Linear growth of quantum circuit complexity

The complexity of quantum states has become a key quantity of interest across various subfields of physics, from quantum computing to the theory of black holes. The evolution of generic quantum systems can be modelled by considering a collection of qubits subjected to sequences of random unitary gates. Here we investigate how the complexity of these random quantum circuits increases by considering how to construct a unitary operation from Haar random two qubit quantum gates. Implementing the unitary operation exactly requires a minimal number of gates this is the operation s exact circuit complexity. We prove a conjecture that this complexity grows linearly, before saturating when the number of applied gates reaches a threshold that grows exponentially with the number of qubits. Our proof overcomes difficulties in establishing lower bounds for the exact circuit complexity by combining differential topology and elementary algebraic geometry with an inductive construction of Clifford circuit

Gamma estimator of Jarzynski equality for recovering binding energies from noisy dynamic data sets

A fundamental problem in thermodynamics is the recovery of macroscopic equilibrated interaction energies from experimentally measured single-molecular interactions. The Jarzynski equality forms a theoretical basis in recovering the free energy difference between two states from exponentially averaged work performed to switch the states. In practice, the exponentially averaged work value is estimated as the mean of finite samples. Numerical simulations have shown that samples having thousands of measurements are not large enough for the mean to converge when the fluctuation of external work is above 4 kBT, which is easily observable in biomolecular interactions. We report the first example of a statistical gamma work distribution applied to single molecule pulling experiments. The Gibbs free energy of surface adsorption can be accurately evaluated even for a small sample size. The values obtained are comparable to those derived from multi-parametric surface plasmon resonance measurements and molecular dynamics simulations

Quantum majorization and a complete set of entropic conditions for quantum thermodynamics

What does it mean for one quantum process to be more disordered than another? Interestingly, this apparently abstract question arises naturally in a wide range of areas such as information theory, thermodynamics, quantum reference frames, and the resource theory of asymmetry. Here we use a quantum-mechanical generalization of majorization to develop a framework for answering this question, in terms of single-shot entropies, or equivalently, in terms of semi-definite programs. We also investigate some of the applications of this framework, and remarkably find that, in the context of quantum thermodynamics it provides the first complete set of necessary and sufficient conditions for arbitrary quantum state transformations under thermodynamic processes, which rigorously accounts for quantum-mechanical properties, such as coherence. Our framework of generalized thermal processes extends thermal operations, and is based on natural physical principles, namely, energy conservation, the existence of equilibrium states, and the requirement that quantum coherence be accounted for thermodynamically

Maximum one-shot dissipated work from Rényi divergences

Thermodynamics describes large-scale, slowly evolving systems. Two modern approaches generalize thermodynamics: fluctuation theorems, which concern finite-time nonequilibrium processes, and one-shot statistical mechanics, which concerns small scales and finite numbers of trials. Combining these approaches, we calculate a one-shot analog of the average dissipated work defined in fluctuation contexts: the cost of performing a protocol in finite time instead of quasistatically. The average dissipated work has been shown to be proportional to a relative entropy between phase-space densities, to a relative entropy between quantum states, and to a relative entropy between probability distributions over possible values of work. We derive one-shot analogs of all three equations, demonstrating that the order-infinity Rényi divergence is proportional to the maximum possible dissipated work in each case. These one-shot analogs of fluctuation-theorem results contribute to the unification of these two toolkits for small-scale, nonequilibrium statistical physics