Complex Paths for Regular-to-Chaotic Tunneling Rates

Tunneling is a fundamental effect of quantum mechanics, which allows waves to penetrate into regions that are inaccessible by classical dynamics. We study this phenomenon for generic non-integrable systems with a mixed phase space, where tunneling occurs between the classically separated phase-space regions of regular and chaotic motion. We derive a semiclassical prediction for the corresponding tunneling rates from the regular region to the chaotic sea. This prediction is based on paths which connect the regular and the chaotic region in complexified phase space. We show that these complex paths can be constructed despite the obstacle of natural boundaries. For the standard map we demonstrate that tunneling rates can be predicted with high accuracy, by using only a few dominant complex paths. This gives the semiclassical foundation for the long-conjectured and often-observed exponential scaling with Planck's constant of regular-to-chaotic tunneling rates

Complex-Path Prediction of Resonance-Assisted Tunneling in Mixed Systems

We present a semiclassical prediction of regular-to-chaotic tunneling in systems with a mixed phase space, including the effect of a nonlinear resonance chain. We identify complex paths for direct and resonance-assisted tunneling in the phase space of an integrable approximation with one nonlinear resonance chain. We evaluate the resonance-assisted contribution analytically and give a prediction based on just a few properties of the classical phase space. For the standard map excellent agreement with numerically determined tunneling rates is observed. The results should similarly apply to ionization rates and quality factors.Comment: 6 pages, 2 figure

Integrable approximation of regular regions with a nonlinear resonance chain

Generic Hamiltonian systems have a mixed phase space where regions of regular and chaotic motion coexist. We present a method for constructing an integrable approximation to such regular phase-space regions including a nonlinear resonance chain. This approach generalizes the recently introduced iterative canonical transformation method. In the first step of the method a normal-form Hamiltonian with a resonance chain is adapted such that actions and frequencies match with those of the non-integrable system. In the second step a sequence of canonical transformations is applied to the integrable approximation to match the shape of regular tori. We demonstrate the method for the generic standard map at various parameters.Comment: 10 pages, 8 figure

Uniform hyperbolicity of a class of scattering maps

In recent years fractal Weyl laws and related quantum eigenfunction hypothesis have been studied in a plethora of numerical model systems, called quantum maps. In some models studied there one can easily prove uniform hyperbolicity. Yet, a numerically sound method for computing quantum resonance states, did not exist. To address this challenge, we recently introduced a new class of quantum maps. For these quantum maps, we showed that, quantum resonance states can numerically be computed using theoretically grounded methods such as complex scaling or weak absorbing potentials. However, proving uniform hyperbolicty for this class of quantum maps was not straight forward. Going beyond that work this article generalizes the class of scattering maps and provides mathematical proofs for their uniform hyperbolicity. In particular, we show that the suggested class of two-dimensional symplectic scattering maps satisfies the topological horseshoe condition and uniform hyperbolicity. In order to prove these properties, we follow the procedure developed in the paper by Devaney and Nitecki. Specifically, uniform hyperbolicity is shown by identifying a proper region in which the non-wandering set satisfies a sufficient condition to have the so-called sector bundle or cone field. Since no quantum map is known where both a proof of uniform hyperbolicity and a methodologically sound method for numerically computing quantum resonance states exist simultaneously, the present result should be valuable to further test fractal Weyl laws and related topics such as chaotic eigenfunction hypothesis.Comment: 50 pages, 23 figure

Consequences of Flooding on Spectral Statistics

We study spectral statistics in systems with a mixed phase space, in which regions of regular and chaotic motion coexist. Increasing their density of states, we observe a transition of the level-spacing distribution P(s) from Berry-Robnik to Wigner statistics, although the underlying classical phase-space structure and the effective Planck constant remain unchanged. This transition is induced by flooding, i.e., the disappearance of regular states due to increasing regular-to-chaotic couplings. We account for this effect by a flooding-improved Berry-Robnik distribution, in which an effectively reduced size of the regular island enters. To additionally describe power-law level repulsion at small spacings, we extend this prediction by explicitly considering the tunneling couplings between regular and chaotic states. This results in a flooding- and tunneling-improved Berry-Robnik distribution which is in excellent agreement with numerical data.Comment: 9 pages, 5 figure

Fractional-Power-Law Level-Statistics due to Dynamical Tunneling

For systems with a mixed phase space we demonstrate that dynamical tunneling universally leads to a fractional power law of the level-spacing distribution P(s) over a wide range of small spacings s. Going beyond Berry-Robnik statistics, we take into account that dynamical tunneling rates between the regular and the chaotic region vary over many orders of magnitude. This results in a prediction of P(s) which excellently describes the spectral data of the standard map. Moreover, we show that the power-law exponent is proportional to the effective Planck constant h.Comment: 4 pages, 2 figure

Dynamic-ADAPT-QAOA: An algorithm with shallow and noise-resilient circuits

The quantum approximate optimization algorithm (QAOA) is an appealing proposal to solve NP problems on noisy intermediate-scale quantum (NISQ) hardware. Making NISQ implementations of the QAOA resilient to noise requires short ansatz circuits with as few CNOT gates as possible. Here, we present Dynamic-ADAPT-QAOA. Our algorithm significantly reduces the circuit depth and the CNOT count of standard ADAPT-QAOA, a leading proposal for near-term implementations of the QAOA. Throughout our algorithm, the decision to apply CNOT-intensive operations is made dynamically, based on algorithmic benefits. Using density-matrix simulations, we benchmark the noise resilience of ADAPT-QAOA and Dynamic-ADAPT-QAOA. We compute the gate-error probability $p_\text{gate}^\star$ below which these algorithms provide, on average, more accurate solutions than the classical, polynomial-time approximation algorithm by Goemans and Williamson. For small systems with $6-10$ qubits, we show that $p_{\text{gate}}^\star>10^{-3}$ for Dynamic-ADAPT-QAOA. Compared to standard ADAPT-QAOA, this constitutes an order-of-magnitude improvement in noise resilience. This improvement should make Dynamic-ADAPT-QAOA viable for implementations on superconducting NISQ hardware, even in the absence of error mitigation.Comment: 15 pages, 9 figure

Variational quantum chemistry requires gate-error probabilities below the fault-tolerance threshold

The variational quantum eigensolver (VQE) is a leading contender for useful quantum advantage in the NISQ era. The interplay between quantum processors and classical optimisers is believed to make the VQE noise resilient. Here, we probe this hypothesis. We use full density-matrix simulations to rank the noise resilience of leading gate-based VQE algorithms in ground-state computations on a range of molecules. We find that, in the presence of noise: (i) ADAPT-VQEs that construct ansatz circuits iteratively outperform VQEs that use "fixed" ansatz circuits; and (ii) ADAPT-VQEs perform better when circuits are constructed from gate-efficient elements rather than physically-motivated ones. Our results show that, for a wide range of molecules, even the best-performing VQE algorithms require gate-error probabilities on the order of $10^{-6}$ to $10^{-4}$ to reach chemical accuracy. This is significantly below the fault-tolerance thresholds of most error-correction protocols. Further, we estimate that the maximum allowed gate-error probability scales inversely with the number of noisy (two-qubit) gates. Our results indicate that useful chemistry calculations with current gate-based VQEs are unlikely to be successful on near-term hardware without error correction.Comment: 17 pages, 8 figure