=0$

In the regime where
$\varepsilon\gg\varepsilon_{\mathrm{err}},\gamma_{1},\gamma_{2}$, the
quantities in Eq. 14 simplify to

$v_{\infty}\approx\eta\frac{\gamma_{1}}{\varepsilon}\
,\quad\omega_{*}\approx\varepsilon_{\mathrm{tot}}=\varepsilon+\varepsilon_{\mathrm{err}},$
(15)

to first order. Moreover, since we are interested in the fidelities after each
DD sequence, i.e., at $t=2nt_{g}$ ($n\in\mathbb{N})$,
$\sin{(2\omega_{*}nt_{g})}\approx\sin{(2n\delta\theta)}$ and
$\cos{(2\omega_{*}nt_{g})}\approx\cos{(2n\delta\theta)}$. The term
corresponding to $\sin{(2\omega_{*}nt_{g})}$ vanishes to first order initially
because it is multiplied by another first-order term and later because it is
exponentially suppressed. This results in a simplified expression for
$v_{x}(2nt_{g})$:

$v_{x}(2nt_{g})=v_{\infty}+(1-v_{\infty})e^{-2nt_{g}\gamma_{*}}\cos{(2n\delta\theta)}$
(16)

The fidelity of the $|{+}\rangle$ state as a function of $YY$ repetitions $n$,
$F_{\delta\phi=0}(n)$, is therefore given by

$\displaystyle F_{\delta\phi=0}(t_{n})$
$\displaystyle=\frac{1}{2}(1+v_{\infty})+\frac{1}{2}(1-v_{\infty})e^{-\gamma_{*}t_{n}}\cos(2n\delta\theta).$
(17)

Eq. 17 is identical in form to Eq. 3, with
$a=v_{\infty}\approx\gamma_{1}/\varepsilon$, $T_{D}=1/\gamma_{*}$ and $2\omega
t_{g}\approx\delta\theta$. Note that the oscillation period of
$2n\delta\theta$ matches that of the fidelity in the closed system case as
discussed below Eq. 4.

####  A.2.3 Perturbation theory for $\Delta_{\mathrm{err}}\neq 0$

If the solution to the Bloch vector equation given in Eq. 12 is denoted by
$\mathbf{v}_{0}(t)$, then in the case where $\Delta_{\mathrm{err}}\neq 0$, let
the deviation of the actual Bloch vector trajectory $\mathbf{v}(t)$ from
$\mathbf{v}_{0}(t)$ be given by $\bm{\xi}(t)\in\mathbb{R}^{3}$, i.e.,
$\mathbf{v}(t)=\mathbf{v}_{0}(t)+\bm{\xi}(t)$. Substituting this form into the
Bloch vector equation $\dot{\mathbf{v}}=G\mathbf{v}+\mathbf{c}$ gives

$\dot{\bm{\xi}}=G\bm{\xi}+\delta G\mathbf{v}_{0}.$ (18)

The solution for $\bm{\xi}(t)$ is then given by

$\bm{\xi}(t)=\int_{0}^{t}e^{G(t-s)}\delta G\mathbf{v}_{0}(s)\,ds.$ (19)

Taking the norm on both sides and moving the norm inside the integral, we
obtain

$\mathinner{\\!\left\lVert\bm{\xi}(t)\right\rVert}\leq\int_{0}^{t}\mathinner{\\!\left\lVert
e^{G(t-s)}\right\rVert}\mathinner{\\!\left\lVert\vphantom{e^{G}}\delta
G\mathbf{v}_{0}(s)\right\rVert}\,ds.$ (20)

Using Eq. 11, $\mathinner{\\!\left\lVert\delta
G\mathbf{v}_{0}(s)\right\rVert}$ simplifies to
$\mathinner{\\!\left\lvert\Delta_{\mathrm{err}}v_{0,x}(s)\right\rvert}$ since
$v_{0,y}(s)=0$. From Eq. 13b, $\mathinner{\\!\left\lVert
v_{0,x}(s)\right\rVert}$ in turn can be strictly upper bounded by
$\mathinner{\\!\left\lvert v_{\infty}\right\rvert}+ke^{-\gamma_{*}s}$ for some
constant $k$. A smaller value of $k$ will result in a tighter bound. In the
approximation regime discussed above, $k$ can be conveniently chosen to be
$2$.

If $G=SDS^{-1}$ where $D$ is the diagonal matrix of $G$’s eigenvalues
$\lambda_{i}$ and $S$ is the similarity matrix with the corresponding
eigenvectors of $G$ as its columns,

$\mathinner{\\!\left\lVert
e^{G(t-s)}\right\rVert}\leq\mathinner{\\!\left\lVert
S\vphantom{S^{-}1}\right\rVert}\mathinner{\\!\left\lVert
e^{D(t-s)}\right\rVert}\mathinner{\\!\left\lVert S^{-1}\right\rVert}\leq\kappa
e^{-\lambda(t-s)}$ (21)

where $\kappa=\mathinner{\\!\left\lVert
S\vphantom{S^{-}1}\right\rVert}\mathinner{\\!\left\lVert S^{-1}\right\rVert}$
and $\lambda=-\max_{i}\Re(\lambda_{i})$ [61]. This gives the inequality

$\mathinner{\\!\left\lVert\bm{\xi}(t)\right\rVert}<\begin{cases}\kappa\mathinner{\\!\left\lvert\Delta_{\mathrm{err}}\right\rvert}\left(\frac{1-e^{-\lambda
t}}{\lambda}\mathinner{\\!\left\lvert
v_{\infty}\right\rvert}+k\frac{e^{-\gamma_{*}t}-e^{-\lambda
t}}{\lambda-\gamma_{*}}\right)\text{ if}\ \lambda\neq\gamma_{*}\\\
\kappa\mathinner{\\!\left\lvert\Delta_{\mathrm{err}}\right\rvert}\left(\frac{1-e^{-\lambda
t}}{\lambda}\mathinner{\\!\left\lvert v_{\infty}\right\rvert}+kte^{-\lambda
t}\right)\qquad\text{otherwise}\end{cases}$ (22)

Note that the first term is $O(\Delta_{\mathrm{err}}v_{\infty}/\lambda)$.

From first-order eigenvalue perturbation theory, we have

$\delta{\lambda_{i}}=\frac{\mathbf{w}_{i}\cdot\delta{G}\mathbf{v}_{i}}{\mathbf{w}_{i}\cdot\mathbf{v}_{i}},$
(23)

where $\mathbf{w}_{i}$ and $\mathbf{v}_{i}$ are the left and right
eigenvectors of $G$ in Eq. 10 with eigenvalues
$\\{\lambda_{i}\\}=\\{\\-\gamma_{*}\pm i\omega_{*},-\gamma_{2}\\}$. Therefore,
using Eq. 11:

$\displaystyle\lambda_{1,2}$ $\displaystyle=(-\gamma_{*}\pm
i\omega_{*})+O(\Delta_{\mathrm{err}})$ (24a) $\displaystyle\lambda_{3}$
$\displaystyle=-\gamma_{2}+O(\Delta_{\mathrm{err}}).$ (24b)

Under the additional assumption $\Delta_{\mathrm{err}}/\gamma_{1,2}=O(1)$, the
first summand in Eq. 22 simplifies to
$O(v_{\infty})=O(\gamma_{1}/\varepsilon)$. Combining this with the fact that
the second summand decays exponentially with time, $\xi(t)$ is a small enough
quantity such that $\mathbf{v}(t)$ can be well approximated by
$\mathbf{v}_{0}(t)$. Therefore, the resulting fidelity expression $F_{YY}(n)$
in a $\Delta_{\mathrm{err}}\neq 0$ system is still well captured by Eq. 17. It
matches Eq. 3 with an error $\mathinner{\\!\left\lvert
F_{YY}(n)-F_{\delta\phi=0}(n)\right\rvert}<\frac{1}{2}\mathinner{\\!\left\lVert\bm{\xi}(2nt_{g})\right\rVert}$,
with the factor $1/2$ due to Eq. 7. In big-O notation,

$\mathinner{\\!\left\lvert
F_{YY}(n)-F_{\delta\phi=0}(n)\right\rvert}=O(\gamma_{1}/\varepsilon)=O(\gamma_{1}t_{g}).$
(25)

###  A.3 $X\overline{X}$ sequence

We solve for a simplified model in which $T_{\phi}=2T_{1}$, that is,
$\gamma_{1}=\gamma_{2}$: all the components of the Bloch vector decay
uniformly to the steady state $(0,0,\eta)$. The resulting Bloch vector
equations are:

$\dot{\mathbf{v}}(t)=\begin{cases}G_{+}\mathbf{v}(t)+\mathbf{c},&\text{if}\
(2k)t_{g}<t\leq(2k+1)t_{g},\,k\in\mathbb{Z}\\\
G_{-}\mathbf{v}(t)+\mathbf{c},&\text{if}\
(2k-1)t_{g}<t\leq(2k)t_{g},\,k\in\mathbb{Z}\\\ \end{cases}$ (26)

where $\mathbf{c}=(0,0,\eta\,\gamma)$, $\gamma=\gamma_{1}=\gamma_{2}$, and the
matrices $G_{\pm}$ are

$G_{\pm}=\begin{pmatrix}-\gamma&-\Delta_{\mathrm{err}}&0\\\
\Delta_{\mathrm{err}}&-\gamma&\mp\varepsilon_{\mathrm{tot}}\\\
0&\pm\varepsilon_{\mathrm{tot}}&-\gamma\end{pmatrix}.$ (27)

The solutions for the Bloch vector after each $X\overline{X}$ sequence (which
corresponds to a duration of $2t_{g}$) are then given recursively by

$\mathbf{v}(2(n+1)t_{g})=A\mathbf{v}(2nt_{g})-\bm{\zeta}$ (28)

where $\bm{\zeta}\in\mathbb{R}^{3}$, and

$\displaystyle A$ $\displaystyle=e^{G_{-}t_{g}}e^{G_{+}t_{g}}$ (29a)
$\displaystyle\bm{\zeta}$
$\displaystyle=(I-e^{G_{-}t_{g}})(G_{-})^{-1}\mathbf{c}+e^{G_{-}t_{g}}(I-e^{G_{+}t_{g}})(G_{+})^{-1}\mathbf{c}.$
(29b)

Unraveling the recursion yields

$\mathbf{v}(2nt_{g})=A^{n}\mathbf{v}(0)-\sum_{k=0}^{n-1}A^{k}\bm{\zeta}.$ (30)

We are interested in calculating only $v_{x}(2nt_{g})$:

$v_{x}(2nt_{g})=(1,0,0)\cdot A^{n}\mathbf{v}(0)-\sum_{k=0}^{n-1}(1,0,0)\cdot
A^{k}\bm{\zeta}$ (31)

The required elements of the matrix $A^{n}$ can be written as:

$\displaystyle(A^{n})_{11}$ $\displaystyle=\frac{1}{2}(p^{2n}+q^{2n})$ (32a)
$\displaystyle(A^{n})_{12}$
$\displaystyle=\frac{-i\omega^{\prime}(1+e^{i\omega^{\prime}t_{g}})}{2\chi}(p^{2n}-q^{2n})$
(32b) $\displaystyle(A^{n})_{13}$
$\displaystyle=\frac{-\varepsilon_{\mathrm{tot}}(1-e^{i\omega^{\prime}t_{g}})}{2\chi}(p^{2n}-q^{2n})$
(32c)

where

$\displaystyle\omega^{\prime}$
$\displaystyle\equiv\sqrt{\varepsilon_{\mathrm{tot}}^{2}+\Delta_{\mathrm{err}}^{2}}$
(33a) $\displaystyle\chi$
$\displaystyle\equiv\sqrt{4e^{i\omega^{\prime}t_{g}}\omega^{\prime
2}+(1-e^{i\omega^{\prime}t_{g}})^{2}\Delta_{\mathrm{err}}^{2}}$ (33b)
$\displaystyle p$ $\displaystyle\equiv\frac{e^{-\gamma
t_{g}-i\omega^{\prime}t_{g}}}{2\omega^{\prime
2}}\left((1+e^{2i\omega^{\prime}t_{g}})\Delta_{\mathrm{err}}^{2}+2e^{i\omega^{\prime}t_{g}}\varepsilon_{\mathrm{tot}}^{2}+\right.$
$\displaystyle\qquad\qquad\left.(1-e^{i\omega^{\prime}t_{g}})\Delta_{\mathrm{err}}\chi\right)$
(33c) $\displaystyle q$ $\displaystyle\equiv\frac{e^{-\gamma
t_{g}-i\omega^{\prime}t_{g}}}{2\omega^{\prime
2}}\left((1+e^{2i\omega^{\prime}t_{g}})\Delta_{\mathrm{err}}^{2}+2e^{i\omega^{\prime}t_{g}}\varepsilon_{\mathrm{tot}}^{2}-\right.$
$\displaystyle\qquad\qquad\left.(1-e^{i\omega^{\prime}t_{g}})\Delta_{\mathrm{err}}\chi\right).$
(33d)

Substituting these into Eq. 31 gives

$\displaystyle
v_{x}(2nt_{g})=\frac{1}{2}(p^{2n}+q^{2n})+\frac{1-p^{2n}}{2(1-p)}\times$ (34)
$\displaystyle\
\left(\zeta_{x}-\frac{i(1+e^{i\omega^{\prime}t_{g}})\omega^{\prime}}{\chi}\zeta_{y}-\frac{(1-e^{i\omega^{\prime}t_{g}})\varepsilon_{\mathrm{tot}}}{\chi}\zeta_{z}\right)+$
$\displaystyle\frac{1-q^{2n}}{2(1-q)}\left(\zeta_{x}+\frac{i(1+e^{i\omega^{\prime}t_{g}})\omega^{\prime}}{\chi}\zeta_{y}+\frac{(1-e^{i\omega^{\prime}t_{g}})\varepsilon_{\mathrm{tot}}}{\chi}\zeta_{z}\right),$

where $\bm{\zeta}$ carries the temperature dependence. Under first order
approximations, $\chi\approx 2e^{i\omega^{\prime}t_{g}/2}\omega^{\prime}$,
$\omega^{\prime}\approx\varepsilon_{\mathrm{tot}}$, and

$\displaystyle p$ $\displaystyle\approx e^{-\gamma t_{g}-i\vartheta}$ (35a)
$\displaystyle q$ $\displaystyle\approx e^{-\gamma t_{g}+i\vartheta}$ (35b)
$\displaystyle\vartheta$
$\displaystyle\equiv\tan^{-1}{\left(2\sin{\left(\frac{\omega^{\prime}t_{g}}{2}\right)}\frac{\Delta_{\mathrm{err}}}{\omega^{\prime}}\right)}.$
(35c)

The values of $\bm{\zeta}$’s components are lengthy expressions and are given
in Eq. 44 below so as not to interrupt the flow. Under first-order
approximations, $\zeta_{x}\approx 0$,
$(1+e^{i\omega^{\prime}t_{g}})\zeta_{y}/\chi\approx 0$ and $\zeta_{z}\approx
0$. Substituting these values gives the following final expression:

$\displaystyle v_{x}(2nt_{g})$
$\displaystyle\approx\frac{1}{2}(p^{2n}+q^{2n})$ (36a)
$\displaystyle=e^{-2n\gamma t_{g}}\cos{2n\vartheta}.$ (36b)

The expression for the fidelity with the $|{+}\rangle$ state is, therefore

$\displaystyle F(t_{n})$ $\displaystyle=\frac{1}{2}+\frac{1}{2}e^{-\gamma
t_{n}}\cos{(2n\vartheta)}.$

This expression matches Eq. 3 with $2\omega t_{g}=\vartheta$, $T_{D}=1/\gamma$
and $a=0$. Moreover, the oscillation frequency matches the closed system
fidelity oscillation frequency [Eq. 5] up to the first order (noting that
$\theta_{\mathrm{err}}=\omega^{\prime}t_{g}$).

Solving the general case where $\gamma_{1}\neq\gamma_{2}$ is difficult: we
cannot apply perturbative techniques directly to bound the deviation of the
Bloch vector trajectory in a $\\{\gamma_{1}\neq\gamma_{2}$,
$\Delta_{\mathrm{err}}\neq 0\\}$ environment (where $v_{x}$ exhibits
oscillating exponential decay) from a Bloch vector trajectory in a
$\\{\gamma_{1}\neq\gamma_{2}$, $\Delta_{\mathrm{err}}=0\\}$ environment (where
$v_{x}$ exhibits exponential decay). We rely instead on numerical simulations
of the Lindblad equation.

##  Appendix B Finite temperature

We discuss numerical, experimental, and analytical approaches for accounting
for finite-temperature effects.

###  B.1 Numerical

In our simulations, we take the zero temperature limit, which fits our
experimental data well and is typically valid for transmons. However, it is
straightforward to generalize to a non-zero temperature. For zero temperature,
we use a Lindblad operator $L_{1}=\sigma^{-}=|{0}\rangle\\!\langle 1|$ with
rate $\gamma_{1}=1/T_{1}$. For a non-zero temperature $T$, we replace this
with two operators $L_{+}=\sigma^{-}=|{0}\rangle\\!\langle 1|$ and
$L_{-}=\sigma^{+}=|{1}\rangle\\!\langle 0|$, with rates $\gamma_{-}$ and
$\gamma_{+}$, respectively. The rates obey
$\gamma_{-}+\gamma_{+}=\gamma_{1}=1/T_{1}$ and the KMS condition
$\gamma_{+}/\gamma_{-}=\mathrm{exp}(-\hbar\omega/k_{B}T)$ [62], where $\omega$
is the qubit frequency. Varying $T$ in our numerical model thus allows us to
fit dynamics at any temperature. In particular, relaxation-sensitive
sequences, such as $Y\overline{Y}$ applied to $|{+}\rangle$, saturate to a
fidelity that strongly depends on the device temperature.

###  B.2 Experimental

The effect of finite temperature can also be seen in our 4-step DB protocol.
In general, state preparation and measurement (SPAM) errors will cause the
initial state fidelity to be less than 1. We can calibrate these errors out by
defining the last point in the $T_{1}$ sequence (i.e., the thermal state) to
be a fidelity of 0, and the first point (i.e., the population-inverted thermal
state) to be a fidelity of 1. Such a definition accounts for all temperature
effects in state preparation and all measurement errors, rescaling all
fidelities by the same multiplicative factor, and thus leaving fitting and
analysis unaffected. If we then compare the saturation fidelities of the $XX$,
$Y\overline{Y}$, and $\overline{Y}Y$ sequences to the midpoint of the $T_{1}$
fidelities, we see that they differ in a way that depends on the temperature,
with all saturating to 0.5 in the limit of infinite temperature. We can thus
use DB to extract the device temperature, although using the $XX$ sequence
(with saturation close to the mixed state) will be less sensitive to small
temperature variations than using $Y\overline{Y}/\overline{Y}Y$ or typical
Rabi-based methods. Nevertheless, our original 4-step protocol captures the
effect of finite temperature. Note that this technique is insensitive to
measurement error, which can be significant and would disrupt temperature
measurements based on single-shot state-assignment readout.

###  B.3 Analytical

Given $a$ in Eq. 3, in fact, the temperature can be computed without
simulation of the Lindblad master equation. This can be shown in the following
way: suppose that the Bloch vector equation is the same as Eq. 26, but with
$G_{+}$ and $G_{-}$ replaced with $G_{1}$ and $G_{2}$, respectively. $G_{1}$
and $G_{2}$ correspond to the two pulses chosen for the DD sequence; e.g., in
an $XX$ sequence, $G_{1}=G_{2}=G_{+}$. Note that $G_{1,2}$ carry no
temperature dependence. Then, similar to the analysis in the previous section,
the Bloch vector at the end of $n$ pulses is given by

$\mathbf{v}(2nt_{g})=B^{n}\mathbf{v}(0)-\eta\sum_{k=0}^{n-1}B^{k}\bm{\zeta}^{\prime},$
(37)

where similar to Eq. 29,

$\displaystyle B$ $\displaystyle=e^{G_{2}t_{g}}e^{G_{1}t_{g}}$ (38a)
$\displaystyle\bm{\zeta}^{\prime}$
$\displaystyle=(I-e^{G_{2}t_{g}})(G_{2})^{-1}\mathbf{c}+e^{G_{2}t_{g}}(I-e^{G_{1}t_{g}})(G_{1})^{-1}\mathbf{c},$
(38b)

and $\mathbf{c}=(0,0,\gamma_{1})$. The fidelity is then

$F(n)=\frac{1}{2}\left(1+\mathbf{v}(0)\cdot
B^{n}\mathbf{v}(0)-\eta\sum_{k=0}^{n-1}\mathbf{v}(0)\cdot
B^{k}\bm{\zeta}^{\prime}\right)$ (39)

where again $\eta$ is the thermal factor defined in Eq. 9. For
$n\rightarrow\infty$, $\mathinner{\\!\left\lVert B^{n}\right\rVert}\rightarrow
0$ from Eq. 21 and

$F(\infty)=\frac{1}{2}\left(1-\eta\mathbf{v}(0)\cdot(I-B)^{-1}\bm{\zeta}^{\prime}\right),$
(40)

where we used $\sum_{k=0}^{\infty}B^{k}=(I-B)^{-1}$. Substituting
$F(\infty)=(1+a)/2$ from Eq. 3 gives

$a=\eta\mathbf{v}(0)\cdot(B-I)^{-1}\bm{\zeta}^{\prime}.$ (41)

The LHS of Eq. 41 is measured experimentally, while the inner product in the
RHS is computed using the parameters
$\\{T_{1},T_{2},\varepsilon_{\mathrm{tot}},\Delta_{\mathrm{err}}\\}$.

Note that $a$ is maximized when $\eta=1$, i.e., at zero temperature [Eq. 9].
Accordingly, substituting
$a_{T=0}=(\mathbf{v}(0)\cdot(B-I)^{-1}\bm{\zeta}^{\prime})$ into Eq. 41 and
rearranging Eq. 9 gives the temperature as

$k_{B}T=\omega_{01}\left(\ln{\left(\frac{a_{T=0}+a}{a_{T=0}-a}\right)}\right)^{-1}.$
(42)

As mentioned in the main text, for transmons $\omega_{01}\in[3-6]\,$GHz and
$T\sim 10$ mK, so $a\lesssim a_{T=0}$. However, for devices where finite
temperature effects are significant, $a<a_{T=0}$; moreover $a\approx 0$ for
$T\rightarrow\infty$.

Since the experimental error in calculating $a$ is largely unaffected by the
type of pulse sequence (i.e., by the value of $a$ itself), it is advisable to
select pulse sequences that have a higher value of $|a_{T=0}|$ to minimize the
fractional uncertainty in temperature $T$. For instance, a $\overline{Y}Y$ or
$Y\overline{Y}$ sequence which gives $\mathinner{\\!\left\lvert
a\right\rvert}=0.22\pm 0.01$ would be preferred over a $YY$ sequence which
gives $\mathinner{\\!\left\lvert a\right\rvert}=0.05\pm 0.01$ in Fig. 2.

##  Appendix C Leakage

Additional analysis of the $T_{1}$ and $YY$ sequence results may reveal
leakage errors induced by higher levels of the transmon, presenting as excess
relaxation, excitation, or depolarization per gate, depending on the readout
signal when the qubit leaks out of the computational subspace. This excess
allows for quantification of the amount of leakage. However, we do not
consider leakage in our analysis, as our use of DRAG [49] with sufficiently
slow gates makes leakage negligible. Leakage concerns complicate the
analytical discussion of temperature concerns above, but numerical analysis
should still be able to account for it once the leakage readout signal is
appropriately added to the Lindblad model.

##  Appendix D Induced rotation error

Note that in the presence of only detuning error $\Delta_{\mathrm{err}}$ in
Eq. 1, the Rabi rate increases from $\varepsilon$ to
$\sqrt{\varepsilon^{2}+\Delta_{\mathrm{err}}^{2}}$. Thus, for a gate time
$t_{g}$, there would be an additional induced rotation error
$\delta\theta_{\rm
ind}=(\sqrt{\varepsilon^{2}+\Delta_{\mathrm{err}}^{2}}-\varepsilon)t_{g}\simeq\Delta_{\mathrm{err}}^{2}t_{g}/(2\varepsilon)$
(assuming a square pulse), even without a drive amplitude error. Using
$\Delta_{\mathrm{err}}=\bar{\varepsilon}\delta\phi$ and
$\bar{\varepsilon}=\theta/t_{g}$, this gives

$\delta\theta_{\rm ind}\simeq\frac{1}{2}\delta\phi^{2}\theta.$ (43)

Looking at the data in Fig. 3 in the main text, we have at most
$\delta\phi=0.90^{\circ}$ with a nominal rotation angle of $\theta=\pi$,
giving $\delta\theta_{\rm ind}\approx 0.02^{\circ}$ which is negligible. We
therefore ignore the induced rotation error in our analysis.

##  Appendix E Device design and experimental setup

The device used in the experiment consists of a simple grounded Xmon geometry
[63] dispersively coupled to its co-planar waveguide (CPW) readout resonator.
It is nominally identical to the device reported in [64], and the design is
posted on the open-source SQuADDS database [65]. We make use of a separate
weakly-coupled microwave line to drive the qubit, which results in reasonably
fast gate times (80 ns) with an average gate fidelity of 99.74%. The device
layout was created using the Qiskit Metal open-source package [66] with the
finite-element (FE) simulations taking place in the Ansys HFSS simulator to
find the various parameters of the device, i.e., the qubit and cavity
frequencies ($\omega_{\mathrm{q}}$ and $\omega_{\mathrm{c}}$), as well as the
electric field distribution across the device of interest. The FE simulation
results are then analyzed using the pyEPR package [67] to extract the qubit
anharmonicity ($\eta$) and the qubit-cavity dispersive shift
($\chi_{\mathrm{qc}}$). Lastly, the resonator linewidth ($\kappa$) is
extracted using the HFSS driven-modal scattering simulations. Table 1 shows
the measured parameters of the device used in this paper.

$\omega_{\mathrm{q}}/2\pi$ |  $|\eta|/2\pi$ |  $\chi_{\mathrm{qc}}/2\pi$ |  $\omega_{\mathrm{c}}/2\pi$
---|---|---|---
4.37 GHz | 215 MHz | 230 kHz | 6.95 GHz

$\kappa/2\pi$ |  $T_{1}$ |  $T_{2}^{*}$
---|---|---
150 kHz | 20 $\mu\mathrm{s}$ | 32 $\mu\mathrm{s}$

Table 1: Measured parameters of the device used in the experiment.

The device is fabricated and packaged in a copper box. We then mount and
thermalize the box in the lowest stage of a dilution refrigerator (DR), which
has a base temperature of $\sim 15\mathrm{mK}$, surrounded by an additional
copper shielding as well as a Cryoperm to further isolate the chip from the
external infrared radiation and magnetic fields. The readout and drive lines
each have 50 and 60 dB of attenuation to reach the single-photon power regime
on the chip. Moreover, we use an 8 GHz low-pass filter on each line to protect
the device from high-frequency noise, as well as high-frequency Eccosorb
filters mounted close to the device in the DR to repel the infrared radiation
from the higher stages of the DR. The readout line is protected by using four
cryogenic microwave circulators, with the signal being amplified using both a
traveling wave parametric amplifier (TWPA) at the mixing chamber and a high-
electron-mobility transistor (HEMT) amplifier at the 4K stage to achieve
single-shot readout.

Figure 5: Randomized benchmarking measurement for the outdated gate
calibration parameters used in Fig. 2 of the main text. The error bars
represent the deviation of the averaged benchmarking results across 30
distinct random sequences with a maximum number of Cliffords
($N_{\mathrm{Cliffords}}$) of 700, which are smaller than the marker sizes.

##  Appendix F Conventional single-qubit gate calibration

In our experiments, the $H_{x}$ Hamiltonian [Eq. 1] is fully calibrated and
the $H_{y}$ Hamiltonian is then obtained simply by applying a phase shift via
an arbitrary waveform generator (AWG), which is equivalent to
$H_{y}=R_{z}(-\frac{\pi}{2})H_{x}R_{z}(\frac{\pi}{2})$, and leaving
$\varepsilon_{\mathrm{err}}$ and $\Delta_{\mathrm{err}}$ unchanged.

We perform the following steps to calibrate the single-qubit gates used in the
experiments (note that we use a cosine envelope pulse shape for all the
experiments conducted in this paper):

  1. 1.

Find the resonance frequencies of the cavity and the qubit using conventional
spectroscopy techniques.

  2. 2.

Look for a monotone signal sent at the frequency of the qubit (cavity) at the
output spectrum of the microwave mixers and try to minimize the leakage and
the mirror signals caused by the mixer’s electrical offset and imbalance.

  3. 3.

Perform simple Rabi experiments to find the right gate length for a specific
gate amplitude. We chose a gate length of $80$ ns and a cosine-shaped gate
envelope for the experiments reported here. Note that there is a padding of
$8$ ns between consecutive pulses, and hence the effective pulse interval is
$88$ ns.

  4. 4.

Characterize the relaxation and decoherence rates of the device and use the
Ramsey experiments to fine-tune the qubit’s frequency.

  5. 5.

Optimize the readout fidelity by using the previously calibrated $R_{x}(\pi)$
pulses and varying the readout tone’s frequency and amplitude.

  6. 6.

Apply consecutive $R_{x}(\pi)$ and $R_{x}(\pi/2)$ pulses to the system
initialized in the ground state, such that after each measurement the qubit
ideally returns to its ground state, i.e., $4N\times R_{x}(\pi/2)$ and
$2N\times R_{x}(\pi)$ pulses with $N\in[1,\dots,20]$. Using this method we can
observe the ground state fidelity of the qubit and pick an amplitude that pins
the qubit to its ground state without exhibiting any oscillations. As
described in the main text, calibrating the pulses along the $x$-axis
automatically confirms the calibration on the $y$-axis.

  7. 7.

Optimize the DRAG weighting parameter ($\alpha$) by applying multiple pairs of
$R_{x}(\pi)R_{x}(-\pi)$ pulses to the system initialized in its ground state
[68], observe the ground state fidelity, and choose the parameter values for
which the qubit remains in its ground state.

##  Appendix G Randomized benchmarking

We need to consider the gate fidelities to verify the calibration techniques
used in the previous section. To do so, we use randomized benchmarking (RB),
where we apply a certain number of random single-qubit Clifford gates to the
qubit and choose the very last pulse to be the inverse of the product of all
previous pulses; in this manner, the population returns to the ground state in
the absence of errors [see Fig. 1(a) in the main text].

Fig. 5 shows the average RB result for 30 distinct experiments with a total
length of 700 Clifford gates for the same outdated gate calibration parameters
shown in Fig. 2. On average, our RB Clifford gate set consists of
approximately $80\%$ $\pi/2$ and $20\%$ $\pi$ gates, with $\pi/2$ pulses
expected to exhibit half the rotation errors and one-fourth of the phase
errors. The RB curve can be fit to a function of the form $Ap^{m}+B$ (Fig. 5).
Even though the gate calibrations are outdated, we still achieve an average
low Clifford error rate of $r_{\mathrm{Clifford}}=(1-p)/2\simeq 0.262\pm
0.003\%$, which shows that RB alone is not sufficient to achieve accurate gate
calibration. Note that this method is limited by the relaxation time of the
qubit ($T_{1}$) as well as the gate duration ($t_{g}$) with the average
single-qubit gate fidelity scaling as $\sim t_{g}/T_{1}$ [69] and in our case,
the gate fidelity is significantly affected by the relatively short $T_{1}$ of
the device.

##  Appendix H $\zeta$’s components

The values of $\bm{\zeta}$’s components [defined in Eq. 29b], along with their
first order approximations
($\Delta_{\mathrm{err}},\gamma,\varepsilon_{\mathrm{err}}\ll\varepsilon=\pi/t_{g}$)
are noted below:

$\displaystyle\zeta_{x}$ $\displaystyle=\frac{-4\eta e^{-2\gamma
t_{g}}\Delta_{\mathrm{err}}\varepsilon_{\mathrm{tot}}}{\omega^{\prime
4}(\omega^{\prime 2}+\gamma^{2})}\left[\omega^{\prime
2}\left(\left(\gamma\sin{\frac{\omega^{\prime}t_{g}}{2}}+\frac{(1-e^{\gamma
t_{g}})}{2}\omega^{\prime}\cos{\frac{\omega^{\prime}t_{g}}{2}}\right)^{2}+\frac{\omega^{\prime
2}}{4}\left((1-e^{\gamma
t_{g}})\sin{\frac{\omega^{\prime}t_{g}}{2}}\right)^{2}\right)\right.$ (44a)
$\displaystyle\qquad-\left.\Delta_{\mathrm{err}}^{2}(1-\cos{(\omega^{\prime}t_{g})})\left(\left(\gamma\sin{\frac{\omega^{\prime}t_{g}}{2}}+\frac{1}{2}\omega^{\prime}\cos{\frac{\omega^{\prime}t_{g}}{2}}\right)^{2}+\frac{\omega^{\prime
2}}{4}\left(2(1-e^{\gamma
t_{g}})-\cos^{2}{\frac{\omega^{\prime}t_{g}}{2}}\right)\right)\right]$
$\displaystyle\approx 0$ $\displaystyle\zeta_{y}$ $\displaystyle=\frac{\eta
e^{-2\gamma t_{g}}\omega^{\prime}\varepsilon_{\mathrm{tot}}}{\omega^{\prime
4}(\omega^{\prime
2}+\gamma^{2})}\left[-4\Delta_{\mathrm{err}}^{2}\sin{(\omega^{\prime}t_{g})}\left(\left(\gamma\sin{\frac{\omega^{\prime}t_{g}}{2}}+\frac{1}{2}\omega^{\prime}\cos{\frac{\omega^{\prime}t_{g}}{2}}\right)^{2}+\frac{\omega^{\prime
2}}{4}\left(2(1-e^{\gamma
t_{g}})-\cos^{2}{\frac{\omega^{\prime}t_{g}}{2}}\right)\right)\right.$ (44b)
$\displaystyle\qquad+\left.\gamma\omega^{\prime 3}\left(\left(e^{\gamma
t_{g}}-\cos{(\omega^{\prime}t_{g})}\right)^{2}+\sin^{2}{(\omega^{\prime}t_{g})}\right)\right]$
$\displaystyle\approx(1+e^{-\gamma t_{g}})^{2}\frac{\gamma}{\varepsilon}$
$\displaystyle\zeta_{z}$ $\displaystyle=\frac{\eta e^{-2\gamma
t_{g}}}{\omega^{\prime 4}(\omega^{\prime
2}+\gamma^{2})}\left[8\Delta_{\mathrm{err}}^{2}\varepsilon_{\mathrm{tot}}^{2}\sin^{2}{\frac{\omega^{\prime}t_{g}}{2}}\left(\left(\gamma\sin{\frac{\omega^{\prime}t_{g}}{2}}+\frac{1}{2}\omega^{\prime}\cos{\frac{\omega^{\prime}t_{g}}{2}}\right)^{2}-\frac{\omega^{\prime
2}}{4}\cos^{2}{\frac{\omega^{\prime}t_{g}}{2}}\right)\right.$ (44c)
$\displaystyle\qquad+\left.e^{\gamma t_{g}}\gamma\omega^{\prime
3}\left(e^{\gamma
t_{g}}\gamma\omega^{\prime}+2\varepsilon_{\mathrm{tot}}^{2}\sin{(\omega^{\prime}t_{g})}\right)+\Delta_{\mathrm{err}}^{2}\omega^{\prime
2}(e^{\gamma t_{g}}-1)\left(e^{\gamma t_{g}}\omega^{\prime
2}+(2\cos{(\omega^{\prime}t_{g})}-1)\varepsilon_{\mathrm{tot}}^{2}+\Delta_{\mathrm{err}}^{2}\right)\right]$
$\displaystyle\approx 0$


# Machine learning based dimension reduction for a stable modeling of periodic
flow phenomena

Hiroshi Omichi
Department of Mechanical Engineering, Keio University;
Department of Mechanical and Aerospace Engineering, University of California,
Los Angeles
<EMAIL_ADDRESS>
&Takeru Ishize
Department of Mechanical Engineering, Keio University
<EMAIL_ADDRESS>
&Koji Fukagata
Department of Mechanical Engineering, Keio University
<EMAIL_ADDRESS>

###### Abstract

In designing efficient feedback control laws for fluid flow, the modern
control theory can serve as a powerful tool if the model can be represented by
a linear ordinary differential equation (ODE). However, it is generally
difficult to find such a linear model for strongly nonlinear and high-
dimensional fluid flow phenomena. In this study, we propose an autoencoder
which maps the periodic flow phenomena into a latent dynamics governed by a
linear ODE, referred to as a pseudo-symplectic Linear system Extracting
AutoEncoder (LEAE). In addition to the normal functionality of autoencoder,
pseudo-symplectic LEAE emulates a symplectic time integration scheme so that
the Hamiltonian (i.e., the pseudo-energy) of the latent variables is
conserved. We demonstrate that the stability of the derived ODE is improved by
considering the integration stepping forward and backward at the same time.
Here, we consider the circular cylinder wake at $Re_{D}=100$ as a typical
periodic flow phenomenon.

_Keywords_ Autoencoder $\cdot$ Linear system extraction $\cdot$ Reduced order
modeling $\cdot$ Pseudo-symplectic

##  1 Introduction

Particularly when a target state of flow control locates far away from its
base state, active flow control (Park et al.,, 1994; Naito and Fukagata,,
2014; Jin et al.,, 2020) is one of the promising techniques in terms of its
efficacy and degrees of freedom in its design. Among the active control
strategies, model-free approaches (Rabault et al.,, 2019; Wang et al.,, 2023)
have strong merits in their capability to follow the shifted dynamics and its
transient process, although it is difficult to interpret or generalize the
constructed controllers (Brunton and Noack,, 2015). Model-based approaches
have strong advantages also in that the existing control theories can be
applied; when the model can be represented by a linear ODE, there is a wide
range of options like linear quadratic regulator even though they are only
valid as long as the controlled states can be linearly approximated around the
model. However, as can easily be imagined, it is difficult to find a proper
model which governs fluid flow phenomena due to their high-dimensionality and
strong nonlinearity. In this context, dimension reduction with machine
learning and modeling for the temporal evolution of low-dimensionalized
variables (i.e.,latent variables) have been investigated (Milano and
Koumoutsakos,, 2002; Brenner et al.,, 2019; Hasegawa et al.,, 2020; Brunton et
al., 2020a, ; Farazmand and Saibaba,, 2023); however, with a naïve dimension
reduction, the extracted latent dynamics will remain highly nonlinear (Fukami
et al.,, 2021).

To address this problem, we propose an enhanced autoencoder named pseudo-
symplectic Linear system Extracting AutoEncoder (LEAE) to compress flow field
data into a latent space such that the latent dynamics is governed by a linear
ODE. Inside the pseudo-symplectic LEAE, the dimension reduction is performed
with an autoencoder, assuming the time integration of latent variables using
the Crank-Nicolson scheme. In particular, we propose a method accounting for
the symplectic property of the latent system by training the temporal
evolution in forward and backward directions simultaneously. By so doing, the
Hamiltonian (i.e., the pseudo-energy) of the latent variables
$H=\frac{1}{2}\sum_{i}\varphi_{i}^{2}$ is conserved, where $\varphi_{i}$
denotes $i$-th component of the latent vector $\varphi$.

In this study, we consider the iconic periodic flow phenomena in this field,
i.e, the periodic circular cylinder wake at the Reynolds number $Re_{D}=100$.

##  2 Methods

###  2.1 Dataset

Figure 1: Computational configuration of DNS.

To prepare the training data, let us consider the two-dimensional flow around
a circular cylinder at the Reynolds number based on its diameter ${\rm
Re}_{D}=100$. We use direct numerical simulation (DNS) (Kor et al.,, 2017),
and the governing equations are the continuity and Navier-Stokes equations for
incompressible flows,

$\displaystyle{\mbox{\boldmath$\nabla$}}\cdot{\mbox{\boldmath$u$}}$
$\displaystyle=0,$ (1)
$\displaystyle\frac{\partial{\mbox{\boldmath$u$}}}{\partial
t}+{\mbox{\boldmath$\nabla$}}\cdot({\mbox{\boldmath$uu$}})$
$\displaystyle=-{\mbox{\boldmath$\nabla$}}p+\frac{1}{{\rm{Re}}_{D}}\nabla^{2}{\mbox{\boldmath$u$}},$
(2)

where ${{\mbox{\boldmath$u$}}}=\\{u,v\\}$ and $p$ are the velocity vector and
pressure, respectively. Both quantities are nondimensionalized by the fluid
density $\rho^{*}$, the diameter of cylinder $D^{*}$, and the uniform velocity
$U_{\infty}^{*}$, where $(\cdot)^{*}$ denotes dimensional quantities.

The left side of Figure 1 indicates the computational domain with the
streamwise $(x)$ and transverse $(y)$ lengths of $25.6D$ and $20.0D$,
respectively. A uniform inflow velocity of $U_{\infty}=1$, free-slip
conditions on the upper and lower boundaries, and a convective boundary
condition on the outflow boundary are applied, while the ghost-cell method
(Kor et al.,, 2017) is utilized to impose the no-slip boundary condition on
the surface. A uniform Cartesian grid system is used, and the number of grid
points in each direction is $\left(N_{x},N_{y}\right)=\left(1024,800\right)$,
and the center of the cylinder locates at $x=9D$ downstream from the inflow
boundary in $x$ direction and at the center in $y$ direction. The time step is
$\Delta t_{\rm DNS}=2.5\times 10^{-3}$.

For training of a machine learning model, the velocity fields
$\left(u,v\right)$ inside the confined region depicted with the blue line in
Figure 1 is used to extract the key feature of the cylinder wake. The size of
a snapshot is $\left(384\times 192\times 2\right)$, and we use the developed
wake shedding of $2000$ snapshots with a time step of $\Delta t_{\rm
Data}=2.5\times 10^{-2}$ as the dataset for the machine learning model.

###  2.2 Pesudo-symplectic linear system extraction autoncoder

####  2.2.1 Autoencoder

Figure 2: Schematic of a typical autoencoder.

The machine learning model of this study is based on the consept of
autoencoder (Hinton and Salakhutdinov,, 2006) to reduce the dimension of data.
In what follows, general description of autoencoders is detailed. As
illustrated in Figure 2, an autoencoder $\cal{F}$ is comprised of an encoder
$\cal{F}_{\rm enc}$ and a decoder $\cal{F}_{\rm dec}$ with a bottleneck in the
middle. As long as the autoencoder precisely replicates the input data
${{\mbox{\boldmath$q$}}}\in{\mathbb{R}}^{\mathit{n}_{\rm H}}$ and output it as
$\hat{{\mbox{\boldmath$q$}}}\in{\mathbb{R}}^{\mathit{n}_{\rm H}}$, it can be
said that the encoder maps the high dimensional data into its low dimensional
representation ${\mbox{\boldmath$\varphi$}}\in\mathbb{R}^{\mathit{n}_{\rm
L}}$, i.e., latent variables (vector), and the decoder has the inverse
function. Note that $n_{\rm H}$ and $n_{\rm L}$ denote the dimension of the
original and the low-dimensionalized data (usually, $n_{\rm H}\gg n_{\rm L}$),
and the latent variables $\varphi$ probably hold essential characteristics of
the original dynamics of $q$. This conversion can be formulated as

$\displaystyle{{\mbox{\boldmath$q$}}}\approx\hat{{\mbox{\boldmath$q$}}}=\cal{F}({\mbox{\boldmath$q$}};{\mbox{\boldmath$w$}})=\cal{F}_{\rm
dec}({\mbox{\boldmath$\varphi$}})=\cal{F}_{\rm dec}(\cal{F}_{\rm
enc}({\mbox{\boldmath${\mbox{\boldmath$q$}}$}})),$ (3)

where $w$ denotes the weights inside the machine learning model to be
optimized so that

$\displaystyle{{\mbox{\boldmath$w$}}}={\rm
argmin}_{{{\mbox{\boldmath$w$}}}}{\parallel{\mbox{\boldmath$q$}}-\hat{{\mbox{\boldmath$q$}}}\parallel}_{2}.$
(4)

In terms of the network architecture, we follow Hasegawa et al., (2020) and
employ nonlinear neural networks, i.e., Convolutional Neural Network (LeCun et
al.,, 1998) (CNN) and Multi-layer perception (Rumelhart et al.,, 1986).

####  2.2.2 Linear ODE layer

After the dimension reduction, the temporal evolution of the latent variables
have to be modeled to construct a reduced order model. In this study, the
temporal evolution is predicted with what we call the linear ODE (LODE) layer,
which follows a temporal discretization of linear ODEs using the Crank-
Nicolson scheme. Considering a linear ODE,

$\displaystyle\dot{{\mbox{\boldmath$\varphi$}}}={{\mbox{\boldmath$A$}}}{{\mbox{\boldmath$\varphi$}}},$
(5)

where ${\mbox{\boldmath$\varphi$}}\in\mathbb{R}^{\mathit{n}_{\rm L}}$
corresponds to the latent vector in section 2.2.1, and $A$ is the coefficient
matrix to be optimized through the training process, equation (5) can be
discretized with Crank-Nicolson scheme as

$\displaystyle\frac{{\mbox{\boldmath$\varphi$}}(t+\Delta
t)-{\mbox{\boldmath$\varphi$}}(t)}{\Delta
t}=\frac{A[{\mbox{\boldmath$\varphi$}}(t+\Delta
t)+{\mbox{\boldmath$\varphi$}}(t)]}{2}$ (6)
$\displaystyle\Rightarrow{\mbox{\boldmath$\varphi$}}(t+\Delta t)=(2I+\Delta
tA)(2I-\Delta tA)^{-1}{\mbox{\boldmath$\varphi$}}(t).$ (7)

Regarding ${\mbox{\boldmath$\varphi$}}(t)$ and
${\mbox{\boldmath$\varphi$}}(t+\Delta t)$ as input and output, respectively,
we can use a single MLP layer corresponding to $n_{\rm L}\times n_{\rm L}$
matrix $A$.

####  2.2.3 Overview of pseudo-symplectic linear system extraction autoencoder

Figure 3: Schematic of pseudo-symplectic linear system extraction autoencoder.

Here, the overview of our machine learning model, pseudo-symplectic linear
system extraction autoencoder (LEAE), is summarized in Figure 3. The pseudo-
symplectic LEAE basically consists of the encoder, the decoder, and the LODE
layer in the middle. Both of the encoders completely share the weights, and
both of the decoders do as well, which means that only a single encoder and
decoder are trained here. Needless to say, the encoder and the decoder are
trained to duplicate the original snapshot as precisely as possible, but what
we want to highlight here is the usage of snapshots mainly to train LODE
layer. LODE layer is trained to not only predict
${{\mbox{\boldmath$\varphi$}}}(t+\Delta t)$ from
${{\mbox{\boldmath$\varphi$}}}(t)$ (the orange arrows) but also
${{\mbox{\boldmath$\varphi$}}}(t)$ from
${{\mbox{\boldmath$\varphi$}}}(t+\Delta t)$ (the purple arrows). By doing so,
LODE layer can be trained so as to satisfy the time reversal symmetry, which
is the important property of symplectic integrators that must be satisfied in
the time integration of Hamiltonian system (Aceto and Trigiante,, 1999). In
sum, the present training process consists of four types of operations:

$\displaystyle\begin{cases}\hat{{\mbox{\boldmath$q$}}}(t)&=\mathcal{F}_{\rm
dec}(\mathcal{F}_{\rm enc}({{\mbox{\boldmath$q$}}}(t)))=\mathcal{F}_{\rm
dec}({{\mbox{\boldmath$\varphi$}}}(t))\\\ \hat{{\mbox{\boldmath$q$}}}(t+\Delta
t)&=\mathcal{F}_{\rm dec}(\mathcal{F}_{\rm
enc}({{\mbox{\boldmath$q$}}}(t+\Delta t)))=\mathcal{F}_{\rm
dec}({{\mbox{\boldmath$\varphi$}}}(t+\Delta t))\\\
\tilde{{\mbox{\boldmath$q$}}}(t+\Delta t)&=\mathcal{F}_{\rm
dec}(\mathcal{F}_{\rm ODE}({{\mbox{\boldmath$\varphi$}}}(t),\Delta t))\\\
\tilde{{\mbox{\boldmath$q$}}}(t)&=\mathcal{F}_{\rm dec}(\mathcal{F}_{\rm
ODE}({{\mbox{\boldmath$\varphi$}}}(t+\Delta t),-\Delta t))\\\ \end{cases},$
(8)

where $\Delta t$ is a time step. While the former two equations contribute to
guaranteeing that $\cal{F}_{\rm enc}$ and $\cal{F}_{\rm dec}$ are providing
the nonlinear mappings at exactly the same time indices, the latter two commit
to learning the temporal evolution stepping forward and backward, and all
operations are taken simultaneously. Therefore, the loss function is defined
as

$\displaystyle{\cal
L}={\parallel{\mbox{\boldmath$q$}}(t)-\hat{{\mbox{\boldmath$q$}}}(t)\parallel}_{2}+{\parallel{\mbox{\boldmath$q$}}(t+\Delta
t)-\hat{{\mbox{\boldmath$q$}}}(t+\Delta
t)\parallel}_{2}+{\parallel{\mbox{\boldmath$q$}}(t+\Delta
t)-\tilde{{\mbox{\boldmath$q$}}}(t+\Delta
t)\parallel}_{2}+{\parallel{\mbox{\boldmath$q$}}(t)-\tilde{{\mbox{\boldmath$q$}}}(t)\parallel}_{2}.$
(9)

##  3 Results

Figure 4: The performances of the naive LEAE (blue plots) and the pseudo-
symplectic LEAE (red plots). $(a)$, $(b)$ reproduced latent trajectories,
$(c)$ $L_{2}$ error of reconstructed fields, and $(d)$ $L_{2}$ error of
decoded fields from the reproduced latent trajectories.

In this section, the efficacy of the pseudo-symplectic manner is summarized by
comparing the pseudo-symplectic LEAE and a naïve LEAE. Here, the naïve LEAE is
a form of LEAE which excludes the temporally inverse procedure denoted as the
purple arrows in figure 3.

Figure 4 visualizes the performances of the naïve LEAE (blue plots) and the
pseudo-symplectic LEAE (red plots). In figure 4 $(a)$ and $(b)$, the latent
variables encoded from flow fields (gray lines) and ones predicted with
numerical integration of the derived ODEs (colored lines) are compared. While
the predicted trajectories blow up in the case of the naïve LEAE, the encoded
flow fields and the result of numerical integration show a nice agreement in
the case of the pseudo-symplectic LEAE. Note that the initial values for the
cases of the numerical integration are given through encoding the initial flow
fields. In figure 4 $(c)$ and $(d)$, both types of LEAEs are evaluated in
terms of $L_{2}$ error of decoded fields at each snapshot, and means and
standard deviations taken over three-fold cross validation are presented.
Shown in figure 4 $(c)$ is the errors of the reconstructed flow fields through
the autoencoder. Please note that no temporal evolution is considered here;
the reconstruction procedure corresponds to the horizontal flow in figure 3,
and the error is exactly the same as the first term of the equation (9). The
errors including those along the temporal evolution are presented in figure 4
$(d)$. Here, decoded fields from the predicted trajectories in figure 4 $(a)$
and $(b)$ are assessed. The above two evaluations demonstrate that the pseudo-
symplectic LEAE is a substantially improved method compared to the naïve one.

##  4 Conclusions

In this study, we proposed a pseudo-symplectic Linear system Extraction
AutoEncoder (LEAE) to exract a low-dimensionalized dynamics which is governed
by a linear ODE. It has been shown that the pseudo-symplectic LEAE has the
ability to derive an ODE which models a periodic latent dynamics more
precisely than a naïve LEAE. However, in the present study, we only considered
the case where the ODE strongly adheres to a limit-cycle corresponding to the
periodic wake shedding, and this modeling is imaginably not applicable for
cases where flow states are shifting. Toward an effective flow control
framework, there is room for further research into a modeling capable of
following transient processes due to control inputs.

## Acknowledgments

This work was supported by JSPS KAKENHI Grant Number 21H05007. The authors
acknowledge Mr. Shoei Kanehira (Keio University) for fruitful discussion and
comments.

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