One T Gate Makes Distribution Learning Hard.

The task of learning a probability distribution from samples is ubiquitous across the natural sciences. The output distributions of local quantum circuits are of central importance in both quantum advantage proposals and a variety of quantum machine learning algorithms. In this work, we extensively characterize the learnability of output distributions of local quantum circuits. Firstly, we contrast learnability with simulatability by showing that Clifford circuit output distributions are efficiently learnable, while the injection of a single T gate renders the density modeling task hard for any depth d=n^{Ω(1)}. We further show that the task of generative modeling universal quantum circuits at any depth d=n^{Ω(1)} is hard for any learning algorithm, classical or quantum, and that for statistical query algorithms, even depth d=ω[log(n)] Clifford circuits are hard to learn. Our results show that one cannot use the output distributions of local quantum circuits to provide a separation between the power of quantum and classical generative modeling algorithms, and therefore provide evidence against quantum advantages for practically relevant probabilistic modeling tasks.

TIMP-1 as a biomarker in obstructive sleep apnea: screening, monitoring, risk stratification, and a step towards precision medicine.

OBJECTIVE: The diagnosis of obstructive sleep apnea (OSA) is a complex time- and resource-intensive diagnostic procedure. Since tissue inhibitors of matrix metalloproteinases (TIMP's) are involved in various pathophysiological processes and are correlated with a high cardiovascular risk, TIMP's appear to be a suitable candidate for an OSA-biomarker. PATIENTS AND METHODS: In a prospective controlled diagnostic study, TIMP-1 serum levels of 273 OSA-patients and controls were analyzed for correlation with OSA severity, BMI, age, sex, cardio-/ cerebrovascular comorbidities. Furthermore, longitudinal medium- and long-term effects of CPAP-treatment (n=15) on TIMP-1-levels were investigated. RESULTS: TIMP-1 was clearly linked to OSA as well as to disease severity (mild, moderate, severe; each p<0.001) and was not influenced by age, gender, BMI, or cardio-/cerebrovascular comorbidities. ROC curve analysis revealed an AUC of 0.91 ± 0.017 SE (p<0.001), suggesting a TIMP-1 cut-off value of 75 ng/ml (sensitivity 0.78; specificity 0.91) being especially sensitive for patients with severe OSA (sensitivity 0.89; specificity 0.91). The likelihood ratio was 8.88, while the diagnostic odds ratio was 37.14. CPAP-treatment led to a significant decrease of TIMP-1 after 6-8 months (p=0.008). CONCLUSIONS: TIMP-1 seems to fulfill the preconditions for a circulating OSA-biomarker: disease-specific with a mandatory presence in affected patients, reversible on treatment, reflects disease severity and provides a cutoff value between the healthy state and disease. In the clinical routine, TIMP 1 may help to stratify the individual OSA-associated cardiovascular risk and to monitor the treatment response to CPAP-therapy as a further step towards providing a personalized therapy.

Efficient Unitary Designs with a System-Size Independent Number of Non-Clifford Gates.

Many quantum information protocols require the implementation of random unitaries. Because it takes exponential resources to produce Haar-random unitaries drawn from the full n-qubit group, one often resorts to t-designs. Unitary t-designs mimic the Haar-measure up to t-th moments. It is known that Clifford operations can implement at most 3-designs. In this work, we quantify the non-Clifford resources required to break this barrier. We find that it suffices to inject O ( t 4 log 2 ( t ) log ( 1 / ε ) ) many non-Clifford gates into a polynomial-depth random Clifford circuit to obtain an ε -approximate t-design. Strikingly, the number of non-Clifford gates required is independent of the system size - asymptotically, the density of non-Clifford gates is allowed to tend to zero. We also derive novel bounds on the convergence time of random Clifford circuits to the t-th moment of the uniform distribution on the Clifford group. Our proofs exploit a recently developed variant of Schur-Weyl duality for the Clifford group, as well as bounds on restricted spectral gaps of averaging operators.

Closing Gaps of a Quantum Advantage with Short-Time Hamiltonian Dynamics.

Demonstrating a quantum computational speed-up is a crucial milestone for near-term quantum technology. Recently, sampling protocols for quantum simulators have been proposed that have the potential to show such a quantum advantage, based on commonly made assumptions. The key challenge in the theoretical analysis of this scheme-as of other comparable schemes such as boson sampling-is to lessen the assumptions and close the theoretical loopholes, replacing them by rigorous arguments. In this work, we prove two open conjectures for a simple sampling protocol that is based on the continuous time evolution of a translation-invariant Ising Hamiltonian: anticoncentration of the generated probability distributions and average-case hardness of exactly evaluating those probabilities. The latter is proven building upon recently developed techniques for random circuit sampling. For the former, we exploit the insight that approximate 2-designs for the unitary group admit anticoncentration. We then develop new techniques to prove that the 2D time evolution of the protocol gives rise to approximate 2-designs. Our work provides the strongest theoretical evidence to date that Hamiltonian quantum simulators are classically intractable.

The Erlangen Questionnaire: a new 5-item screening tool for obstructive sleep apnea in a sleep clinic population - A prospective, double blinded study.

OBJECTIVE: The aim of this study was to develop and validate a simplified screening tool for identifying obstructive sleep apnea (OSA) in a sleep clinic population. PATIENTS AND METHODS: A total of 160 patients from a sleep clinic population was enrolled in the prospective, double-blinded study. OSA was defined by using diagnostic criteria of the ICSD-3 after overnight polysomnography (n=96) or polygraphy (n=64). The first 60 patients filled out a multi-item questionnaire and formed the development group. Subsequently, the five most predictive factors were selected to create the Erlangen Questionnaire (EQ). For validation of the EQ, the next 100 patients formed the validation group. RESULTS: The following factors were incorporated into the 5-item EQ: (1) ESS > 10, (2) age > 60, (3) gasping and (4) cardiovascular risk factors, (5) witnessed apneas. The EQ had sensitivities of 94.3%, 92.7% and 92.3%, specificities of 50.0%, 33.3% and 22.9%, positive predictive values of 81.5%, 62.9% and 29.6%, and negative predictive values of 78.9%, 78.9% and 89.5% with respect to mild, moderate and severe OSA. CONCLUSIONS: EQ is a compact 5-item-based, concise and easy-to-use screening tool to identify both male and female patients with OSA in a sleep clinic-population and exhibits all essential factors of internal and external validity. The results of the EQ are comparable to the best-validated and most commonly used STOP-Bang questionnaire regarding sensitivity and specificity in a sleep clinic population.