Enhancing the Quantum Linear Systems Algorithm Using Richardson Extrapolation

We present a quantum algorithm to solve systems of linear equations of the form Ax = b , where A is a tridiagonal Toeplitz matrix and b results from discretizing an analytic function, with a circuit complexity of O (1/√ε, poly (log κ, log N )), where N denotes the number of equations, ε is the accuracy, and κ the condition number. The repeat-until-success algorithm has to be run O (κ/(1-ε)) times to succeed, leveraging amplitude amplification, and needs to be sampled O (1/ε 2 ) times. Thus, the algorithm achieves an exponential improvement with respect to N over classical methods. In particular, we present efficient oracles for state preparation, Hamiltonian simulation, and a set of observables together with the corresponding error and complexity analyses. As the main result of this work, we show how to use Richardson extrapolation to enhance Hamiltonian simulation, resulting in an implementation of Quantum Phase Estimation (QPE) within the algorithm with 1/√ε circuits that can be run in parallel each with circuit complexity 1/√ ε instead of 1/ε. Furthermore, we analyze necessary conditions for the overall algorithm to achieve an exponential speedup compared to classical methods. Our approach is not limited to the considered setting and can be applied to more general problems where Hamiltonian simulation is approximated via product formulae, although our theoretical results would need to be extended accordingly. All the procedures presented are implemented with Qiskit and tested for small systems using classical simulation as well as using real quantum devices available through the IBM Quantum Experience.

Towards Compact Modeling of Noisy Quantum Computers: A Molecular-Spin-Qubit Case of Study

Classical simulation of Noisy Intermediate Scale Quantum computers is a crucial task for testing the expected performance of real hardware. The standard approach, based on solving Schrödinger and Lindblad equations, is demanding when scaling the number of qubits in terms of both execution time and memory. In this article, attempts in defining compact models for the simulation of quantum hardware are proposed, ensuring results close to those obtained with standard formalism. Molecular Nuclear Magnetic Resonance quantum hardware is the target technology, where three non-ideality phenomena—common to other quantum technologies—are taken into account: decoherence, off-resonance qubit evolution, and undesired qubit-qubit residual interaction. A model for each non-ideality phenomenon is embedded into a MATLAB simulation infrastructure of noisy quantum computers. The accuracy of the models is tested on a benchmark of quantum circuits, in the expected operating ranges of quantum hardware. The corresponding outcomes are compared with those obtained via numeric integration of the Schrödinger equation and the Qiskit’s QASMSimulator. The achieved results give evidence that this work is a step forward towards the definition of compact models able to provide fast results close to those obtained with the traditional physical simulation strategies, thus paving the way for their integration into a classical simulator of quantum computers.

Parallel approach of Schrödinger-based quantum corrections for ultrascaled semiconductor devices

In the current technology node, purely classical numerical simulators lack the precision needed to obtain valid results. At the same time, the simulation of fully quantum models can be a cumbersome task in certain studies such as device variability analysis, since a single simulation can take up to weeks to compute and hundreds of device configurations need to be analyzed to obtain statistically significative results. A good compromise between fast and accurate results is to add corrections to the classical simulation that are able to reproduce the quantum nature of matter. In this context, we present a new approach of Schrödinger equation-based quantum corrections. We have implemented it using Message Passing Interface in our in-house built semiconductor simulation framework called VENDES, capable of running in distributed systems that allow for more accurate results in a reasonable time frame. Using a 12-nm-gate-length gate-all-around nanowire FET (GAA NW FET) as a benchmark device, the new implementation shows an almost perfect agreement in the output data with less than a 2% difference between the cases using 1 and 16 processes. Also, a reduction of up to 98% in the computational time has been found comparing the sequential and the 16 process simulation. For a reasonably dense mesh of 150k nodes, a variability study of 300 individual simulations can be now performed with VENDES in approximately 2.5 days instead of an estimated sequential execution of 137 days.

Efficient verification of Boson Sampling

The demonstration of quantum speedup, also known as quantum computational supremacy, that is the ability of quantum computers to outperform dramatically their classical counterparts, is an important milestone in the field of quantum computing. While quantum speedup experiments are gradually escaping the regime of classical simulation, they still lack efficient verification protocols and rely on partial validation. Here we derive an efficient protocol for verifying with single-mode Gaussian measurements the output states of a large class of continuous-variable quantum circuits demonstrating quantum speedup, including Boson Sampling experiments, thus enabling a convincing demonstration of quantum speedup with photonic computing. Beyond the quantum speedup milestone, our results also enable the efficient and reliable certification of a large class of intractable continuous-variable multimode quantum states.

Variational Quantum Optimization with Multi-Basis Encodings

Despite extensive research efforts, few quantum algorithms for classical optimization demonstrate realizable quantum advantage. The utility of many quantum algorithms is limited by high requisite circuit depth and nonconvex optimization landscapes. We tackle these challenges by introducing a new variational quantum algorithm that utilizes multi-basis graph encodings and nonlinear activation functions. Our technique results in increased optimization performance, a factor of two increase in effective quantum resources, and a quadratic reduction in measurement complexity. While the classical simulation of many qubits with traditional quantum formalism is impossible due to its exponential scaling, we mitigate this limitation with exact circuit representations using factorized tensor rings. In particular, the shallow circuits permitted by our technique, combined with efficient factorized tensor-based simulation, enable us to successfully optimize the MaxCut of the nonlocal 512-vertex DIMACS library graphs on a single GPU. By improving the performance of quantum optimization algorithms while requiring fewer quantum resources and utilizing shallower, more error-resistant circuits, we offer tangible progress for variational quantum optimization.

On the extremal points of the Lambda polytopes and classical simulation of quantum computation with magic states

We investigate the $\Lambda$-polytopes, a convex-linear structure recently defined and applied to the classical simulation of quantum computation with magic states by sampling. There is one such polytope, $\Lambda_n$, for every number $n$ of qubits. We establish two properties of the family $\{\Lambda_n, n\in \mathbb{N}\}$, namely (i) Any extremal point (vertex) $A_\alpha \in \Lambda_m$ can be used to construct vertices in $\Lambda_n$, for all $n>m$. (ii) For vertices obtained through this mapping, the classical simulation of quantum computation with magic states can be efficiently reduced to the classical simulation based on the preimage $A_\alpha$. In addition, we describe a new class of vertices in $\Lambda_2$ which is outside the known classification. While the hardness of classical simulation remains an open problem for most extremal points of $\Lambda_n$, the above results extend efficient classical simulation of quantum computations beyond the presently known range.

Parallel Approach of Schr¨Odinger Based Quantum Corrections for Ultrascaled Semiconductor Devices

In the current technology node, purely classical numerical simulators lack the precision needed to obtain valid results. At the same time, the simulation of fully quantum models can be a cumbersome task in certain studies such as device variability analysis, since a single simulation can take up to weeks to compute and hundreds of device configurations need to be analyzed to obtain statistically significative results. A good compromise between fast and accurate results is to add corrections to the classical simulation that are able to reproduce the quantum nature of matter. In this context, we present a new approach of Schrödinger equation-based quantum corrections. We have implemented it using Message Passing Interface (MPI) in our in-house built semiconductor simulation framework called VENDES, capable of running in distributed systems that allow for more accurate results in a reasonable time frame. Using a 12 nm gate length Gate-AllAround Nanowire FET (GAA NW FET) as an application example, the new implementation shows an almost perfect agreement in the output data with less than a 2% difference between the cases using 1 and 16 processes. Also, a reduction of up to 98% in the computational time has been found comparing the sequential and the 16 process simulation. For a reasonably dense mesh of 150k nodes, a variability study of 300 individual simulations, can be now performed with VENDES in approximately 2.5 days instead of an estimated sequential execution of 137 days.