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 minimizing 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 optimizations of paramount importance to classical machine learning while we also offload significant but tractable work onto the classical processor.As we demonstrate numerically, the approach shares features with phase-estimation protocols that prepare eigenstates with a dominant initial fidelity contribution.
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 minimizing 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 optimizations of paramount importance to classical machine learning while we also offload significant but tractable work onto the classical processor.As we demonstrate numerically, the approach shares features with phase-estimation protocols that prepare eigenstates with a dominant initial fidelity contribution.
Abstract CoVariance Root finding with classical shadows (CoVaR) was recently introduced as a new paradigm for training variational quantum circuits. Common approaches, such as variants of the variational quantum eigensolver, aim to optimise a non-linear classical cost function and thus suffer from, e.g. poor local minima, high shot requirements and barren plateaus. In contrast, CoVaR fully exploits powerful classical shadows and finds joint roots of a very large number of covariances using only a logarithmic number of shots and linearly scaling classical computing resources. As a result, CoVaR has been demonstrated to be particularly robust against local traps, however, its main limitation has been that it requires a sufficiently good initial state. We address this limitation by introducing an adiabatic morphing of the target Hamiltonian and demonstrate in a broad range of application examples that CoVaR can successfully prepare eigenstates of the target Hamiltonian when no initial warm start is known. CoVaR succeeds even when Hamiltonian energy gaps are very small—this is in stark contrast to adiabatic evolution and phase estimation algorithms where circuit depths scale inversely with the Hamiltonian energy gaps. On the other hand, when the energy gaps are relatively small then adiabatic CoVaR may converge to higher excited states as opposed to a targeted specific low-lying state. Nevertheless, we exploit this feature of adiabatic CoVaR and demonstrate that it can be used to map out the low lying spectrum of a Hamiltonian which can be useful in practical applications, such as estimating thermal properties or in high-energy physics.
Gradient-based optimizers have been proposed for training variational quantum circuits in settings such as quantum neural networks (QNNs). The task of gradient estimation, however, has proven to be challenging, primarily due to distinctive quantum features such as state collapse and measurement incompatibility. Conventional techniques, such as the parameter-shift rule, necessitate several fresh samples in each iteration to estimate the gradient due to the stochastic nature of state measurement. Owing to state collapse from measurement, the inability to reuse samples in subsequent iterations motivates a crucial inquiry into whether fundamentally more efficient approaches to sample utilization exist. In this paper, we affirm the feasibility of such efficiency enhancements through a novel procedure called quantum shadow gradient descent (QSGD), which uses a single sample per iteration to estimate all components of the gradient. Our approach is based on an adaptation of shadow tomography that significantly enhances sample efficiency. Through detailed theoretical analysis, we show that QSGD has a significantly faster convergence rate than existing methods under locality conditions. We present detailed numerical experiments supporting all of our theoretical claims.
In this paper, we focus on the task of optimizing the parameters in Parametrized Quantum Circuits (PQCs). While popular algorithms, such as Simultaneous Perturbation Stochastic Approximation (SPSA), limit the number of circuit-execution to two per iteration, irrespective of the number of parameters in the circuit, they have their own challenges. These methods use central-differences to calculate biased estimates of directional derivatives. We show, both theoretically and numerically, that this may lead to instabilities in \emph{training} the PQCs. To remedy this, we propose Stochastic Shadow Descent (\texttt{SSD}), which uses random-projections (or \emph{shadows}) of the gradient to update the parameters iteratively. We eliminate the bias in directional derivatives by employing the Parameter-Shift Rule, along with techniques from Quantum Signal Processing, to construct a quantum circuit that parsimoniously computes \emph{unbiased estimates} of directional derivatives. Finally, we prove the convergence of the \texttt{SSD} algorithm, provide worst-case bounds on the number of iterations, and numerically demonstrate its efficacy.
The variational approach is a cornerstone of computational physics, considering both conventional and quantum computing computational platforms. The variational quantum eigensolver algorithm aims to prepare the ground state of a Hamiltonian exploiting parametrized quantum circuits that may offer an advantage compared to classical trial states used, for instance, in quantum Monte Carlo or tensor network calculations. While, traditionally, the main focus has been on developing better trial circuits, we show that the algorithm's success, if optimized within stochastic gradient descent (SGD) or quantum natural gradient descent (QNGD), crucially depends on other parameters such as the learning rate, the number Ns of measurements to estimate the gradient components, and the Hamiltonian gap $\mathrm{\ensuremath{\Delta}}$. Within the standard SGD or QNGD, we first observe the existence of a finite Ns value below which the optimization is impossible, and the energy variance resembles the behavior of the specific heat in second-order phase transitions. Second, when Ns is above such threshold level, and learning is possible, we develop a phenomenological model that relates the fidelity of the state preparation with the optimization hyperparameters and $\mathrm{\ensuremath{\Delta}}$. More specifically, we observe that the computational resources scale as $1/{\mathrm{\ensuremath{\Delta}}}^{2}$, and we propose a symmetry enhancement of the variational ansatz as a way to increase the closing gap. We test our understanding on several instances of two-dimensional frustrated quantum magnets, which are believed to be the most promising candidates for near-term quantum advantage through variational quantum simulations.
Quantum Machine Learning models typically require expensive on-chip training procedures and often lack efficient gradient estimation methods. By employing Pauli propagation, it is possible to derive a symbolic representation of observables as analytic functions of a circuit's parameters. Although the number of terms in such functional representations grows rapidly with circuit depth, suitable choices of ansatz and controlled truncations on Pauli weights and trigonometric degree components yield accurate yet tractable estimators of the target observables. With the right ansatz design, this approach can be extended to system sizes beyond the reach of classical statevector simulation, enabling scalable training for larger quantum systems. This also enables a form of classical pre-training through gradient-based optimization prior to deployment on quantum hardware. The proposed approach is demonstrated on the Variational Quantum Eigensolver for obtaining the ground state of the ANNNI spin model on 32 qubits, showing that accurate results can be achieved with a scalable and computationally efficient procedure.
The analysis of quantum state properties is of fundamental importance in quantum physics and serves as a primary motivation for the development of quantum computing. In particular, low-temperature and ground states are of significant interest, as they most prominently exhibit phenomena unique to quantum mechanics. However, extracting such properties typically requires costly numerical simulations or laboratory experiments, while simulation on quantum hardware remains limited by the capabilities of current devices. This thesis proposes two complementary approaches to address these challenges.The first contribution presents an algorithm for predicting ground-state properties across an entire family of quantum states within the same phase of matter, based on training data. Under slightly stronger assumptions, we improve upon the sample complexity of existing methods, prove the same rigorous guarantees for a deep neural network-based approach, and demonstrate its practical advantages through extensive numerical experiments.The second contribution addresses the challenge of preparing quantum states of interest using Variational Quantum Algorithms (VQAs), a framework based on the classical optimization of parametrized quantum circuits. We introduce multi-armed stochastic bandits into this setting, propose an instance-optimal algorithm for continuous single-parameter optimization, and and motivate further exploration of the framework we introduced with an array of experiments.