Measurement cost of metric-aware variational quantum algorithms

Variational quantum algorithms are promising tools for near-term quantum computers as their shallow circuits are robust to experimental imperfections. Their practical applicability, however, strongly depends on how many times their circuits need to be executed for sufficiently reducing shot-noise. We consider metric-aware quantum algorithms: variational algorithms that use a quantum computer to efficiently estimate both a matrix and a vector object. For example, the recently introduced quantum natural gradient approach uses the quantum Fisher information matrix as a metric tensor to correct the gradient vector for the co-dependence of the circuit parameters. We rigorously characterise and upper bound the number of measurements required to determine an iteration step to a fixed precision, and propose a general approach for optimally distributing samples between matrix and vector entries. Finally, we establish that the number of circuit repetitions needed for estimating the quantum Fisher information matrix is asymptotically negligible for an increasing number of iterations and qubits.Comment: 17 pages, 3 figure

Continuous phase-space representations for finite-dimensional quantum states and their tomography

Continuous phase spaces have become a powerful tool for describing, analyzing, and tomographically reconstructing quantum states in quantum optics and beyond. A plethora of these phase-space techniques are known, however a thorough understanding of their relations was still lacking for finite-dimensional quantum states. We present a unified approach to continuous phase-space representations which highlights their relations and tomography. The infinite-dimensional case from quantum optics is then recovered in the large-spin limit.Comment: 15 pages, 9 figures, v4: extended tomography analysis, added references and figure

Corporate social responsibility and its effect on image and reputation: The case of L’Oréal and its acquisition of The Body Shop

A Work Project, presented as part of the requirements for the Award of a Masters Degree in Management from the NOVA – School of Business and EconomicsThis paper studies the effect of Corporate Social Responsibility (CSR) on image and reputation in the case of L’Oréal and its acquisition of The Body Shop. L’Oréal was suffering from negative headlines claiming that its products were animal tested and they expected that the acquisition of The Body Shop in 2006 would change this image. This study investigates whether CSR practices of The Body Shop helped L’Oréal to rebound from the negative headings. Using a sample of 321 respondents for The Body Shop and a sample of 289 respondents for L’Oréal, it is concluded that CSR is an important factor for a company’s success since the results show that CSR campaigns have a statistically significant impact on a company’s reputation and image. Since a CSR strategy is the key to success, this paper ends with recommendations to successfully implement CSR strategies

Exponential Error Suppression for Near-Term Quantum Devices

As quantum computers mature, quantum error correcting codes (QECs) will be adopted in order to suppress errors to any desired level $E$ at a cost in qubit-count $n$ that is merely poly-logarithmic in $1/E$. However in the NISQ era, the complexity and scale required to adopt even the smallest QEC is prohibitive. Instead, error mitigation techniques have been employed; typically these do not require an increase in qubit-count but cannot provide exponential error suppression. Here we show that, for the crucial case of estimating expectation values of observables (key to almost all NISQ algorithms) one can indeed achieve an effective exponential suppression. We introduce the Error Suppression by Derangement (ESD) approach: by increasing the qubit count by a factor of $n\geq 2$, the error is suppressed exponentially as $Q^n$ where $Q<1$ is a suppression factor that depends on the entropy of the errors. The ESD approach takes $n$ independently-prepared circuit outputs and applies a controlled derangement operator to create a state whose symmetries prevent erroneous states from contributing to expected values. The approach is therefore `NISQ-friendly' as it is modular in the main computation and requires only a shallow circuit that bridges the $n$ copies immediately prior to measurement. Imperfections in our derangement circuit do degrade performance and therefore we propose an approach to mitigate this effect to arbitrary precision due to the remarkable properties of derangements. a) they decompose into a linear number of elementary gates -- limiting the impact of noise b) they are highly resilient to noise and the effect of imperfections on them is (almost) trivial. In numerical simulations validating our approach we confirm error suppression below $10^{-6}$ for circuits consisting of several hundred noisy gates (two-qubit gate error $0.5\%$) using no more than $n=4$ circuit copies.Comment: 34 pages, 12 figure

Analytic gradient techniques for investigating the complex-valued potential energy surfaces of electronic resonances

Electronic resonances are metastable atomic or molecular systems that can decay by electron detachment. They play an important role in biological processes such as DNA fragmentation induced by slow electrons or in interstellar reactions as in the formation of neutral molecules and molecular anions. As opposed to bound states, resonances do not correspond to discrete eigenstates of a Hermitian Hamiltonian, and therefore their theoretical description requires special methods. The complex absorbing potential (CAP) method can be used to calculate both the energy and the lifetime of a resonance as a discrete eigenstate in a non-Hermitian time-independent framework. The CAP method allows for applying well-known bound-state electronic structure methods to resonances as well. In this work, the applicability of CAP-augmented equation-of-motion coupled-cluster (CAP-EOM-CC) methods is extended for locating equilibrium structures and crossings on complex-valued potential energy surfaces of electronic resonances by introducing analytic energy gradients. The structure and energy of these points are needed for, e.g., estimating the importance of a specific dissociation route or deactivation process. The accuracy of structural parameters, vertical and adiabatic electron affinities, and resonance widths obtained with approximate methods and various diffuse basis sets is investigated. Applications of optimization methods are also presented for systems that are relevant in interstellar or biological processes. Properties of the complex-valued potential energy surfaces of anionic resonances of acrylonitrile and methacrylonitrile are connected to experimental observations. Dissociative electron attachment to chlorosubstituted ethylenes is also investigated. This can help in understanding detoxification processes of these compounds and might facilitate the exploration of DEA pathways for other halogenated molecules as well

Analytic gradient techniques for investigating the complex-valued potential energy surfaces of electronic resonances

Electronic resonances are metastable atomic or molecular systems that can decay by electron detachment. They play an important role in biological processes such as DNA fragmentation induced by slow electrons or in interstellar reactions as in the formation of neutral molecules and molecular anions. As opposed to bound states, resonances do not correspond to discrete eigenstates of a Hermitian Hamiltonian, and therefore their theoretical description requires special methods. The complex absorbing potential (CAP) method can be used to calculate both the energy and the lifetime of a resonance as a discrete eigenstate in a non-Hermitian time-independent framework. The CAP method allows for applying well-known bound-state electronic structure methods to resonances as well. In this work, the applicability of CAP-augmented equation-of-motion coupled-cluster (CAP-EOM-CC) methods is extended for locating equilibrium structures and crossings on complex-valued potential energy surfaces of electronic resonances by introducing analytic energy gradients. The structure and energy of these points are needed for, e.g., estimating the importance of a specific dissociation route or deactivation process. The accuracy of structural parameters, vertical and adiabatic electron affinities, and resonance widths obtained with approximate methods and various diffuse basis sets is investigated. Applications of optimization methods are also presented for systems that are relevant in interstellar or biological processes. Properties of the complex-valued potential energy surfaces of anionic resonances of acrylonitrile and methacrylonitrile are connected to experimental observations. Dissociative electron attachment to chlorosubstituted ethylenes is also investigated. This can help in understanding detoxification processes of these compounds and might facilitate the exploration of DEA pathways for other halogenated molecules as well

Quantum natural gradient generalised to non-unitary circuits

Variational quantum circuits are promising tools whose efficacy depends on their optimisation method. For noise-free unitary circuits, the quantum generalisation of natural gradient descent was recently introduced. The method can be shown to be equivalent to imaginary time evolution, and is highly effective due to a metric tensor reconciling the classical parameter space to the device's Hilbert space. Here we generalise quantum natural gradient to consider arbitrary quantum states (both mixed and pure) via completely positive maps; thus our circuits can incorporate both imperfect unitary gates and fundamentally non-unitary operations such as measurements. Whereas the unitary variant relates to classical Fisher information, here we find that quantum Fisher information defines the core metric in the space of density operators. Numerical simulations indicate that our approach can outperform other variational techniques when circuit noise is present. We finally assess the practical feasibility of our implementation and argue that its scalability is only limited by the number and quality of imperfect gates and not by the number of qubits.Comment: 20 pages, 6 figure

Can shallow quantum circuits scramble local noise into global white noise?

Shallow quantum circuits are believed to be the most promising candidates for achieving early practical quantum advantage—this has motivated the development of a broad range of error mitigation techniques whose performance generally improves when the quantum state is well approximated by a global depolarising (white) noise model. While it has been crucial for demonstrating quantum supremacy that random circuits scramble local noise into global white noise—a property that has been proved rigorously—we investigate to what degree practical shallow quantum circuits scramble local noise into global white noise. We define two key metrics as (a) density matrix eigenvalue uniformity and (b) commutator norm that quantifies stability of the dominant eigenvector. While the former determines the distance from white noise, the latter determines the performance of purification based error mitigation. We derive analytical approximate bounds on their scaling and find in most cases they nicely match numerical results. On the other hand, we simulate a broad class of practical quantum circuits and find that white noise is in certain cases a bad approximation posing significant limitations on the performance of some of the simpler error mitigation schemes. On a positive note, we find in all cases that the commutator norm is sufficiently small guaranteeing a very good performance of purification-based error mitigation. Lastly, we identify techniques that may decrease both metrics, such as increasing the dimensionality of the dynamical Lie algebra by gate insertions or randomised compiling

Training variational quantum circuits with CoVaR: covariance root finding with classical shadows

Exploiting near-term quantum computers and achieving practical value is a considerable and exciting challenge. Most prominent candidates as variational algorithms typically aim to find the ground state of a Hamiltonian by minimising a single classical (energy) surface which is sampled from by a quantum computer. Here we introduce a method we call CoVaR, an alternative means to exploit the power of variational circuits: We find eigenstates by finding joint roots of a polynomially growing number of properties of the quantum state as covariance functions between the Hamiltonian and an operator pool of our choice. The most remarkable feature of our CoVaR approach is that it allows us to fully exploit the extremely powerful classical shadow techniques, i.e., we simultaneously estimate a very large number $>10^4-10^7$ of covariances. We randomly select covariances and estimate analytical derivatives at each iteration applying a stochastic Levenberg-Marquardt step via a large but tractable linear system of equations that we solve with a classical computer. We prove that the cost in quantum resources per iteration is comparable to a standard gradient estimation, however, we observe in numerical simulations a very significant improvement by many orders of magnitude in convergence speed. CoVaR is directly analogous to stochastic gradient-based optimisations of paramount importance to classical machine learning while we also offload significant but tractable work onto the classical processor.Comment: 25 pages, 9 figure