# Performance of QUIC Implementations Over Geostationary Satellite Links

Sebastian Endres, Jörg Deutschmann, Kai-Steffen Hielscher, and Reinhard German
Computer Networks and Communication Systems
University of Erlangen-Nuremberg
{basti.endres, joerg.deutschmann, kai-steffen.hielscher,
<EMAIL_ADDRESS>
###### Abstract

QUIC was recently standardized as RFC 9000, but the performance of QUIC over
geostationary satellite links is problematic due to the non-applicability of
Performance Enhancing Proxies. As of today, there are more than a dozen of
different QUIC implementations. So far performance evaluations of QUIC over
satellite links were limited to specific QUIC implementations. By deploying a
modified version of the IETF QUIC-Interop-Runner, this paper evaluates the
performance of multiple QUIC implementations over multiple geostationary
satellite links. This includes two emulated ones (with and without packet
loss) and two real ones. The results show that the goodput achieved with QUIC
over geostationary satellite links is very poor in general, and especially
poor when there is packet loss. Some implementations fail completely and the
performance of the other implementations varies greatly. The performance
depends on both client and server implementation.

##  I Introduction

Figure 1: Partial Screenshot of the QIR-SE Website Showing the Measurement
Results

Thanks to their wide-area coverage, satellites are the only option for
Internet access for some people. Today’s modern geostationary high throughput
satellites (GEO) provide data rates up to
$100\text{\,}\mathrm{Mbit}\text{/}\mathrm{s}$ in the downlink and even more in
the future [1]. However, compared to other Internet access technologies,
geostationary satellites suffer from high propagation delays. Together with
other delays, Round-Trip Times of $600\text{\,}\mathrm{ms}\text{/}$ and more
are typical [2]. Obviously, latency-sensitive applications are inherently
problematic for GEO satellites, but also the performance of transport
protocols like TCP is problematic over high-delay links because of slow-start
and long feedback loops. In order to mitigate poor TCP performance,
Performance Enhancing Proxies are deployed in satellite networks [3]. With
encrypted transport layer headers, as it is the case for VPNs and QUIC [4],
PEP can not be applied anymore. This leads to significant performance
degradation described in earlier publications [5, 6, 2, 7, 8, 9, 10] and this
work. Research, like [11, 12], that neglects the fact that PEP are usually
used to accelerate TCP connections, conclude that QUIC outperforms TCP on
these satellite links. This is also confirmed by some of the before mentioned
references, which compare QUIC’s performance with TCP used over VPNs. We
prepared a more detailed work in progress literature survey which is available
online111https://github.com/sedrubal/QUIC_HIGH_BDP/blob/master/research_overview.md
.

The QUIC-Interop-Runner (QIR)222https://interop.seemann.io [13] is used by the
IETF QUIC Working Group333https://datatracker.ietf.org/wg/quic and
https://quicwg.org for interoperability testing of QUIC implementations. It
already supports numerous implementations uniformly packaged into
Docker444https://docker.com containers and available on Docker
Hub555https://hub.docker.com . The runner tests every client implementation
with every server implementation three times per day and performs several
checks, like if a handshake completes successfully, or if 0-RTT works. The
following implementations, grouped by their role, were part of the QIR at the
time of writing:

  * •

Client & Server: aioquic, kwik, lsquic, msquic, mvfst, neqo, ngtcp2, picoquic,
quant, quic-go, quiche, quicly, xquic

  * •

Client only: chrome

  * •

Server only: nginx

For a detailed description of the implementations, please refer to the QIR
website or the QUIC Working Group666https://github.com/quicwg/base-
drafts/wiki/Implementations . In the official QIR, chrome was not supported
and quicly always failed. These two implementations are thus also ignored in
our tests.

The focus of the QIR is on testing interoperability, functionalities, and
standards compliance. There are also two performance measurement tests:
Goodput uses an emulated low-latency channel with a data rate of
$10\text{\,}\mathrm{Mbit}\text{/}\mathrm{s}$ to measure the average goodput
for the transfer of a large file, and CrossTraffic measures the goodput with a
competing iperf777https://iperf.fr transmission. These non-challenging link
parameters lead to good results for almost all implementations, i.e., the
achieved goodput is close to the physical layer link rate of
$10\text{\,}\mathrm{Mbit}\text{/}\mathrm{s}$. However, real Internet access
links, especially geostationary satellite links with high delays, are much
more challenging [14]. This motivated us to adapt the QIR to include
geostationary satellite links, which we then call QIR-
SE888https://interop.cs7.tf.fau.de . To the best of our knowledge, this is the
first time that the performance of a broad range of QUIC implementations over
geostationary satellite links is evaluated.

The architecture of QIR and QIR-SE is described in Section II. To gain further
insights into realistic use cases, we also conducted tests over real satellite
links. The configuration of the emulated satellite links and the used real
satellite accesses are described in Section III. Similar to QIR, all results
obtained with the QIR-SE are presented on a website8, which is shown in Fig.
1. In Section IV results are elaborated, which in addition to the original QIR
also include auto-generated time-offset diagrams. Section V concludes this
paper and suggests future research.

##  II Architecture

The architecture of QIR is shown in Fig. 2a. It employs Docker
Compose999Docker Compose: https://docs.docker.com/compose with virtual
networks to deploy the entire test environment on a single host machine. Most
test cases require a setup of three containers: The QUIC client
implementation, the QUIC server implementation and a container containing the
ns-3 Network Simulator101010https://nsnam.org in between. Additional
containers can be deployed for special test cases. For the link emulator,
different ns-3 scenarios are used, which are available online111111Repository
of ns-3 scenarios: https://github.com/marten-seemann/quic-network-simulator .
The runner passes TLS keys and certificates [15] through the Docker volume
/certs into the end point containers. The implementations have to generate log
files in the Docker volume /logs. Servers have to serve files with random
content from the volume /www and clients have to download them to the volume
/downloads, preferably by using HTTP/0.9, which is very simple and has a
minimal overhead121212Discussion about using HTTP/0.9 as protocol:
https://github.com/marten-seemann/quic-interop-runner/issues/267 . During the
run, ns-3 is used to capture packets on the wire into Pcap files. The traces
are then analyzed using pyshark131313https://kiminewt.github.io/pyshark , a
Python wrapper for tshark from the Wireshark protocol
analyzer141414https://wireshark.org tool set. As each client implementation is
tested with each server implementation, a two-dimensional matrix is filled
with test results, which is visualized in Figs. 1 and 4. It is stored as JSON
file. We use this information and the captured traces to automatically
generate time-offset plots for each combination of implementation afterwards,
as shown in Section IV-D.

qlog, keylogpcapqlog, keylogDocker ComposeXQUIC-Interop-
RunnerHostClientCns-3CServerdeploydeploydeployrun

(a) Architecture of the Original QIR used for Emulations

qlog, keylogpcapqlog, keylogServerTcpdumppcap, qlog,
keylogClientTcpdumplogsModemInternetLinkQUIC-Interop-
RunnerDockercalldeployServer HostSSHDockerdeployClient Hostcontrol

(b) Distributed Architecture of QIR used for Real Satellite Links

Figure 2: Architecture of The QIR & QIR-SE

###  II-A Emulation

To emulate satellite links, which have usually different properties in the
forward and return path, we added a new asymmetric ns-3
scenario151515%****␣paper.tex␣Line␣275␣****https://github.com/sedrubal/quic-
network-simulator/tree/feature-asymmetric-p2p/sim/scenarios/asymmetric-p2p .
The data rate, emulated queue size and Packet Loss Rate (PLR) can be
configured individually per direction. The artificial delay applies to both
directions.

###  II-B Real Satellite Links

Additionally, we modified the runner to use real network links between the
implementations under test. We replaced the deployment mechanism, which was
built upon Docker Compose, with direct calls to the Docker API to possibly
distributed Docker deamons, as visualized in Fig. 2b. In our setup, we have
one dedicated machine per satellite access. QIR-SE picks the according machine
per test case to deploy the client. The server is executed on the same machine
as the QIR-SE itself. As ns-3 can no longer be used to capture the Pcap
traces, we run a tcpdump container per end point, which joins the network of
the implementation container on the correspondent host.

##  III Test Setup

All measurements have been performed on the following systems:

The emulated scenarios have been executed on an Ubuntu 20.04.3 LTS server with
Kernel 5.4.0 and Docker 20.10.10. The server is powered by an _Intel® Xeon®
X5650_ CPU and has $16\text{\,}\mathrm{GiB}\text{/}$ of RAM installed.

For measurements with real satellite connections, the server implementations
have been executed on the aforementioned machine. The clients have been run on
other computers, which are connected to the modems of the satellite link.
These computers run Ubuntu 18.04.6 LTS with Kernel 5.4.0 and Docker 20.10.7.
Both computers are powered by an Intel® Core™ i5 4590 CPU and
$8\text{\,}\mathrm{GiB}\text{/}$ RAM each.

###  III-A Emulation

We use the Terrestrial scenario, which equals the Goodput scenario in the
original QIR, but with an increased data rate of
$20\text{\,}\mathrm{Mbit}\text{/}\mathrm{s}$, to compare the satellite
scenarios with a terrestrial link. The RTT is $30\text{\,}\mathrm{ms}\text{/}$
and the queue size is 25 packets. Additionally, we added two performance
measurement tests with path properties similar to satellite links: Sat and
SatLoss. They differ in the artificial PLR, which is set to
$0\text{\,}\mathrm{\char 37\relax}\text{/}$ in the first scenario and to
$1\text{\,}\mathrm{\char 37\relax}\text{/}$ in the latter one. In both cases,
we set the data rate to 20/$2\text{\,}\mathrm{Mbit}\text{/}\mathrm{s}$
(forward / return link) and the one-way delay to
$300\text{\,}\mathrm{ms}\text{/}$, which results in an RTT of
$600\text{\,}\mathrm{ms}\text{/}$. All emulated test cases have been repeated
ten times per combination of implementation.

###  III-B Real Satellite Links

We also ran measurements using two real end-user satellite accesses: Astra and
Eutelsat, which are available in Europe. The satellite dishes and terminals
are located at the University of Erlangen-Nürnberg, Martensstr. 3, Erlangen,
Germany. Details about the tariffs are listed in Table I. The first one
advertises a data rate less than half of the data rate of the second one.
While Eutelsat prioritizes traffic up to $60\text{\,}\mathrm{GB}\text{/}$, we
did not run into that limit during the measurements. Both operators provide
IPv4 connectivity through carrier-grade NAT, and IPv6 is not available.
Further testimonials of the products can be found in a previous measurement
campaign161616https://www.cs7.tf.fau.eu/research/quality-of-service/qos-
research-projects/sat-internet-performance , where measurements assessed a
stable performance of the real satellite links, and the throughput achieves
the advertised link rates. To be on the safe side, our measurements were
carried out during constantly good weather conditions and paused between
potentially congested times (i.e., between 6 p.m. and 11 p.m. local time).
Within one run, the execution of client and server combinations was shuffled,
and each combination was repeated five times.

TABLE I: Measurements with Real Satellite Access Test Case |  Provider & Tariff |  Advertised Data Rate |  Traffic Limit |  Modem
---|---|---|---|---
Astra |  Novostream Astra Connect L+ |  20/$2\text{\,}\mathrm{Mbit}\text{/}\mathrm{s}$ |  — |  Gilat SkyEdge II-c
Eutelsat |  Konnect Zen |  50/$5\text{\,}\mathrm{Mbit}\text{/}\mathrm{s}$ |  priorit. up to $60\text{\,}\mathrm{GB}\text{/}$ |  Hughes HT2000W

##  IV Evaluation

Before we start with the evaluation, we would like to emphasize that not all
implementations might strive for high performance. Some might be only proof of
concept implementations, others might have simplicity or resource efficiency
as primary goal, and others might not have been optimized for high latency
links yet. Yet, the performance degradation compared to the emulated
Terrestrial scenario will be clearly visible.

The evaluation is split into four parts. First, we present heatmaps inspired
by the QIR-SE website and give an overview considering all results, followed
by a discussion whether client or server implementation influences the
outcomes more. Next, we try to analyze the impact of different Congestion
Control Algorithms. Lastly, we pick specific combinations and present their
behavior by means of time-offset graphs.

###  IV-A Overview

To visualize the result matrices of the measurements generated with the QIR-SE
in more detail, we rendered a heatmap for each scenario (Fig. 4). The columns
belong to the server implementation and the rows to the clients. We omitted
non-functioning implementations. The color scale is normalized to the minimum
and maximum observed goodput value of each scenario. The size of the circles
are synchronized between all charts. Currently, we only distinguish between
unspecified errors (marked with ) and timeouts (marked with $T$).

TABLE II:  Overview of Measurement Results
For the columns Mean and Maximum, the absolute goodput values are in the left column and the relative efficiency values in the right column.  Measurement |  Mean |  Maximum |  Timeout $T$ |  Failed
---|---|---|---|---
|  $[$\mathrm{Mbit}\text{/}\mathrm{s}$]$ |  $[$\mathrm{\char 37\relax}\text{/}$]$ |  $[$\mathrm{Mbit}\text{/}\mathrm{s}$]$ |  $[$\mathrm{\char 37\relax}\text{/}$]$ |  $[$\mathrm{\char 37\relax}\text{/}$]$ |  $[$\mathrm{\char 37\relax}\text{/}$]$
Terrestrial | 15.11 | 76 | 19.2 | 96 | 12 | 1
Sat | 5.05 | 25 | 12.0 | 60 | 3 | 6
SatLoss | 3.06 | 15 | 11.5 | 57 | 13 | 17
Astra | 4.91 | 25 | 13.5 | 68 | 23 | 15
Eutelsat | 6.98 | 14 | 17.5 | 35 | 20 | 11

qlog, keylogpcapqlog, keylogServerTcpdumppcap, qlog,
keylogClientTcpdumplogs$0\text{\,}\mathrm{\char
37\relax}\text{/}$$20\text{\,}\mathrm{\char
37\relax}\text{/}$$40\text{\,}\mathrm{\char
37\relax}\text{/}$$60\text{\,}\mathrm{\char
37\relax}\text{/}$$80\text{\,}\mathrm{\char
37\relax}\text{/}$$100\text{\,}\mathrm{\char
37\relax}\text{/}$Efficiency$0\text{\,}\mathrm{\char
37\relax}\text{/}$$20\text{\,}\mathrm{\char
37\relax}\text{/}$$40\text{\,}\mathrm{\char
37\relax}\text{/}$$60\text{\,}\mathrm{\char
37\relax}\text{/}$$80\text{\,}\mathrm{\char
37\relax}\text{/}$$100\text{\,}\mathrm{\char
37\relax}\text{/}$ProportionMeasurementTerrestrialSatSatLossAstraEutelsat

Figure 3: CDFs Using Normalized Efficiency Values as Defined in Equation 1

qlog, keylogpcapqlog, keylogServerTcpdumppcap, qlog,
keylogClientTcpdumplogsaioquickwiklsquicmsquicmvfstneqonginxngtcp2picoquicquantquic-
goquichexquicServerxquicquichequic-
goquantpicoquicngtcp2neqomvfstmsquiclsquickwikaioquicClientGoodput$2.5\text{\,}\mathrm{Mbit}\text{/}\mathrm{s}$$5.0\text{\,}\mathrm{Mbit}\text{/}\mathrm{s}$$7.5\text{\,}\mathrm{Mbit}\text{/}\mathrm{s}$$10.0\text{\,}\mathrm{Mbit}\text{/}\mathrm{s}$

(a) Average Goodput of Measurement Sat

qlog, keylogpcapqlog, keylogServerTcpdumppcap, qlog,
keylogClientTcpdumplogsaioquickwiklsquicmsquicmvfstneqonginxngtcp2picoquicquantquic-
goquichexquicServerxquicquichequic-
goquantpicoquicngtcp2neqomvfstmsquiclsquickwikaioquicClientGoodput$2.5\text{\,}\mathrm{Mbit}\text{/}\mathrm{s}$$5.0\text{\,}\mathrm{Mbit}\text{/}\mathrm{s}$$7.5\text{\,}\mathrm{Mbit}\text{/}\mathrm{s}$$10.0\text{\,}\mathrm{Mbit}\text{/}\mathrm{s}$

(b) Average Goodput of Measurement SatLoss

qlog, keylogpcapqlog, keylogServerTcpdumppcap, qlog,
keylogClientTcpdumplogsaioquickwiklsquicmsquicmvfstneqonginxngtcp2picoquicquantquic-
goquichexquicServerxquicquichequic-
goquantpicoquicngtcp2neqomvfstmsquiclsquickwikaioquicClientGoodput$2.5\text{\,}\mathrm{Mbit}\text{/}\mathrm{s}$$5.0\text{\,}\mathrm{Mbit}\text{/}\mathrm{s}$$7.5\text{\,}\mathrm{Mbit}\text{/}\mathrm{s}$$10.0\text{\,}\mathrm{Mbit}\text{/}\mathrm{s}$

(c) Average Goodput of Measurement Astra

qlog, keylogpcapqlog, keylogServerTcpdumppcap, qlog,
keylogClientTcpdumplogsaioquickwiklsquicmsquicmvfstneqonginxngtcp2picoquicquantquic-
goquichexquicServerxquicquichequic-
goquantpicoquicngtcp2neqomvfstmsquiclsquickwikaioquicClientGoodput$3.0\text{\,}\mathrm{Mbit}\text{/}\mathrm{s}$$6.0\text{\,}\mathrm{Mbit}\text{/}\mathrm{s}$$9.0\text{\,}\mathrm{Mbit}\text{/}\mathrm{s}$$12.0\text{\,}\mathrm{Mbit}\text{/}\mathrm{s}$$15.0\text{\,}\mathrm{Mbit}\text{/}\mathrm{s}$

(d) Average Goodput of Measurement Eutelsat

Figure 4: Result Matrices (: Unspecified Error, $T$: Timeout)

In addition to the heatmaps, we gathered the measurement results in Table II
and as Cumulative Distribution Functions in Fig. 3. For each scenario, the
table shows the mean and maximum value of all experiments and the amount of
combinations that either ran into a timeout or failed for an unknown reason.
On the left side of each column, the absolute goodput values are used (as in
the heatmaps). On the right side, as well as in the CDF, we use the
_efficiency_ , which is defined as the goodput value divided by the emulated
or advertised link data rate:

$\text{Efficiency}=\frac{\text{Goodput}}{\text{Link Data Rate}}$ (1)

Besides the emulated satellite scenarios Sat and SatLoss and the real
satellite links Astra and Eutelsat, the table and the CDF plot also contain
the results for the Terrestrial scenario as reference. Unsuccessful runs are
not considered in the calculation of the mean values in Table II but plotted
as $0\text{\,}\mathrm{\char 37\relax}\text{/}$ efficiency in the CDF in Fig.
3.

First, we take a look at the high number of failed experiments for all
scenarios. The number of failures and timeouts is rather high in the
Terrestrial scenario because of neqo, which did not work as a client in this
scenario for unknown reasons, although it worked in our satellite scenarios.
The reason lsquic fails as a server when measuring with real satellite links
is technical and caused by the Docker image provided by the maintainers. In
the SatLoss, Astra and Eutelsat scenarios we can see a lot of implementations,
that were not able to transfer the test file of
$10\text{\,}\mathrm{MiB}\text{/}$ via the corresponding links. A more detailed
analysis of the reasons of failures is subject to future work.

As one can see in Table II and Fig. 3, the Terrestrial scenario achieves
goodput values close to the link rate. This is not the case for geostationary
satellite scenarios. Instead, there are significant differences between
implementations.

Although Sat, shown in Fig. 4a, has stable link emulation and no artificial
loss, there is a lot of variance in the achieved goodput. Some implementations
perform significantly better than the others. Yet, the achieved goodput is far
below the advertised link rate of
$20\text{\,}\mathrm{Mbit}\text{/}\mathrm{s}$. In half of the measurements, it
is even below $25\text{\,}\mathrm{\char 37\relax}\text{/}$ of the link data
rate. When all entries of a row (respectively column) show poor results, the
implementation performs poor as client (respectively server). A more detailed
analysis of the client vs. server performance of the Sat scenario is given in
Section IV-B.

SatLoss performs worst among all scenarios when considering the absolute
goodput values, see Table II. Figure 4b shows which implementations suffer
most from the $1\text{\,}\mathrm{\char 37\relax}\text{/}$ artificial PLR added
to the emulated satellite link171717With QUIC, a packet loss on any path
segment will impact the end to end transmission, while PEP are able to
retransmit TCP segments locally. . Compared to the no-loss scenario, many
implementations do either not finish a transmission successfully or achieve
considerably lower goodput rates. The efficiency decreases from only
$25\text{\,}\mathrm{\char 37\relax}\text{/}$ in the Sat scenario to even less
in the SatLoss scenario ($15\text{\,}\mathrm{\char 37\relax}\text{/}$). There
are only a few combinations that perform well in the SatLoss scenario, with
picoquic as client and server for example being more thoroughly analyzed later
in Section IV-D1.

The performance of the real satellite operators is visualized in more breadth
in Fig. 4c (Astra) and Fig. 4d (Eutelsat). The PLR on these real satellite
links is usually very low16, but higher dynamics can be expected on real
links. Compared to the Sat scenario, this results in less combinations
finishing successfully and lower goodput values. There is a correlation
between the results for Astra and Eutelsat, i.e., most combinations perform
similarly on both systems. Eutelsat reaches the highest mean and maximum
goodput rates in our tests (note different scale in Fig. 4d). However, given
the forward link data rates (Eutelsat
$50\text{\,}\mathrm{Mbit}\text{/}\mathrm{s}$ vs. other satellite scenarios
with $20\text{\,}\mathrm{Mbit}\text{/}\mathrm{s}$), the efficiency of Eutelsat
is with about $14\text{\,}\mathrm{\char 37\relax}\text{/}$ on average actually
worse than Astra ($25\text{\,}\mathrm{\char 37\relax}\text{/}$). This shows
that more bandwidth does not automatically result in a better performance by
the same factor. The link utilization on both real satellite links is very
low, similar to the results of our simulated test cases.

###  IV-B Client vs. Server Implementation

qlog, keylogpcapqlog, keylogServerTcpdumppcap, qlog,
keylogClientTcpdumplogsaioquickwiklsquicmsquicmvfstneqongtcp2picoquicquantquic-
goquichexquicImplementation02.55.07.510.012.515.017.520.0Goodput
$[$\mathrm{Mbit}\text{/}\mathrm{s}$]$SATRoleServerClient020406080100Efficiency
$[$\mathrm{\char 37\relax}\text{/}$]$

Figure 5: Violin Plots for Each Implementation in the Sat Scenario. Horizontal
Lines represent the Quartiles $Q_{1}$, $Q_{2}$ and $Q_{3}$.

As the server has to send most of the data and therefore has to estimate the
channel parameters and especially the bottleneck bandwidth, it seems likely
that the performance of the transmission is mainly determined by the server.
However, the result matrices in Fig. 4 show no pattern that confirms this
assumption, instead both the client and the server contribute to the overall
performance of a QUIC connection.

In order to verify this statement, we draw violin plots181818Seaborn
Documentation for violin plots that employ kernel density estimation (KDE).
https://seaborn.pydata.org/generated/seaborn.violinplot.html as shown in Fig.
5. Due to space limitations, we only present the plot for the Sat scenario.
Other scenarios lead to similar results. The orange graphs show the
performance of a specific server implementation tested with all clients, and
the blue graph shows the performance of a specific client implementation
tested with all servers.

Again, it can be seen that the achieved goodput is far off the emulated link
rate. It also becomes clearer, that kwik does neither perform well as client
nor as server, aioquic performs well if used as client but not as server, and
mvfst and ngtcp2 perform well if used as server but not as client. An
interesting observation from Fig. 5 is that results are often concentrated
towards one end of the distribution. Especially when looking at the server
role, this means that these server implementations either result in very good
or very poor results. Thus, we conclude that both the client and the server
contribute to the overall performance of a QUIC connection.

###  IV-C Impact of the Congestion Control Algorithm

TABLE III: Implementations and their CCAs.
Algorithms are determined by hand and must be taken with caution. The default CCA is printed bold.  Name |  CCA |  HyStart
---|---|---
aioquic |  NewReno |
chrome |  BBRv2, CUBIC |
kwik |  NewReno |
lsquic |  BBR, CUBIC |
msquic |  CUBIC |
mvfst |  BBR, CUBIC, NewReno, … |
neqo |  CUBIC, NewReno |
nginx |  ?⃝ |
ngtcp2 |  BBRv2, BBR, CUBIC, Reno |
picoquic |  BBR, CUBIC |
quant |  NewReno |
quic-go |  CUBIC ?⃝, Reno ?⃝ |
quiche |  CUBIC |
quicly |  CUBIC, Reno, pico |
xquic |  BBR, CUBIC, Reno |

qlog, keylogpcapqlog, keylogServerTcpdumppcap, qlog,
keylogClientTcpdumplogs$0\text{\,}\mathrm{\char
37\relax}\text{/}$$10\text{\,}\mathrm{\char
37\relax}\text{/}$$20\text{\,}\mathrm{\char
37\relax}\text{/}$$30\text{\,}\mathrm{\char
37\relax}\text{/}$$40\text{\,}\mathrm{\char
37\relax}\text{/}$$50\text{\,}\mathrm{\char
37\relax}\text{/}$$60\text{\,}\mathrm{\char
37\relax}\text{/}$$70\text{\,}\mathrm{\char 37\relax}\text{/}$Efficiency in
Sat$0\text{\,}\mathrm{\char 37\relax}\text{/}$$10\text{\,}\mathrm{\char
37\relax}\text{/}$$20\text{\,}\mathrm{\char
37\relax}\text{/}$$30\text{\,}\mathrm{\char
37\relax}\text{/}$$40\text{\,}\mathrm{\char
37\relax}\text{/}$$50\text{\,}\mathrm{\char
37\relax}\text{/}$$60\text{\,}\mathrm{\char
37\relax}\text{/}$$70\text{\,}\mathrm{\char 37\relax}\text{/}$Efficiency in
SatLossServer CCARenoNewRenoCUBICBBRunknown

Figure 6: Relation of Measurement Results Between Sat & SatLoss
Points are colored according to the CCA of the server implementation.

As we assume that the CCA of the server implementation has a large impact on
the performance of a connection, we determined the supported algorithms and
collected them in Table III. For each implementation, we tried to identify all
implemented algorithms and the default one by inspecting the available command
line flags of the implementations in the Docker images used by the runner or
the corresponding code if available. As there is no database or standardized
interface to query the CCAs, it is not guaranteed that the information in
Table III is correct. Additionally, it should be noted that the coding and
parametrization of the same CCAs might differ substantially among
implementations.

Figure 6 displays a relational plot of the measurement results from the Sat
scenario (horizontal) and the SatLoss scenario (vertical). Again, we use the
normalized efficiency values as defined in Eq. 1. Each point corresponds to
the mean of the efficiencies achieved by the corresponding implementation
tuple over all ten iterations in the corresponding scenario. Two clusters of
measurement results can be identified: The first ([regular] brown) one shows
no correlation between both experiments. While there are medium results in the
Sat scenario, the results achieved in the SatLoss scenario (with a PLR of
$1\text{\,}\mathrm{\char 37\relax}\text{/}$) are altogether very poor. The
other group ([regular] turquoise) shows a correlation between both
experiments, and the results in SatLoss are quite similar and only slightly
poorer than in Sat.

We use the previously determined CCAs to color the points in that plot
according to the algorithm used by the server implementation. In the turquoise
group of data points achieving a link utilization of more than
$10\text{\,}\mathrm{\char 37\relax}\text{/}$ in the SatLoss scenario, there
are only combinations where the server uses CUBIC or BBR. The other group
contains combinations with Reno, NewReno and CUBIC.

This could be an indication for CUBIC and BBR working better in lossy
scenarios than Reno and NewReno. However, it might also be possible that
implementations that employ one of the much more complex CCAs are more
optimized in general. It is hard to draw a more precise conclusion without
deeper knowledge of the code of the implementations.

###  IV-D Time-Offset Graphs

While we try to render and annotate time-offset plots using captured Pcap
traces, this does not succeed for every combination for various reasons. For
example, some combinations do not use UDP port 443, some use HTTP/3 instead of
HTTP/0.9 as requested by the runner, or there are issues with standard
compliance. In this section, we chose five plots that show interesting
behavior and discuss them in more detail. More plots can be found on our
website8. In every plot there are the traces of all iterations of the same
implementation tuple in the according measurement scenario (ten for Sat and
SatLoss, five for Astra and Eutelsat). That one with the medium time to
completion is highlighted colored. The others are gray. If an offset number is
transmitted the first time, the corresponding point in the plot is colored
blue. Retransmissions are colored orange.

####  IV-D1 picoquic—picoquic—SatLoss

Compared to the other implementations, picoquic often achieves good results.
It seems to be built with satellite use-cases in mind. Even in challenging
scenarios, the transmission makes steady progress. Figure 7 shows the
transmission of picoquic as client and as server in the SatLoss scenario—the
simulated scenario with $1\text{\,}\mathrm{\char 37\relax}\text{/}$ PLR.
According to our analysis in Table III, BBR was used in this transmission. As
soon as the bottleneck bandwidth is estimated, the data is transmitted
sequentially at a fairly constant rate and single lost packets are re-
transmitted out-of-order. Picoquic seems to automatically retransmit the last
bytes of the file multiple times to avoid waiting one RTT until the client can
signal a loss. The entire transmission takes slightly more than
$7\text{\,}\mathrm{s}\text{/}$. This equals a goodput of slightly more than
$11\text{\,}\mathrm{Mbit}\text{/}\mathrm{s}$ which equals a link utilization
of slightly more than $50\text{\,}\mathrm{\char 37\relax}\text{/}$.

Figure 7: Time-Offset Plot of Picoquic—Picoquic—SatLoss

####  IV-D2 kwik—msquic—Sat

The example in Fig. 8, with kwik as server using NewReno as CCA, shows a slow
startup phase of about $11\text{\,}\mathrm{s}\text{/}$. Most implementations
that are not optimized for satellite scenarios have this in common. Some
implementations, like aioquic, even never reach a steady state. In this
example there is a second striking behavior: In some iterations, the final
transmission rate is set to a much smaller value than in other iterations.
This results in very different time to completions between iterations of the
same experiment—even though no nondeterministic packet losses are emulated in
the Sat test case. Additionally, the jagged curves indicate bad pacing.

Figure 8: Time-Offset Plot of Kwik—Msquic—Sat

####  IV-D3 lsquic—xquic—Sat

The analysis of the transmissions shown in Fig. 9 reveals remarkably many
retransmissions. Since no artificial losses are introduced in the Sat
scenario, the retransmissions might indicate a bug in either of the
implementations. However, some sequence numbers in the range between 3 and
$4\text{\,}\mathrm{MiB}\text{/}$ seem not to be used sequentially. The second
almost vertical line in this range is not drawn in orange, which indicates
that the according packets are no retransmitted ones. Such behavior should be
analyzed in the future.

Figure 9: Time-Offset Plot of Lsquic—Xquic—Sat

####  IV-D4 msquic—xquic—SatLoss

The plot in Fig. 10 shows very different but high values for the time to
completion in the SatLoss scenario. According to Table III, loss-based CUBIC
was (most likely) used in this scenario. The losses lead to high drops of the
data rate and recovery seems to be very slow probably due to the high RTT.
When many losses occur in quick succession, the data rate is throttled more
heavily. When losses indicate congestion, it is fine to throttle the
transmission rate. However, when packets get lost due to temporary effects on
a path segment, there is no benefit in reducing the transmission data rate.
Instead, doing so has a severe impact on the performance via the satellite
link.

Figure 10: Time-Offset Plot of Msquic—Xquic—SatLoss

####  IV-D5 picoquic—ngtcp2—Astra

A way to avoid dreaded retransmissions is sending each packet multiple times
even when it was not lost. This can speed up the transmission because it is
not necessary to wait for the ACK of the recipient, which has proven to be
beneficial in previous Section IV-D1 where picoquic was used as server, too.
On the other hand, this optimization causes additional traffic on the link and
thus should be used carefully. In Fig. 11, it can be observed that such
optimistic retransmissions leads to lots of redundantly sent data, because
every packet is retransmitted six to eight times. Although only a file of
$10\text{\,}\mathrm{MiB}\text{/}$ is transferred, this leads to approximately
$6080\text{\,}\mathrm{MiB}\text{/}$ of data being sent.

Figure 11: Time-Offset Plot of Picoquic—Ngtcp2—Astra

##  V Conclusion and Future Work

Due to the non-applicability of PEP, the performance of QUIC via geostationary
satellite links depends mainly on the end points. We presented a tool based on
the QIR that allows running measurements via real and emulated satellite links
with numerous QUIC implementations. It also automatically generates time-
offset plots for more detailed analysis. Our results show that differences
between implementations are very high. Many implementations even fail to
transfer a medium-sized file over satellite links. The successful ones usually
achieve a very poor goodput. The performance is even poorer when artificial
packet losses are introduced. Increasing the link data rate does not
automatically increase the achieved goodput in the same ratio. According to
the results of the combinations of implementations, we can conclude that both
the server and the client contribute to the overall performance of the
transmission.

Regarding future work, multiple directions can be thought of. Additional
performance test cases (e.g., short transfers of small objects, enabling
explicit congestion notification, fairness tests, etc.) would provide more
insights into the performance of QUIC over satellite networks. This should
also include different parameters for QUIC protocol stacks [14] and extensions
like the 0-RTT-BDP draft [16]. Testing could include more satellite systems,
including both geostationary and Low Earth Orbit (LEO) megaconstellation
satellite systems (e.g., Starlink). Finally, long-term measurements would be
helpful to track changes of QUIC implementations. This should also include
investigations regarding the high number of unsuccessful runs observed in the
QIR-SE.

## Acknowledgement

We would like to thank Marten Seemann and all contributors for developing and
maintaining the QIR.

This work has been funded by the Federal Ministry for Economic Affairs and
Climate Action in the project QUICSAT.

## References

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All Internet links were last accessed on 2022-02-12.

  *[RFC]: Request for Comments
  *[QIR-SE]:  Satellite Edition
  *[TCP]: Transmission Control Protocol
  *[VPNs]: Virtual Private Network
  *[PEP]: Performance Enhancing Proxy
  *[QIR]: QUIC-Interop-Runner
  *[IETF]: Internet Engineering Task Force
  *[TLS]: Transport Layer Security
  *[HTTP]: Hypertext Transfer Protocol
  *[JSON]: JavaScript Object Notation
  *[PLR]: Packet Loss Rate
  *[API]: Application Programming Interface
  *[CPU]: Central Processing Unit
  *[RAM]: Random Access Memory
  *[RTT]: Round-Trip Time
  *[NAT]: Network Address Translation
  *[CDFs]: Cumulative Distribution Function
  *[CDF]: Cumulative Distribution Function
  *[KDE]: kernel density estimation
  *[CCAs]: Congestion Control Algorithm
  *[CCA]: Congestion Control Algorithm
  *[BBR]: Bottleneck Bandwidth and Round-trip propagation time
  *[UDP]: User Datagram Protocol
  *[ACK]: Acknowledgement
  *[LEO]: Low Earth Orbit


# Experimental demonstration of quantum cooling engine powered by entangled
measurement

Ning-Ning Wang  Key Laboratory of Quantum Information, University of Science
and Technology of China, CAS, Hefei, 230026, People’s Republic of China  CAS
Center For Excellence in Quantum Information and Quantum Physics, University
of Science and Technology of China, Hefei, 230026, People’s Republic of China
Hefei National Laboratory, University of Science and Technology of China,
Hefei 230088, China  Huan Cao  Key Laboratory of Quantum Information,
University of Science and Technology of China, CAS, Hefei, 230026, People’s
Republic of China  University of Vienna, Faculty of Physics, Vienna Center for
Quantum Science and Technology (VCQ) and Research platform TURIS,
Boltzmanngasse 5, 1090 Vienna, Austria  Chao Zhang<EMAIL_ADDRESS>Key
Laboratory of Quantum Information, University of Science and Technology of
China, CAS, Hefei, 230026, People’s Republic of China  CAS Center For
Excellence in Quantum Information and Quantum Physics, University of Science
and Technology of China, Hefei, 230026, People’s Republic of China  Hefei
National Laboratory, University of Science and Technology of China, Hefei
230088, China  Xiao-Ye Xu  Key Laboratory of Quantum Information, University
of Science and Technology of China, CAS, Hefei, 230026, People’s Republic of
China  CAS Center For Excellence in Quantum Information and Quantum Physics,
University of Science and Technology of China, Hefei, 230026, People’s
Republic of China  Hefei National Laboratory, University of Science and
Technology of China, Hefei 230088, China  Bi-Heng Liu  Key Laboratory of
Quantum Information, University of Science and Technology of China, CAS,
Hefei, 230026, People’s Republic of China  CAS Center For Excellence in
Quantum Information and Quantum Physics, University of Science and Technology
of China, Hefei, 230026, People’s Republic of China  Hefei National
Laboratory, University of Science and Technology of China, Hefei 230088, China
Yun-Feng Huang<EMAIL_ADDRESS>Key Laboratory of Quantum Information,
University of Science and Technology of China, CAS, Hefei, 230026, People’s
Republic of China  CAS Center For Excellence in Quantum Information and
Quantum Physics, University of Science and Technology of China, Hefei, 230026,
People’s Republic of China  Hefei National Laboratory, University of Science
and Technology of China, Hefei 230088, China  Chuan-Feng Li<EMAIL_ADDRESS>Key Laboratory of Quantum Information, University of Science and Technology of
China, CAS, Hefei, 230026, People’s Republic of China  CAS Center For
Excellence in Quantum Information and Quantum Physics, University of Science
and Technology of China, Hefei, 230026, People’s Republic of China  Hefei
National Laboratory, University of Science and Technology of China, Hefei
230088, China  Guang-Can Guo  Key Laboratory of Quantum Information,
University of Science and Technology of China, CAS, Hefei, 230026, People’s
Republic of China  CAS Center For Excellence in Quantum Information and
Quantum Physics, University of Science and Technology of China, Hefei, 230026,
People’s Republic of China  Hefei National Laboratory, University of Science
and Technology of China, Hefei 230088, China

###### Abstract

Traditional refrigeration is driven either by external force or an
information-feedback mechanism. Surprisingly, the quantum measurement and
collapse, which are generally detrimental, can also be used to power a cooling
engine even without requiring any feedback mechanism. In this work, we
experimentally demonstrate quantum measurement cooling (QMC) powered by
entangled measurement by using a novel linear optical simulator. In the
simulator, different thermodynamic processes can be simulated by adjusting the
energy-level spacing of working substance and the temperature of thermal bath.
We show experimentally that, without prior knowledge about the measurement to
be made, QMC remains likely to occur. We also demonstrate that QMC is robust
against measurement noise. Those experimental results show that quantum
measurement is not always detrimental but can be a valuable thermodynamic
resource.

Introduction.– Quantum measurement is one of the most peculiar concepts of
quantum mechanics. Contrary to its classical counterpart, the wave function of
the measured system will collapse to an eigenstate of the observable with a
certain probability. Von Neumann describes the quantum measurement process to
be an interaction between system and measurement apparatus von , the combined
system evolves unitarily according to the Schrödinger equation while the
evolution of the system alone is nonunitary. Unlike some other quantum effects
such as coherence coherence1 ; coherence2 ; coherence3 ; coherence4 ;
coherence5 , quantum correlation entanglement1 ; entanglement2 ; cor1 ; cor2 ,
or squeezed bath squeezed1 ; squeezed2 ; squeezed3 , acting as enhancement
roles in some thermodynamic tasks, the quantum measurement is generally
thought to be detrimental backaction1 ; backaction2  because its induced
backaction destroys the quantum coherence of measured system between different
measurement bases. However, this genuine quantum effect can also be helpful in
some cases, e.g. to fuel a quantum engine, which indicates a quantum
measurement can also be viewed as a useful thermodynamic resource res .

Previously, discussions on the role of measurement in thermodynamic engines
have focused mostly on the aspects of extracting information inf1 ; inf2 ;
inf3 ; inf4 . For example, in the mechanism of Maxwell’s demon, information
gained from the measurement of the working substance can be used to design
appropriate feedback to achieve thermodynamic tasks demon1 ; demon2 ; demon3 ;
demon4 ; demon5 ; demon6 ; demon7 ; demon8 . In this scenario, the thermal
bath is usually required to play the role of energy source. However, in the
quantum scenario, the measurement can be invasive, meaning that energy
exchange between the working substance and measurement apparatus occurs when
the observable is not commutative with the Hamiltonian of working substance.
Specifically, the post-measurement state can contain more energy than its
initial state. In this sense, the measurement itself can also be an energy
source. Based on this principle, heat engines with feedback (quantum Maxwell’s
demon) or without feedback (ignore the measurement results) have been proposed
QEM1 ; QEM2 ; QEM3 ; QEM4 ; QEM5 ; QEM6 ; QEM7 ; QEM8 . In these engines,
working substance no longer interacts with hot baths during the energizing
strokes but gains energy from invasive quantum measurement directly.

A natural question is whether all measurement processes can increase the
average energy of a system. The answer is definite for a passive state which
contains no ergotropy QEM6 . For example, a single qubit working substance in
equilibrium with a thermal bath always gains energy from the measurement
process. However, things become different when considering how an entangled
measurement affects subsystems due to its inseparability. Even if the
subsystem is in a passive state, its energy can be lowered during the
entangled measurement.

Lately, a quantum mechanical refrigeration concept–quantum measurement cooling
(QMC)–was proposed QMC1 ; QMC2 . QMC is implemented in a two-stroke two-qubit
cooling engine fueled only by entangled measurement. The collective
measurement reduces the energy of the qubit in thermal equilibrium with a cold
bath, thus allowing this qubit to take heat away from the cold bath during the
engine cycle. But overall, the quantum measurement provides energy for the
reverse flow of heat. Interestingly, this protocol requires no information on
measurement results and no feedback operations; the demon that performs the
quantum measurement only needs to know which measurement basis can be used for
refrigeration. This means that QMC is robust against experimental noise.
Despite its robustness, however, the significant energy level spacing between
two qubits required in QMC protocol is challenging, so such a proposal has yet
to be realized QMC2 .

In this Letter, we verify QMC experimentally in a two-qubit two-stroke quantum
measurement cooling engine constructed by a linear optical simulator. First,
by using various working substances with different energy levels, we verify
the predicted QMC and three other thermodynamic operations that can be
implemented by the cooling engine. Next, in the spirit of no prior knowledge
(i.e., random measurement settings), we evaluate the QMC frequency. Finally,
we show that this refrigeration process is robust against noise.

Figure 1: Two strokes of the cooling engine. The arrows show the direction of
energy transfer. (a) The first stroke. The working substance interacts and
exchanges energy with the measurement apparatus during the measurement
process, but the measurement results are not read. (b) The second stroke. The
collapsed working substance contacts the thermal baths and reaches equilibrium
to close the engine cycle.

Theory and model.– Considering a two-qubit working substance with total
Hamiltonian $\mathit{H}=\mathit{H}_{1}+\mathit{H}_{2}$, where
$\mathit{H}_{i}=-\omega_{i}\sigma_{z}^{i}/2$ is the Hamiltonian of qubit $i$
expressed by the Pauli matrix $\sigma_{z}^{i}$ and $\omega_{i}>0$ is the
energy-level spacing (in units of $\hbar\cdot\mathrm{Hz}$), its energy
eigenstates can be written as $|0_{i}\rangle$ and $|1_{i}\rangle$,
corresponding to the energy eigenvalues $-\omega_{i}/2$ and $\omega_{i}/2$,
respectively. The system has two thermal baths with inverse temperatures
$\beta_{1}$ and $\beta_{2}$ (in units of $k_{b}^{-1}\cdot\mathrm{K}^{-1}$),
where $0<\beta_{1}<\beta_{2}$, which means that thermal bath $1$ is hotter
than thermal bath $2$. Initially, each of the two qubits makes contact with
its corresponding thermal bath to reach an equilibrium state; their joint
state is denoted $\rho=\rho_{1}\otimes\rho_{2}$, where

$\rho_{i}=\frac{1}{\mathit{Z}_{i}}e^{-\beta_{i}\mathit{H}_{i}},\quad\mathit{Z}_{i}={\rm
Tr}(e^{-\beta_{i}\mathit{H}_{i}})$ (1)

$\mathit{Z}_{i}$ is the partition function. The energy of each qubit is
perfectly anti-correlated with its corresponding population, but $\rho$, which
is the direct product of $\rho_{1}$ and $\rho_{2}$, is probably not this case
because the two qubits have different energy-level spacings and their thermal
equilibrium temperatures also differ.

The cooling engine cycle contains two strokes, see Fig. 1. In the first
stroke, the working substance interacts with the measurement apparatus and is
projected into a set of two-qubit bases $\\{|\psi_{k}\rangle\\}_{k=1}^{4}$.
Compared with Maxwell’s demon, no information about the measurement results is
gained. That is, we do not assume post-selection on the measurement output or
any feedback control. Thus the post-measurement state $\rho^{\prime}$ becomes
the mixture of measurement bases:
$\rho^{\prime}=\sum_{k=1}^{4}\langle\psi_{k}|\rho|\psi_{k}\rangle|\psi_{k}\rangle\langle\psi_{k}|$.
During the interaction between working substance and measurement apparatus,
energy exchanges, with the net energy flow being $\langle\Delta
E\rangle=\langle\Delta E_{1}\rangle+\langle\Delta E_{2}\rangle$, where
$\langle\Delta E_{i}\rangle={\rm Tr}((\rho^{\prime}-\rho)\mathit{H}_{i})$
refers to the energy change of qubit $i$. Meanwhile, the second law of
thermodynamics compels $\beta_{1}\langle\Delta
E_{1}\rangle+\beta_{2}\langle\Delta E_{2}\rangle\geq 0$ 2thlaw .

In the second stroke, qubit 1 (2) makes contact with thermal bath 1 (2) to
reach thermal equilibrium again. This exchanges energy between the working
substance and the thermal baths, which is equivalent to a cooling process
transferring heat from cold bath 2 to hot bath 1. This completes the
initialization for the next cycle.

Constrained by conservation of energy and the second law of thermodynamics,
four operations are possible QMC1 : (1) $[R]$, refrigeration, in which
$\langle\Delta E_{1}\rangle\geq 0,\langle\Delta E_{2}\rangle\leq
0,\langle\Delta E\rangle\geq 0$, and the working substance gains energy from
the measurement apparatus and thereby transports thermal energy from the cold
bath to the hot bath. (2) $[E]$, energy extraction, in which $\langle\Delta
E_{1}\rangle\leq 0,\langle\Delta E_{2}\rangle\geq 0,\langle\Delta E\rangle\leq
0$, and energy flow from the hot bath to the cold bath while being partially
extracted by the measurement apparatus. (3) $[A]$, thermal acceleration, in
which $\langle\Delta E_{1}\rangle\leq 0,\langle\Delta E_{2}\rangle\geq
0,\langle\Delta E\rangle\geq 0$, and more thermal energy flow from the hot
bath to the cold bath with the energy injection from the measurement
apparatus, as compared with the natural flow of heat. (4) $[H]$, heater, in
which $\langle\Delta E_{1}\rangle\geq 0,\langle\Delta E_{2}\rangle\geq
0,\langle\Delta E\rangle\geq 0$, and the measurement apparatus heats both
thermal baths by consuming its energy. Note that QMC is related to $[R]$,
which could occur only if some of the measurement projectors are entangled.

Figure 2: Experiment setup. The component abbreviations are HWP, half-wave
plate; QWP, quarter-wave plate; RM, reflective mirror; BS, beam splitter; BD,
beam displacer; SLM, spatial light modulator. The simulator contains three
parts: (a) Initialize (cyan region). Depending on the energy-level spacings of
the two qubits to be simulated, two different BDs are used in the setup. The
BD does not change the path of vertically polarized photons but translates
horizontally polarized photons downward by a distance that depends on the
thickness of BD. The HWP placed behind the BD is set to $45^{o}$ to complete
the initialization. (b) Measurement (yellow region). BDs are used to combine
photons from different paths into one path or vice versa. Three cascaded wave
plates are used to implement arbitrary polarization unitary transformation.
The unitaries before and after the BS are Hermitian with respect to each other
along each path. A Mach-Zehnder (M-Z) interferometer is composed of two BDs
and two HWPs on each arm. Two M-Z interferometers are arranged symmetrically
with respect to the BS on one path. (c) Thermalizing channel ($\beta_{1}$ in
the red region and $\beta_{2}$ in the blue region). The hologram displayed on
the SLM reflects the temperature of the thermal bath. All the wave plates are
mounted on motorized computer-controlled rotation stages.

Linear-optical simulator.– The optical version of QMC is now described (see
Fig. 2).

(1) Initialize. The two-qubit working substance is simulated by twin photons
generated by spontaneous parametric down-conversion. Specifically, the ground
(excited) state $|0_{i}\rangle$ ($|1_{i}\rangle$) of qubit $i$ is encoded in
photonic path-polarization mode $|H\rangle_{\textnormal{-}d_{i}/2}$
($|V\rangle_{d_{i}/2}$), where $|H\rangle_{\textnormal{-}d_{i}/2}$
($|V\rangle_{d_{i}/2}$) denotes the state of photon $i$ with horizontal
(vertical) polarization at the position $\textnormal{-}d_{i}/2$ ($d_{i}/2$),
and the energy level spacing of the qubit is simulated by the spatial spacing
$d_{i}$ of the two states supp . Applying a beam displacer (BD) with suitable
thickness, the qubit with arbitrary energy level spacing can, in principle, be
prepared. The initial state $\rho$ of the working substance is prepared by
time-mixing its ground and excited states, and the time intervals are set
proportionally according to the population of energy levels.

(2) Measurement. The measurement process is a vital ingredient of the cooling
engine. As a demonstration in the photonic testbed, we convert the measurement
of energy space into the photonic polarization space. This is done by
inserting BDs of the same specifications as in the initialization part.

In the spirit of removing access to any information of measurement output, we
mix four projective operations of a measurement process classically. The two-
qubit polarization projection can be achieved by the singlet projection and
local operations projector ; supp . In our setup, the singlet projector is
realized with a beam splitter (BS), at which the twin photons meet and undergo
Hong-Ou-Mandel (HOM) interference. By post-selecting the case of each output
port containing one photon, two photons are projected to a polarization
singlet state. The local operations are local unitary transformations that are
performed with QWP-QWP-HWP waveplate structures, and polarization-dependent
adjustable coefficients modules which are based on the Mach-Zehnder
interferometer (the dotted boxes labeled Bias in Fig. 2). Finally, the two
qubits are restored to their energy space by using the same BDs as before. We
thus complete the first stroke of our cooling engine.

To verify the validity of our setup, we performed the quantum measurement
tomography meato  on several randomly selected bases, with an average fidelity
of about 0.98, verifying the implementation of the proposed measurement
processes supp .

(3) Thermalizing. The interaction between a two-level qubit and a thermal bath
can be modeled by a generalized amplitude damping channel (GAD) which contains
four Kraus operators$\\{K_{1},K_{2},K_{3},K_{4}\\}$ channel1 ; channel2 ;
channel3 . As the interaction time increases and tends to be infinite qubit
reaches thermal equilibrium with its thermal bath, and GAD is referred to as
the thermalizing channel.

The red and blue region in Fig. 2 shows the implementation of the thermalizing
channel. As an example, let’s consider the qubit contacting with a cold bath.
The core device is a spatial light modulator (SLM). The imprinted phase by SLM
is related to the position $z$, and its distribution, i.e.
$\phi_{\beta_{2}}(z)$, reflects the temperature $\beta_{2}$ of the cold bath.
When the photonic qubit with two path modes enters the cold bath and is
incident on the $\pm d/2$ position of SLM, these two modes will evolve
differently according to their path position and translate into each other
supp . This process is like the transition between two energy levels of the
working substance when contacting with a thermal bath, thus simulating the
interaction between working substance and cold bath. Operationally, two
settings with specific angles of (HWP1, HWP2, HWP3) corresponding to two pairs
$\\{K_{1},K_{3}\\}$ or $\\{K_{2},K_{4}\\}$ of Kraus operators are switched
randomly to realize the Kraus representation of thermalizing channel.

Process tomography of the thermalizing channel is also performed proto , for
which we choose the baths at different temperatures and working substances
with different energy gaps. The result shows an average fidelity exceeding
$0.99$, verifying the high performance of our channel simulation supp .

Experiment result.– The first result is that we verify the operations allowed
to realize during the engine cycle, in terms of working substance with
different energy level spacings. The parameter ranges of bath temperatures and
working substance energy level spacings can be divided into three classes: For
$0\leq\omega_{2}/\omega_{1}\leq\beta_{1}/\beta_{2}$, [R]-range, only [R] and
[H] are allowed. For $\beta_{1}/\beta_{2}\leq\omega_{2}/\omega_{1}\leq 1$,
[E]-range, only [E], [A], and [H] are allowed. For $\omega_{2}/\omega_{1}\geq
1$, [A]-range, only [A] and [H] are allowed. Fig. 3(a) plots the regions of
these three different ranges with different colors. As a proof-of-principle
demonstration, we prepare the baths with inverse temperature fixed at
$\beta_{1}=1/2.5$ and $\beta_{2}=1$ by embedding the corresponding hologram
$\phi_{\beta_{i}}(z)$ on the SLM. Moreover, we fix the energy gap of qubit 1
at $\omega_{1}=1.02$ and vary the energy-level spacings of the second qubit
over the set of $\omega_{2}=\\{0.02,0.06,0.14,0.18,0.46,0.86,1.10\\}$ by
replacing BDs with different deflecting distances. This parameter
configuration corresponds to a horizontal line with $\beta_{1}/\beta_{2}=0.4$
in Fig. 3(a). The measurement consists of four projector bases
$\\{|\psi_{k}\rangle\\}$, with two of them maximally entangled:

$\displaystyle|\psi_{1}\rangle$ $\displaystyle=|00\rangle,$
$\displaystyle|\psi_{2}\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}(|01\rangle+|10\rangle),$ (2)
$\displaystyle|\psi_{3}\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}(|01\rangle-|10\rangle),$
$\displaystyle|\psi_{4}\rangle$ $\displaystyle=|11\rangle$

This measurement can be verified to be invasive and not separable. After the
first stroke, the energy of $\rho^{\prime}$, which is the post-measurement
state, is recorded, because the energy exchange occurs during the interaction
between the working substance and the measurement apparatus. In the second
stroke, the working substance makes contact with the thermal bath to exchange
energy and reach equilibrium, which terminates the engine cycle. The energy of
the equilibrium state is measured and compared with the energy of
$\rho^{\prime}$ to determine which process occurs.

For each $\omega_{2}$ we calculate $\langle\Delta E\rangle$, $\langle\Delta
E_{1}\rangle$, and $\langle\Delta E_{2}\rangle$, and the results are shown in
Fig. 3(b). Three different processes are possible according to different
$\omega_{2}$: as $\omega_{2}$ increases, $[R]$, $[E]$, and $[A]$ are
successively realized. In the $[R]$ range, $\left\rvert\langle\Delta
E_{2}\rangle\right\rvert$, which is the heat extracted from the cold bath,
does not monotonically increase as $\omega_{2}$ increases. However,
$\left\rvert\langle\Delta E\rangle\right\rvert$, which is the pump energy
provided by the measurement apparatus, decreases as $\omega_{2}$ increases,
which indicates a greater refrigeration efficiency when
$\omega_{2}/\omega_{1}$ reaches $\beta_{1}/\beta_{2}$. In the $[E]$ range,
$\rho$ is not a passive state, that is, the energy-level population is not
inversely related to its energy. Some measurement processes, therefore, exist
that can extract energy from working substances. We see that the energy
$\left\rvert\langle\Delta E\rangle\right\rvert$ flow to the measurement
apparatus, first increasing, and then decreasing to zero at
$\omega_{1}=\omega_{2}$. However, the energy $\left\rvert\langle\Delta
E_{1}\rangle\right\rvert$ extracted from the hot bath is always increasing,
which means that a small energy-level spacing for qubit 2 makes the operation
$[E]$ more efficient. The last non-trivial process $[A]$ occurs when
$\omega_{2}\geq\omega_{1}$. The larger $\omega_{2}$ is, the faster the energy
exchange occurs between the hot bath and the cold bath.

Figure 3: (a) Parameter regions. The parameters used in our experiment are
marked with crosses. (b) Different thermodynamic operations of our measurement
fueled engine when using working substances with different energy spacings.
(c) The frequency of successful realization of the [R] operation, when the
measurement is performed randomly. The solid line is a result of Monte Carlo
sampling and the blue circles are experimental results. All the quoted errors
are derived from Poissonian statistics on the raw data.

The second result is that the frequency $P_{[R]}$ of the QMC is realized with
no prior knowledge of what measurement to do, illustrating the robustness of
QMC. Experimentally, the measurement basis is randomly selected. This is done
by randomly picking several unitaries $U$ from the SU(4) invariant (Haar)
measure. The unitaries are then applied to the projectors in Eq. (2),
producing the new basis $\\{U|\psi_{k}\rangle\\}_{k=1}^{4}$. We generated 200
unitary matrices from Haar measures with a computer and implement 200
measurement settings for each type of working substance. During the engine
cycles, we measure $\langle\Delta E\rangle$, $\langle\Delta E_{1}\rangle$, and
$\langle\Delta E_{2}\rangle$ and calculate the direction of heat flow to
determine whether QMC occurs for each measurement setting. Fig. 3(c) shows
that QMC is possible only when $\omega_{2}/\omega_{1}\leq\beta_{1}/\beta_{2}$.
The frequency of the $[R]$ operation increases and tends to $0.5$ as
$\omega_{2}/\omega_{1}$ decreases and tends to zero, showing that QMC becomes
more robust against the randomness of measurement with decreasing
$\omega_{2}/\omega_{1}$. These results also indicate that a significant
difference in energy level spacings between the two qubits is necessary to
implement QMC. If we average all the measurement effects, the most useless
operation $[H]$ is implemented supp .

Finally, note that the realization of QMC is irrelevant to the measurement
efficiency. During the measurement process, we consider the measurement
efficiency of $\nu$, meaning that the measurement apparatus interacts with the
working substance with probability $\nu$, while $1-\nu$ is the probability of
no interaction. No matter how inefficient the measurement process is, the QMC
can still be realized because the energy of the working substance does not
change if it does not interact with the measuring apparatus–only a successful
measurement contributes to the energy exchange. In addition, the experimental
noise of our simulator is mainly due to imperfect HOM interference. Although
this may eventually prevent the refrigerator from working, we find that our
setup remains noise-tolerant, especially when the energy-level spacing between
two qubits is significant (see the Supplemental Material for more details supp
).

Conclusion.– In summary, we propose herein a simulation protocol in which the
energy level of working substance and the temperature of the thermal baths are
easily adjusted. In addition, we experimentally demonstrate QMC with a quantum
cooling engine built by this simulator, which means that we transfer heat from
the cold bath to the hot bath by pumping with an invasive quantum entangled
measurement. We show that this type of measurement-fueled cooling engine is
robust against experimental noise, and the results are consistent with the
theory. This evidence indicates that quantum measurements can be a useful
thermodynamic resource.

Although the thermodynamic quantities obtained are only from the simulation of
an actual situation channel3 ; thermometer1 ; otto ; simu , our experiment
still deepens our understanding of quantum measurements and provides a
convenient setup for future research into quantum thermodynamics, such as
measurement-based quantum engines QEM1 ; QEM2 ; QEM3 ; QEM4 ; QEM5 ; QEM6 ;
QEM7 ; QEM8 , environment-system interaction thermometer2 ; thermometer3 ;
chan1 ; chan2 , quantum algorithm alg1 ; alg2 , and so on.

Acknowledgments.– We thank Sheng Liu for the beneficial discussions. This
research was supported by the Innovation Program for Quantum Science and
Technology (No. 2021ZD0301604), the National Natural Science Foundation of
China (Nos. 11821404, 11734015, 62075208, 12022401, 62075207). This work was
partially carried out at the USTC Center for Micro and Nanoscale Research and
Fabrication.

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## Supplemental Material

###  .1 1\. Simulation of working substance

In this section, we introduce the simulation protocol of the energy level of
the working substance in detail. As described in the main text, we use the
photonic path-polarization mode $|H\rangle_{\textnormal{-}d/2}$ and
$|V\rangle_{d/2}$ to encode the ground state $|0\rangle$ with the energy of
$-\omega/2$ and the excited state $|1\rangle$ with the energy of $\omega/2$
respectively. Practically, since both the photonic beam and the pixels on the
SLM are of a certain size, the energy level spacing $\omega$ is set to have
the same value when photonic qubits incident in a certain area, see Fig.
S1(a). The specification of the SLM is 512 $\times$ 512 pixels, and each pixel
is $15\mu m\times 15\mu m$ in size. To make the most use of the screen of SLM,
and considering that the beam impinging on the screen is focused to a size of
about 4 pixels, we specify that the minimum change unit of $\omega(d)$ is
0.02. The relationship between the spatial spacing $d$ of two path modes and
the energy level spacing $\omega$ is set as increasing $\omega$ by 0.02 for
every 8-pixel distance increasing of $d$.

Figure S1: (a) Part of the hologram displayed on SLM. The lower half $(z\leq
0)$ is just the image of the upper half plus the image imparted the phase of
$\pi$. Every four pixels load the image with the same color. Photonic qubits
incident into the same color area and its symmetric areas to $z=0$ are
considered to simulate the working substance with the same energy level
spacing. The maximum energy level spacing in this setup is
$\omega_{max}=1.28$. (b) The experimental setup of a thermalizing channel.

###  .2 2\. Thermalizing channel

There we introduce the details of the implementation of the thermalizing
channel. When a qubit system is in contact with a heat bath and reaches
equilibrium, its final state becomes a classical mixed state satisfying the
Boltzmann distribution, regardless of the initial state. This process is well
described by the thermalizing channel, and its four Kraus operators
$\\{K_{1},K_{2},K_{3},K_{4}\\}$ can be written as:

$\displaystyle K_{1}$ $\displaystyle=\sqrt{p}\begin{pmatrix}1&0\\\
0&0\end{pmatrix},$ $\displaystyle K_{2}=\sqrt{p}\begin{pmatrix}0&1\\\
0&0\end{pmatrix},\quad K_{3}$ $\displaystyle=\sqrt{1-p}\begin{pmatrix}0&0\\\
0&1\end{pmatrix},$ $\displaystyle K_{4}=\sqrt{1-p}\begin{pmatrix}0&0\\\
1&0\end{pmatrix}$ (S1)

where $p$ is the equilibrium population of the ground state, related to the
bath temperature $\beta$ and the qubit energy level spacing $\omega$. The
setup is depicted in Fig. S1 (b), which consists of an SLM, three HWPs, and
two BDs. The specification of BD corresponds to the energy level spacing of
the qubit. The hologram $\phi_{\beta}(z)$ corresponding to the temperature
$\beta$ of thermal bath will be loaded on the SLM. Photonic qubit incident on
the SLM at the position of $\pm d/2$. Then two different settings are
performed to complete the thermalizing channel: in the first setting, angles
of HWP1-3 are set to be $22.5^{o}$, $22.5^{o}$, $0^{o}$ respectively, the
transformation of this configuration can be expressed as
$|V\rangle_{d/2}\rightarrow\cos(\phi_{\beta}(d/2)/2)|V\rangle_{d/2}$,
$|H\rangle_{\textnormal{-}d/2}\rightarrow\cos(\phi_{\beta}(\textnormal{-}d/2)/2)|H\rangle_{\textnormal{-}d/2}$;
in the second setting, angles of HWP1-3 are set to be $22.5^{o}$,
$\textnormal{-}22.5^{o}$, $45^{o}$ respectively, and the transformation of
this configuration can be expressed as
$|V\rangle_{d/2}\rightarrow\sin(\phi_{\beta}(d/2)/2)|H\rangle_{\textnormal{-}d/2}$,
$|H\rangle_{\textnormal{-}d/2}\rightarrow-\sin(\phi_{\beta}(\textnormal{-}d/2)/2)|V\rangle_{d/2}$.
In addition, the upper half of the hologram is applied $\pi$ more than the
lower half, i.e. $\phi_{\beta}(\textnormal{-}z)=\phi_{\beta}(z)-\pi$, and the
coherence between those two encoded states is removed due to the difference
between their optical paths. Corresponding them to four Kraus operators, we
get

$\displaystyle
K_{1}:|H\rangle_{\textnormal{-}d/2}\rightarrow\sin(\phi_{\beta}(d/2)/2)|H\rangle_{\textnormal{-}d/2}\quad$
$\displaystyle
K_{3}:|V\rangle_{d/2}\rightarrow\cos(\phi_{\beta}(d/2)/2)|V\rangle_{d/2}$
$\displaystyle
K_{2}:|V\rangle_{d/2}\rightarrow\sin(\phi_{\beta}(d/2)/2)|H\rangle_{\textnormal{-}d/2}\quad\quad$
$\displaystyle
K_{4}:|H\rangle_{\textnormal{-}d/2}\rightarrow\cos(\phi_{\beta}(d/2)/2)|V\rangle_{d/2}$

Compared to Eq. S1, we find $p=\sin^{2}(\phi_{\beta}(d/2)/2)$. Remember that
the qubit system is in equilibrium so that
$p=e^{\beta\omega/2}/(e^{\beta\omega/2}+e^{-\beta\omega/2})$. With the
functional relationship between $\omega(d)$ and $d$ in the Fig. S1 (a) and
taking $d$ through the entire SLM screen, the $\phi_{\beta}(z)$ can be solved.
During our experiment, we solved the holograms corresponding to
$\beta_{1}=1/2.5$ and $\beta_{2}=1$. The process tomography results of the
interaction between different thermal baths and different working substances
are shown in Fig. S2.

Figure S2: Reconstructed matrices of the results of process tomography, left
is the real part and right is the imaginary part in the Pauli basis. The
corresponding fidelity is $F$.

###  .3 3\. Two-qubit polarization measurement

The two-qubit polarization measurement is based on the Hong-Ou-Mandel (HOM)
effect. Two incident photons overlap with each other at BS. In the case of two
photons exiting from different ports of BS, the input state is thus projected
onto singlet state $|\psi^{-}\rangle=(|HV\rangle-|VH\rangle)/\sqrt{2}$. An
arbitrary projector $|\varphi_{p}\rangle\langle\varphi_{p}|$, in which
$|\varphi_{p}\rangle$ can be written as
$|\varphi_{p}\rangle=U_{1}U_{2}(a|HV\rangle-b|VH\rangle)$ according to Schmidt
decomposition theorem, is performed by local unitaries and coefficients bias
on the singlet projector. The bias of two real coefficients $a$ and $b$ is
realized by introducing polarization-dependent loss to one of two photons,
which is a Mach-Zehnder (M-Z) interferometer with an HWP placed on each arm.
The angle of HWP in the V-arm is $45^{o}$ while in H-arm is $\theta$. $\theta$
can vary in the range of $0^{o}$ to $45^{o}$ (see Fig.2 in the main text). The
M-Z interferometer also flips the polarization of the input state, but this
can be rectified by the Pauli X operation, which is absorbed into the local
unitary performed by three wave plates. In total, the transmission of $V$
polarized state is unit but $H$ is $\sin 2\theta$. The transformation of the
bias act on $|\psi^{-}\rangle$ can be expressed as follows

$|\psi^{-}\rangle\quad\rightarrow\quad\eta(a|HV\rangle-b|VH\rangle),\quad\eta=\frac{\sin^{2}2\theta+1}{2},\quad
a=\frac{\sin 2\theta}{\sqrt{1+\sin^{2}2\theta}},\quad
b=\frac{1}{\sqrt{1+\sin^{2}2\theta}}$ (S2)

where $\eta$ is the projection efficiency. Together with local unitaries that
are performed by three cascade wave plates, we can get
$\sqrt{\eta}|\varphi_{p}\rangle$. In the measurement setup, the unitaries and
bias modules are symmetrically placed to BS. In this way, the setup is
equivalent to $\eta|\varphi_{p}\rangle\langle\varphi_{p}|$. For a measurement
setting containing four projectors, $\theta$ is calculated according to Eq.
S2, and four projection results are renormalized by their efficiency
respectively. By mixing those results we complete the measurement process
finally.

Figure S3: The five results of measurement tomography. In each measurement
setting, four top pictures show the real part of four reconstructed projectors
and four bottom pictures are the corresponding imaginary part respectively.
All the reconstructed matrices are shown in their measurement basis
respectively.

To characterize our measurement, we performed measurement tomography for five
randomly selected measurement settings, the experimental fidelity is
$0.9803\pm 0.0016,0.9800\pm 0.0013,0.9816\pm 0.0012,0.9796\pm 0.0013,0.9791\pm
0.0012$ respectively, the results are shown in Fig. S3.

Below we consider the case where the measurement is inefficient. The
projective measurement with the basis of Eq.(3) in the main text thus turns
into positive operator-valued measurement (POVM). The POVM elements
$\\{M_{k}\\}_{k=1}^{4}$ satisfy
$M_{i}^{\dagger}M_{i}=\nu|\psi_{i}\rangle\langle\psi_{i}|+(1-\nu)I/4$, and
$\sum_{k=1}^{4}M_{k}^{\dagger}M_{k}=I$. Therefore, we get

$\displaystyle|\psi_{1}\rangle\langle\psi_{1}|$ $\displaystyle\rightarrow
M_{1}=\frac{\sqrt{1+3\nu}-\sqrt{1-\nu}}{2}|\psi_{1}\rangle\langle\psi_{1}|+\frac{\sqrt{1-\nu}}{2}I,\quad$
$\displaystyle|\psi_{2}\rangle\langle\psi_{2}|$ $\displaystyle\rightarrow
M_{2}=\frac{\sqrt{1+3\nu}-\sqrt{1-\nu}}{2}|\psi_{2}\rangle\langle\psi_{2}|+\frac{\sqrt{1-\nu}}{2}I,$
(S3) $\displaystyle|\psi_{3}\rangle\langle\psi_{3}|$ $\displaystyle\rightarrow
M_{3}=\frac{\sqrt{1+3\nu}-\sqrt{1-\nu}}{2}|\psi_{3}\rangle\langle\psi_{3}|+\frac{\sqrt{1-\nu}}{2}I,\quad$
$\displaystyle|\psi_{4}\rangle\langle\psi_{4}|$ $\displaystyle\rightarrow
M_{4}=\frac{\sqrt{1+3\nu}-\sqrt{1-\nu}}{2}|\psi_{4}\rangle\langle\psi_{4}|+\frac{\sqrt{1-\nu}}{2}I$

The post-measurement state of the working substance becomes

$\rho_{\nu}^{\prime}=\sum_{k=1}^{4}M_{k}\rho
M_{k}^{\dagger}=(\frac{\sqrt{1+3\nu}-\sqrt{1-\nu}}{2})^{2}\rho^{\prime}+\frac{(\sqrt{1+3\nu}+\sqrt{1-\nu})\sqrt{1-\nu}}{2}\rho$
(S4)

where $\rho$ is the input state and $\rho^{\prime}$ is the projective post-
measurement state. So that only the original projective measurement can change
the energy of working substances, the white noise has no effect on the
system’s energy. For working substances with $\omega_{1}=1.02$ and
$\omega_{2}=0.18$, thermal baths at temperatures of $\beta_{1}=1/2.5$ and
$\beta_{2}=1$, we plot the energy change of working substances before and
after measurement in Fig. S4(a).

Then we consider the measurement noise of our simulator. This is due to the
imperfection of HOM interference at BS. In this scenario, we think that the
$\nu$ ratio pairs of photons interfere without defects, where $\nu$ is the
interference visibility. But the remaining photons are just divided into two
different paths by BS, and only half of them are detected due to post-
selection configuration. As a result, the four project operations of
measurement become four noisy channels. With the measurement basis of Eq.(3),
for working substances with $\omega_{1}=1.02$ and $\omega_{2}=0.18$ and
thermal baths at temperatures of $\beta_{1}=1/2.5$ and $\beta_{2}=1$, we
decrease $\nu$ from the best value of our simulator. In Fig. S4(b), we see
that qubit 2 is transferring less heat from the cold bath. When $\nu$
decreases to the critical visibility $\nu_{c}\approx 0.44$, the demon can not
realize QMC longer. In Fig. S4(c), we plot the critical visibility $\nu_{c}$,
as $\omega_{2}/\omega_{1}$ goes down, $\nu_{c}$ gets smaller and tends to 0
when $\omega_{2}/\omega_{1}$ tends to 0. The analysis shows that our simulator
is noise-robust, especially when the energy level spacing difference between
two qubits is significant.

Figure S4: (a) Energy changes of working substances under the white noise of
measurement. (b) Energy change of working substances under the measurement
noise of our simulator. (c) The change in critical visibility $\nu_{c}$ with
respect to $\omega_{2}/\omega_{1}$.

###  .4 4\. Other results of measurement cooling engine

Suppose the demon has no knowledge about what measurements were performed
during the engine strokes. In that case, $[H]$ operation is realized on
average, i.e. $\overline{\langle\Delta E_{1}\rangle}\geq 0$,
$\overline{\langle\Delta E_{2}\rangle}\geq 0$, and $\overline{\langle\Delta
E\rangle}\geq 0$. During all the measurement settings of different working
substances, our experimental results confirm this conclusion (see Table. S1).
We also evaluate the frequency of the other three thermodynamic operations on
200 measurement basis, see Fig. S5.

$\overline{\langle E\rangle}$ $\omega_{2}$ | 0.02 | 0.06 | 0.14
---|---|---|---
$\overline{\langle\Delta E_{1}\rangle}$ | 0.09277 $\pm$ 0.00023 | 0.08217 $\pm$ 0.00025 | 0.07593 $\pm$ 0.00028
$\overline{\langle\Delta E_{2}\rangle}$ | 0.00009 $\pm$ 0.00001 | 0.00073 $\pm$ 0.00002 | 0.00413 $\pm$ 0.00004
$\overline{\langle\Delta E\rangle}$ | 0.09286 $\pm$ 0.00023 | 0.08290 $\pm$ 0.00024 | 0.08006 $\pm$ 0.00029
$\overline{\langle E\rangle}$ $\omega_{2}$ | 0.18 | 0.46 | 0.86 | 1.10
---|---|---|---|---
$\overline{\langle\Delta E_{1}\rangle}$ | 0.09159 $\pm$ 0.00027 | 0.08999 $\pm$ 0.00013 | 0.07371 $\pm$ 0.00031 | 0.05855 $\pm$ 0.00028
$\overline{\langle\Delta E_{2}\rangle}$ | 0.00853 $\pm$ 0.00005 | 0.05016 $\pm$ 0.00027 | 0.12741 $\pm$ 0.00026 | 0.20197 $\pm$ 0.00027
$\overline{\langle\Delta E\rangle}$ | 0.10012 $\pm$ 0.00027 | 0.14015 $\pm$ 0.00027 | 0.20112 $\pm$ 0.00043 | 0.26052 $\pm$ 0.00039
Table S1: Experimental results of $\overline{\langle\Delta E_{1}\rangle}$,
$\overline{\langle\Delta E_{2}\rangle}$, and $\overline{\langle\Delta
E\rangle}$. The temperatures of thermal baths are $\beta_{1}=1/2.5$ and
$\beta_{2}=1$ respectively, and the energy level spacing of qubit 1 is
$\omega_{1}=1.02$. Figure S5: Measured frequency of (a) $[H]$, (b) $[A]$, and
(c) $[E]$ operations, when the temperatures of two baths are $\beta_{1}=1/2.5$
and $\beta_{2}=1$ respectively. The probability of realizing $[E]$ operation
is extremely small, but $[H]$ operation occurs most frequently.