01837nas a2200169 4500008004100000245009700041210006900138260001300207520131000220100001301530700001301543700002001556700001801576700002001594700001601614856003701630 2022 eng d00aEfficient quantum algorithm for nonlinear reaction-diffusion equations and energy estimation0 aEfficient quantum algorithm for nonlinear reactiondiffusion equa c5/2/20223 a

Nonlinear differential equations exhibit rich phenomena in many fields but are notoriously challenging to solve. Recently, Liu et al. [1] demonstrated the first efficient quantum algorithm for dissipative quadratic differential equations under the condition R<1, where R measures the ratio of nonlinearity to dissipation using the ℓ2 norm. Here we develop an efficient quantum algorithm based on [1] for reaction-diffusion equations, a class of nonlinear partial differential equations (PDEs). To achieve this, we improve upon the Carleman linearization approach introduced in [1] to obtain a faster convergence rate under the condition RD<1, where RD measures the ratio of nonlinearity to dissipation using the ℓ∞ norm. Since RD is independent of the number of spatial grid points n while R increases with n, the criterion RD<1 is significantly milder than R<1 for high-dimensional systems and can stay convergent under grid refinement for approximating PDEs. As applications of our quantum algorithm we consider the Fisher-KPP and Allen-Cahn equations, which have interpretations in classical physics. In particular, we show how to estimate the mean square kinetic energy in the solution by postprocessing the quantum state that encodes it to extract derivative information.

1 aAn, Dong1 aFang, Di1 aJordan, Stephen1 aLiu, Jin-Peng1 aLow, Guang, Hao1 aWang, Jiasu uhttps://arxiv.org/abs/2205.0114101880nas a2200133 4500008004100000245008500041210006900126260001500195520144100210100001701651700002101668700002001689856003701709 2022 eng d00aEstimating gate complexities for the site-by-site preparation of fermionic vacua0 aEstimating gate complexities for the sitebysite preparation of f c07/04/20223 a

An important aspect of quantum simulation is the preparation of physically interesting states on a quantum computer, and this task can often be costly or challenging to implement. A digital, ``site-by-site'' scheme of state preparation was introduced in arXiv:1911.03505 as a way to prepare the vacuum state of certain fermionic field theory Hamiltonians with a mass gap. More generally, this algorithm may be used to prepare ground states of Hamiltonians by adding one site at a time as long as successive intermediate ground states share a non-zero overlap and the Hamiltonian has a non-vanishing spectral gap at finite lattice size. In this paper, we study the ground state overlap as a function of the number of sites for a range of quadratic fermionic Hamiltonians. Using analytical formulas known for free fermions, we are able to explore the large-N behavior and draw conclusions about the state overlap. For all models studied, we find that the overlap remains large (e.g. >0.1) up to large lattice sizes (N=64,72) except near quantum phase transitions or in the presence of gapless edge modes. For one-dimensional systems, we further find that two N/2-site ground states also share a large overlap with the N-site ground state everywhere except a region near the phase boundary. Based on these numerical results, we additionally propose a recursive alternative to the site-by-site state preparation algorithm.

1 aSewell, Troy1 aBapat, Aniruddha1 aJordan, Stephen uhttps://arxiv.org/abs/2207.0169201826nas a2200145 4500008004100000245009400041210006900135260001300204300001200217490000700229520136600236100002101602700002001623856003701643 2019 eng d00aBang-bang control as a design principle for classical and quantum optimization algorithms0 aBangbang control as a design principle for classical and quantum c8/1/2019 a424-4460 v193 a

Physically motivated classical heuristic optimization algorithms such as simulated annealing (SA) treat the objective function as an energy landscape, and allow walkers to escape local minima. It has been argued that quantum properties such as tunneling may give quantum algorithms advantage in finding ground states of vast, rugged cost landscapes. Indeed, the Quantum Adiabatic Algorithm (QAO) and the recent Quantum Approximate Optimization Algorithm (QAOA) have shown promising results on various problem instances that are considered classically hard. Here, we argue that the type of control strategy used by the optimization algorithm may be crucial to its success. Along with SA, QAO and QAOA, we define a new, bang-bang version of simulated annealing, BBSA, and study the performance of these algorithms on two well-studied problem instances from the literature. Both classically and quantumly, the successful control strategy is found to be bang-bang, exponentially outperforming the quasistatic analogues on the same instances. Lastly, we construct O(1)-depth QAOA protocols for a class of symmetric cost functions, and provide an accompanying physical picture.

1 aBapat, Aniruddha1 aJordan, Stephen uhttps://arxiv.org/abs/1812.0274602638nas a2200265 4500008004100000245009500041210006900136260001500205300001200220490000800232520186000240100002102100700001902121700001802140700001802158700001502176700002002191700001902211700001602230700002502246700001802271700002402289700002202313856003702335 2018 eng d00aExperimentally Generated Randomness Certified by the Impossibility of Superluminal Signals0 aExperimentally Generated Randomness Certified by the Impossibili c2018/04/11 a223-2260 v5563 a

From dice to modern complex circuits, there have been many attempts to build increasingly better devices to generate random numbers. Today, randomness is fundamental to security and cryptographic systems, as well as safeguarding privacy. A key challenge with random number generators is that it is hard to ensure that their outputs are unpredictable. For a random number generator based on a physical process, such as a noisy classical system or an elementary quantum measurement, a detailed model describing the underlying physics is required to assert unpredictability. Such a model must make a number of assumptions that may not be valid, thereby compromising the integrity of the device. However, it is possible to exploit the phenomenon of quantum nonlocality with a loophole-free Bell test to build a random number generator that can produce output that is unpredictable to any adversary limited only by general physical principles. With recent technological developments, it is now possible to carry out such a loophole-free Bell test. Here we present certified randomness obtained from a photonic Bell experiment and extract 1024 random bits uniform to within 10−12. These random bits could not have been predicted within any physical theory that prohibits superluminal signaling and allows one to make independent measurement choices. To certify and quantify the randomness, we describe a new protocol that is optimized for apparatuses characterized by a low per-trial violation of Bell inequalities. We thus enlisted an experimental result that fundamentally challenges the notion of determinism to build a system that can increase trust in random sources. In the future, random number generators based on loophole-free Bell tests may play a role in increasing the security and trust of our cryptographic systems and infrastructure.

1 aBierhorst, Peter1 aKnill, Emanuel1 aGlancy, Scott1 aZhang, Yanbao1 aMink, Alan1 aJordan, Stephen1 aRommal, Andrea1 aLiu, Yi-Kai1 aChristensen, Bradley1 aNam, Sae, Woo1 aStevens, Martin, J.1 aShalm, Lynden, K. uhttps://arxiv.org/abs/1803.0621901418nas a2200145 4500008004100000245007200041210006900113260001500182300001100197490000600208520097500214100002601189700002001215856003701235 2018 eng d00aFaster Quantum Algorithm to simulate Fermionic Quantum Field Theory0 aFaster Quantum Algorithm to simulate Fermionic Quantum Field The c2018/05/04 a0123320 vA3 a

In quantum algorithms discovered so far for simulating scattering processes in quantum field theories, state preparation is the slowest step. We present a new algorithm for preparing particle states to use in simulation of Fermionic Quantum Field Theory (QFT) on a quantum computer, which is based on the matrix product state ansatz. We apply this to the massive Gross-Neveu model in one spatial dimension to illustrate the algorithm, but we believe the same algorithm with slight modifications can be used to simulate any one-dimensional massive Fermionic QFT. In the case where the number of particle species is one, our algorithm can prepare particle states using O(ε−3.23…) gates, which is much faster than previous known results, namely O(ε−8−o(1)). Furthermore, unlike previous methods which were based on adiabatic state preparation, the method given here should be able to simulate quantum phases unconnected to the free theory.

1 aMoosavian, Ali, Hamed1 aJordan, Stephen uhttps://arxiv.org/abs/1711.04006