Sudarsana Reddy Kadiri, Manila Kodali, Paavo Alku

# Severity Classification of Parkinson's Disease from Speech using
Single Frequency Filtering-based Features

###### Abstract

Developing objective methods for assessing the severity of Parkinson's disease
(PD) is crucial for improving the diagnosis and treatment. This study proposes
two sets of novel features derived from the single frequency filtering (SFF)
method: (1) SFF cepstral coefficients (SFFCC) and (2) MFCCs from the SFF
(MFCC-SFF) for the severity classification of PD. Prior studies have
demonstrated that SFF offers greater spectro-temporal resolution compared to
the short-time Fourier transform. The study uses the PC-GITA database, which
includes speech of PD patients and healthy controls produced in three speaking
tasks (vowels, sentences, text reading). Experiments using the SVM classifier
revealed that the proposed features outperformed the conventional MFCCs in all
three speaking tasks. The proposed SFFCC and MFCC-SFF features gave a relative
improvement of 5.8% & 2.3% for the vowel task, 7.0% & 1.8% for the sentence
task, and 2.4% & 1.1% for the read text task, in comparison to MFCC features.

Index Terms: Parkinson's disease, severity classification, biomarking, support
vector machine.

##  1 Introduction

Parkinson's disease (PD) is the second most common neurodegenerative disease
after Alzheimer’s disease, and it is characterized by the progressive
degeneration of dopaminergic neurons in the brain, resulting in motor and non-
motor symptoms. Speech impairment is a common non-motor symptom of PD, and
this impairment can significantly impact the patient's quality of life.
Presence of PD and its progression is typically evaluated by a neurologist or
a movement disorder specialist who assess the patient's symptoms, and overall
condition over time. Regular clinical evaluations, imaging studies (such as
magnetic resonance imaging or positron emission tomography scans), and other
tests are utilized to monitor the progression of the disease and track the
effectiveness of treatment [1, 2, 3]. However, these assessments are costly,
laborious and prone to bias due to the neurologist's/specialist's familiarity
and experience with the patients. Therefore, developing objective methods for
detecting and assessing the severity of PD is crucial for improving the
diagnosis, monitoring, and treatment of the disease. Speech-based severity
assessment methods can be more accessible and cost-effective than traditional
clinical evaluations, making them useful tools for both healthcare
professionals and patients [3, 4].

Automatic detection and severity level classification of PD from speech is
facilitated by data-driven approaches based on supervised learning. This
involves constructing machine learning models that are trained using speech
data collected from patients and labeled by neurologists. The detection of PD
(i.e., healthy $vs.$ parkinsonian) from speech has been investigated in many
studies [5, 6, 7, 8, 9, 10, 11, 12, 13]. More details about the various types
of features and approaches used in the literature can be found in [14, 15, 16,
17, 18]. The present study focuses on speech-based severity level
classification of PD (i.e., healthy $vs.$ mild $vs.$ severe).

Compared to the detection task, much less research has been conducted in the
severity level classification of PD [19, 20, 21, 22]. In [19], the severity
level (mild $vs.$ moderate $vs.$ severe) was classified using spectral,
prosody, and glottal features from speech signals produced in various speaking
tasks (syllable repetition tasks, read sentences, and paragraphs, as well as
monologues) with the support vector machine (SVM) classifier. In [20], the
severity level of PD was studied by grouping patients and healthy talkers into
a 3-class classification problem (healthy $vs.$ mild $vs.$ affected) using
perturbation, spectral, cepstral, and complexity features with deep neural
network (DNN) and convolutional neural network (CNN) as classifiers. In [21],
automatic multi-class assessment was studied using a multi-class SVM following
a one $vs.$ all strategy using prosody features and the monologue speaking
task. In [22], authors used onset and offset transitions from various speech
sounds in continuous speech with CNNs as classifiers.

The results of [15, 23, 19, 20] suggest that spectral features such as MFCCs
perform better than conventional features such as phonatory/glottal and
prosodic features in detection and severity classification of PD. Motivated by
this, the current study investigates cepstral coefficients derived using a
recently proposed signal processing method, single frequency filtering (SFF)
[24, 25]. Two feature sets are derived: (1) single frequency filtering
cepstral coefficients (denoted as SFFCC) and (2) MFCCs computed from the SFF
spectrum (denoted as MFCC-SFF). The SFF method was shown in [25, 26, 27] to
provide higher spectral and temporal resolution for deriving speech features
compared to the short-time Fourier transform (STFT), which is used in the
computation of conventional MFCCs. Illustrations of spectrograms obtained with
SFF and STFT are shown in Figs. 1 and 2, respectively, for healthy speech and
for PD speech of mild and severe levels. From the figures, it can be clearly
seen that the SFF spectrogram highlights spectral information clearly (with
sharper harmonics) compared to the STFT spectrogram.

The major contributions of this study are:

  * •

The effectiveness of the cepstral coefficients derived from the SFF spectrum
(SFFCC and MFCC-SFF) is studied for severity level classification of PD from
speech.

  * •

A systematic comparison is conducted in severity level classification of PD
between three speaking tasks (production of vowels, production of sentences as
well as a text reading task).

The paper is organized as follows. Section 2 describes the SFF method and
feature extraction from SFF method. Section 3 describes the experimental
protocol including the database, baseline features and prior studies for
comparison, classifier and evaluation metrics. Results and discussion on
classification experiments are presented in Section 4. A summary of the paper
is given in Section 5.

##  2 Single Frequency Filtering (SFF)-based Features

SFF is a time-frequency analysis technique, which provides an amplitude
envelope of the speech signal as a function of time for a selected frequency
[24, 25]. In this method, the speech signal $s[n]$ is first frequency-shifted
(i.e., modulated) by multiplying it with an exponential function:

$\hat{s}[n,k]=s[n]e^{-j{2\pi\bar{f}_{k}}{n}/{f_{s}}},$ (1)

where $f_{s}$ is the sampling frequency,
$\bar{f_{k}}=\frac{f_{s}}{2}-{f_{k}}$, and $f_{k}$ is the $k^{th}$ desired
frequency. Then the frequency-shifted signal is filtered through a single-pole
filter, whose root ($z=-r$) is located on the negative real axis. The transfer
function of the filter is given by:

$H(z)=\frac{1}{1+rz^{-1}}.$ (2)

The output of the filter can be expressed as:

$y[n,k]=-ry[n-1,k]+\hat{s}[n,k].\vspace{-0.1cm}$ (3)

The magnitude or amplitude envelope $v[n,k]$ and the phase $\psi[n,k]$ of the
signal $y[k,n]$ at $k^{th}$ frequency can be written as:

$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}v[n,k]=\sqrt{y_{r}^{2}[n,k]+y_{i}^{2}[n,k]},$
(4)

and

$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\psi[n,k]=tan^{-1}(\frac{y_{i}[n,k]}{y_{r}[n,k]}).$
(5)

Here $y_{r}{[n,k]}$ and $y_{i}{[n,k]}$ are the real and imaginary parts of
$y[n,k]$, respectively. The amplitude envelope of the signal can be computed
for several frequencies at intervals of $\Delta{f}$ by defining ${f_{k}}$ as
follows:

$f_{k}=k\Delta{f},~{}~{}~{}~{}~{}~{}~{}~{}~{}k=1,2,\ldots,K,$ (6)

where $K=\frac{({f_{s}}/2)}{\Delta{f}}.$ The SFF magnitude spectrum can be
obtained for each instant of time from $v[n,k]$.

Figure 1: Illustration of SFF spectrograms for (a) healthy speech, and for PD
speech of (b) mild and (c) severe severity level. Figure 2: Illustration of
STFT spectrograms for (a) healthy speech, and for PD speech of (b) mild and
(c) severe severity level.

###  2.1 Extraction of the SFFCC features

SFFCCs are extracted by computing the cepstrum ($C[n,k]$). Cepstrum is derived
from the SFF spectrum ($v[n,k]$) as follows:

$C[n,k]={\rm IFFT}(\log(v[n,k])),\vspace{-0.1cm}$ (7)

where IFFT is the inverse Fourier transform. From cepstrum $C[n,k]$, the first
13 cepstral coefficients (including $0^{th}$) are considered and they are
referred to as SFFCC. Apart from the static coefficients, the delta ($\Delta$)
and double-delta ($\Delta\Delta$) coefficients are also computed, which
results in a 39-dimensional feature vector. A schematic block diagram
describing the steps involved in the extraction of SFFCC is shown in Fig. 3.

Figure 3: Block diagram of the extraction of single frequency filtering
cepstral coefficients (SFFCC).

###  2.2 Extraction of the MFCC-SFF features

A schematic block diagram describing the steps involved in the extraction of
MFCCs from the SFF spectrum is shown in Fig. 4. The MFCC extraction consists
of mel filter$-$bank analysis on the SFF spectrum followed by logarithm and
discrete cosine transform (DCT) operations. This is expressed as follows:

$MFCC_{SFF}[n,k]=DCT(\log(Mel({S_{SFF}[n,k]}^{2}))),$ (8)

where $MFCC_{SFF}[n,k]$ denotes the mel-cepstrum. The resulting cepstral
coefficients are referred as MFCC$-$SFF, and they represent compactly the
spectral characteristics of speech. From the mel-cepstrum, the first 13
cepstral coefficients (including the $0^{th}$) are considered. Apart from the
static coefficients, the $\Delta$ and $\Delta\Delta$ coefficients are also
computed, which results in a 39-dimensional feature vector.

Figure 4: Block diagram of the extraction of MFCCs from the SFF spectrum
(MFCC-SFF) [28].

##  3 Experimental Protocol

This section describes the speech database used in the experiments, the
reference features that were selected for comparison and the classifier.

###  3.1 Database

This study uses publicly available PC-GITA, a repository of Spanish
parkinsonian speech [29]. The database is balanced with respect to gender and
age, and it consists of speech recordings from 50 PD patients (25 female and
25 male) and 50 healthy control speakers (HCs) (25 female and 25 male). For
the male PD patients, the age range is between 33 and 77 years (mean 62.3
years), for the female PD patients, the age range is between 44 and 75 years
(mean 60.2 years). For the male HCs, the age range is between 31 and 86 years
(mean 61.3 years), and for the female HCs, the age range is between 43 and 76
years (mean 60.8 years). The database consists of different speaking tasks
including productions of vowels, sentences, diadochokinetic words, as well as
text reading and a monologue task. In this study, we considered the vowel
task, the sentence task and the read text task. The vowel task includes three
repetitions of five Spanish vowels. The sentence task includes productions of
six different sentences. The read text task consists of a dialog between a
doctor and a speaker, and this task is phonetically balanced (36 words). All
the patients were diagonsed by neurologists, and their disease severity was
labeled according to the modified H & Y scale and the MDS-UPDRS-III. The MDS-
UPDRS-III scale is based on the motor examination and it consists of 33 items,
where each of the item ranges from 0 (normal) to 4 (severe). The database was
partitioned into three severity classes (healthy, mild, and severe) according
to the MDS-UPDRS-III scores. Figure 5 shows the distribution of the MDS-UPDRS-
III and the number of speakers per each class. This study uses the balanced
number of speakers for each class. Further details of the database can be
found in [29, 30].

|
---|---
Figure 5: Distribution of the MDS-UPDRS III (left) and the number of speakers
available in the database for each severity class (right).

###  3.2 Baseline features and reference classifiers of a prior study

The most popular speech features, namely MFCCs [31, 32, 15, 19], are used as
the baseline features. The first 13 MFCCs (including $0^{th}$) are extracted
using a Hamming window size of $30$ ms and a shift of $10$ ms. Apart from the
above static coefficients, the $\Delta$ and $\Delta\Delta$ coefficients are
also computed resulting in a 39-dimensional feature vector. In addition to the
baseline MFCCs, we compared the results of the current study also with the
results reported in [20], where a protocol similar to this study was utilized
using only the vowel task. In [20], perturbation, spectral, cepstral and
complexity features were used together with a DNN, and the modulation spectrum
was used with a CNN and multi-modal architectures.

###  3.3 Parameters used for the SFF-based features

For the SFF spectrum estimation, $r=0.99$ and $\Delta{f}=31.25$ Hz (resulting
in 512 amplitude envelopes) are used. SFFCC and MFCC-SFF are extracted with an
interval of $10$ ms rather than considering every time instant. The number of
mel-filters used is 80, and the first 39 cepstral coefficients are extracted
for both of the features. Frame-wise features (for baseline and proposed) are
merged into a 1-dimensional feature vector by taking the mean.

###  3.4 Classifier and evaluation metrics

Severity classification experiments were carried out using the SVM classifier
(linear kernel in the one-vs-one architecture, with c and gamma parameters in
the range of $10^{-4}$ to $10^{4}$), which is known to be an effective
classifier when the amount of training data is limited. The experiments were
conducted with leave-one-speaker-out (LOSO) cross-validation, where one
speaker's data was considered as a test set and the remaining speakers' data
was used for training the classifier. In each fold, the evaluation metrics
were saved, and the procedure was repeated for all the speakers, and finally
the evaluation metrics were averaged.

The metrics chosen are the balanced classification accuracy (also known as
unweighted average recall (UAR)), class-wise precision, class-wise recall, and
class-wise F1 score. We also used confusion matrices for assessing the
performance of the classification systems.

##  4 Results and Discussion

Table 1: Results of Parkinson's disease severity classification for the
reference MFCC features, the proposed SFFCC features and the MFCC-SFF features
in three speaking tasks (vowels, sentences and read text). Precision, recall
and F1 score are given for individual classes (0 for healthy, 1 for mild, and
2 for severe).

Vowel task
---
Feature | Accuracy | Precision-0 | Recall-0 | F1 score-0 | Precision-1 | Recall-1 | F1 score-1 | Precision-2 | Recall-2 | F1 score-2
MFCC | 48.7 ± 7 | 0.53 | 0.57 | 0.55 | 0.40 | 0.38 | 0.39 | 0.45 | 0.45 | 0.45
SFFCC | 51.5 ± 4 | 0.50 | 0.56 | 0.52 | 0.43 | 0.40 | 0.42 | 0.49 | 0.46 | 0.47
MFCC-SFF | 49.8 ± 5 | 0.49 | 0.54 | 0.52 | 0.42 | 0.42 | 0.42 | 0.45 | 0.42 | 0.44
Sentence task
MFCC | 54.5 ± 8 | 0.53 | 0.61 | 0.57 | 0.52 | 0.45 | 0.48 | 0.63 | 0.62 | 0.62
SFFCC | 58.3 ± 7.8 | 0.60 | 0.63 | 0.62 | 0.53 | 0.51 | 0.52 | 0.62 | 0.61 | 0.62
MFCC-SFF | 55.5 ± 8 | 0.58 | 0.60 | 0.59 | 0.54 | 0.53 | 0.54 | 0.62 | 0.61 | 0.61
Read text task
MFCC | 61.5 ± 10 | 0.59 | 0.79 | 0.68 | 0.45 | 0.42 | 0.43 | 0.50 | 0.38 | 0.43
SFFCC | 63.0 ± 12 | 0.55 | 0.75 | 0.63 | 0.44 | 0.33 | 0.38 | 0.62 | 0.54 | 0.58
MFCC-SFF | 62.2 ± 13 | 0.58 | 0.62 | 0.60 | 0.46 | 0.46 | 0.46 | 0.50 | 0.46 | 0.48

MFCC |  SFFCC |  MFCC-SFF
---|---|---
Actual class |  |

Predicted class

Figure 6: Confusion matrices for the MFCC (left), SFFCC (middle) and MFCC-SFF (right) features in the read text task. Rows correspond to the actual Parkinson's disease severity classes and columns correspond to the predicted classes. Table 2: Classification accuracy in the vowel task based on the features proposed in the current study and based on classifiers studied in a previous study.  |  | Class-wise accuracies
---|---|---
| Accuracy | Healthy | Mild | Severe
Current study
SFFCC (SVM) | 51.5 $\pm$ 4 | 55.9 | 40.4 | 45.6
MFCC-SFF (SVM) | 49.8 $\pm$ 5 | 54.2 | 40.9 | 42.5
Previous study [20]
DNN | 50.0 $\pm$ 9 | 30.9 | 43.6 | 70.0
CNN | 41.0 $\pm$ 5 | 68.0 | 21.8 | 33.3
Multi-modal | 52.0 $\pm$ 7 | 63.9 | 40.0 | 46.7

This section reports the results of the experiments for the three speaking
tasks (vowels, sentences and read text) by first describing the classification
accuracies obtained and then describing the confusion matrices. Table 1 shows
the results in terms of the mean and standard deviation of accuracy along with
class-wise precision, recall and F1-score. From the results in the table, it
can be observed that the performance of both of the proposed features (SFFCC
and MFCC-SFF) is clearly better than the baseline MFCCs features for all the
three speaking tasks. From the speaking tasks, the performance of the read
text task is better that of the sentence and vowel tasks. Between the sentence
and vowel tasks, the former task gave better results. Taken together, the
order of preference for the speaking tasks is:

$Read~{}Text$ $>$ $Sentences$ $>$ $Vowels$.

This trend was expected as the read text task included richer articulation
information (due to phonetically balanced sounds) which is helpful for
severity classification. From the metrics based on precision, recall and F1
score, it is evident that the MFCC features are more biased towards the
healthy class (specifically in the vowel and read text tasks) compared to the
proposed features. Overall, the proposed SFFCC and MFCC-SFF features gave an
absolute improvement of 2.8% and 1.1% (relative improvement of 5.8% and 2.3%)
for the vowel task, 3.8% and 1.0% (relative improvement of 7.0% and 1.8%) for
the sentence task, and 1.5% and 0.7% (relative improvement of 2.4% and 1.1%)
for the read text task, in comparison to the baseline MFCC features. Between
the two proposed features, SFFCC performed better than MFCC-SFF.

Figure 6 shows the confusion matrices for the MFCC (left), SFFCC (middle) and
MFCC-SFF (right) features in the read text task. This figure shows clearly
that all the features are biased towards healthy class. Among the features,
the proposed features are, however, less biased towards health class compared
to the MFCC features. This is also in conformity with the performances
reported in Table 1.

Table 2 reports the performance of the proposed SFFCC and MFCC-SFF features
for the vowel task by showing the results of the previous reference study
[20]. It can be seen that the performance of the proposed features is better
than in the reference study (except for the multi-modal system). In addition,
we would like to note that the standard deviations in accuracy are lower for
the proposed features in comparison to the prior study. The results of the
experiments indicate that the proposed features effectively capture speech
articulation variations which reflect changes in the disease severity.

##  5 Summary

In this study, we proposed two sets of features derived from the SFF spectrum
(SFFCC and MFCC-SFF) for the severity classification of PD from speech. The
classification experiments were carried out with three speaking tasks
(sustained vowels, sentences and read text) of the well-known PC-GITA
database. The experiments with the SVM classifier revealed that the proposed
SFFCC and MFCC-SFF features outperformed the conventional MFCCs features in
all three speaking tasks. Among the speaking tasks, the classification
performance was highest for the text reading task and lowest for the vowel
production task. This trend is due to the larger diversity of articulation
information in speech signals produced in the text reading task (due to
phonetically balanced sounds), which is helpful for severity classification.

##  6 Acknowledgements

This study was funded by the Academy of Finland (project no. 330139).

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# DiVincenzo-like criteria for autonomous quantum machines

José Antonio Marín Guzmán  Joint Center for Quantum Information and Computer
Science, NIST and University of Maryland, College Park, MD 20742, USA  Paul
Erker  Atominstitut, Technische Universität Wien, 1020 Vienna, Austria
Institute for Quantum Optics and Quantum Information (IQOQI), Austrian Academy
of Sciences, Boltzmanngasse 3, 1090 Vienna, Austria  Simone Gasparinetti
Department of Microtechnology and Nanoscience, Chalmers University of
Technology, 412 96 Gothenburg, Sweden  Marcus Huber  Atominstitut, Technische
Universität Wien, 1020 Vienna, Austria  Institute for Quantum Optics and
Quantum Information (IQOQI) Vienna, Austrian Academy of Sciences,
Boltzmanngasse 3, 1090 Vienna, Austria  Nicole Yunger Halpern
<EMAIL_ADDRESS>Joint Center for Quantum Information and Computer Science,
NIST and University of Maryland, College Park, MD 20742, USA  Institute for
Physical Science and Technology, University of Maryland, College Park, MD
20742, USA

###### Abstract

Controlled quantum machines have matured significantly. A natural next step is
to grant them autonomy, freeing them from timed external control. For example,
autonomy could unfetter quantum computers from classical control wires that
heat and decohere them; and an autonomous quantum refrigerator recently reset
superconducting qubits to near their ground states, as is necessary before a
computation. What conditions are necessary for realizing useful autonomous
quantum machines? Inspired by recent quantum thermodynamics and chemistry, we
posit conditions analogous to DiVincenzo’s criteria for quantum computing. Our
criteria are intended to foment and guide the development of useful autonomous
quantum machines.

Automata are machines that operate independently of external control. In an
early example, the ancient Greek mathematician Archytas of Tarentum supposedly
built a wooden pigeon that spun around in a circle [1]. For centuries,
automata served as curiosities for entertainment and for impressing visitors.
Not until the 18th-century Industrial Revolution did engineers harness
automata for practical purposes on large scales. (We use an expansive
definition of _automaton_ , not restricting the term to machines that resemble
humans or animals.) Today, automata speed up manufacturing, deliver packages,
drive car passengers, and even clean kitchen floors.

Autonomous quantum machines have embarked upon a trajectory that we hope will
unfold analogously. Quantum thermodynamicists have designed autonomous engines
[2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 12, 13, 14, 15, 16, 17, 18, 19],
refrigerators [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35,
36, 37, 38, Hewgill_20_Three, 39, 40, 41, 42, 43, 44, 45, 46, 47], clocks [48,
49, 50, 51, 52, 53, 54, 54], and more [55, 56, 57, 58, 59, 60] without
external drives [61, 62]. Some of these machines do not even require
thermodynamic-work inputs. Instead, the refrigerators and clocks siphon off
heat flowing between different-temperature baths nearby. Nature, too, has
designed autonomous quantum switches [63, 64, 65, 66], energy transducers
[67], and catalysts [68].

Experimentalists have just begun building autonomous quantum machines. For
example, autonomous quantum refrigerators have been realized with trapped ions
and superconducting qubits [37, 69]. Such experiments have initiated the
bridge from theory to reality. Yet they largely resemble Archytas’s pigeon:
They are impressive curiosities, not practical tools. For example, a single-
atom engine should extract only $\approx 1$ eV of work per cycle. Cooling and
trapping an atom costs far more work, so the engine cannot earn its keep. One
exception was initiated recently: the autonomous quantum refrigerator formed
from superconducting qubits [69]. The quantum refrigerator cooled a target
qubit to below the temperatures realizable via passive thermalization with the
dilution-refrigerator environment. This target qubit, reset to near its ground
state, could serve in a later quantum computation. In such an application, the
quantum refrigerator could draw energy from the temperature gradient between
the dilution refrigerator’s inner and outer plates. Cooling the autonomous
refrigerator to the quantum regime would cost no extra work, the dilution
refrigerator already being cold to support the upcoming computation.

One can envision other useful autonomous quantum machines.111 Practicality has
motivated the design of nonautonomous quantum thermal machines [70, 71, 72,
73, 74], too. First, an autonomous quantum computer would apply its own gates.
Autonomy could pare down the control wires in superconducting-qubit
architectures. Control wires limit the number of superconducting qubits that
fit on a chip [75, App. C.4.I]. Also, being classical, control wires heat
qubits up [76, Sec. 2.1]. Eliminating control wires could therefore benefit
quantum computers’ scalability and coherence times.222 One might object that
qubits cannot rotate without external control—without experiencing classical
fields. However, autonomous quantum computers could implement Brownian
circuits [77]. Brownian circuits consist of gates effected only with
qubit–qubit interactions.

To apply gates at the proper times, autonomous quantum computers would need
autonomous quantum machines of a second type: clocks. Theorists have designed
autonomous quantum clocks that tick by emitting photons [48]. Such clocks
differ from the quantum clocks used today, formed from ultracold atoms
interacting with lasers [78]: Today’s quantum clocks receive external feedback
when the atoms are measured, the lasers are tuned, etc.

Third, autonomous quantum machines can detect, transduce, and transmit energy.
_Photoisomers_ are molecules operative across nature and technologies [63, 64,
65, 66]. Examples include retinal, found in the protein rhodopsin in our eyes
[79]. A photoisomer has one shape in thermal equilibrium at biological
temperatures. Absorbing a photon—say, from the sun—enables the molecule to
switch configurations. The switch can galvanize chemical reactions leading to
the experience of sight. Photoisomers exhibit quantum phenomena including
coherence relative to the energy eigenbasis, as well as nonadiabatic
evolution. Beyond photoisomers, chemistry features other autonomous quantum
machines: Photosynthetic complexes transform light into chemical energy [67],
enzymes interconvert molecules with help from tunneling [68], etc.

Third, autonomy could benefit quantum sensors [80]. Quantum sensors detect
small magnetic fields and temperature gradients. External control includes
microwave pulses and placement near the to-be-detected source. For instance,
experimentalists have injected nanodiamond sensors into embryo tissues whose
temperatures need measuring [81, 82]. One can envision quantum sensors that
reach and report about sources autonomously.

More fancifully, one can imagine autonomous quantum machines assembling
bespoke molecules, servicing quantum computers, delivering atoms as drones, or
building other quantum devices. Such machines may seem like castles in the
air. But so did controlled quantum computers, decades ago; and quantum
computers have been built and are being scaled up. DiVincenzo’s criteria have
guided quantum computers from castle-in-the-air status to reality [83]. He
posited five criteria necessary for building a quantum computer, plus two
optional criteria necessary for information transmission. Analogously, we
posit eight criteria necessary for building a useful autonomous quantum
machine. Two optional criteria concern transportation and information
transmission. We hope that these criteria, analogously to DiVincenzo’s, guide
experimentalists in realizing useful autonomous machines.

First, we stipulate what we mean by _machine_ , _autonomous_ , and _quantum_.
Alternative definitions may exist, as the terms are broad. Still, we believe
our definitions to be reasonable. Specifying them will clarify our criteria:

  1. (a)

_Machine: A physical device, potentially formed from components working
together, that harnesses energy to accomplish a task._

  2. (b)

_Autonomous: A machine is autonomous if its microscopic Hamiltonian remains
constant in time._ Microscopic Hamiltonians contrast with effective
Hamiltonians. An effective Hamiltonian can result from shifting a microscopic
Hamiltonian into a rotating reference frame, then dropping small terms. A
microscopic Hamiltonian remains constant in the absence of time-dependent
external drives.

  3. (c)

_Quantum: A system is quantum if the axioms of quantum theory describe the
system usefully._ Quantum systems can, but need not, exhibit quantum phenomena
such as entanglement, coherence relative to relevant bases, discretized
spectra, measurement disturbance, contextuality [84, 85, 86], and the quantum-
computational resource called magic [87]. Classical systems can approximate
discretized spectra, and classical waves exhibit coherence. Quantum theory’s
axioms describe classical systems, but not usefully; classical theories offer
greater calculational efficiency and physical insight. We do not mean
_nonclassical_ by _quantum_. We call a phenomenon _nonclassical_ if no
classical theory (technically, no noncontextual ontological model [84, 85,
86]) reproduces the phenomenon.

##  I Criteria

This section contains our DiVincenzo-like criteria for realizing useful
autonomous quantum machines. Table 1 summarizes the criteria. We denote by
$\sigma_{a}$ the Pauli-$a$ operator, for $a=x,y,z$; by $\lvert 1\rangle$, the
eigenvalue-1 eigenstate of $\sigma_{z}$; and by $\lvert 0\rangle$, the
eigenvalue-$(-1)$ eigenstate.

Subsection | Criterion
---|---
A | Access to useful energy
B | Processing unit or target
C | Interactions among the machine’s components
D | Timekeeping mechanism
E | Structural integrity
F | Sufficient purity
G | Output worth the input
H | Ability to switch off after completing assignment
I | Mobility (optional)
J | Interoperability (optional)
Table 1: Summary: DiVincenzo-like criteria for autonomous quantum machines.

###  A Access to useful energy

Useful energy enables one to perform thermodynamic work. Work empowers a
machine to direct its motion—to overcome its momentum and random buffets from
its environment. Free energy, or a nonequilibrium generalization thereof,
offers the capacity to perform work. A machine can access this energy directly
or indirectly, as we now discuss.

We label as a _battery_ any system that reliably stores (the capacity to
perform) work and from which work can reliably be retrieved. Small-scale
batteries include adenosine triphosphate (ATP), a molecule that powers
chemical reactions in cells. Autonomous classical nanowalkers leverage ATP, as
discussed under criterion I. Quantum thermodynamics features multiple battery
models (e.g., [88, 89, 90, 91, 92, 93, 94]), many idealized. Examples include
a _work bit_ , a two-level system governed by a Hamiltonian
$\Delta\sigma_{z}$, for $\Delta>0$ [88, 89]. A work bit charges during $\lvert
0\rangle\mapsto\lvert 1\rangle$ and provides work during $\lvert
1\rangle\mapsto\lvert 0\rangle$.

Other energy sources do not provide work directly; they are nonequilibrium
systems from which the machine harvests free energy (or a generalization
thereof). Often, such sources have gradients of temperature, chemical
potential, or other thermodynamic forces. The commonest example consists of
two heat baths. One, at a temperature $T_{\mathcal{C}}$, is out of equilibrium
with a heat bath at a temperature $T_{\mathcal{H}}>T_{\mathcal{C}}$. Heat
flows between the baths. A machine can siphon off part of the current via a
three-body interaction (discussed under criterion C). Such machines are called
_absorption machines_ [62]. Despite providing energy, baths can threaten the
purity of a machine’s state. Yet as discussed under criterion F, non-Markovian
baths, which retain memories, can revive lost purity.

###  B Processing unit or target

A processing unit receives and uses the energy accessed by the machine
(criterion A). For example, a quantum computer’s processing units are qubits
that store quantum information that undergoes logic gates. In an autonomous
quantum clock (described under criterion D), a processing unit ticks, emitting
excitations.

A machine’s purpose can be to operate on a target. For instance, a
refrigerator’s target is the system that undergoes cooling. An engine’s target
may be a battery that stores the energy extracted by the engine. A drone’s
target is the package to be delivered.

###  C Interactions among the machine’s components

The interacting components can include the energy source (criterion A), a
processing unit (criterion B), a target (criterion B), and a timekeeping
device (criterion D). We illustrate interactions with two examples: a quantum
absorption refrigerator and a switch.

A simple quantum absorption refrigerator consists of three qubits: a hot, a
cold, and a target qubit (Fig. 1(a)) [20, 62]. The hot qubit, $\mathcal{H}$,
evolves under a Hamiltonian
$H_{\mathcal{H}}=\Delta_{\mathcal{H}}\,\sigma_{z}$, wherein
$\Delta_{\mathcal{H}}>0$. This qubit exchanges heat with a thermal bath at a
temperature $T_{\mathcal{H}}=1/\beta_{\mathcal{H}}$. (We set Boltzmann’s
constant to $k_{\mathrm{B}}=1$.) Hence $\mathcal{H}$ begins in the thermal
state $e^{-\beta_{\mathcal{H}}H_{\mathcal{H}}}/Z_{\mathcal{H}}$. The partition
function is $Z_{\mathcal{H}}\coloneqq{\rm
Tr}(e^{-\beta_{\mathcal{H}}H_{\mathcal{H}}})$. The cold qubit, $\mathcal{C}$,
evolves under a Hamiltonian $H_{\mathcal{C}}=\Delta_{\mathcal{C}}\,\sigma_{z}$
with a larger gap: $\Delta_{\mathcal{C}}>\Delta_{\mathcal{H}}$. $\mathcal{C}$
thermalizes with a bath at the lower temperature
$T_{\mathcal{C}}<T_{\mathcal{H}}$, to the state
$e^{-\beta_{\mathcal{C}}H_{\mathcal{C}}}/Z_{\mathcal{C}}$. The refrigerator
cools the target qubit, $\mathcal{T}$, which evolves under the Hamiltonian
$H_{\mathcal{T}}=\Delta_{\mathcal{T}}\,\sigma_{z}$, wherein
$\Delta_{\mathcal{T}}>0$.

(a) Quantum absorption refrigerator.

(b) Autonomous quantum clock.

Figure 1: Interactions amongst components: The quantum absorption refrigerator
and clock evolve under similar Hamiltonians. However, the machines’ qudits
have different energy gaps: the refrigerator’s
$\Delta_{\mathcal{H}}<\Delta_{\mathcal{C}}$, whereas the clock’s
$\Delta_{\mathcal{H}}>\Delta_{\mathcal{C}}$. Also, the number of the target’s
energy gaps can differ between machines. The gaps, with the baths’
temperatures ($T_{\mathcal{H}}$ and $T_{\mathcal{C}}$), determine the
directions in which energy flows.

To analyze the absorption refrigerator, we invoke the density operator’s
statistical interpretation: We can imagine running the refrigerator many
times. Each time, $\mathcal{H}$ begins in an energy eigenstate, $\lvert
1\rangle$ or $\lvert 0\rangle$, selected according to the Boltzmann
distribution
$\\{e^{-\beta_{\mathcal{H}}\Delta_{\mathcal{H}}}/Z_{\mathcal{H}},\,e^{\beta_{\mathcal{H}}\Delta_{\mathcal{H}}}/Z_{\mathcal{H}}\\}$.
The cold qubit begins in an energy eigenstate selected analogously. In an
illustrative “trial,” $\mathcal{H}$ begins excited (in $\lvert 1\rangle$),
while $\mathcal{C}$ and $\mathcal{T}$ begin de-excited (in $\lvert 0\rangle$).
The hot and target qubits emit their excitations into $\mathcal{C}$, via the
three-body interaction

$\displaystyle\lvert 1\rangle_{\mathcal{H}}\lvert 0\rangle_{\mathcal{C}}\lvert
1\rangle_{\mathcal{H}}\leftrightarrow\lvert 0\rangle_{\mathcal{H}}\lvert
1\rangle_{\mathcal{C}}\lvert 0\rangle_{\mathcal{H}}.$ (1)

Three-body interactions are necessary for autonomous cooling [20]. They can be
effected perturbatively with simultaneous two-body interactions [27, 29,
Hewgill_20_Three, 69]. The exchange (1) occurs only if
$\Delta_{\mathcal{H}}+\Delta_{\mathcal{T}}=\Delta_{\mathcal{C}}$—only upon
satisfying the first law of thermodynamics. The thermal qubits’ gaps and
temperatures bias the interaction (1) to the right. Ending in its ground
state, $\mathcal{T}$ has undergone cooling by the refrigerator.

The second interaction example involves a _switch_ , a component that keeps
time and controls the rest of the machine. Denote by $\mathcal{S}$ a switch
whose Hilbert space has an eigenbasis $\\{\lvert\varphi_{j}\rangle\\}$ [95,
Suppl. Note VIII]. If $\mathcal{S}$ is in state $\lvert\varphi_{j}\rangle$,
then a Hamiltonian $H_{j}$ evolves the rest of the machine. Under different
$H_{j}$’s, the machine performs different tasks. For example, $H_{j}$ can
denote the $j^{\rm th}$ gate available to an autonomous quantum computer. A
Hamiltonian $H_{\mathcal{S}}$ evolves the switch’s state between $\lvert
j\rangle$’s. If the rest of the machine has a constant Hamiltonian
$H_{\mathcal{R}}$, the total Hamiltonian is

$\displaystyle H_{\rm
tot}=\sum_{j}\left(H_{j}\otimes\lvert\varphi_{j}\rangle\\!\langle\varphi_{j}\rvert\right)+H_{\mathcal{S}}+H_{\mathcal{R}}.$
(2)

Identity operators $\mathbbm{1}$ are tensored on wherever necessary for each
operator to act on the full Hilbert space.

Photoisomers were argued to contain switches [App. D][96].333 Photoisomers are
often called molecular switches. Molecular switches should not be confused
with the switches described in the previous paragraph—timekeeping switches
that control the rest of a machine. Molecular switches contain timekeeping
switches, according to [96]. A photoisomer has two degrees of freedom (DOFs),
one nuclear and one electronic. The nuclear DOF, being heavy and slow,
determines the electronic DOF’s potential landscape. After photoexcitation,
some nuclei rotate away from others under
$H_{\mathcal{S}}=\ell_{\varphi}^{2}/(2I)$. $\ell_{\varphi}$ denotes the
angular-momentum operator, and $I$ denotes the moment of inertia. $\varphi$
labels the nuclei’s relative angular position (Fig. 2). The angles form a
continuous set; so the sum in Eq. (2) becomes an integral, and $H_{j}$ becomes
$H(\varphi)$. $H_{\mathcal{S}}$ evolves the nuclear configuration from some
initial angular position $\lvert\varphi_{0}\rangle$ to other
$\lvert\varphi\rangle$’s. Rotating, the nuclear DOF changes the potential
landscape and so the $H(\varphi)$ experienced by the electronic DOF, which
forms the rest of the machine.

Figure 2: Photoisomer: The molecule can switch between _cis_ and _trans_
configurations as some of its nuclei rotate. Reproduced from Fig. 1 of [96].

We have discussed switches under criterion C to elucidate their interactions
with the rest of their machines. As timekeepers, though, switches belong also
under the following criterion.

###  D Timekeeping mechanism

Timekeeping devices include the switches introduced in the previous
subsection; clocks, which tick regularly; and timers, which announce when
programmable amounts of time have passed. Below, we review a simple autonomous
quantum clock. Then, we discuss timekeeping mechanisms not formed from
physical devices. Finally, we delineate three purposes of autonomous quantum
machines’ timekeeping mechanisms.

An autonomous quantum machine may carry an autonomous quantum clock to avoid
decoherence by a classical clock. Reference [48] introduced a simple,
idealized autonomous quantum clock (Fig. 1(b)). The clock contains a hot qubit
$\mathcal{H}$ and a cold qubit $\mathcal{C}$, like the absorption refrigerator
under criterion C. $\mathcal{H}$ has the larger gap here, however:
$\Delta_{\mathcal{H}}>\Delta_{\mathcal{C}}$. The clock contains also a ladder
$\mathcal{L}$ of $d$ energy levels:
$H_{\mathcal{L}}=\sum_{j=0}^{d-1}j\Delta\,\lvert j\rangle\\!\langle j\rvert$.
$\mathcal{L}$ begins in its ground state, $\lvert 0\rangle$. The qudits
undergo a three-body interaction similar to the quantum absorption
refrigerator’s Eq. (1):

$\displaystyle\lvert 1\rangle_{\mathcal{H}}\lvert 0\rangle_{\mathcal{C}}\lvert
0\rangle_{\mathcal{L}}\leftrightarrow\lvert 0\rangle_{\mathcal{H}}\lvert
1\rangle_{\mathcal{C}}\lvert j\rangle_{\mathcal{L}}.$ (3)

The ladder state $\lvert j\rangle$ satisfies the first law through
$\Delta_{\mathcal{H}}=\Delta_{\mathcal{C}}+j\Delta$. The condition
$\Delta_{\mathcal{H}}>\Delta_{\mathcal{C}}$ biases the interaction (3)
rightward. The hot qubit, de-exciting (undergoing $\lvert
1\rangle\mapsto\lvert 0\rangle$), drives the cold qubit upward in energy
(through $\lvert 0\rangle\mapsto\lvert 1\rangle$) and drives the ladder system
upward (through $\lvert 0\rangle\mapsto\lvert j\rangle$). The baths reset
$\mathcal{H}$ and $\mathcal{C}$, and the process repeats. Upon reaching its
top rung, the ladder system ticks—emits an excitation—returning to $\lvert
0\rangle$.

Figures of this clock’s merit include its _accuracy_ , $N$ [50]. Denote by
$\bar{t}$ the average time interval between ticks; and by $\Delta t$, the
standard deviation in that time interval. The accuracy is
$N\coloneqq(\bar{t}/\Delta t)^{2}$. To interpret it, we consider the limit as
the number of ticks, assumed to be distributed independently and identically,
approaches infinity. $N$ equals the number of ticks that pass, on average,
before the clock is off by one tick. One can improve $N$ and other figures of
merit by complicating the clockwork [50]. We discuss figures of merit further
under criterion G.

Not all autonomous quantum machines need timekeeping devices as physical
components. The finiteness of a machine’s energy resources can induce a timer,
as can the machine’s coherence, we explain under criterion H. Furthermore,
some autonomous quantum engines operate continuously: They undergo fixed
dynamics, rather than cycles formed from discrete (timed) strokes [97].

An autonomous quantum machine’s timekeeper fulfills three purposes. First, the
timekeeper ensures that the machine initiates an action at the right time. For
example, a quantum computer should begin each gate during the appropriate part
of a computation. A Rydberg-atom computer [98] might autonomously perform a
Rydberg-blockade entangling gate [99] as follows. Let the computer have an
autonomous quantum clock that ticks by emitting an excitation. Two excitations
can boost two atoms to their Rydberg (high-energy) states. The excited atoms
will repel each other, entangling. Two more excitations may stimulate
emissions from the atoms, which will return to their ground states.

Second, the timekeeper ensures that the machine performs an action for the
desired amount of time. For example, a quantum computer implements a gate by
effecting some Hamiltonian for the correct time interval. Xuereb _et al._
calculated the average fidelity $\bar{\mathcal{F}}$ of a CNOT gate $U$
approximated with an autonomous quantum clock of accuracy $N$ [100]. The
evolution implemented is a channel $\mathcal{E}$. Denote by $d\psi$ the Haar
measure (loosely speaking, the uniform measure) over the set of two-qubit pure
states $\lvert\psi\rangle$. The average fidelity is defined as
$\bar{\mathcal{F}}\coloneqq\int
d\psi\langle\psi|U^{\dagger}\,\mathcal{E}(\lvert\psi\rangle\\!\langle\psi\rvert)\,U\lvert\psi\rangle$.
According to [100], $\bar{\mathcal{F}}=(2+e^{-\pi^{2}/(2N)})/3$. The clock
accuracy $N$ exponentially influences the correction to the constant $2/3$.
Third, a timekeeper ensures that the machine turns off, satisfying criterion
H, upon completing its task.

###  E Structural integrity

Imagine placing three atoms beside each other, as with optical tweezers. If
left unattended, the atoms will drift apart. A machine’s components must not
separate, lest they cease to satisfy the interaction criterion C. Denote by
$r$ the distance between two components. Any Coulombic interaction potential
between them weakens as $r^{-1}$; and any Rydberg-blockade potential, as
$r^{-6}$ [98]. An autonomous quantum machine formed from multiple atoms, ions,
etc. requires trapping, as by lasers. Laser light, forming a wave, provides a
time-dependent external potential. Hence a laser-trapped machine can be
autonomous only if the trapped spatial DOFs play no role in the machine’s
functioning. Those DOFs’ Hamiltonian must commute with the microscopic
Hamiltonian that governs the machine’s operation.

Yet a machine need not consist of physically separated components. A
photoisomer functions as an autonomous quantum energy transmitter, as
explained in the introduction. The molecule contains an autonomous quantum
clock, according to [96]. Similarly, photosynthetic complexes [67] and enzymes
[68] are autonomous machines modeled usefully with quantum theory. A molecule
can therefore form an autonomous quantum machine. So can an atom, arguably.444
The atom interacts with heat baths whose frequencies are filtered. One might
count the filters as parts of the machine. For example, three atomic energy
levels can form an autonomous engine or refrigerator [2, 3, 20]. Not all self-
contained autonomous quantum machines need be natural: Superconducting qubits
cannot separate if printed on the same chip.

###  F Sufficient purity

In this subsection, we define purity and compare it with coherence. Next, we
explain the _sufficient_ in the subsection’s title. We then discuss the
tradeoff between purity and accessible energy. Non-Markovianity may soften the
tradeoff.

Purity is defined as follows. Denote by $\rho$ an arbitrary quantum state
(density operator) defined on a $d$-dimensional Hilbert space $\mathscr{H}$.
$\rho$ has an amount $\mathcal{P}(\rho)\coloneqq{\rm Tr}(\rho^{2})$ of purity.
If $\rho$ is pure (if $\rho=\lvert\psi\rangle\\!\langle\psi\rvert$ for some
$\lvert\psi\rangle\in\mathscr{H}$), then $\mathcal{P}(\rho)=1$. If $\rho$ is
the maximally mixed state $\mathbbm{1}/d$, then $\mathcal{P}(\rho)=1/d$. A
machine’s processing unit, target, timekeeping device, and/or output might
require purity at various times.

One might expect our criteria to include coherence, rather than purity. In
quantum thermodynamics, coherence relative to the energy eigenbasis often
serves as a resource (e.g., [102, 103, 104, 105, 106, 107, 108, 109, 49, 96,
110]). To quantify this coherence, we suppose that $\rho$ evolves under a
Hamiltonian $H=\sum_{j}E_{j}\lvert j\rangle\\!\langle j\rvert$. For $j\neq k$,
the off-diagonal element $\rho_{jk}\coloneqq\langle j\rvert\rho\lvert
k\rangle$ is a coherence. Denote the one-norm of an operator $\mathcal{O}$ by
$||\mathcal{O}||_{1}\coloneqq{\rm
Tr}\big{(}\sqrt{\mathcal{O}\mathcal{O}^{\dagger}})$. The mode-$\omega$
coherence can be quantified with
$\sum_{j,k:E_{j}-E_{k}=\omega}||\rho_{jk}||_{1}$ [104] (other measures exist).
Despite its applications in timekeeping and work extraction, such coherence
can be undesirable. For instance, qubits are often initialized to near their
ground states, $\lvert 0\rangle$, before a quantum computation. Resetting
computational qubits, an autonomous quantum refrigerator destroys coherences
relative to the energy eigenbasis. Still, the refrigerator enhances the
qubits’ purities. Hence our criteria’s including purity, rather than
coherence.

Our criteria include _sufficient_ purity for two reasons: (i) Different
machine components may require different amounts of purity. (ii) One component
may require different amounts of purity at different times. We illustrate with
the quantum absorption refrigerator described under criterion C. The hot and
cold qubits, interacting with finite-temperature baths, are always mixed. The
target requires purity—closeness to $\lvert 0\rangle$—but only when the
protocol ends. In contrast, suppose that an autonomous quantum clock’s ladder
(Fig. 1(b)) is mixed at any time. The clock’s accuracy seems likely to suffer
[48]. Similarly, a quantum computer must retain enough purity to meet the
threshold for fault-tolerant quantum error correction throughout its
computation [111, 112, 113]. The error rate is 0.6%–1% for the two-dimensional
surface code [114, 115].

The sufficient-purity criterion trades off with the accessible-energy
criterion A. As described below A, an autonomous quantum machine may extract
energy from heat baths. The baths reduce the machine’s $\mathcal{P}(\rho)$.
However, non-Markovianity can revive a machine’s purity [116]. A non-Markovian
bath retains a memory; information, upon entering the bath from the machine,
can recollect and act back on the machine. Non-Markovianity can arise from
strong system–bath couplings, small baths, low temperatures, and initial
system–environment couplings. One can engineer non-Markovianity from a
Markovian environment: One would mediate machine–environment interactions
through a memory-retaining interface [117]. Appendix A illustrates non-
Markovianity’s potential for reviving a machine’s purity. Non-Markovianity,
the appendix shows, can help baths achieve the energy criterion A without
endangering the purity criterion F as much as thermal baths do.

###  G Output worth the input

A machine’s output is intended to fulfill the machine’s purpose. How
effectively the output fulfills the purpose depends on figures of merit. We
illustrate with four examples. First, a refrigerator outputs a cooled target,
$\mathcal{T}$, whose final temperature quantifies the refrigerator’s
effectiveness. So does the refrigerator’s steady-state coefficient of
performance (COP): Denote by $\dot{Q}_{\mathcal{T}}$ the current of heat
extracted from $\mathcal{T}$ and by $\dot{Q}_{\mathcal{H}}$ the current of
heat flowing from the hot bath. The steady-state COP is
$\dot{Q}_{\mathcal{T}}/\dot{Q}_{\mathcal{H}}$ [62].

Second, an engine outputs work. Denote by $\dot{W}$ the power and by
$\dot{Q}_{\mathcal{H}}$ the current of heat flowing from the engine’s hot
bath. Figures of merit include the steady-state efficiency,
$\eta=\dot{W}/\dot{Q}_{\mathcal{H}}$; the power, $\dot{W}$; and the power at
maximum efficiency, $\max_{\eta}\\{\dot{W}\\}$.

Third, a clock outputs ticks (emitted excitations). Its figures of merit
include the accuracy, $N\coloneqq(\bar{t}/\Delta t)^{2}$, described under
criterion D. Another figure of merit is the resolution [48]. Recall that
$\bar{t}$ denotes the average time between successive ticks. The resolution is
$1/\bar{t}$.

Fourth, a quantum computer outputs a state $\sigma$ that approximates an ideal
$\rho$. Figures of merit can quantify the distance between the states. For
example, the fidelity is $\big{(}{\rm
Tr}\sqrt{\sqrt{\sigma}\,\rho\,\sqrt{\sigma}}\big{)}^{2}$.

Each figure of merit above measures an output’s quality. Any agent running a
machine can choose their threshold for an acceptable output figure of merit.
Yet a more comprehensive figure of merit assesses not only the output, but
also the input [118]. Inputs can include energy, time, control resources, and
funding. As discussed in the introduction, one might expect from an autonomous
atomic engine $\approx 1$ eV per cycle. However, cooling the engine to the
quantum regime and setting the engine up can cost orders of magnitude more
energy. A useful autonomous quantum machine’s output merits the input.

###  H Ability to switch off after completing assignment

At least three mechanisms can spur an autonomous quantum machine to shut down.
First, consider a machine that contains an autonomous quantum clock (criterion
D). The clock can galvanize not only steps in the rest of the machine’s
operation (e.g., gates implemented at the right times), but also a halt.
Second, the machine accesses only a finite amount of energy (criterion A)—for
instance, finite-size hot and cold baths. Depleting the energy source winds
the machine down.

Third, components of the machine can require varying degrees of purity to
operate (criterion F). Hence the components’ coherence times limit the
operation time. We illustrate with a qubit with a ground state $\lvert
0\rangle$, excited state $\lvert 1\rangle$, and time-evolving density operator
$\rho(t)=\sum_{j,k=0}^{1}\rho_{jk}(t)\lvert j\rangle\\!\langle k\rvert$, for
$t\geq 0$ [119]. Over the amplitude-damping time $T_{1}$, the excited-state
weight $\rho_{11}(t)$ decays to $1/e$ of its initial value:
$\rho_{11}(T_{1})=\rho_{11}(0)/e$. Over the phase-damping time $T_{2}<T_{1}$,
the off-diagonal elements decay similarly: $\rho_{jk}(T_{2})=\rho_{jk}(0)/e$
$\forall j\neq k$.

###  I Mobility (optional)

DiVincenzo listed two optional criteria necessary for transmitting information
from place to place. Our optional criteria, I and J, concern the transmission
of machines and of messages between machines. Mobility would benefit
autonomous quantum machines including delivery drones, sensors that navigate
to their targets, and machines that build molecules (or other machines).
External potentials and accessible energy (criterion A) can help machines
achieve directionality.

We illustrate with autonomous classical nanowalkers, which may inspire
builders of autonomous quantum walkers. Reference [120] reports on a
nanowalker, formed from a DNA strip, walking along a track consisting of more
DNA strips. The nanowalker burns ATP as fuel. Enzymes ensure the nanowalker’s
directionality: One enzyme ligates (joins together) the walker and the next
site, and another enzyme cleaves the walker from the previous site.

###  J Interoperability (optional)

Autonomous quantum machines may communicate with each other and work together.
Communications may occur via (at least) two mechanisms.

First, machine $\mathcal{A}$ may emit a signal for machine $\mathcal{B}$ to
absorb. For example, we envisioned an autonomous quantum clock emitting an
excitation absorbed by an autonomous Rydberg-atom computer (under criterion
D).555 Under criterion D, we cast the clock as a part of the computer. Here,
we cast the clock as separate but as aiding the computer. $\mathcal{A}$ and
$\mathcal{B}$ must satisfy three requirements:

  1. (1)

$\mathcal{B}$ must be physically able to detect $\mathcal{A}$’s signal. For
example, denote by $\hbar\omega$ the excitation’s energy, by
$\Omega_{\mathcal{B}}$ the bandwidth of $\mathcal{B}$, and by $E_{0}$ the
center of $\mathcal{B}$’s energy spectrum. The energy must lie in the
bandwidth:
$\hbar\omega\in[E_{0}-\Omega_{\mathcal{B}}/2,\,E_{0}+\Omega_{\mathcal{B}}/2]$.

  2. (2)

The signal must, given condition (1), have a sufficiently high probability of
affecting $\mathcal{B}$. The excitation’s momentum should direct the signal
toward $\mathcal{B}$, and no intervening medium should swallow the signal.
Once the excitation arrives, $\mathcal{B}$ should have a high probability of
absorbing it. Fermi’s golden rule governs the rate
$\Gamma_{\text{i}\mapsto\text{f}}$ at which $\mathcal{B}$ jumps from a state
$\lvert\rm i\rangle$ to any energy-$E_{\rm f}$ state $\lvert\rm f\rangle$:
$\Gamma_{\text{i}\mapsto\text{f}}=\frac{2\pi}{\hbar}\,|\langle\text{f}\rvert
H^{\prime}\lvert\text{i}\rangle|^{2}\,\mu(E_{\rm f})$. $H^{\prime}$ denotes
the excitation-induced perturbation to $\mathcal{B}$’s Hamiltonian. $\mu(E)$
denotes the density of $\mathcal{B}$’s energy-$E$ states. If $\mathcal{B}$
absorbs the signal through a photodetector, the external efficiency quantifies
how effectively $\mathcal{B}$ satisfies requirement (2).

  3. (3)

After absorbing a signal, $\mathcal{B}$ might take time to reset before being
able to absorb another signal. Denote this _dead time_ by $\tau_{\rm D}$.
Suppose that $\mathcal{A}$ sends multiple signals. The time interval $\tau$
between them should be $\tau\geq\tau_{\rm D}$. Photodetectors have dead times.
Also, consider a photoisomer that has absorbed a photon and rotated into a
metastable configuration. The molecule must re-equilibrate before undergoing
another photoexcitation. The metastable configuration can have a half-life of
two days, in solution at room temperature, if the photoisomer is azobenzene
[64].

Under the second signaling mechanism, machine $\mathcal{A}$ arrives near
$\mathcal{B}$, changing the external potential experienced by $\mathcal{B}$.
Again, we illustrate with a photoisomer. Some of its nuclei rotate through
many $\lvert\varphi\rangle$’s, in the notation under criterion C. Advancing
so, the nuclei change the electronic DOF’s potential landscape—change
$H(\varphi)$.

In another example, a qubit and a bosonic mode undergo a dispersive
interaction. Such an interaction is effective (approximate), and we omit the
derivation [121]. During it, one attributes to the qubit an effective gap
$\Delta$; to the mode an effective frequency $\omega$, a creation operator
$a^{\dagger}$, and an annihilation operator $a$; and to the coupling a
strength $\chi$. We set $\hbar=1$. The dispersive Hamiltonian is

$\displaystyle\Delta\,\sigma_{z}+\omega
a^{\dagger}a+\chi\sigma_{z}a^{\dagger}a.$ (4)

If the qubit is excited (in $\lvert 1\rangle$), it effectively adds $\chi$ to
the mode’s frequency. Analogously, if the mode is occupied, it effectively
adds $\chi$ to the qubit’s gap. Each subsystem therefore affects the other’s
Hamiltonian.

##  II Outlook

We have proposed DiVincenzo-like criteria for autonomous quantum machines.
Eight criteria, we regard as necessary for most autonomous machines’ useful
operation. Satisfying the remaining two criteria, machines can move and
interface. We emphasize the _useful_ in the penultimate sentence: The
literature has demonstrated that autonomous quantum machines can be designed
and, with effort, realized experimentally. Autonomous quantum machines should
now progress from curiosities to tools, like their classical counterparts. We
hope that these criteria guide the progression.

Certain autonomous quantum machines appear to call for another criterion:
instructions. Instructions could guide an autonomous quantum drone to walk
some distance forward, turn leftward, and then turn off. In another example,
an autonomous quantum computer should implement one circuit, rather than
another. Upon satisfying our criteria, however, one can effect instructions,
which form a kind of emergent criterion. One could guide the drone with a
track covered by the mobility criterion (I). The switching-off criterion (H)
would enforce the final instruction. One can feed an autonomous quantum
computer instructions by leveraging a clock (criterion D), interactions
(criterion C), interoperability (criterion J), etc. Single-purpose machines,
such as autonomous quantum engines, do not require such instructions.

###### Acknowledgements.

N.Y.H. thanks Kenneth Brown for input about castle-in-the-air autonomous
quantum machines, David Limmer for input about chemical machines, and John
Preskill for input about superconducting-qubit quantum computers. N.Y.H. and
J.A.M.G. acknowledge support from the National Science Foundation (QLCI grant
OMA-2120757), the John Templeton Foundation (award no. 62422), and NIST grant
70NANB21H055_0. P.E. and M.H. acknowledge support from the European Research
Council (Consolidator grant “Co-coquest” 101043705), the European flagship on
quantum technologies (“ASPECTS” consortium 101080167), and FQXi (FQXi-
IAF19-03-S2, within the project “Fueling quantum field machines with
information”). S.G. acknowledges support from the Knut and Alice Wallenberg
foundation via the Wallenberg Centre for Quantum Technology (WACQT), from the
European Research Council (Grant 101041744 ESQuAT), from the European Union
(Grant 101080167 ASPECTS), and from the Swedish Research Council (Grant
2021-05624).

##  Appendix A Non-Markovianity’s potential for reviving a machine’s purity

We illustrate the title’s claim by following [116]. That reference presents a
setup common in cavity quantum electrodynamics: Consider a qubit $\mathcal{S}$
governed by a Hamiltonian $H_{\mathcal{S}}=\Delta\,\sigma_{z}$ with an energy
splitting $\Delta>0$. A bosonic environment $\mathcal{E}$ evolves under a
Hamiltonian
$H_{\mathcal{E}}=\sum_{\ell}\omega_{\ell}\,a_{\ell}^{\dagger}a_{\ell}$. Mode
$\ell$ corresponds to energy $\omega_{\ell}$, to an annihilation operator
$a_{\ell}$, and to a creation operator $a_{\ell}^{\dagger}$. In the rotating-
wave approximation, $S$ couples to $\mathcal{E}$ via the Jaynes–Cummings
Hamiltonian $H_{\rm
int}=\sum_{\ell}\big{(}g_{\ell}\,\sigma_{+}a_{\ell}+g_{\ell}^{*}\sigma_{-}a_{\ell}^{\dagger}\big{)}$.
The qubit has raising and lowering operators $\sigma_{+}\coloneqq\lvert
1\rangle\\!\langle 0\rvert$ and $\sigma_{-}\coloneqq\lvert 0\rangle\\!\langle
1\rvert$, and the $g_{\ell}$’s denote coupling constants. Denote by
$\rho_{\mathcal{S}/\mathcal{E}}(t)$ the time-$t$ reduced state of
$\mathcal{S}/\mathcal{E}$ in the interaction picture. Let $\mathcal{S}$ and
$\mathcal{E}$ begin in a product state
$\rho_{\mathcal{S}}(0)\otimes\rho_{\mathcal{E}}(0)$, $\rho_{\mathcal{E}}(0)$
being the bath’s vacuum state.

$\mathcal{S}$ evolves under a linear, completely positive map $\Phi_{t}$ as
$\rho_{\mathcal{S}}(t)=\Phi_{t}\,\rho_{\mathcal{S}}(0)$. The evolved state has
off-diagonal coherences $\rho_{jk}(t)\coloneqq\langle
j\rvert\rho_{\mathcal{S}}(t)\lvert k\rangle$ determined by the complex
_decoherence function_ $G(t)$: $\rho_{jk}(t)=G(t)\rho_{jk}(0)$ $\forall j\neq
k$. To specify $G(t)$, we assume that $\mathcal{E}$ has a Lorentzian spectral
density function (SDF) of width $\lambda$. Define the function
$\lambda^{\prime}\coloneqq\sqrt{\lambda^{2}-2\gamma_{0}\lambda}\,,$ whose
$\gamma_{0}$ depends on the couplings $g_{\ell}$. Suppose that the mode is on
resonance with the qubit: The SDF is centered at $\Delta$. The decoherence
function becomes

$\displaystyle G(t)=e^{-\lambda
t/2}\left[\cosh\left(\frac{\lambda^{\prime}t}{2}\right)+\frac{\lambda}{\lambda^{\prime}}\,\sinh\left(\frac{\lambda^{\prime}t}{2}\right)\right].$
(A1)

If $\gamma_{0}<\lambda/2$, the coupling is weak, and the bath is Markovian.
$G(t)$ is real and decreases monotonically with $t$. If
$\gamma_{0}>\lambda/2$, the coupling is strong, and the bath is non-Markovian.
Figure 3 shows the implications for a qubit initialized to $\lvert 1\rangle$.
The complex $G(t)$ has a magnitude that oscillates over time (Fig. 3(a)). The
coherences $|\rho_{jk}|$ revive, and so does $\rho_{\mathcal{S}}(t)$’s purity
(Fig. 3(b)). Hence non-Markovianity can help baths achieve the energy
criterion A without threatening the purity criterion F as much as thermal
baths do.

(a) Decoherence function vs. time.

(b) Purity vs. time.

Figure 3: Coherence and purity at strong coupling: A bosonic mode decoheres a
qubit initialized in $\lvert 1\rangle$. The Lorentzian width $\lambda=1$. The
coupling $\gamma_{0}=50\lambda/2$ is strong, so the bath is non-Markovian. For
all $t$, the decoherence function $G(t)\in\mathbb{R}$. $G(t)$ and the purity,
$P\bm{(}\rho_{S}(t)\bm{)}$, oscillate. Hence non-Markovianity can revive the
purity. ($P$ eventually reaches 1 because the decoherence maps all states to
$\lvert 0\rangle$.)

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