Approximating dynamical correlation functions with constant depth quantum circuits

One of the most important quantities characterizing the microscopic properties of quantum systems are dynamical correlation functions. These correlations are obtained by time-evolving a perturbation of an eigenstate of the system, typically the ground state. In this work, we study approximations of these correlation functions that do not require time dynamics. We show that having access to a circuit that prepares an eigenstate of the Hamiltonian, it is possible to approximate the dynamical correlation functions up to exponential accuracy in the complex frequency domain $\omega=\Re(\omega)+i\Im(\omega)$, on a strip above the real line $\Im(\omega)=0$. We achieve this by exploiting the continued fraction representation of the dynamical correlation functions as functions of frequency $\omega$, where the level $k$ approximant can be obtained by measuring a weight $O(k)$ operator on the eigenstate of interest. In the complex $\omega$ plane, we show how this approach allows to determine approximations to correlation functions with accuracy that increases exponentially with $k$. We analyse two algorithms to generate the continuous fraction representation in scalar or matrix form, starting from either one or many initial operators. We prove that these algorithms generate an exponentially accurate approximation of the dynamical correlation functions on a region sufficiently far away from the real frequency axis. We present numerical evidence of these theoretical results through simulations of small lattice systems. We comment on the stability of these algorithms with respect to sampling noise in the context of quantum simulation using quantum computers.Comment: 29 pages, 10 figure

Efficient Quantum Algorithm for Filtering Product States

We introduce a quantum algorithm to efficiently prepare states with a small energy variance at the target energy. We achieve it by filtering a product state at the given energy with a Lorentzian filter of width $\delta$. Given a local Hamiltonian on $N$ qubits, we construct a parent Hamiltonian whose ground state corresponds to the filtered product state with variable energy variance proportional to $\delta\sqrt{N}$. We prove that the parent Hamiltonian is gapped and its ground state can be efficiently implemented in $\mathrm{poly}(N,1/\delta)$ time via adiabatic evolution. We numerically benchmark the algorithm for a particular non-integrable model and find that the adiabatic evolution time to prepare the filtered state with a width $\delta$ is independent of the system size $N$. Furthermore, the adiabatic evolution can be implemented with circuit depth $\mathcal{O}(N^2\delta^{-4})$. Our algorithm provides a way to study the finite energy regime of many body systems in quantum simulators by directly preparing a finite energy state, providing access to an approximation of the microcanonical properties at an arbitrary energy.Comment: 10 pages, 8 figure

Approximating dynamical correlation functions with constant depth quantum circuits

One of the most important quantities characterizing the microscopic properties of quantum systems are dynamical correlation functions. These correlations are obtained by time-evolving a perturbation of an eigenstate of the system, typically the ground state. In this work, we study approximations of these correlation functions that do not require time dynamics. We show that having access to a circuit that prepares an eigenstate of the Hamiltonian, it is possible to approximate the dynamical correlation functions up to exponential accuracy in the complex frequency domain $\omega=\Re(\omega)+i\Im(\omega)$, on a strip above the real line $\Im(\omega)=0$. We achieve this by exploiting the continued fraction representation of the dynamical correlation functions as functions of frequency $\omega$, where the level $k$ approximant can be obtained by measuring a weight $O(k)$ operator on the eigenstate of interest. In the complex $\omega$ plane, we show how this approach allows to determine approximations to correlation functions with accuracy that increases exponentially with $k$. We analyse two algorithms to generate the continuous fraction representation in scalar or matrix form, starting from either one or many initial operators. We prove that these algorithms generate an exponentially accurate approximation of the dynamical correlation functions on a region sufficiently far away from the real frequency axis. We present numerical evidence of these theoretical results through simulations of small lattice systems. We comment on the stability of these algorithms with respect to sampling noise in the context of quantum simulation using quantum computers

Ultrasound guided needle biopsy of axilla to evaluate nodal metastasis after preoperative systemic therapy in cohort of 106 breast cancers enriched with BRCA1/2 pathogenic variant carriers.

BACKGROUND: Aim of the study is to evaluate the role of ultrasound guided fine needle aspiration cytology (FNAC) in the restaging of node positive breast cancer after preoperative systemic therapy (PST). METHODS: From January 2016 - October 2020 106 node positive stage IIA-IIIC breast cancer cases undergoing PST were included in the study. 18 (17 %) were carriers of pathogenic variant in BRCA1/2. After PST restaging of axilla was performed with ultrasound and FNAC of the marked and/or the most suspicious axillary node. In 72/106 cases axilla conserving surgery and in 34/106 cases axillary lymph node dissection (ALND) was performed. RESULTS: False Positive Rate (FPR) of FNAC after PST in whole cohort and BRCA1/2 positive subgroup is 8 and 0 % and False Negative Rate (FNR) - 43 and 18 % respectively. Overall Sensitivity - 55 %, specificity- 93 %, accuracy 70 %. CONCLUSION: FNAC after PST has low FPR and is useful to predict residual axillary disease and to streamline surgical decision making regarding ALND both in BRCA1/2 positive and negative subgroups. FNR is high in overall cohort and FNAC alone are not able to predict ypCR and omission of further axillary surgery. However, FNAC performance in BRCA1/2 positive subgroup is more promising and further research with larger number of cases is necessary to confirm the results

Quantum Simulation of Z2 Lattice Gauge theory with minimal resources

The quantum simulation of fermionic gauge field theories is one of the anticipated uses of quantum computers in the NISQ era. Recently work has been done to simulate properties of the fermionic Z2 gauge field theory in (1+1) D and the pure gauge theory in (2+1) D. In this work, we investigate various options for simulating the fermionic Z2 gauge field theory in (2+1) D. To simulate the theory on a NISQ device it is vital to minimize both the number of qubits used and the circuit depth. In this work we propose ways to optimize both criteria for simulating time dynamics. In particular, we develop a new way to simulate this theory on a quantum computer, with minimal qubit requirements. We provide a quantum circuit, simulating a single first order Trotter step, that minimizes the number of 2-qubit gates needed and gives comparable results to methods requiring more qubits. Furthermore, variational approaches are investigated that allow to further decrease the circuit depth.Comment: 11 pages, 3 figures. Improved numerical results, added references, fixed typo

Ultrasound guided needle biopsy of axilla to evaluate nodal metastasis after preoperative systemic therapy in cohort of 106 breast cancers enriched with BRCA1/2 pathogenic variant carriers

Abstract Background Aim of the study is to evaluate the role of ultrasound guided fine needle aspiration cytology (FNAC) in the restaging of node positive breast cancer after preoperative systemic therapy (PST). Methods From January 2016 – October 2020 106 node positive stage IIA-IIIC breast cancer cases undergoing PST were included in the study. 18 (17 %) were carriers of pathogenic variant in BRCA1/2. After PST restaging of axilla was performed with ultrasound and FNAC of the marked and/or the most suspicious axillary node. In 72/106 cases axilla conserving surgery and in 34/106 cases axillary lymph node dissection (ALND) was performed. Results False Positive Rate (FPR) of FNAC after PST in whole cohort and BRCA1/2 positive subgroup is 8 and 0 % and False Negative Rate (FNR) – 43 and 18 % respectively. Overall Sensitivity − 55 %, specificity- 93 %, accuracy 70 %. Conclusion FNAC after PST has low FPR and is useful to predict residual axillary disease and to streamline surgical decision making regarding ALND both in BRCA1/2 positive and negative subgroups. FNR is high in overall cohort and FNAC alone are not able to predict ypCR and omission of further axillary surgery. However, FNAC performance in BRCA1/2 positive subgroup is more promising and further research with larger number of cases is necessary to confirm the results. </jats:sec