Bounds on entanglement assisted source-channel coding via the Lovasz theta number and its variants

We study zero-error entanglement assisted source-channel coding (communication in the presence of side information). Adapting a technique of Beigi, we show that such coding requires existence of a set of vectors satisfying orthogonality conditions related to suitably defined graphs $G$ and $H$. Such vectors exist if and only if $\vartheta(\overline{G}) \le \vartheta(\overline{H})$ where $\vartheta$ represents the Lov\'asz number. We also obtain similar inequalities for the related Schrijver $\vartheta^-$ and Szegedy $\vartheta^+$ numbers. These inequalities reproduce several known bounds and also lead to new results. We provide a lower bound on the entanglement assisted cost rate. We show that the entanglement assisted independence number is bounded by the Schrijver number: $\alpha^*(G) \le \vartheta^-(G)$. Therefore, we are able to disprove the conjecture that the one-shot entanglement-assisted zero-error capacity is equal to the integer part of the Lov\'asz number. Beigi introduced a quantity $\beta$ as an upper bound on $\alpha^*$ and posed the question of whether $\beta(G) = \lfloor \vartheta(G) \rfloor$. We answer this in the affirmative and show that a related quantity is equal to $\lceil \vartheta(G) \rceil$. We show that a quantity $\chi_{\textrm{vect}}(G)$ recently introduced in the context of Tsirelson's conjecture is equal to $\lceil \vartheta^+(\overline{G}) \rceil$. In an appendix we investigate multiplicativity properties of Schrijver's and Szegedy's numbers, as well as projective rank.Comment: Fixed proof of multiplicativity; more connections to prior work in conclusion; many changes in expositio

Hamiltonian simulation algorithms for near-term quantum hardware.

The quantum circuit model is the de-facto way of designing quantum algorithms. Yet any level of abstraction away from the underlying hardware incurs overhead. In this work, we develop quantum algorithms for Hamiltonian simulation "one level below" the circuit model, exploiting the underlying control over qubit interactions available in most quantum hardware and deriving analytic circuit identities for synthesising multi-qubit evolutions from two-qubit interactions. We then analyse the impact of these techniques under the standard error model where errors occur per gate, and an error model with a constant error rate per unit time. To quantify the benefits of this approach, we apply it to time-dynamics simulation of the 2D spin Fermi-Hubbard model. Combined with new error bounds for Trotter product formulas tailored to the non-asymptotic regime and an analysis of error propagation, we find that e.g. for a 5 × 5 Fermi-Hubbard lattice we reduce the circuit depth from 1, 243, 586 using the best previous fermion encoding and error bounds in the literature, to 3, 209 in the per-gate error model, or the circuit-depth-equivalent to 259 in the per-time error model. This brings Hamiltonian simulation, previously beyond reach of current hardware for non-trivial examples, significantly closer to being feasible in the NISQ era

Systematic meta-analyses and field synopsis of genetic association studies in colorectal adenomas

BackgroundLow penetrance genetic variants, primarily single nucleotide polymorphisms, have substantial influence on colorectal cancer (CRC) susceptibility. Most CRCs develop from colorectal adenomas (CRA). Here we report the first comprehensive field synopsis that catalogues all genetic association studies on CRA, with a parallel online database [http://www.chs.med.ed.ac.uk/CRAgene/].MethodsWe performed a systematic review, reviewing 9750 titles, and then extracted data from 130 publications reporting on 181 polymorphisms in 74 genes. We conducted meta-analyses to derive summary effect estimates for 37 polymorphisms in 26 genes. We applied the Venice criteria and Bayesian False Discovery Probability (BFDP) to assess the levels of the credibility of associations.ResultsWe considered the association with the rs6983267 variant at 8q24 as 'highly credible', reaching genome-wide statistical significance in at least one meta-analysis model. We identified 'less credible' associations (higher heterogeneity, lower statistical power, BFDP > 0.02) with a further four variants of four independent genes: MTHFR c.677C>T p.A222V (rs1801133), TP53 c.215C>G p.R72P (rs1042522), NQO1 c.559C>T p.P187S (rs1800566), and NAT1 alleles imputed as fast acetylator genotypes. For the remaining 32 variants of 22 genes for which positive associations with CRA risk have been previously reported, the meta-analyses revealed no credible evidence to support these as true associations.ConclusionsThe limited number of credible associations between low penetrance genetic variants and CRA reflects the lower volume of evidence and associated lack of statistical power to detect associations of the magnitude typically observed for genetic variants and chronic diseases. The CRA gene database provides context for CRA genetic association data and will help inform future research directions

Functional antibody and T-cell immunity following SARS-CoV-2 infection, including by variants of concern, in patients with cancer: the CAPTURE study

Patients with cancer have higher COVID-19 morbidity and mortality. Here we present the prospective CAPTURE study (NCT03226886) integrating longitudinal immune profiling with clinical annotation. Of 357 patients with cancer, 118 were SARS-CoV-2-positive, 94 were symptomatic and 2 patients died of COVID-19. In this cohort, 83% patients had S1-reactive antibodies, 82% had neutralizing antibodies against WT, whereas neutralizing antibody titers (NAbT) against the Alpha, Beta, and Delta variants were substantially reduced. Whereas S1-reactive antibody levels decreased in 13% of patients, NAbT remained stable up to 329 days. Patients also had detectable SARS-CoV-2-specific T cells and CD4+ responses correlating with S1-reactive antibody levels, although patients with hematological malignancies had impaired immune responses that were disease and treatment-specific, but presented compensatory cellular responses, further supported by clinical. Overall, these findings advance the understanding of the nature and duration of immune response to SARS-CoV-2 in patients with cancer

Author Correction: Scalable and robust SARS-CoV-2 testing in an academic center.

An amendment to this paper has been published and can be accessed via a link at the top of the paper

Genomic–transcriptomic evolution in lung cancer and metastasis

Intratumour heterogeneity (ITH) fuels lung cancer evolution, which leads to immune evasion and resistance to therapy. Here, using paired whole-exome and RNA sequencing data, we investigate intratumour transcriptomic diversity in 354 non-small cell lung cancer tumours from 347 out of the first 421 patients prospectively recruited into the TRACERx study. Analyses of 947 tumour regions, representing both primary and metastatic disease, alongside 96 tumour-adjacent normal tissue samples implicate the transcriptome as a major source of phenotypic variation. Gene expression levels and ITH relate to patterns of positive and negative selection during tumour evolution. We observe frequent copy number-independent allele-specific expression that is linked to epigenomic dysfunction. Allele-specific expression can also result in genomic–transcriptomic parallel evolution, which converges on cancer gene disruption. We extract signatures of RNA single-base substitutions and link their aetiology to the activity of the RNA-editing enzymes ADAR and APOBEC3A, thereby revealing otherwise undetected ongoing APOBEC activity in tumours. Characterizing the transcriptomes of primary–metastatic tumour pairs, we combine multiple machine-learning approaches that leverage genomic and transcriptomic variables to link metastasis-seeding potential to the evolutionary context of mutations and increased proliferation within primary tumour regions. These results highlight the interplay between the genome and transcriptome in influencing ITH, lung cancer evolution and metastasis

Phase estimation of local Hamiltonians on NISQ hardware

In this work we investigate a binned version of quantum phase estimation (QPE) set out by Somma (2019 New J. Phys. 21 123025) and known as the quantum eigenvalue estimation problem ( QEEP ). Specifically, we determine whether the circuit decomposition techniques we set out in previous work, Clinton et al (2021 Nat. Commun. 12 1–10), can improve the performance of QEEP in the noisy intermediate scale quantum (NISQ) regime. To this end we adopt a physically motivated abstraction of NISQ device capabilities as in Clinton et al (2021 Nat. Commun. 12 1–10). Within this framework, we find that our techniques reduce the threshold at which it becomes possible to perform the minimum two-bin instance of this algorithm by an order of magnitude. This is for the specific example of a two dimensional spin Fermi-Hubbard model. For example, we estimate that the depolarizing single qubit error rate required to implement a minimum two bin example of QEEP —with a $5\times5$ Fermi-Hubbard model and up to a precision of $10\%$ —can be reduced from 10 ^−7 to 10 ^−5 . We explore possible modifications to this protocol and propose an application, which we dub randomized quantum eigenvalue estimation problem ( rQEEP ). rQEEP outputs estimates on the fraction of eigenvalues which lie within randomly chosen bins and upper bounds the total deviation of these estimates from the true values. One use case we envision for this algorithm is resolving density of states features of local Hamiltonians

Phase estimation of local Hamiltonians on NISQ hardware

In this work we investigate a binned version of quantum phase estimation (QPE) set out by Somma (2019 New J. Phys. 21 123025) and known as the quantum eigenvalue estimation problem (QEEP). Specifically, we determine whether the circuit decomposition techniques we set out in previous work, Clinton et al (2021 Nat. Commun. 12 1–10), can improve the performance of QEEP in the noisy intermediate scale quantum (NISQ) regime. To this end we adopt a physically motivated abstraction of NISQ device capabilities as in Clinton et al (2021 Nat. Commun. 12 1–10). Within this framework, we find that our techniques reduce the threshold at which it becomes possible to perform the minimum two-bin instance of this algorithm by an order of magnitude. This is for the specific example of a two dimensional spin Fermi-Hubbard model. For example, we estimate that the depolarizing single qubit error rate required to implement a minimum two bin example of QEEP—with a 5×5 Fermi-Hubbard model and up to a precision of 10% —can be reduced from 10−7 to 10−5. We explore possible modifications to this protocol and propose an application, which we dub randomized quantum eigenvalue estimation problem (rQEEP). rQEEP outputs estimates on the fraction of eigenvalues which lie within randomly chosen bins and upper bounds the total deviation of these estimates from the true values. One use case we envision for this algorithm is resolving density of states features of local Hamiltonians

Phase estimation of local Hamiltonians on NISQ hardware

In this work we investigate a binned version of quantum phase estimation (QPE) set out by Somma (2019 New J. Phys. 21 123025) and known as the quantum eigenvalue estimation problem (QEEP). Specifically, we determine whether the circuit decomposition techniques we set out in previous work, Clinton et al (2021 Nat. Commun. 12 1–10), can improve the performance of QEEP in the noisy intermediate scale quantum (NISQ) regime. To this end we adopt a physically motivated abstraction of NISQ device capabilities as in Clinton et al (2021 Nat. Commun. 12 1–10). Within this framework, we find that our techniques reduce the threshold at which it becomes possible to perform the minimum two-bin instance of this algorithm by an order of magnitude. This is for the specific example of a two dimensional spin Fermi-Hubbard model. For example, we estimate that the depolarizing single qubit error rate required to implement a minimum two bin example of QEEP—with a 5×5 Fermi-Hubbard model and up to a precision of 10% —can be reduced from 10−7 to 10−5. We explore possible modifications to this protocol and propose an application, which we dub randomized quantum eigenvalue estimation problem (rQEEP). rQEEP outputs estimates on the fraction of eigenvalues which lie within randomly chosen bins and upper bounds the total deviation of these estimates from the true values. One use case we envision for this algorithm is resolving density of states features of local Hamiltonians