Quantum Computing

Quantum computation is based on implementing selected unitary transformations representing algorithms. A generalized optimal control theory is used to find the driving field that generates a prespecified unitary transformation. The approach is independent of the physical implementation of the quantum computer and it is illustrated for one and two qubit gates in model molecular systems, where only part of the Hilbert space is used for computation.

We present two quantum algorithms based on evolution randomization, a simple variant of adiabatic quantum computing, to prepare a quantum state |x⟩ that is proportional to the solution of the system of linear equations Ax[over →]=b[over →]. The time complexities of our algorithms are O(κ^{2}log(κ)/ε) and O(κlog(κ)/ε), where κ is the condition number of A and ε is the precision. Both algorithms are constructed using families of Hamiltonians that are linear combinations of products of A, the projector onto the initial state |b⟩, and single-qubit Pauli operators. The algorithms are conceptually simple and easy to implement. They are not obtained from equivalences between the gate model and adiabatic quantum computing. They do not use phase estimation or variable-time amplitude amplification, and do not require large ancillary systems. We discuss a gate-based implementation via Hamiltonian simulation and prove that our second algorithm is almost optimal in terms of κ. Like previous methods, our techniques yield an exponential quantum speed-up under some assumptions. Our results emphasize the role of Hamiltonian-based models of quantum computing for the discovery of important algorithms.

Abstract Digital quantum computers provide a computational framework for solving the Schrödinger equation for a variety of many‐particle systems. Quantum computing algorithms for the quantum simulation of these systems have recently witnessed remarkable growth, notwithstanding the limitations of existing quantum hardware, especially as a tool for electronic structure computations in molecules. In this review, we provide a self‐contained introduction to emerging algorithms for the simulation of Hamiltonian dynamics and eigenstates, with emphasis on their applications to the electronic structure in molecular systems. Theoretical foundations and implementation details of the method are discussed, and their strengths, limitations, and recent advances are presented. This article is categorized under: Quantum Computing > Algorithms Electronic Structure Theory > Ab Initio Electronic Structure Methods Quantum Computing > Theory Development

For a practical quantum computer to operate, it is essential to properly manage decoherence. One important technique for doing this is the use of "decoherence-free subspaces" (DFSs), which have recently been demonstrated. Here we present the first use of DFSs to improve the performance of a quantum algorithm. An optical implementation of the Deutsch-Jozsa algorithm can be made insensitive to a particular class of phase noise by encoding information in the appropriate subspaces; we observe a reduction of the error rate from 35% to 7%, essentially its value in the absence of noise.

In 1994 Peter Shor [65] published a factoring algorithm for a quantum computer that finds the prime factors of a composite integer N more efficiently than is possible with the known algorithms for a classical com­ puter. Since the difficulty of the factoring problem is crucial for the se­ curity of a public key encryption system, interest (and funding) in quan­ tum computing and quantum computation suddenly blossomed. Quan­ tum computing had arrived. The study of the role of quantum mechanics in the theory of computa­ tion seems to have begun in the early 1980s with the publications of Paul Benioff [6]' [7] who considered a quantum mechanical model of computers and the computation process. A related question was discussed shortly thereafter by Richard Feynman [35] who began from a different perspec­ tive by asking what kind of computer should be used to simulate physics. His analysis led him to the belief that with a suitable class of "quantum machines" one could imitate any quantum system.

We present a stochastic quantum computing algorithm that can prepare any eigenvector of a quantum Hamiltonian within a selected energy interval [E-ε,E+ε]. In order to reduce the spectral weight of all other eigenvectors by a suppression factor δ, the required computational effort scales as O[|logδ|/(pε)], where p is the squared overlap of the initial state with the target eigenvector. The method, which we call the rodeo algorithm, uses auxiliary qubits to control the time evolution of the Hamiltonian minus some tunable parameter E. With each auxiliary qubit measurement, the amplitudes of the eigenvectors are multiplied by a stochastic factor that depends on the proximity of their energy to E. In this manner, we converge to the target eigenvector with exponential accuracy in the number of measurements. In addition to preparing eigenvectors, the method can also compute the full spectrum of the Hamiltonian. We illustrate the performance with several examples. For energy eigenvalue determination with error ε, the computational scaling is O[(logε)^{2}/(pε)]. For eigenstate preparation, the computational scaling is O(logΔ/p), where Δ is the magnitude of the orthogonal component of the residual vector. The speed for eigenstate preparation is exponentially faster than that for phase estimation or adiabatic evolution.

The recent biotechnological progress has allowed life scientists and physicians to access an unprecedented, massive amount of data at all levels (molecular, supramolecular, cellular and so on) of biological complexity. So far, mostly classical computational efforts have been dedicated to the simulation, prediction or de novo design of biomolecules, in order to improve the understanding of their function or to develop novel therapeutics. At a higher level of complexity, the progress of omics disciplines (genomics, transcriptomics, proteomics and metabolomics) has prompted researchers to develop informatics means to describe and annotate new biomolecules identified with a resolution down to the single cell, but also with a high-throughput speed. Machine learning approaches have been implemented to both the modelling studies and the handling of biomedical data. Quantum computing (QC) approaches hold the promise to resolve, speed up or refine the analysis of a wide range of these computational problems. Here, we review and comment on recently developed QC algorithms for biocomputing, with a particular focus on multi-scale modelling and genomic analyses. Indeed, differently from other computational approaches such as protein structure prediction, these problems have been shown to be adequately mapped onto quantum architectures, the main limit for their immediate use being the number of qubits and decoherence effects in the available quantum machines. Possible advantages over the classical counterparts are highlighted, along with a description of some hybrid classical/quantum approaches, which could be the closest to be realistically applied in biocomputation.

The interpretation of measurements of high-energy particle collisions relies heavily on the performance of full event generators, which include the calculation of the hard process and the subsequent parton shower step. With the continuous improvement of quantum devices, dedicated algorithms are needed to exploit the potential quantum that computers can provide. We propose general and extendable algorithms for quantum gate computers to facilitate calculations of helicity amplitudes and the parton shower process. The helicity amplitude calculation exploits the equivalence between spinors and qubits and the unique features of a quantum computer to compute the helicities of each particle involved simultaneously, thus fully utilizing the quantum nature of the computation. This advantage over classical computers is further exploited by the simultaneous computation of s- and t-channel amplitudes for a $2\ensuremath{\rightarrow}2$ process. The parton shower algorithm simulates collinear emission for a two-step, discrete parton shower. In contrast to classical implementations, the quantum algorithm constructs a wave function with a superposition of all shower histories for the whole parton shower process, thus removing the need to explicitly keep track of individual shower histories. Both algorithms utilize the quantum computers ability to remain in a quantum state throughout the computation and represent a first step towards a quantum computing algorithm describing the full collision event at the LHC.

Quantum computers provide new avenues to access ground and excited state properties of systems otherwise difficult to simulate on classical hardware. New approaches using subspaces generated by real-time evolution have shown efficiency in extracting eigenstate information, but the full capabilities of such approaches are still not understood. In recent work, we developed the variational quantum phase estimation (VQPE) method, a compact and efficient real-time algorithm to extract eigenvalues on quantum hardware. Here we build on that work by theoretically and numerically exploring a generalized Krylov scheme where the Krylov subspace is constructed through a parametrized real-time evolution, which applies to the VQPE algorithm as well as others. We establish an error bound that justifies the fast convergence of our spectral approximation. We also derive how the overlap with high energy eigenstates becomes suppressed from real-time subspace diagonalization and we visualize the process that shows the signature phase cancellations at specific eigenenergies. We investigate various algorithm implementations and consider performance when stochasticity is added to the target Hamiltonian in the form of spectral statistics. To demonstrate the practicality of such real-time evolution, we discuss its application to fundamental problems in quantum computation such as electronic structure predictions for strongly correlated systems.

For cyber-physical systems (CPS), ensuring process and data security is critically important since the corresponding infrastructure needs to have high operational efficiency with no downtime. There are many techniques available that make communications in CPS environments secure – such as enabling traffic encryption between sensors and the computers processing the sensor’s data, incorporating message authentication codes to achieve integrity, etc. However, most of these techniques are dependent on some form of symmetric or asymmetric cryptographic algorithms like AES and RSA. These algorithms are under threat because of the emerging quantum computing paradigm: with quantum computing, these encryption algorithms can be potentially broken. It is therefore desirable to explore the use of quantum cryptography – which cannot be broken by quantum computing – for securing the classical communications infrastructure deployed in CPS. In this paper, we discuss possible consequences of this option. We also explain how quantum computers can help even more: namely, they can be used to maximize the system’s security where scalability is never a constraint, and to ensure we are not wasting time cycles on communicating and processing irrelevant information.