# Digital control of a superconducting qubit using a Josephson pulse generator
at 3 K

L. Howe<EMAIL_ADDRESS><EMAIL_ADDRESS>National Institute of
Standards and Technology, Boulder CO 80305, USA  M. Castellanos-Beltran
<EMAIL_ADDRESS>National Institute of Standards and
Technology, Boulder CO 80305, USA  A. J. Sirois  National Institute of
Standards and Technology, Boulder CO 80305, USA  D. Olaya  National Institute
of Standards and Technology, Boulder CO 80305, USA  University of Colorado,
Boulder CO 80309, USA  J. Biesecker  National Institute of Standards and
Technology, Boulder CO 80305, USA  P. D. Dresselhaus  National Institute of
Standards and Technology, Boulder CO 80305, USA  S. P. Benz  National
Institute of Standards and Technology, Boulder CO 80305, USA  P. F. Hopkins
National Institute of Standards and Technology, Boulder CO 80305, USA

###### Abstract

Scaling of quantum computers to fault-tolerant levels relies critically on the
integration of energy-efficient, stable, and reproducible qubit control and
readout electronics. In comparison to traditional semiconductor control
electronics (TSCE) located at room temperature, the signals generated by
Josephson junction (JJ) based rf sources benefit from small device sizes, low
power dissipation, intrinsic calibration, superior reproducibility, and
insensitivity to ambient fluctuations. Previous experiments to co-locate
qubits and JJ-based control electronics resulted in quasiparticle poisoning of
the qubit; degrading the qubit’s coherence and lifetime. In this paper, we
digitally control a 0.01 K transmon qubit with pulses from a Josephson pulse
generator (JPG) located at the 3 K stage of a dilution refrigerator. We
directly compare the qubit lifetime $T_{1}$, coherence time $T_{2}^{*}$, and
thermal occupation $P_{th}$ when the qubit is controlled by the JPG circuit
versus the TSCE setup. We find agreement to within the daily fluctuations on
$\pm 0.5~{}\mu$s and $\pm 2~{}\mu$s for $T_{1}$ and $T_{2}^{*}$, respectively,
and agreement to within the 1% error for $P_{th}$. Additionally, we perform
randomized benchmarking to measure an average JPG gate error of $2.1\times
10^{-2}$. In combination with a small device size ($<25$ mm2) and low on-chip
power dissipation ($\ll 100~{}\mu$W), these results are an important step
towards demonstrating the viability of using JJ-based control electronics
located at temperature stages higher than the mixing chamber stage in highly-
scaled superconducting quantum information systems.

Quantum computing, scalable, cryogenic control, Josephson junction (JJ),
Single Flux Quantum (SFQ), transmon, superconductor, voltage metrology

††preprint: APS/123-QED

##  I Introduction

Error-corrected quantum computers are projected to require large numbers,
$\mathcal{O}(10^{6})$, of qubits [1, 2, 3, 4]; placing stringent requirements
on the per-qubit hardware overhead. Superconducting quantum circuits are a
leading technology for scaling existing systems into the noisy intermediate-
scale quantum (NISQ) era of $\gtrsim 1000$ qubits. In present systems, qubit
gates and entangling operations are performed using shaped microwave pulses
synthesized using instrumentation at room temperature [5] – here referred to
as traditional semiconductor control electronics (TSCE). Signals are routed
into a dilution refrigerator (DR) to the $\sim 0.01$ K qubits and typically
attenuated by 40–60 dB to suppress thermal noise on drive lines [6].

Limitations in cryogenic cooling power, TSCE power instability [7], and system
complexity mandate a shift to miniaturize and enhance the stability/precision
of waveform generation in superconducting quantum information systems.
Recently, the pulse-shaping DAC/mixers used to generate qubit control signals
have been successfully integrated at 3 K [8, 9] and 100 mK [10] using
cryogenic CMOS (cryoCMOS) technology. While an impressive step towards
miniaturization and large-scale integration, a significant gap exists between
these devices and scalable qubit control. Specifically, gate fidelity; power
dissipation; and the accuracy, stability, and repeatability of the signals
need improvement [11, 7, 12].

The scalability constraints of physical size and power consumption per channel
may be satisfied by superconducting Josephson junction (JJ) signal generator
circuits or aforementioned cryoCMOS controllers. Comparable to cryoCMOS
devices, JJ circuits have small device sizes ($<1\times 1$ cm2) and very low
on-chip power dissipation ($\ll 100~{}\mu$W); while also leveraging the
intrinsically-calibrated nature of single flux quantum (SFQ) pulses. This
feature provides avenues for improving waveform quality and repeatability
beyond what is achievable using semiconductor-based generators. Capitalizing
on pulse area quantization enables use of JJ arrays to construct exceptionally
stable and repeatable voltage sources from dc to a few gigahertz [13, 14, 15,
16]. Similar devices are used to realize intrinsically accurate voltages for
the international system of units, and are disseminated worldwide as primary
dc and ac voltage standards [17]. Furthermore, the use of SFQ pulses has been
proposed as a scalable paradigm for digitally controlling qubits [18, 19, 20],
and was recently demonstrated with a SFQ driver and qubit circuit co-
fabricated on the same chip [21].

Figure 1: (Color) Digital qubit control with a 3 K JPG constructed using JJs
with amorphous silicon ($\alpha$Si) barriers and palladium-gold (PdAu) shunt
resistors. (a) Circuit schematic of the JPG containing $N_{JJ}=4650$ series JJ
cells and their shunt resistors $R_{n}$. Low-frequency inductive taps are used
to provide a dc bias current $I_{b}$ and permit measurement of $I$-$V$ curves.
A dc block ($C_{in}$) is used on the drive input. (b) Rendering of the input
portion of the JPG circuit layout showing approximately 250 JJs. Light blue
(red) is the base (top) niobium metal layer, green is the $\alpha$Si JJ
barrier, and yellow the PdAu shunts, $R_{n}$. The bowtie structure (left)
forms $C_{in}$ and the square washer inductor network (top) forms the low-
frequency current and voltage taps. The dashed region shows a magnified view
to display the first 20 JJs and shunt resistors in the array. (c) Photo of the
packaged JPG chip prior to installation in the DR. The drive input (JPG
output) is the left (right) coplanar waveguide microwave launch. (d) Cartoon
qubit state evolution on the Bloch sphere under an artificially large coupling
($\delta\theta=\pi/10$) with $\omega_{d}=\omega_{10}$. (e) Schematic example
of sinusoidal JPG drive and corresponding output. Under proper conditions, one
JPG pulse is output per drive period, which may be done at the qubit frequency
($\omega_{d}=\omega_{10}$) or at any subharmonic $\omega_{10}/m$. The
orthogonal $\hat{y}$ control axis is realized by phase-shifting, by
$\phi_{d}^{Y}$, the drive tone with respect to the timing reference
established via the $\hat{x}$ drive. The JPG emits no pulses when the drive is
inactive.

The primary limitation of SFQ operation proximal to quantum arrays is
degradation of qubit lifetimes from quasiparticles created during pulse
generation [22, 23]. A solution that mitigates these quasiparticles must be
implemented – such as physically separating the SFQ elements and qubits. In
this work we locate the JJ control circuitry on the 3 K DR stage to interrupt
quasiparticle-qubit propagation. Similar to [21], we deliver sparse trains of
pulses subresonantly to enact control – giving our device its name of the
Josephson pulse generator (JPG). A 0.01 K bump-bonded multi-chip configuration
[24, 25] and/or introduction of normal-metal quasiparticle traps can also be
effective for quasiparticle mitigation [26, 27].

Location of the cryogenic control electronics at a higher temperature stage
liberates physical volume at 0.01 K – commonly monopolized by the quantum
array and readout hardware – and leverages higher cooling powers. This
approach may also benefit from integration with cryoCMOS circuits by
exploiting the advantages of cryoCMOS-implemented logic/memory elements [9, 8,
10]. Location of the control electronics at 3 K does increase wiring
complexity and the parasitic heat loads to the $<3$ K stages; however,
solutions are under development which demonstrate low thermal loading and
crosstalk [28, 29, 30].

While the aforementioned merits of JJ-based sources [17] are expected to apply
for qubit control, this work is the first validation of using JJ-based pulse
generation at 3 K to control a 0.01 K qubit. Here we show that the JPG does
not adversely affect the qubit by separately measuring the qubit energy
relaxation time $T_{1}$, coherence time $T_{2}^{*}$, and thermal occupancy
$P_{th}$ with both a TSCE setup and the JPG. Our findings show good agreement
in all three metrics with each control setup. Additionally, we measure the JPG
gate fidelity to be within an order of magnitude of the qubit coherence limit,
and provide discussion of future devices expected to yield coherence-limited
gates.

##  II JPG-based Qubit Control

An input current evolving the JJ superconducting phase difference by $2\pi$
generates a voltage pulse whose time-integrated area equals the magnetic flux
quantum $\Phi_{0}\equiv h/2e$:

$\int Vdt=\Phi_{0}.$ (1)

The duration of this SFQ pulse is approximately $\tau=\Phi_{0}/I_{c}R_{n}$,
where $\tau$ is the JJ characteristic time, $I_{c}$ is its critical current,
and $R_{n}$ is its normal resistance [31]. SFQ signal amplification can be
achieved by connecting a series of $N_{JJ}$ junctions whose pulses add
coherently. This yields a larger pulse of area $N_{JJ}\Phi_{0}$, which we call
a JPG pulse. Depending on qubit coupling to the control line, arrays with
$N_{JJ}\sim 10^{2}-10^{4}$ are required if located at 3 K. In this work our
JPG has $N_{JJ}=4650$, $I_{c}=3.05$ mA, and $R_{n}=6.93$ m$\Omega$ – resulting
in a characteristic frequency of $f_{c}=1/\tau=10.2$ GHz. Fig. 1(a)–(c) show a
schematic, portion of the JPG layout, and image of the packaged device.

If $f_{c}$ is much larger than the qubit transition frequency,
$\omega_{10}/2\pi$, then during pulse arrival the qubit undergoes a discrete
rotation

$\delta\theta=N_{JJ}AC_{c}\Phi_{0}\sqrt{\frac{2\omega_{10}}{\hbar C_{T}}},$
(2)

where $A$ is the JPG-qubit amplitude attenuation, $C_{c}$ is the control line-
qubit coupling capacitance, and $C_{T}$ is the qubit capacitance [18].
$N_{JJ}$, $A$, and $C_{c}$ may be treated as free design parameters to realize
a combination of adequate control line thermalization and tip-angle per pulse,
$\delta\theta$. For 3D readout cavity configurations $C_{c}$ also encapsulates
attenuation from pulse transit of the cavity resonance. A train of sharp
pulses arriving resonantly to the qubit ($\omega_{d}=\omega_{10}$), or at a
subharmonic ($\omega_{d}=\omega_{10}/k$, where $k\geq 2$ is an integer),
discretely rotate the qubit around the Bloch sphere during pulse arrival,
while between pulses the qubit precesses for $k$ periods at fixed $\theta$
(see Fig. 1(d)).

In our implementation the JPG must be driven using a sinusoidal signal at
$k\geq 2$ because there is no isolation between the drive input and device
output. Otherwise, the large drive signal dominates and induces spurious qubit
rotations. Generation of an integer number of JPG pulses $\ell$ is performed
by sending an integer number of sinusoidal drive periods, $\nu$. Under the
correct bias parameters there is a one-to-one correspondence between the
number of JPG pulses generated and the number of drive periods ($\nu=\ell$).
Orthogonal axis control, realized by phasing the drive signal relative to a
timing reference, is depicted in Fig. 1(e). More details are found in
Supplementary Material XI.

Figure 2: (Color) Simplified schematic of the experimental setup. JPG pulses
are routed to the qubit through the weakly coupled cavity port. The
conventional TSCE qubit control/readout synthesizers, and cold readout
components (JPA, isolators, filters, etc.) are attached to the strongly
coupled port. JPG pulse generation is driven by a commercial 65 GSa/s
arbitrary waveform generator, which is also semiconductor-based, located at
ambient temperature. Control line thermalization is achieved with attenuators
or lossy microwave components at the {3 K, 1 K, 0.05 K, 0.01 K} DR temperature
stages. The attenuation stacks (in dB units) are {20, 10, 10, 20} and {9, 3,
6, 10} for the TSCE and JPG control lines respectively. See Fig. S1 for more
details.

##  III Experimental Details

In this work we use a transmon qubit dispersively coupled to a 3D aluminum
readout cavity possessing two control lines with different coupling strengths.
A simplified experimental schematic is shown in Fig 2. The JPG is connected to
the cavity’s weakly-coupled port (0.175 MHz coupling rate) and the TSCE
control and readout line to its strongly-coupled port (2.01 MHz coupling
rate). With this setup, a direct comparison of qubit performance with both
control schemes is possible during the same cooldown. This qubit was measured
for a previous publication; for other parameters see [32].

Qubit state readout is performed by probing the qubit-state-dependent
frequency shift of the cavity. The dressed cavity frequency is
$\omega_{\ket{0},\ket{1}}=\omega_{r}\pm\chi$, where $\omega_{r}$ is the bare
frequency and $\chi$ is the shift due to cavity-qubit coupling [33]. A
Josephson parametric amplifier [34] is operated with a gain of 20 dB (phase-
insensitive) to enable single-shot measurements. To minimize measurement-
induced transitions the cavity probe tone amplitude is typically $n_{r}\approx
6\lesssim n_{crit}/20$, where $n_{r}$ and $n_{crit}$ are the readout and
critical photon numbers [35]. We perform passive qubit state reset via
relaxation over a period $\geq 15T_{1}$. The same readout procedure and
instrumentation is used for both TSCE and JPG measurements.

Figure 3: (Color) JPG $X_{\pi}$ calibration procedure. (a) JPG $I$-$V$ curves
– shown here with $\omega_{d}/2\pi=2.679$ GHz – are first used to establish
rough bounds on the range of $I_{b}$ giving a constant Rabi oscillation period
with respect to the number of drive periods $\nu$. The inset shows behavior on
smaller scales near the calculated Shapiro voltage (green dotted line, given
by Eq. 3). Red dashed lines correspond to the Shapiro voltage if there were
one missing or one additional JJ. From this dc measurement we extract a
locking range of 1.6–2.25 mA, which is indicated in all plots with blue dashed
lines. (b) Next, we perform a JPG Rabi oscillation scan in which both the JPG
current bias $I_{b}$ and the number of drive periods, $\nu$, are swept. (c)
Single $I_{b}=1.9$ mA JPG Rabi oscillation measurement. A fit to the
functional form $\sin(2\pi\nu/2\nu_{\pi})$ of the oscillation period yields
the corresponding value for $\nu_{\pi}$ of 351.7 for $I_{b}=1.9$ mA – which is
rounded to 352 as we can only send an integer number of pulses. For these data
the JPG is driven at 2.685 GHz, resulting in an $X_{\pi}$ gate time of
$t_{gate}=131$ ns. (d) Fitted $\nu_{\pi}$ at each $I_{b}$ from data in (b)
emphasizing that at low $I_{b}$ the JPG does not pulse efficiently (or at all
for $I_{b}=0$). (e) Analyzed data of a finer resolution Rabi scan around the
locking range as determined in (a). The inset again shows smaller scale
features with red dashed lines bounding $\nu_{\pi}$ by $\pm 1$ pulse. Here the
Rabi oscillation period of the qubit is insensitive to variations in $I_{b}$
over the range of 1.82–1.97 mA and $\nu_{\pi}=352$ is fixed.

###  III.1 JPG Operation and $X_{\pi}$ Calibration

After characterization with the TSCE setup, we establish operating parameters,
specifically the rf drive power and dc current bias $I_{b}$, for the JPG in
which the number of output JPG pulses is equal to the number of input drive
periods – called the locking range. Under sinusoidal rf drive at frequency
$f_{d}$, a constant voltage Shapiro step manifests at

$V=N_{JJ}\Phi_{0}f_{d}$ (3)

When the measured voltage is constant and equal to Eq. 3 over a range of
$I_{b}$, then for any $I_{b}$ on the Shapiro step the device is locked. Thus,
we first maximize the locking range by determining the drive power giving the
largest Shapiro steps. Fig. 3(a) shows the JPG $I$-$V$ curve with maximized
locking range.

For all measurements in this work we drive at subharmonic $k=2$
($\omega_{d}=\omega_{10}/2$) and use the second harmonic power of the JPG
pulse train to control the qubit. As we are restricted to subharmonic drive,
$k=2$ maximizes the locking range by making $f_{d}$ as close as possible to
$f_{c}$ [36] and provides the highest fidelity (fastest) gates. Next, we
measure JPG-induced Rabi oscillations to characterize the JPG-qubit
interaction. At the optimal drive power we measure the number of drive periods
$\nu$ required for a $\pi$ rotation, $\nu_{\pi}$, versus $I_{b}$. Results of
this procedure are shown in Fig. 3(b)–(e). Fitting these Rabi oscillations at
constant $I_{b}$ yields $\nu_{\pi}(I_{b})$ and we look for regions where
$\nu_{\pi}$ is insensitive to the number of Rabi periods (i.e. drive time).
This demonstrates locking of the JPG, where $\nu=\ell$, and a stable JPG-qubit
interaction as the drive pattern is lengthened.

One may expect the entire locking range in Fig. 3(a) to give a constant
$\nu_{\pi}$, however, this is not observed in Fig 3(e). This is because the
pulse width for Josephson devices operated at $f_{d}\sim f_{c}$ varies as
$I_{b}$ traverses the Shapiro step. Widening of the pulses results in a
reduction in $\delta\theta$ and is discussed further in the following section,
and in Supplementary Material XII. Our simulations for pulse width variation
across the Shapiro step agree with previous work [37, 38] and a variation of
$<10\%$ is expected. This restricts the region of constant $\nu_{\pi}$ in the
Rabi measurements relative to the dc locking range measurement. Despite these
effects, Fig. 3(e) nevertheless demonstrates a range of 150 $\mu$A where
$\nu_{\pi}$ is constant to within one pulse.

Figure 4: (Color) Simulations of a qubit digitally driven by a Gaussian pulse
train to extract the expected $X_{\pi}$ fidelity as a function of the pulse
standard deviation $\sigma$. Here $\sigma$ is in units of qubit periods
$T_{q}$. Pulses are delivered at $\omega_{d}=\omega_{10}/2$ so a pulse arrives
at every other qubit period. We use energy relaxation and dephasing rates
consistent with our qubit (see Sec. V). Fits to single JPG pulses measured at
room temperature yield an upper bound of $\sigma_{\mathrm{JPG}}=0.19~{}T_{q}$
(35 ps) for the width of pulses delivered to the qubit when parameterized as a
Gaussian. (a) Qubit excitation probability as a function of number of pulses
for increasing $\sigma$ and for the first half of the first Rabi oscillation.
(b) Extracted value of $\nu_{\pi}$ versus $\sigma$ and the corresponding total
$X_{\pi}$ total infidelity, $1-\mathcal{F}_{\mathrm{tot}}$. The pulse-qubit
coupling is normalized such that for the Dirac delta function limit
$\nu_{\pi}=100$. With the coupling re-adjusted to match our experimental
values of $\nu_{\pi}=352$ and $\sigma_{\mathrm{JPG}}=0.19~{}T_{q}$, we obtain
$1-\mathcal{F}_{tot}=2\times 10^{-3}$ and $1-\mathcal{F}_{pulse}=1\times
10^{-4}$. (c) Digital-pulse-only $X_{\pi}$ infidelity,
$1-\mathcal{F}_{\mathrm{pulse}}$, as function of $\nu_{\pi}$ – which is
parameterized by $\sigma$ in (b). $1-\mathcal{F}_{\mathrm{pulse}}$ is
calculated by subtracting the coherence limit contribution to
$1-\mathcal{F}_{\mathrm{tot}}$.

###  III.2 Finite-Width Pulses

Production of perfectly sharp pulses is not possible, so $\delta\theta$ also
depends on JPG pulse width. Indeed, we have demonstrated this by broadening
the JPG pulses (we heat the JPG to reduce $I_{c}$) and observe $\nu_{\pi}$ to
increase by approximately the same factor $I_{c}$ decreases. To explore the
pulse width dependence of $\delta\theta$ we perform simulations [39] of a
qubit driven at $\omega_{d}=\omega_{10}/2$ by Gaussian pulses whose width
(standard deviation $\sigma$ in units of the qubit period, $T_{q}$) we
control. Fig. 4 shows our simulation results and illustrates a strongly
nonlinear dependence on the qubit response for $\sigma>0.25~{}T_{q}$. For
short pulses in the Dirac delta function limit $\sigma<0.01~{}T_{q}$, the
qubit response is independent of $\sigma$ ($\nu_{\pi}$ changes by less than
one pulse). Supplementary Material XIV discusses the relationship between JJ
$\tau$ and $\sigma$ of a Gaussian fitted to the pulses. For our JPG with
$\tau=98$ ps the (on-chip) Gaussian-parameterized pulse width is $\sigma=17$
ps.

To measure the pulse width, the JPG output is split (see Fig. S1) and recorded
with an oscilloscope at room temperature. We find $\sigma=35$ ps which, for
our 5.37 GHz qubit, gives $\sigma=0.19~{}T_{q}$. This is an upper bound for
the widths of pulses delivered to the qubit due to added dispersion in the
additional 2 m of JPG-oscilloscope cabling compared to the JPG-qubit cable
length. From the simulations (after adjusting coupling so $\nu_{\pi}=352$) and
pulse width measurements we obtain a lower bound of our expected JPG $X_{\pi}$
infidelity of $2\times 10^{-3}$. While the total infidelity is coherence-
limit-dominated, subtraction of this contribution gives the infidelity due
only to the nature of control via digital pulses,
$1-\mathcal{F}_{\mathrm{pulse}}$. Fig. 4(c) demonstrates that, for our current
gate times, this digital-pulse-only infidelity is $\sim 10\%$ of the total
infidelity. Furthermore, this is competitive with state-of-the-art TSCE
techniques, reaching $\sim 10^{-4}$ infidelity, and shows there is no
fundamental limitation imposed by digital control with sharp pulses 111These
simulations were repeated using simulated SFQ pulses with the JPG $\tau=98$ ps
(Supplementary Material XIV) and show no significant change in fidelity.. See
Supplementary Material XII for more discussion.

When driven at a frequency below $f_{c}$, our prototype JPG’s locking range
decreases by a factor $\propto f_{d}/f_{c}$ but a lower $f_{c}$ also
compromises ideal digital qubit control dynamics. More ideal control may be
realized at the expense of locking range (or vice versa) by tuning
$f_{c}=I_{c}R_{n}/\Phi_{0}$ – with the caveat that $\sigma\gtrsim 0.3~{}T_{q}$
pulses are too wide for efficient digital control. We keep
$\sigma<0.2~{}T_{q}$ to balance locking range and optimal qubit dynamics.
Future devices (see Sec. VI) will not possess this limitation.

Figure 5: (Color) Comparison of the measured qubit lifetime $T_{1}$ and Ramsey
coherence time $T_{2}^{*}$ using a TSCE setup and a JPG at 3 K. For each each
setup 500 individual measurements of $T_{1}$ and $T_{2}^{*}$ are taken. Data
in blue (circles) are data taken with the TSCE setup, while data in red
(squares) are with the JPG. Observed daily fluctuations in the $T_{1}$ and
$T_{2}^{*}$ distribution means – i.e. the uncertainty in the means – are $\pm
0.5~{}\mu$s and $\pm 2~{}\mu$s, respectively. Solid lines are fits to the
$T_{1}$ decay curves and Ramsey fringes. (a) $T_{1}$ decay curve averaged
across all measurements and fitted $T_{1}$ value for each setup. (b)
Histograms and Gaussian fits of the $T_{1}$ distributions showing excellent
agreement in the mean ($\mu$) and standard deviation ($\sigma$) of the
distributions. (c) Ramsey fringe averaged across all measurements and fitted
$T_{2}^{*}$ value for each setup. (d) Histograms and Gaussian fits for
$T_{2}^{*}$, again showing good agreement with each setup.

##  IV Qubit Performance Comparison

Here we describe the side-by-side comparison of the TSCE and JPG setups
through measurements of $T_{1}$, $T_{2}^{*}$, and $P_{th}$. For the $T_{1}$
comparison, a JPG $X_{\pi}$ rotation is constructed of $\nu_{\pi}=352$ drive
periods (131 ns drive time) obtained with the calibration shown in Fig. 3. For
the $T_{2}^{*}$ comparison a JPG $X_{\pi/2}$ rotation is created with a
$\nu_{\pi}/2=176$ period drive waveform. We gather statistics on 500
measurements of $T_{1}$ and $T_{2}^{*}$ with each setup. Data are compiled in
Fig. 5 and show energy decay curves and Ramsey fringes averaged over all
measurements, as well as histograms of the extracted $T_{1}$ and $T_{2}^{*}$.
The small discrepancy in the distribution means are well within the expected
variation in $T_{1}$ and $T_{2}^{*}$ for superconducting qubits [41, 42, 27,
43, 4, 44] and within the observed daily fluctuations for this device of $\pm
0.5~{}\mu$s and $\pm 2~{}\mu$s for the $T_{1}$ and $T_{2}^{*}$ mean values,
respectively. Indeed, excellent agreement is found in $T_{1}$ and $T_{2}^{*}$
as measured with each setup – showing that JPG operation does not enhance
relaxation or dephasing from quasiparticle poisoning or larger cavity photon
number fluctuations.

An important validation of JPG-qubit compatibility is to demonstrate adequate
thermalization when controlled with the JPG. State inversions from elevated
qubit thermal occupancy can be $\gtrsim 10$% in transmon qubits with 3D
aluminum readout cavities [45, 46, 47]. We define the qubit thermal occupancy
$P_{th}$ as the probability of incorrect state identification based on the
desired preparation.

Measurement of $P_{th}$ is performed in a two-part experiment [48]. First, no
qubit rotation is applied and the state is simply measured. Second, we apply
an $X_{\pi}$ rotation to invert the qubit population, and then measure.
Measurements are single-shot and we do not perform heralding. The total state
preparation and measurement (SPAM) fidelity is

$\mathcal{F}=1-P(1|0)-P(0|1),$ (4)

where $P(i|j)$ is the probability of measuring state $\ket{i}$ when the qubit
was intended to be prepared in $\ket{j}$. Eq. (4) describes the combined
preparation fidelity and ability for the single-shot measurement to correctly
distinguish between $\ket{0}$ or $\ket{1}$. Ideally, each $P(i|j)$ contains
contributions only from thermal occupancy. In reality, both $P(i|j)$ include
decays from correctly prepared $\ket{1}$ and spuriously excited $\ket{0}$
initial states, and the $\ket{0}$ and $\ket{1}$ distribution overlap. The SPAM
fidelity thus gives an upper bound on $P_{th}$ and we minimize the effects of
overlap infidelity and decays during measurement to improve our estimate of
$P_{th}$.

We limit these decays to $<1\%$, which become the dominant uncertainty in the
$P_{th}$ measurement, by shortening the readout pulse to 400 ns and combat the
corresponding reduction of the single-shot SNR using an optimal mode-matching
integration weight function [48]. Overlap infidelity is minimized by
increasing the cavity drive strength to separate the primary $\ket{0}$ and
$\ket{1}$ distribution lobes. For the 400 ns readout pulse this occurs at
$n_{r}=50\approx n_{crit}/2.3$, which still avoids measurement-induced
transitions [35].

Figure 6: (Color) Measurement of the qubit thermal population, $P_{th}$, to
compare control with the TSCE (top) and JPG (bottom). Solid lines are a
bimodal Gaussian fit to the data. The $P_{th}$ estimate is obtained by
integrating the fit to the no-$X_{\pi}$ population with voltage levels greater
than zero. These data points correspond to instances in which a spurious state
inversion occurred; spoiling the desired $\ket{0}$ preparation. The primary
uncertainty in $P_{th}$ using this method is due to decays during measurement,
i.e. a $\sim 1\%$ error.

Fig. 6 shows data from $10^{4}$ measurements with each setup and bimodal
Gaussian fits to the data. As the no-$X_{\pi}$ case (desired preparation in
$\ket{0}$) additionally removes decays during preparation, we choose this
population to bound $P_{th}$ with the most accuracy. Integration of all
spurious $\ket{1}$ outcomes for this case yields excellent agreement in
$P_{th}$ of $0.036\pm 0.01$ and $0.032\pm 0.01$ for the TSCE and JPG setups,
respectively. These results demonstrate that qubit thermalization is not
affected when the JPG is used – our final compatibility metric of digital
control using JJ-based pulses from 3 K.

##  V JPG Randomized Benchmarking

We now characterize JPG gate fidelities through a randomized benchmarking (RB)
routine [49, 50, 51, 52, 53] where we apply a sequence of $m$ random gates
followed by a single sequence-inverting gate. The sequence fidelity is an
exponential decay

$\mathcal{F}(m)=ap^{m}+b,$ (5)

where the constants $a$ and $b$ encapsulate SPAM errors and errors on the
final gate. For single qubit gates, the depolarizing parameter $p$ is related
to the per-gate error $r$ by

$r=\frac{1}{2}(1-p).$ (6)

Results of the same routine using the TSCE setup are provided as a reference
gate error.

We choose the set of primitive and Pauli gates: {$I$, $\pm X_{\pi/2}$, $\pm
Y_{\pi/2}$, $\pm X_{\pi}$, $\pm Y_{\pi}$}, where the idle $I$ and $X_{\pi}$
gates lengths are equal [54, 55]. Given that $\pi$ gates are twice as long as
$\pi/2$ gates, and the fact we use a Clifford group subset, rescaling $m$ by a
factor of 1.125 permits comparison to full-Clifford-group RB [52, 21]. JPG
gates are constructed as described above, while TSCE gates use a $\sigma=35$
ns Gaussian pulse truncated at $\pm 2\sigma$ to closely match the JPG gate
time.

Figure 7: (Color) Depolarizing curve for single qubit RB using the TSCE and 3
K JPG qubit control setups. Solid lines are a fit to Eq. 5. TSCE rotations are
performed with Gaussian pulses with $\sigma=35$ ns and truncated at $\pm
2\sigma$ to match the JPG $X_{\pi}$ gate length of 131 ns. We extract an
average error per gate of $r_{TSCE}=(4.8\pm 0.5)\times 10^{-3}$ and
$r_{JPG}=(2.1\pm 0.1)\times 10^{-2}$. The former is just over twice the
coherence limit of the qubit, while the latter is a factor of 10 higher than
the $X_{\pi}$ fidelity of $2\times 10^{-3}$ as determined in simulations –
which is dominated by qubit coherence. Uncertainties in $r$ are determined as
the standard error of the fits to Eq. 5.

In Fig. 7 we show the results of the RB routine with both setups giving
$r_{TSCE}=(4.8\pm 0.5)\times 10^{-3}$ and $r_{JPG}=(2.1\pm 0.1)\times
10^{-2}$, where the uncertainties are from the Eq. 5 fit standard error. The
JPG $r$ is approximately a factor of 10 higher than the simulated single
$X_{\pi}$ gate error of $2.1\times 10^{-3}$ 222Note that a small measurement
error is present in all the JPG RB data, which were taken first and on a
separate cooldown than the TSCE RB. This error was remedied before the TSCE
measurement was performed and is responsible for the offset in the JPG curve
(Fig. 7). We emphasize this has no influence on the extracted $r$ for either
setup.. Performing a detailed accounting of known possible errors
(Supplementary Material XII) from digitization, finite pulse widths, higher
state leakage, and pulse timing jitter gives an estimated infidelity of
$6\times 10^{-3}$. This is a factor of three below the RB result. We attribute
the remaining error to possible systematic or coherent errors which are
presently under investigation. Regardless, these measurements serve as an
excellent proof-of-concept demonstration of qubit control using 3 K JJ-based
digital pulses.

##  VI Scalability and Discussion of Future Devices

The scalability of digital qubit control using JJ devices at 3 K is promising
even with the current device and configuration – which were not optimized for
size or power dissipation. The JPG circuit is $<15$ mm2 and the power
dissipated at duty cycle $\eta_{d}$ is approximately

$P=I_{c}V=\Phi_{0}N_{JJ}I_{c}f_{d}\eta_{d}.$ (7)

For the maximum $\eta_{d}$ in our experiment of 0.02, this yields
$P=1.6~{}\mu$W. Commercial cryocoolers delivering $\sim 1$ W of cooling power
at 3 K [57] permit use of over $\sim$500k similar devices; occupying $\sim
10^{-2}$ m3. The JPG pulse output power is $\sim-56$ dBm (at $\eta_{d}=0.02$),
giving off-chip dissipation of the qubit drive signal of {2 nW, 0.3 nW, 0.02
nW, 0.008 nW} at the {3 K, 1 K, 0.05 K, 0.01 K} stages (see Supplementary
Material VIII for details of the JPG control line attenuation stack). Of
larger consideration is dissipation of the large drive signal, which is
$\sim-6$ dBm at the JPG input at 3 K, however the new devices discussed below
offer techniques to circumvent this limitation.

Plans for NISQ systems require $\sim 10$ m3 of cryostat volume and numerous
cryocoolers. We conclude that neither power dissipation, nor device size
(volume of silicon) present significant obstacles in scaling the number of 3 K
JJ devices to control NISQ-era quantum arrays. Furthermore, our experimental
and JPG architectures can both be adjusted in a straightforward manner to
reduce device size and on-chip power dissipation each by more than a factor of
10. Thus, the primary obstacle in scaling a 3 K JJ-based qubit control
architecture, at least in the near-term, is one shared by many competing qubit
control technologies: wiring and signal routing logistics. Multi-chip modules
[58, 59, 60], high-density/bandwidth interconnects [28, 30], and out-of-plane
coupling [61] make overcoming these challenges feasible.

The present experiment architecture, where JPG pulses are heavily attenuated
by the cavity resonance before reaching the qubit, necessitates use of a large
$N_{JJ}$ to yield appropriate signal levels. This increases device size, power
dissipation, and can reduce qubit coherence by allowing the higher pulse train
harmonics to populate the cavity (not observed here). These limitations can be
eliminated using 2D readout cavity qubit devices and an independent control
line. With such a device we expect to reduce the array factor to
$N_{JJ}\approx 500$ without sacrificing thermalization or gate time.

Next-generation devices will implement an SFQ logic shift register and voltage
multiplier pulse amplification with no increase in JJ count [16, 62]. Use of a
high speed clock far above the qubit spectrum eliminates the qubit-drive
interaction and permits pulse delivery at $\omega_{d}\geq\omega_{10}$ or at
variable timing. The latter has been theoretically shown to reach 99.99%
fidelity with under 10 ns gates [19]. Voltage multiplier amplification
minimizes on-chip dispersion and permits narrower output pulses; enabling more
ideal digital qubit dynamics. Finally, these devices permit signal routing
which eliminates dissipation of the drive signal in cryogenic attenuators,
which is a major consideration for the present JPG device and would strongly
limit scaling of these prototype devices.

##  VII Conclusion

In this article we have demonstrated, for the first time, successful digital
control of a transmon qubit at 0.01 K using a superconducting Josephson pulse
generator located at 3 K. Through dual characterization of the system, using
both traditional semiconductor control electronics [5] and the 3 K JPG, we see
no reduction in intrinsic qubit performance. Specifically we measured no
negative impact to $T_{1}$, $T_{2}^{*}$, or $P_{th}$ – indicative that
quasiparticle propagation is effectively broken by locating the JJ elements
and quantum circuits on separate temperature stages. Additionally, we measure
an average JPG gate error of $r=2.1\times 10^{-2}$ which, considering the
improvements of the future JJ devices discussed in the previous section, are
expected to reach the simulated coherence-dominated infidelity of
$r_{min}=2\times 10^{-3}$.

These results enable scaled quantum information systems which leverage the
merits of Josephson-based sources for qubit control: signal stability,
reproducibility, SFQ pulse self-calibration, small device size, and low power
dissipation. Straightforward alterations in the qubit and JPG architectures
enable factors of $\geq 10$ reduction in dissipation and size, and future
devices are expected to bring JJ-based digital gates into competition with
contemporary TSCE gates. Such improvements further increase the potential
value of 3 K JJ-based qubit control, as current device sizes and dissipation
are commensurate with operation of over $\sim$500k devices. Integration with
cryoCMOS devices [9, 8, 10] is also possible; potentially yielding a hybrid
cryogenic controller exploiting the advantages of both technologies.

###### Acknowledgements.

We acknowledge the NIST Advanced Microwave Photonics Group for the qubit, JPA,
and readout microwave components. We thank N. Flowers-Jacobs for readout
microwave components. We thank F. Lecocq, R. McDermott, and B Plourde for
fruitful discussions, and A. Babenko and A. Fox for design and fabrication of
initial devices. L. Howe was funded through the NRC Postdoctoral Research
Associateship Program.

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Supplementary Material: Digital control of a superconducting qubit using a
Josephson pulse generator at 3 K

##  VIII Full Experimental Schematic

Figure S1: (Color) Detailed schematic of the setup used to evaluate qubit
control using a Josephson Pulse Generator (JPG) at 3 K in a commercial DR. The
dc connection to the bias T (via twisted pair) is used to provide a dc flux
bias to the JPA. Both the JPA dc flux bias and the $I$ and $V$ JPG taps are
low-pass filtered at 1.9 MHz and the JPG taps are further filtered with an on-
chip inductor network (see Fig 1(b) in the main body). We insert an attenuator
between the JPG splitter output and the 3 K plate to minimize reflections from
signals exiting the cryostat and prevent them from affecting the rf JPG signal
on-chip.

We present the full experimental diagram in Fig. S1. The JPG has two sets of
twisted pair leads connected to on-chip inductive taps to permit low-frequency
$I$-$V$ characterization of the JPG. A splitter is also used on the JPG rf
output line for analysis of high frequency characteristics using ambient
instrumentation. A 2.85 GHz low pass filter with over 60 dB insertion loss at
5 GHz is used at 300 K to fully prevent any higher harmonic content of the JPG
clock signal from driving the qubit.

##  IX Rabi and Ramsey Frequency Sweeps

Figure S2: (Color) (a) JPG Rabi chevron at fixed $I_{b}$. (b) Generalized
Ramsey scan. In both measurements the JPG drive frequency $\omega_{d}$ is
swept in a small region $\delta\omega_{d}=\omega_{d}-\omega_{10}/2$.

In Fig. S2 we present generalized Rabi and Ramsey scans where we sweep the JPG
drive frequency $\omega_{d}+\delta\omega_{d}=\omega_{10}/2$. This is simply a
demonstration that we may make connection with various routines and protocols
with the JPG that are common with TSCE control setups.

##  X Ramsey Fringe Compilation

Figure S3: (Color) Compilation colorplot of all Ramsey fringes gathered with
both control setups for statistical evaluation of the insensitivity of the
qubit coherence time to the control setup used. The three dark, horizontal,
stripe-like features in the TSCE compilation are due to instances where the
pulse-shaping AWG loses locking during the course of the measurement of a
single fringe.

In Fig. S3 we compile all Ramsey fringes gathered from the $T_{2}^{*}$
measurements in Sec. IV. The consistent level of fluctuations in the fringe
frequency indicate a similar level of stability in $\omega_{10}$ with both
qubit control setups.

##  XI Orthogonal Axis Control and Construction of JPG Drive Waveforms

Here we provide details of JPG drive waveforms and the procedure to establish
and verify orthogonal axis control of the qubit. For orthogonal control, first
a timing reference is established to define the $\hat{x}$ control axis
($\phi_{d}^{\hat{x}}=0$). At subharmonic drive $k\geq 2$ the $\hat{y}$ control
axis is realized by phasing the drive signal to allow the qubit to precess by
$\phi_{q}=\pi/2$ before the control pulse train arrives [21]. Intuitively,
there are $k$ qubit periods per drive period and during each qubit period the
qubit phase advances relative to the drive by a factor $2\pi k$. Therefore, to
accumulate the required $\phi_{q}=\pi/2$ in a single drive period, the
$\hat{y}$ drive signal must be phased by $\phi_{d}^{\hat{y}}=\pi/2k$ relative
to the $\hat{x}$ drive timing reference.

Drive waveforms are constructed from sine waves with an integer number of
samples per sine period with the integer chosen to be compatible with the DAC
sample rate of the AWG and also results in synthesis at the proper drive
frequency. For all qubit control experiments we generate JPG pulses at the
qubit subharmonic $k=2$ and use sine periods of length 24 samples, leading to
an AWG clock frequency of 64.32 GSa/s to synthesize 2.68 GHz clocking sine
pulses. For $\omega_{d}=\omega_{10}/2$ the expected delay required for
$\hat{y}$ rotations is $\phi_{d}^{y}=\pi/4$.

Phasing between the timing reference (in-phase $\hat{x}$ drive) and the
$\hat{y}$ control axis is verified by performing a Ramsey-like experiment. The
JPG clock frequency is swept, as well as the delay between two “$X_{\pi/2}$”
pulses. I.e. we begin with zero delay, yielding an $X_{\pi}$ pulse, and
increase the delay by inserting a number $n_{\phi}$ of zero delay samples
until the rotation sequence $X_{\pi/2}+Y_{\pi/2}$ is observed. Thus the
signature of the $\hat{y}$ axis calibration experiment will be a measurement
of the qubit in the $\ket{1}$ state ($P_{1}=1$) and decreasing to an equal
$\ket{0}+\ket{1}$ ($P_{1}=0.5$) superposition when the proper $\hat{y}$ phase
($n_{\phi}$) is reached. Oscillations then proceed with a period of
$4n_{\phi}$ delay samples as the required delay in the qubit frame is one-
fourth of a period. An example of this calibration routine in shown in Fig.
S4.

Figure S4: (Color) Calibration of the delay samples $n_{\phi}$, with respect
to the $\hat{x}$ timing reference, which manifests the orthogonal $\hat{y}$
control. In this measurement the clock frequency and number of zero delay
samples $n_{\phi}$ between sine patterns of $\nu_{\pi}/2$ periods are swept.
When $P_{1}$ progresses from 1 (at $n_{\phi}=0$) to 0.5 we have realized an
$X_{\pi/2}$+$Y_{\pi/2}$ gate sequence; determining the correct value for
$n_{\phi}$ for $\hat{y}$ control. Using sine patterns with 24 samples per
period and with $\omega_{d}=\omega_{10}/2$ we expect proper phasing to occur
at $n_{\phi}=3$ ($\phi_{d}=\pi/4$). The black trace (horizontal slice at
$\delta\omega_{d}=0$) and vertical red line demonstrate correct calibration of
$n_{\phi}$.

For the JPG RB tests we note that there are often subsets of the full pattern
resembling $X_{\pi}+Y_{\pi}$, or $Y_{\pi}+X_{\pi}$ gate combinations.
Referring to Fig. 1(d), we see that to construct a $\hat{y}$ rotation, the
phase of the drive must be delayed relative to the initial $\hat{x}$ timing
reference by $\phi_{d}^{y}=\pi/2k$. Similarly, to achieve an $\hat{x}$
rotation immediately following a $\hat{y}$ rotation, the drive phase must be
advanced by the same amount to re-establish the null timing reference phase.
To this end, every $X_{\pi}$ and $Y_{\pi}$ JPG waveform is $\nu_{\pi}+2$
periods in length. There are $\nu_{\pi}$ periods of active drive, with the
additional two periods of idle divided appropriately at the beginning and end
of the waveform to permit delaying and advancing the drive phase as needed.
The same procedure is followed for $X_{\pi/2}$ and $Y_{\pi/2}$ rotations. Due
to the unidirectional nature of digital qubit control using a JPG, $-\pi/2$
rotations are achieved using $\phi_{d}^{-x}=3\pi/2$ and
$\phi_{d}^{-Y}=\pi/2n+3\pi/2$ phases relative to the timing reference (we take
$X_{\pi}=-X_{\pi}$ and $Y_{\pi}=-Y_{\pi}$ as the ideal final states are
identical to those from positive gates). Note that, for both positive and
negative $\pi/2$ rotations, the drive signals are equal in length as described
above.

##  XII Digital Qubit Control Fidelity Limits

Multiple factors arise in a digital qubit control architecture which affect
fidelity and are not present in systems using TSCE-based microwave pulses.
They are: digitization error, error due to the finite temporal width of the
digital pulses, higher state leakage of the pulse train spectrum due to finite
qubit anharmonicity, and pulse arrival timing jitter.

Digitization error arises from eliminating the continuous variable
corresponding to the $X_{\pi}$ pulse power and manifests as a maximal over- or
under-rotation by (at most) $\pm 1/2$ pulse. The digitization error is

$1-\mathcal{F}_{dig}=1-\sin^{2}\left(\pi\frac{\nu_{\pi}\pm
1/2}{\nu_{\pi}}\right),$ (S1)

which decreases as $\nu_{\pi}$ increases. With finite qubit coherence time a
sweet spot arises which minimizes the infidelity, as is shown in Fig. S5. Note
that as the qubit lifetime increases the value of $\nu_{\pi}$ which minimizes
the combined digitization and coherence-limited infidelity also increases. For
$\nu_{\pi}=352$ the digitization infidelity of $3\times 10^{-5}$ is much
smaller than the infidelity due to qubit coherence.

The second affect on the gate fidelity is due to the finite temporal width of
the control pulses. As discussed in the main text, Sec. III.2, we simulate
these effects [39] using a train of Gaussian pulses delivered at
$\omega_{d}=\omega_{10}/2$ to evolve the qubit state and study the fidelity of
$X_{\pi}$ rotations as a function of the pulse standard deviation $\sigma$.
The Hamiltonian for the system is

$\hat{H}=\hat{H}_{0}+\hat{H}_{d}=\frac{1}{2}\hat{\sigma}_{z}-\Omega_{d}s_{d}(t)\hat{\sigma}_{x}$
(S2)

where $\Omega_{d}$ and $s_{d}(t)$ describe the coupling strength and the pulse
train time-dependent amplitude, respectively. Note that due to the pulse
drive’s time-dependent nature we cannot make a rotating wave approximation.
Since our simulations use a true two-level system, there is no leakage outside
of {$\ket{0}$, $\ket{1}$} so $P_{1}$ can be used to calculate the $X_{\pi}$
fidelity $\mathcal{F}$. To determine $\mathcal{F}$ we evaluate $P_{1}$ only at
the center of the idle qubit period following a pulse and take the resultant
maximum value as $\mathcal{F}$. This also defines the value of
$\nu_{\pi}(\sigma)$, i.e. the number of pulses which maximize the following
idle qubit period $P_{1}$ as a function of $\sigma$. As $\sigma$ is increased,
the pulse train power at the second harmonic, and thus the tip angle per
pulse, $\delta\theta$, diminishes so more pulses are required to realize a
$\pi$ rotation with wider pulses (this is in conjunction with distributed
pulse energy delivery as the qubit precesses, discussed below). In turn,
$\nu_{\pi}(\sigma)$ increases so we must recalculate $\nu_{\pi}(\sigma)$ at
each pulse width. For these simulations we include loss with realistic values
of $T_{1}=T_{2}^{*}=34~{}\mu$s ($T_{\phi}=68~{}\mu$s) for our qubit. Fig. 4 in
the main body shows the $X_{\pi}$ Rabi oscillation curves,
$\nu_{\pi}(\sigma)$, and the corresponding $X_{\pi}(\sigma)$ fidelity.

Figure S5: (Color) JPG $X_{\pi}$ digitization infidelity as a function of
$\nu_{\pi}$, the number of pulses, and thus the gate time, required for an
$X_{\pi}$ rotation. For small $\nu_{\pi}$ (i.e. fast gates) the digitization
error from the finite single pulse rotation angle $\delta\theta$ dominates,
whereas for large $\nu_{\pi}$ the qubit coherence limit becomes dominant. We
use representative values for our qubit of $T_{1}=T_{2}^{*}=34~{}\mu$s (i.e.
$T_{\phi}=68~{}\mu$s). The qubit frequency is $\omega_{10}/2\pi=5.37$ GHz and
drive at $\omega_{d}=\omega_{10}/2$.

If the Gaussian pulse duration ($\sim 4\sigma$) approaches $2\pi/\omega_{10}$,
the pulses no longer deliver the majority of their power when the qubit is
aligned with the desired control axes. Rather, the pulses turn on and off
slowly relative to the qubit period and their action is distributed over a
non-negligible portion of the qubit precession. For a wide pulse near its
peak, the pulse delivers the majority of its energy and rotates the qubit
state $\ket{\psi}$ along the target control axis. However, the long tails
before and after the peak impart lesser rotations while the qubit advances
phase at $\omega_{10}$. As the qubit precesses, $\ket{\psi}$ moves through
having a component anti-aligned with the target control axis, to being
aligned, and then anti-aligned, etc. In the anti-alignment phase the leading
and trailing pulse tails induce small negative (back towards $\ket{0}$)
rotations on either side of the main positive rotation; corresponding to the
pulse maximum and complete alignment of $\ket{\psi}$ and the target control
axis. This behavior manifests as a dip in $P_{1}$ when the pulse begins to
arrive, a larger increase in $P_{1}$ as the pulse peak arrives, and another
small reduction in $P_{1}$ for the trailing pulse tail.

To demonstrate these effects, Fig. S6 shows a magnified view of the qubit
excited state evolution for the first few pulses and as the pulse $\sigma$ is
swept, as well as the reduction in $\delta\theta$ as $\sigma$ increases. We
find that for wide pulses ($\sigma\gtrsim 0.1$) the $X_{\pi}(\sigma)$
infidelity, after subtraction of the coherence limit contribution, roughly
corresponds to the rotation lost due to pulse energy delivery while the qubit
is anti-aligned with the desired control axis. In the main text we refer to
this as the digital-pulse-only infidelity, $1-\mathcal{F}_{pulse}$. As stated
in Fig 4(c), for $\sigma_{\mathrm{JPG}}=0.19T_{q}$, we expect
$1-\mathcal{F}_{pulse}=1\times 10^{-4}$. This is over an order of magnitude
smaller than the coherence-limit contribution as we find
$1-\mathcal{F}_{tot}=2\times 10^{-3}$ with finite coherence.

Figure S6: (Color) (a) Depiction of the simulated qubit excited state
evolution during delivery of the first few Gaussian pulses and as a function
of the pulse $\sigma$. The $x$-axis is scaled to the $X_{\pi}$ time and each
curve is offset relative to their departure from the ideal delta function case
of $\nu_{\pi}=100$ to better show dynamics. For wide pulses ($4\sigma\sim
2\pi/\omega_{10}$) the effects of distributed delivery of the pulse energy
during qubit precession can be seen as dips in $P_{1}$ at the beginning and
end of the pulses. (b) Tip angle per pulse $\delta\theta$ (evaluated at the
center of the qubit idle period) as a function of $\sigma$. Reduction in
$\delta\theta$ is due primarily to pulse energy delivery when the qubit is out
of phase of the target control axis, but is also affected by the decreased
harmonic tone power for increasing $\sigma$. Fast pulses do not suffer from
these effects and are seen to approach the Dirac delta function limit for
$\sigma<0.02T_{q}$.

Leakage to the higher transmon level arises from the finite length pulse train
used to evolve the qubit state. To estimate the infidelity due to higher state
leakage we follow analysis in [18], which uses $\nu_{\pi}=100$ and true Dirac
delta function pulses. Since $(1-\mathcal{F})\propto\nu_{\pi}^{-2}$ we rescale
by a factor $(352/100)^{-2}$. For our qubit, with an anharmonicity
$\alpha=5\%$, this results in higher state leakage infidelity of $7\times
10^{-4}$. We note that for our case of moderate-width pulses, the ringing in
the pulse train spectrum will be further suppressed relative to the Dirac
delta function pulse train in [18].

For our configuration we consider the case of pulse timing jitter when the JPG
is driven with a stable external clock [18]. Gaussian-distributed thermal
jitter in pulse generation in SFQ circuitry has been measured to have a
standard deviation of 3 ps or less at 4.2 K [63]. However, for an
$N_{JJ}=4650$ array, the effect of thermal jitter at each JJ results only in a
slight broadening of the output JPG pulse with little effect on the pulse
arrival time. Timing jitter from the JPG drive signal, however, will cause
jitter of the JPG output signal. Measurements of jitter in the JPG drive
signal at 300 K indicate jitter of 3 ps (standard deviation), corresponding to
a jitter-sourced infidelity estimate of $3\times 10^{-3}$.

Combining all contributions above we obtain a total expected gate infidelity
of $1-\mathcal{F}_{tot}=6\times 10^{-3}$, a factor of three lower than the
observed JPG RB error per gate of $2\times 10^{-2}$. We attribute the
unaccounted infidelity to possible systematic and coherent errors in the RB
routine and are actively investigating these effects.

##  XIII JPG Fabrication details and Reduced $I_{c}$ Temperature Sensitivity

Figure S7: (Color) (a) Rendering of the JJ (red) and shunt resistor $R_{n}$
(yellow) layout within the center conductor of the coplanar waveguide. The top
wiring layer is not shown. (b) Cross-sectional diagram of the JPG layer stack
(non-planarized) where the view into the page for (b) corresponds to looking
left along the array in (a). Figure S8: (Color) Reduced critical current
$i=I/I_{c}^{3.1~{}\mathrm{K}}$ of NbSi and $\alpha$Si barrier JPG devices
(top), and the temperature-sensitivity ($i^{\prime}=di/dT$) ratio of the NbSi
and $\alpha$Si devices (bottom). The drastically-reduced 3 K - 4 K temperature
sensitivity of the $\alpha$Si JPG provides the necessary stability of the
Shapiro step when the JPG is driven with both short patterns for $T_{1}$,
$T_{2}^{*}$, and $P_{th}$ measurements, and with long patterns for measuring
many Rabi oscillations and performing randomized benchmarking.

Gating pulses from a JJ array with our scheme of driving with an integer
number of pure sine periods means the rf dissipation is toggled rapidly
between null and 100%. Depending on the $I_{c}$ temperature sensitivity, this
can cause an unstable locking range for a JJ array and make calibrations such
as those detailed in Fig. 3 difficult or impossible.

Figure S9: (Color) Simulation of the output of a single-JJ using the RSJ
model, driven by an ac drive at frequency $f_{d}/f_{c}=0.2$ and with a dc
current bias which places the JJ on the first Shapiro step and results in a
periodic train of SFQ pulses. (a) The first seven SFQ pulses and the ac drive
signal (b) Magnification of the second and third SFQ pulses with a Gaussian
fit to the former. Here we also show pictorially the relative size of the
fitted Gaussian $\sigma$ and the JJ characteristic time $\tau$.

To reduce the temperature sensitivity of the JPG we have departed from the
superconductor-normal metal-superconductor (SNS) JJ technology [64] based on
triple-stacked self-shunted niobium-doped-silicon barriers (NbSi) embedded in
the center conductor of a niobium (Nb) superconducting coplanar waveguide.
Instead, we form the JPG array from externally shunted superconductor-
insulator-superconductor (SIS) junctions, as can be seen in Fig. S7(a). The
JJs were fabricated using amorphous silicon ($\alpha$Si) for the barriers and
palladium-gold (PdAu) alloy for shunting resistors. Due to the small value of
the shunting resistors (25 m$\Omega$) required to achieve the target
characteristic frequency of $f_{c}=10$ GHz, the usual method of fabricating
planar resistors could not be used. That method relies on resistor films with
sheet resistances of 1–2 $\Omega/\Box$. The form factor to achieve 25
m$\Omega$ entails very wide and long arrays to accommodate interconnecting
vias to minimize contact resistance. Instead, (vertical) stud resistors were
used with a height similar to the junction counter electrode thickness. See a
schematic of the cross-section in Fig. S7(b). This enables significant
reduction in the JJ cell size and results in a much more compact array.

The circuits were fabricated on 3-inch silicon wafers with 150 nm of thermal
oxide. A trilayer of Nb/$\alpha$Si/Nb was deposited by sputtering. The
thickness of the silicon was chosen to obtain a junction critical current
density of 12 kA/cm2. PdAu resistors were electron-beam evaporated with a
thickness of 240 nm. Silicon oxide insulator was then deposited by electron
cyclotron plasma enhanced chemical vapor deposition (EC-PECVD). Lithography
followed by dry etching opened vias to the junction counter electrodes,
resistors, and base electrodes. The wiring layer was deposited by niobium
sputtering and then patterned by dry etching. Resulting devices display
$I_{c}=3.05$ mA and $R_{n}=6.93$ m$\Omega$, giving $f_{c}=10.2$ GHz. These
devices exploit the much more stable $I_{c}$ of $\alpha$Si barrier JJs with
respect to temperature. Fig S8 shows a comparison of the temperature stability
of $I_{c}$ for NbSi and $\alpha$Si devices, demonstrating that in our region
of interest of 3–3.3 K the $\alpha$Si barrier device is $10$–$20$ times less
sensitive.

##  XIV Relation Between JPG $\tau$ and Gaussian-fitted $\sigma$

Simulations of JPG dynamics can be performed by numerically integrating the
resistively shunted junction (RSJ) model with an ac drive term
$i_{\mathrm{ac}}\sin(f_{d}\Theta/f_{c})$

$\frac{d\phi}{d\Theta^{2}}=\frac{1}{\beta_{c}}\left[i+i_{\mathrm{ac}}\sin\left(\frac{f_{d}}{f_{c}}\Theta\right)-\sin\phi-\frac{d\phi}{d\Theta}\right].$
(S3)

Here $\phi$ is the superconducting phase difference across the JJ, $\Theta$ is
the dimensionless time variable $\Theta=(2\pi I_{c}R_{n}/\Phi_{0})t=2\pi
t/\tau$, $\beta_{c}=2\pi f_{c}R_{n}C$ is the Stewart-McCumber damping
parameter with $C$ the intrinsic JJ shunting capacitance, and $i=I_{d}/I_{c}$
and $i_{\mathrm{ac}}=I_{\mathrm{ac}}/I_{c}$ are the normalized dc and ac
(amplitude) bias currents, respectively. Note that the ac drive frequency
$f_{d}$ has units of time scaled to the Josephson time constant
$\tau=1/f_{c}=\Phi_{0}/I_{c}R_{n}$ and not to $\Theta$, as these differ by a
factor of $2\pi$.

With the second Josephson equation

$V(t)=\frac{\Phi_{0}}{2\pi}\frac{d\phi}{dt}$ (S4)

and Eq. S3, we can solve for the time-dependent voltage response of the JJ. We
choose $i_{\mathrm{ac}}=0.6$, and $f_{d}/f_{c}=0.2$, which approximately
reflects the relation between the JPG $f_{c}$ of 10.2 GHz and the typical
drive frequency of $\omega_{d}/2\pi=2.68$ GHz, and ramp $i_{\mathrm{dc}}$
until pulses are observed. For $\beta_{c}\ll 1$, the JJ response is overdamped
and non-hysteretic so we simulate a heavily overdamped regime of
$\beta_{c}=0.01$. For our JPG, $\beta_{c}\lesssim 10^{-4}$. Results are
displayed in Fig S9 and we perform a Gaussian fit to one SFQ pulse. Time units
are re-scaled from the $\Theta$ time of Eq S3 to units of the JJ
characteristic time $\tau$. For these parameters we extract
$\sigma/\tau=1.08/2\pi$. This indicates that, given the JPG $\tau$ of 98 ps,
the pulses at 3 K (on-chip, prior to any significant dispersion and
broadening) have an expected Gaussian sigma of 17 ps. Fits at room temperature
of the JPG pulses captured on the oscilloscope (Sec. III.2 of main body) thus
display approximately a factor of two in broadening compared to the expected
on-chip value. As mentioned in Sec. III.2, this is the reason we take
$\sigma=35$ ps to be a conservative upper bound on the approximate width of
pulses delivered to the qubit, even after traversal of the readout cavity
resonance.


# Calculating Max-eigenvalue and Max-eigenvector with Jumps of Matrices

Ali Ebadian  Department of Mathematics, Payame Noor University (PNU), P.O.
Box, 19395-3697, Tehran, Iran<EMAIL_ADDRESS>,  Saeed Hashemi Sababe
Department of Mathematics, Malard Branch, Islamic Azad University, Tehran,
Iran.  Hashemi<EMAIL_ADDRESS>and  Hojr Shokoh Saljoghi  Department of
Mathematics, Isfahan University of Technology, Isfahan, Iran.  h
<EMAIL_ADDRESS>
###### Abstract.

The eigenvector problem for an irreducible non negative matrix $A=[a_{ij}]$ in
the max algebras is the form $A\otimes x=\lambda x$, where $(A\otimes
x)_{i}=max(a_{ij}x_{j})$, $x=(x_{1},x_{2},\dots,x_{n})^{T}$ and $\lambda$
refers to the maximum cycle geometric mean $\mu(A)$. In this paper, we exhibit
a method to compute $\mu(A)$ and max-eigenvector by using mutations of
matrices. Since the order of power method algorithm is $O(n^{3})$, the
advantage of this paper present a faster procedure.

###### Key words and phrases:

max-eigenvector; max-eigenvalue; jump of a amatrix

###### 1991 Mathematics Subject Classification:

15A18; 15A48; 91B06

##  1\. Introduction

The max-algebra or ($\max,\times$) semi ring, is the set $\mathbb{R}_{+}$ ,
equipped with $\max$ as addition, and ordinary multiplication as its
multiplication. It is traditional to use the notation $\oplus$ for $\max$, and
$\otimes$ for $\times$ . For vectors $x=(x_{i}),y=(y_{i})$ in
$\mathbb{R}_{+}^{n}$ and $c\in\mathbb{R}_{+}$ the vectors $x\oplus
y=(\max\\{x_{i},y_{i}\\})$ and $cx=(cx_{i})$ are defined element wise. The sum
$A\oplus B$ of two matrices is defined analogously. This structure satisfies
all the semi ring axioms, i.e. $\oplus$ is associative, commutative, with zero
element, $\otimes$ is associative, has a unit, distributes over $\oplus$, and
zero is absorbing. This semi ring is commutative $(a\oplus b=b\oplus a)$ ,
idempotent $(a\oplus a=a)$ , and non zero elements have an inverse for
$\otimes$. The term diode is sometimes used for an idempotent semi ring. Using
the new symbols $\oplus$ and $\otimes$ instead of the familiar $\max$ and
$\times$ notation is the price to pay to easily handle all the familiar
algebraic constructions. For instance, we will write, in the Max semi ring:

(1) $\left[\begin{array}[]{lr}2&0\\\
11&15\end{array}\right]\left[\begin{array}[]{lr}10\\\
13\end{array}\right]=\left[\begin{array}[]{lr}2\otimes 10\oplus 0\otimes 13\\\
11\otimes 10\oplus 15\otimes
13\end{array}\right]=\left[\begin{array}[]{lr}20\\\ 195\end{array}\right],$
(2) $(3+x)^{2}=(3+x)(3+x)=6\oplus 3x\oplus x^{2}=6\oplus x^{2}.$

Obviously with this multiplication for matrices, we can define max eigenvalues
and eigenvectors of a matrix. If for some matrix $A$ a vector $x\geq 0$ and a
number $\mu(A)$ exist such that

(3) $A\otimes x=\mu(A)x,\quad(A\otimes x)_{i}=\max_{j}a_{ij}x_{j},$

Then $x$ is called an max eigenvector of $A$, and $\mu(A)$ an max eigenvalue
of $A$. For abbreviation we simply use eigenvalue and eigenvector instead of
max eigenvalue and eigenvector.

There are several applications of max algebras one can found in [1, 2, 5, 10,
14]. As it clearly showed in application, max algebras and especially
eigenvalues and eigenvectors can describe some important properties.

One of important application of max algebras is discrete events dynamical
systems (DEDS). In control engineering, a discrete event dynamic system (DEDS)
is a discrete-state, event-driven system of which the state evolution depends
entirely on the occurrence of asynchronous discrete events over time. Although
similar to continuous-variable dynamic systems (CVDS), DEDS consists solely of
discrete state spaces and event-driven state transition mechanisms. We can
correspond to the graph of every such systems, a matrix $A$ which every entry
$A_{ij}$ is the weight of arc $(j,i)$ [4].

A path of length $l$ in a graph is a sequence of arcs

(4) $(i_{1},i_{2}),(i_{2},i_{3}),\dots,(i_{l},i_{l+1}).$

A circuit of length $l$ is a closed path of length $l$. A circuit which its
first and last elements coincide is called a Cycle.

######  Definition 1.1.

A graph is called strongly connected if from any node $i$ to any node $j$ a
path exists. If a graph is strongly connected, the corresponding matrix will
be called irreducible. A non negative matrix $A\in M_{n}$ is said to be
primitive if it is irreducible and has only one eigenvalue of maximum modules.

Irreducible matrices have a special matrix form. Indeed, a reducible matrix
$A\in M_{n}$ of size $1$ is $0$ and for $n\geq 2$, there is a perjump matrix
$P\in M_{n}$ and an integer $r$ with $1\leq r\leq n-1$ such that

(5) $P^{T}AP=\left[\begin{array}[]{lr}B&C\\\ 0&D\end{array}\right],$

Where $B\in M_{r}$, $D\in M_{n-r}$, $C\in M_{r,n-r}$ and $0\in M_{n-r,r}$ is a
zero matrix. A matrix $A$ is irreducible if it is not reducible.

######  Definition 1.2.

The weight of a path $i_{1}\rightarrow i_{2}\rightarrow\dots i_{l}\rightarrow
i_{l+1}$ is the sum of the weights of the individual arcs. The average weight
of a path is its weight divided by the number of arcs:
$(A_{i_{2},i_{1}}+A_{i_{3},i_{2}}+\dots+A_{i_{l},i_{l-1}})/(l-1)$. The circuit
mean is the average weight of a circuit. Any circuit of maximum average weight
is called a critical circuit. Similarly, we can define circuit geometrical
mean by

(6) $\sqrt[n]{A_{i_{2},i_{1}}\times A_{i_{3},i_{2}}\times\dots\times
A_{i_{l},i_{l-1}}}.$

######  Definition 1.3.

A matrix $A\in M_{n}$ is said to have property SC if for every pair of
distinct integers $p,q$ with $1<p,q<n$ there is a sequence of distinct
integers $k_{1}=p,k_{2},k_{3},\dots,k_{m-1},k_{m}=q,~{}1<m<n$, such that all
of the matrix entries $a_{k_{1}k_{2}},a_{k_{2}k_{3}},\dots,a_{k_{m-1}k_{m}}$
are non zero.

Several related subjects about property $SC$ can be found in [12]. We just
mention to following theorem without proof.

######  Theorem 1.4.

Let $A\in M_{n}$ corresponding to graph $\Gamma(A)$. The following are
equivalent:

  * (a)

$A$ is irreducible;

  * (b)

$\Gamma(A)$ is strongly connected;

  * (c)

$A$ has property $SC$.

Braker and Olsder present in [4] a power algorithm to find max eigenvalue.
They also found some useful properties. Esner and Van Den Driessche try to
modify Braker’s method. In section 2, we try to present a method, based on
concept of jump of matrices which is faster and has smaller absolute value of
error.
In section 3, we present a new method, based on max algebra, corresponding an
optimization problem to a SR matrix such that minimize following function:

(7) $e(w)=\max|\frac{a_{ik}-w_{i}/w_{k}}{a_{ik}}|.$

We also, introduce jumps of a matrix and its applications in symmetric,
reciprocal and trasitive matrices and in the third section, we calculate the
error refered to jumps and study behavior of jumps of transitive matrices.

##  2\. Main Theorems

To evaluate the determinant of a matrix $A\in M_{n}$ by Laplace expansion, we
use the following formula:

(8)
$det~{}A=\sum_{j=1}^{n}(-1)^{i+j}a_{ij}~{}det~{}A_{ij}=\sum_{i=1}^{n}(-1)^{i+j}a_{ij}~{}det~{}A_{ij},$

Where $A_{ij}$ is a matrix which obtain by eliminating the $i-$th row and
$j-$th column of matrix $A$. This process is continued to evaluate
$det~{}A_{ij}$ while the size of remain matrix becomes equal to $2$. To
evaluate the determinant of a matrix with size $2$, we simply differ the
multiplication of entries on main diagonal and other diagonal. In fact in this
process, we multiply on entry of every row and column. Determinant of the
matrix is a special kind of linear combination of these multiplications.
We can have an other formula to exhibit determinant of matrix $A\in M_{n}$.

(9) $det~{}A=\sum_{\sigma}sgn~{}\sigma\prod_{i=1}^{n}a_{i\sigma(i)},$

where the sum runs over all $n!$ perjumps a of the $n$ items $\\{1,\dots,n\\}$
and the ”sign” or ”signum” of a perjump $\sigma$, $sgn~{}\sigma$, is $+1$ or
$-1$, according to whether the minimum number of transpositions, or pair wise
interchanges, necessary to achieve it starting from $\\{1,2,\dots,n\\}$ is
even or odd. Thus, each product $a_{1\sigma(1)}a_{2\sigma(2)}\dots
a_{n\sigma(n)}$ enters into the determinant with a $+$ sign if the perjump
$\sigma$ is even or a $-$ sign if it is odd.

######  Definition 2.1.

For matrix $A\in M_{n}$, every product of $n$ entries of different rows and
columns is called a Jump. A Jump with no main diagonal entry as its
multipliers is called a Principal Jump. Other wise, we called it a Subordinate
Jump.

In this view, determinant of a matrix is a special linear combination of its
Jumps.

######  Example 2.2.

Consider the following $3\times 3$ matrix

(10) $A=\left[\begin{array}[]{ccc}1&2&3\\\ 4&5&6\\\ 7&8&9\end{array}\right].$

We have

(11) $det~{}A=1\times 5\times 9-1\times 6\times 8+2\times 4\times 9-2\times
6\times 7+3\times 4\times 8-3\times 5\times 7.$

In this example, $3\times 4\times 8$ and $2\times 6\times 7$ are principal
Jumps and others are subordinate Jumps.

######  Theorem 2.3.

Let $A$ be a $n\times n$ non negative matrix. The part of jump that hasn’t any
entry of principal diagonal form a cycle.

###### Proof.

Let $\alpha$ is a Jumps of $A$. Then entries of $\alpha$ are
$a_{i_{1}i_{2}},a_{i_{2}i_{3}},\dots,a_{i_{k}i_{1}}$ which diagonal entries of
$A$ are eliminated. By definition, $\alpha$ is a cycle. ∎

######  Theorem 2.4.

Let $A$ be a $n\times n$ non negative matrix. A part of a Jumps with non zero
production of elements and without any element of a principal Jump form a
cycle.

###### Proof.

It is not hard to show, using lemma 2. ∎

######  Definition 2.5.

Let $A$ be a square matrix. The set of all Jumps of $A$ is denoted by $M(A)$.
Let $\alpha$ be a Jump of $A$. The product of non zero elements of $\alpha$ is
denoted by $p(\alpha)$ and the number of non zero elements of $\alpha$ is
denoted by $S_{\alpha}$.

######  Theorem 2.6.

Let $A$ be a non negative square matrix and $\alpha\in M(A)$. Then

(12) $\mu(A)=\max\\{\sqrt[S_{\alpha}]{p(\alpha)}~{},~{}\alpha\in M(A)\\}.$

###### Proof.

By using of two primer theorem and knowing that a circuit can appear on a
vertex therefore element of principal diagonal can yield a circuit then whit
attention to define of $\mu(A)$ proof is evident. ∎

######  Example 2.7.

The eigenvalue problem for an irreducible non negative matrix $A=[a_{ij}]$ in
the max algebras is

(13) $A\otimes x=\lambda x\quad(A\otimes\lambda)_{i}=\max_{j}a_{ij}x_{j},$

Where $\lambda$ denotes the maximum cycle mean $\mu(A)$. But by properties of
max-algebras

(14) $A^{*}=[a_{ij}]=\left\\{\begin{array}[]{lr}a_{ij}&\qquad\textit{if
$(i,j)$ is in jump $\mu(A)$}\\\ 0&\qquad other~{}wise\end{array}\right.,$

And $A^{*}x=\mu(A)x$ we can simply obtain eigenvalues of $A$.

######  Example 2.8.

Consider matrix $A$ as follows:

(15) $A=\left[\begin{array}[]{ccc}0&8&1\\\ 3&0&2\\\ 4&1&1\end{array}\right].$

Then

(16) $det(A)=0\cdot 0\cdot 1-0\cdot 2\cdot 1-8\cdot 3\cdot 1+8\cdot 2\cdot
4+1\cdot 3\cdot 1-1\cdot 0\cdot 4,$ (17) $\mu(A)=\max\\{\sqrt[2]{2\cdot
1},\sqrt[3]{8\cdot 2\cdot 4},\sqrt[3]{1\cdot 3\cdot 1}\\}=4.$

Where $\mu(A)$ is corresponding to subordinate jump $a_{12},a_{21},a_{33}$.
Thus by equation $A^{*}x=\mu(A)x$ we have

(18) $\left[\begin{array}[]{ccc}0&8&0\\\ 0&0&2\\\
4&0&0\end{array}\right]\left[\begin{array}[]{ccc}x_{1}&x_{2}&x_{3}\end{array}\right]=\left[\begin{array}[]{ccc}x_{1}&x_{2}&x_{3}\end{array}\right]^{T}4.$

So

(19)
$\left[\begin{array}[]{ccc}x_{1}&x_{2}&x_{3}\end{array}\right]=\left[\begin{array}[]{ccc}2&1&2\end{array}\right].$

######  Theorem 2.9.

Let $A$ be a $n\times n$ non negative matrix. Then

(20) $\prod a_{ii}>\max\\{p(\alpha)~{},~{}\alpha\in
M(A)\\}\Longrightarrow\mu(A)=\max\\{a_{ii}\\}.$

###### Proof.

When $\Pi a_{ii}$ is greater than all $p(\alpha)$ ’s, then there one of the
main diagonal entry has the greatest module. Then the maximum of geometric
mean holds on the main diagonal. ∎

######  Theorem 2.10.

Let $A$ be a non negative square matrix. Then

$\forall\alpha\in M(A),p(\alpha)<1\Longleftrightarrow\mu(A)<1.$

###### Proof.

It is not hard to see, since the maximum of geometric mean is equal to
$S_{\alpha}-th$ root of some jump $\alpha$. ∎

######  Theorem 2.11.

Let $A$ be a non negative square matrix. If maximum geometric mean $\mu(A)$
holds on a jump with only elements of principal diagonal, then max-eigenvector
corresponding with $\mu(A)$ is $e_{i}$, where $i$ is the index of
$\mu(A)=a_{ii}$, else $x\neq e_{i}$.

###### Proof.

By the condition of theorem, all entries of $A^{*}$ is equal to zero unless
the entry corresponding to $\mu(A)=a_{ii}$. Then by $A^{*}x=\mu(A)x$, $e_{i}$
is the eigenvector corresponding to $\mu(A)$. If the maximum geometric mean
$\mu(A)$ holds on any other jump, there is a non zero entry in every row of
$A^{*}$ since there is an entry of every row and column of $A$ in this jump.
So $A^{*}x=\mu(A)x$ concludes $\mu(A)>0$. ∎

The following theorem shows the relation between critical matrix and jumps.

######  Theorem 2.12.

Let $A$ be a non negative matrix. The critical matrix of $A$ is corresponding
with a non zero jump which holds on its Jump cyclic geometric mean $\mu(A)$,
therefore

(21) ${a_{ij}}^{c}=\left\\{\begin{array}[]{lr}a_{ij}&\qquad\textit{if
~{}$(i,j)$~{} exist~{} in~{} Jump~{} of~{} $\mu(A)$}\\\ 0&\textit{ else
}\end{array}\right..$

###### Proof.

It is not hard to see by 12 and definition of critical matrix. ∎

##  3\. Application of jumps in Producing Transitive Matrices

An (entrywise) positive $n\times n$ matrix $A=(a_{ij})$ is called a
symmetrically reciprocal matrix (SR- matrix) if $a_{ij}a_{ji}=1$ for all
i,j=1,…,n, thus $a_{ii}=1$. The firest time were introduced by Saaty, used in
the analytic hierarchy process(AHP method) for multicriteria decision making.
And An SR- matrix $B=(b_{ij})$ is called transitive if there is a positive
n-vector $W=(w_{1},w_{2},\dots,w_{n})$ such that $b_{ij}=\dfrac{w_{i}}{w_{j}}$
for $i,j=1,\dots,n$.thus sometimes it is required to deduce positive weights
$w_{1},w_{2},dots,w_{n}$ attached to the alternatives $A_{1},\dots,A_{n}$
respectively, from the SR-matix A. Approximating an SR-matrix A by a
transitive matrix B is an importance step in the analytic hierarchy
process(AHP) for decision making by attaching a ranking to the SR- matrix. In
this way the alternatives can be ranked. For the ideal case,
$a_{ij}=\dfrac{w_{i}}{w_{j}}$ for $i,j=1,\dots,n$. it should be noted a
transitive matrix is SR- matrix of rank one. However, in a realistic case
$a_{ij}$ is only approximately given by $\dfrac{w_{i}}{w_{j}}$. one of the
most important subjects in section is , costructing a weight vector
$W=(w_{1},w_{2},\dots,w_{n})$ that there are several suggstions in literature
for constructing a weight vector . Some of these methods is given below. Saaty
proposes taking the perron vector of A . the vector W is chosen in such a way
that the matrix with entries $\dfrac{w_{i}}{w_{j}}$ has minimal distance from
A in the Euclidean matix norm, i.e., that

$\sum(a_{ik}-\dfrac{w_{i}}{w_{j}})^{2}$

is minimal. in the for costructing a weight vector
$W=(w_{1},w_{2},\dots,w_{n})$ using that of max-eginvector of A that solves a
useful optimization problem, namely minimizing the relative error useful
optimization problem, namely minimizing relative error

$e(W)=\max|\dfrac{a_{ik}-(w_{i}/w_{j})}{a_{ik}}|\qquad(*)$

but we in this peaper given new method for costructing a weight vector
$W=(w_{1},w_{2},\dots,w_{n})$ with using of jumps of matrix . This method has
the following advantages compared to other methods that are Short out the
answer and Makes much more efficient error (*). the firest we mention several
theorem about relation between jumps and SR-marix and transitive matrix.

######  Theorem 3.1.

A matrix $A\in M_{n}(n\geq 3)$ is symmetrically reciprocal if and only if the
production of all its subordinate jumps equals 1.

###### Proof.

Let $A$ be a symmetrically reciprocal matrix. By definition, a subordinate
jump contain some elements like $a_{ij},a_{ji}$ and $a_{ii}$ and by the
properties of $SR$ matrices on such a subordinate jump, $\prod a_{ij}=1$.
Other side of the theorem is obvious. ∎

######  Theorem 3.2.

A symmetrically reciprocal matrix $A$ is transitive if and only if the
production of all its jumps equals 1.

###### Proof.

Let $A$ be a transitive matrix, then there exist a wight vector
$w=(w_{1},w_{2},\dots,w_{n})$ such that $a_{ij}=w_{i}/w_{j}$. Form of
principal jumps is $a_{ij},a_{jk},a_{ki}$ and form of subordinated jump is
$a_{ij},a_{ji},a_{ii}$. So we have

(22)
$\frac{w_{i}}{w_{j}}\times\frac{w_{j}}{w_{k}}\times\frac{w_{k}}{w_{i}}=1\hskip
14.22636ptand\hskip
14.22636pt\frac{w_{i}}{w_{j}}\times\frac{w_{j}}{w_{i}}\times\frac{w_{i}}{w_{i}}=1.$

Other side of the theorem is obvious. ∎

######  Lemma 3.3.

For three integers $a,b,c$, following statements are equivalent:

  * (a)

$\frac{1}{1+c}\leq ab\leq 1+c,$

  * (b)

$|a-b|\leq ca$ and $|\frac{1}{a}-b|\leq\frac{c}{a}.$

###### Proof.

See [12]. ∎

######  Theorem 3.4.

Let $A\in M_{n}(\mathbb{R}_{+})$ and we have its determinant by Laplace
expansion. If $\prod a_{ij}$ be the production of elements of each jump,
$w=(w_{1},w_{2},\dots,w_{n})$ be the weight vector and $B=[b_{ij}]$ is a
matrix such that $b_{ij}=\frac{w_{i}}{w_{j}}$, then following statements are
equivalent:

  * (a)

$\frac{1}{1+c}\leq a_{ij}b_{ij}\leq 1+c,$

  * (b)

$|a_{ij}-b_{ij}|\leq ca_{ij}$ and
$|\frac{1}{a_{ij}}-b_{ij}|\leq\frac{c}{a_{ij}}.$

In special case, if the maximum production is of a principal jump, then

(23) $c=\sqrt[k]{\prod a_{ij}}-1,$

Where $\prod a_{ij}$ is the maximum product of jumps and $k$ is the number of
elements of it.

###### Proof.

Let $A$ be a $SR$ matrix. By lemma 3.3, set $a=a_{ij}$ and
$b=\frac{w_{j}}{w_{i}}$. So $(a)$ and $(b)$ are equivalent.
But by (7) we have

(24) $e(w)=\max|\frac{a_{ik}-w_{i}/w_{k}}{w_{ik}}|\hskip
14.22636pt\forall~{}w>0,$

And for every row of $A$ and $B$ we have

(25) $1+e(w)=\max(a_{ik}\frac{w_{i}}{w_{k}}).$

So

(26) $(1+e(w))^{k}=(\prod\max(a_{ik}~{}\frac{w_{i}}{w_{k}})=\max(\prod
a_{ij}~{}\prod\frac{w_{i}}{w_{k}}).$

But $\max\frac{w_{i}}{w_{k}}=1$. Therefore

(27) $(1+e(w))^{k}=\max\prod a_{ik}\Rightarrow(1+e(w))=\sqrt[k]{\prod
a{ik}}\Rightarrow e(w)=\sqrt[k]{\prod a{ik}}-1.$

∎

Let $A$ be a positive SR matrix and $\tau>0$. We define
$A^{\tau}=[a_{ij}^{\tau}]$ as follow:

(28)
$a_{ij}^{\tau}=\left\\{\begin{array}[]{lr}a_{i_{1}j_{1}}\tau&\textit{$(i_{1},j_{1})$
is the in the jump which $\mu(A)$ holds}\\\
a_{j_{1}i_{1}}/\tau&\qquad\textit{$(j_{1},i_{1})$ is the in the jump which
$1/\mu(A)$ holds}\\\ a_{ij}&\textit{other wise}\end{array}\right..$

######  Lemma 3.5.

Let $A$ be the matrix defined in 28 with $n\geq 2$. There exist positive
numbers $\tau_{1}$ and $\tau_{2}$ such that $\mu(\tau)$ is strictly decreasing
on $(0,\tau_{1})$, is constant on $(\tau_{1},\tau_{2})$ and is strictly
increasing on $(\tau_{2},+\infty)$. Moreover for
$\tau\in(0,\tau_{1})\cup(\tau_{2},+\infty)$, eigenvector is unique.

###### Proof.

Since $\mu(\tau)$ is constructed by elements of a principal jump and the
product of principal jumps increases on interval $(-\infty,\tau_{1})$, so
$\mu(\tau)$ decreases on this interval and increases on $(\tau_{2},\infty)$. ∎

######  Lemma 3.6.

By lemma 3.5, if we define $\mu_{0}=\min\\{\mu(\tau),\tau>0\\}$ then following
statements hold:

  * (a)

$\mu_{0}=1,\tau=\prod a_{ij},$

  * (b)

$\tau_{1}<\prod a_{ij},\tau_{2}>\prod a_{ij},$

  * (c)

$\tau=\emptyset,$

  * (d)

$\tau=\tau_{0}.$

###### Proof.

  * (a)

Since $A$ is a $SR$ matrix and by changing the elements of jump, the product
of its element changes, but this minimum equals $1$ if and only if $A$ is
transitive. So

(29) $\mu(\tau)=\sqrt[k]{\prod a_{ij}}=1\Rightarrow\mu(\tau)=\mu_{0}=1.$

  * (b)

Let $(i_{0},j_{0})$ be a coordinate which $a_{i_{0}j_{0}}$ is an element of
$\mu(\tau)=\sqrt[k]{\prod a_{ij}}$. Since by definition, changing the elements
$a_{ij}$ to $\frac{a_{ij}}{\tau}$ and changing the coordinates to $(j,i)$
cause change of elements to $a_{ij}\tau$, while $a_{ij}$ are in principal
jump, so the maximum change holds when it occures on a principal jump.

  * (c)

It is clear by definitions.

  * (d)

If $\tau=\tau_{0}$ Then $A$ is a transitive matrix with $\prod a_{ij}=1$. So
$rank(A)=1$.

∎

######  Theorem 3.7.

Let $\mu(\tau)$ be the eigenvector defined on 28and element that changes
$a_{ij}$, $0<\tau_{0}<\tau_{1}$ and $x,y$ be two max eigenvector of $A(\tau)$
such that

(30) $A(\tau_{0})\otimes x=\mu(\tau)x\hskip 14.22636ptA(\tau_{1})\otimes
y=\mu(\tau)y.$

Then

(31)
$\mu(\tau_{0})<\mu(\tau_{1})\Rightarrow\frac{y_{1}}{x_{1}}>\frac{y_{i}}{x_{i}}\quad
i\geq 2\quad
and\quad\mu(\tau_{0})>\mu(\tau_{1})\Rightarrow\frac{y_{2}}{x_{2}}<\frac{y_{i}}{x_{i}}\quad
i\neq 2.$

###### Proof.

Let A is a SR matrix and $a_{1,2}$ is changed to $a_{1,2}t$ than since
$\tau_{0}<\tau_{1}$ and by attention of propety of max algebra and by helping
of matix $A(\tau_{0})^{*}$ and $A(\tau_{1})^{*}$ that introduced in second
section we have

$\displaystyle\tau_{0}a_{1,2}x_{2}=\mu_{0}x_{1}$ $\displaystyle
a_{ik}x_{k}=\mu_{0}x_{i}$ $\displaystyle\vdots$ $\displaystyle
a_{ps}x_{s}=\mu_{0}x_{p}$

and

$\displaystyle\tau_{1}a_{1,2}x_{2}=\mu_{1}x_{1}$ $\displaystyle
a_{ik}x_{k}=\mu_{1}x_{i}$ $\displaystyle\vdots$ $\displaystyle
a_{ps}x_{s}=\mu_{1}x_{p}$

So we have the first equations

$\tau_{0}a_{1,2}x_{2}=\mu_{0}x_{1}\Rightarrow\mu_{0}=\tau_{0}a_{1,2}\dfrac{x_{1}}{x_{2}}$

and

$\tau_{1}a_{1,2}x_{2}=\mu_{1}x_{1}\Rightarrow\mu_{1}=\tau_{0}a_{1,2}\dfrac{x_{1}}{x_{2}}$

now since

$\mu_{0}<\mu_{1}\Rightarrow\tau_{0}a_{1,2}\dfrac{x_{1}}{x_{2}}<\tau_{0}a_{1,2}\dfrac{y_{1}}{y_{2}}$

therefor $\dfrac{x_{1}}{x_{2}}<\dfrac{x_{1}}{x_{2}}$ and other Equations we
can easily obtain Other inequality .
The second result follows from (i) by interchanging the roles of $A(\tau_{0})$
and $A(\tau_{1})$. ∎

######  Corollary 3.8.

With condition of Theorem 3.7, if $\mu(\tau_{0})=\mu(\tau_{1})$ then

(32) $\frac{y_{1}}{x_{1}}\geq\frac{y_{i}}{x_{i}}\geq\frac{y_{2}}{x_{2}}.$

###### Proof.

It is a direct conclusion of previews theromes. ∎

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