Quantum Physics

Motivated by the recent experimental demonstrations of quantum supremacy, proving the hardness of the output of random quantum circuits is an imperative near term goal. We prove under the complexity theoretical assumption of the non-collapse of the polynomial hierarchy that approximating the output probabilities of random quantum circuits to within exp(?��?mlogm)) additive error is hard for any classical computer, where m is the number of gates in the quantum computation. More precisely, we show that the above problem is #P -hard under BPP NP reduction. In the recent experiments, the quantum circuit has n -qubits and the architecture is a two-dimensional grid of size n ??????? n ??????. Indeed for constant depth circuits approximating the output probabilities to within 2 ?��?nlogn) is hard. For circuits of depth logn or n ??????for which the anti-concentration property holds, approximating the output probabilities to within 2 ?��?n log 2 n) and 2 ?��? n 3/2 logn) is hard respectively. We made an effort to find the best proofs and proved these results from first principles, which do not use the standard techniques such as the Berlekamp--Welch algorithm, the usual Paturi's lemma, and Rakhmanov's result.

Quantum chemistry calculations are important applications of quantum annealing. For practical applications in quantum chemistry, it is essential to estimate a ground state energy of the Hamiltonian with chemical accuracy. However, there are no known methods to guarantee the accuracy of the estimation of the energy calculated by quantum annealing. Here, we propose a way to improve the accuracy of the estimate of the ground state energy by combining quantum annealing with classical computation. In our scheme, before running the QA, we need a pre-estimation of the energies of the ground state and first excited state with some error bars (corresponding to possible estimation error) by performing classical computation with some approximations. We show that, if an expectation value and variance of the energy of the state after the QA are smaller than certain threshold values (that we can calculate from the pre-estimation), the QA provides us with a better estimate of the ground state energy than that of the pre-estimation. Since the expectation value and variance of the energy can be experimentally measurable by the QA, our results pave the way for accurate estimation of the ground state energy with the QA.

The variational quantum eigensolver (VQE) is a hybrid quantum-classical algorithm for finding the minimum eigenvalue of a Hamiltonian that involves the optimization of a parameterized quantum circuit. Since the resulting optimization problem is in general nonconvex, the method can converge to suboptimal parameter values which do not yield the minimum eigenvalue. In this work, we address this shortcoming by adopting the concept of variational adiabatic quantum computing (VAQC) as a procedure to improve VQE. In VAQC, the ground state of a continuously parameterized Hamiltonian is approximated via a parameterized quantum circuit. We discuss some basic theory of VAQC to motivate the development of a hybrid quantum-classical homotopy continuation method. The proposed method has parallels with a predictor-corrector method for numerical integration of differential equations. While there are theoretical limitations to the procedure, we see in practice that VAQC can successfully find good initial circuit parameters to initialize VQE. We demonstrate this with two examples from quantum chemistry. Through these examples, we provide empirical evidence that VAQC, combined with other techniques (an adaptive termination criteria for the classical optimizer and a variance-based resampling method for the expectation evaluation), can provide more accurate solutions than "plain" VQE, for the same amount of effort.

In this paper we introduce a measure of genuine quantum incompatibility in the estimation task of multiple parameters, that has a geometric character and is backed by a clear operational interpretation. This measure is then applied to some simple systems in order to track the effect of a local depolarizing noise on the incompatibility of the estimation task. A semidefinite program is described and used to numerically compute the figure of merit when the analytical tools are not sufficient, among these we include an upper bound computable from the symmetric logarithmic derivatives only. Finally we discuss how to obtain compatible models for a general unitary encoding on a finite dimensional probe.

Single molecules in solid-state matrices have been proposed as sources of single-photon Fock states back 20 years ago. Their success in quantum optics and in many other research fields stems from the simple recipes used in the preparation of samples, with hundreds of nominally identical and isolated molecules. Main challenges as of today for their application in photonic quantum technologies are the optimization of light extraction and the on-demand emission of indistinguishable photons. We here present Hong-Ou-Mandel experiments with photons emitted by a single molecule of dibenzoterrylene in an anthracene nanocrystal at 3 K, under continuous wave and also pulsed excitation. A detailed theoretical model is applied, which relies on independent measurements for most experimental parameters, hence allowing for an analysis of the different contributions to the two-photon interference visibility, from residual dephasing to spectral filtering.

Counterfactual quantum communication is unique in its own way that allows remote parties to transfer information without sending any message carrier in the channel. Although no message carrier travels in the channel at the time of successful information transmission, it is impossible to transmit information faster than the speed of light, thus without an information carrier. In this paper, we address an important question What carries the information in counterfactual quantum communication? and optimize the resource efficiency of the counterfactual quantum communication in terms of the number of channels used, time consumed to transmit 1-bit classical information between two remote parties, and the number of qubits required to accomplish the counterfactual quantum communication.

A beautiful idea about the incompatibility of Physical Context(IPC) was introduced in [Phys. Rev. A 102, 050201(R) (2020)]. Here, a context is defined as a set of a quantum state and two sharp rank-one measurements, and the incompatibility of physical context is defined as the leakage of information while implementing those two measurements successively in that quantum state. In this work, we show the limitations in their approach. The three primary limitations are that, (i) their approach is not generalized for POVM measurements and (ii), they restrict information theoretic agents Alice, Eve and Bob to specific quantum operations and do not consider most general quantum operations i.e., quantum instruments and (iii), their measure of IPC can take negative values in specific cases in a more general scenario which implies the limitation of their information measure. Thereby, we have introduced a generalization and modification to their approach in more general and convenient way, such that this idea is well-defined for generic measurements, without these limitations. We also present a comparison of the measure of the IPC through their and our method. Lastly, we show, how the IPC reduces in the presence of memory using our modification, which further validates our approach.

This paper studies the application of the Quantum Approximate Optimization Algorithm (QAOA) to spin-glass models with random multi-body couplings in the limit of a large number of spins. We show that for such mixed-spin models the performance of depth 1 QAOA is independent of the specific instance in the limit of infinite sized systems and we give an explicit formula for the expected performance. We also give explicit expressions for the higher moments of the expected energy, thereby proving that the expected performance of QAOA concentrates.

Recent work by Bravyi et al. constructs a relation problem that a noisy constant-depth quantum circuit (QNC 0 ) can solve with near certainty (probability 1?�o(1) ), but that any bounded fan-in constant-depth classical circuit (NC 0 ) fails with some constant probability. We show that this robustness to noise can be achieved in the other low-depth quantum/classical circuit separations in this area. In particular, we show a general strategy for adding noise tolerance to the interactive protocols of Grier and Schaeffer. As a consequence, we obtain an unconditional separation between noisy QNC 0 circuits and AC 0 [p] circuits for all primes p?? , and a conditional separation between noisy QNC 0 circuits and log-space classical machines under a plausible complexity-theoretic conjecture. A key component of this reduction is showing average-case hardness for the classical simulation tasks -- that is, showing that a classical simulation of the quantum interactive task is still powerful even if it is allowed to err with constant probability over a uniformly random input. We show that is true even for quantum tasks which are ??L-hard to simulate. To do this, we borrow techniques from randomized encodings used in cryptography.

We show that general time-local quantum master equations admit an unravelling in quantum trajectories with jumps. The sufficient condition is to weigh state vector Monte Carlo averages by a probability pseudo-measure which we call the "influence martingale". The influence martingale satisfies a 1d stochastic differential equation enslaved to the ones governing the quantum trajectories. Our interpretation is that the influence martingale models interference effects between distinct realizations of the quantum trajectories at strong system-environment coupling. If the master equation generates a completely positive dynamical map, there is no interference. In such a case the influence martingale becomes positive definite and the Monte Carlo average straightforwardly reduces to the well known unravelling of completely positive divisible dynamical maps.