Learning to measure: adaptive informationally complete POVMs for near-term quantum algorithms
Guillermo García-Pérez, Matteo A. C. Rossi, Boris Sokolov, Francesco Tacchino, Panagiotis Kl. Barkoutsos, Guglielmo Mazzola, Ivano Tavernelli, Sabrina Maniscalco
Variational quantum algorithms stand as the most promising approaches towards practical applications of near-term quantum computers. However, these methodologies usually require a large number of measurements, which represents an important roadblock for future real-world use cases. We introduce a novel approach to tackle this problem: a variational measurement scheme. We present an algorithm that optimises informationally complete POVMs on the fly in order to minimise the statistical fluctuations in the estimation of relevant cost functions. We use it in combination with the Variational Quantum Eigensolver to calculate ground-state energies of molecular Hamiltonians in numerical simulations and show that it is competitive with state-of-the-art measurement reduction approaches. We also highlight the potential of the informational completeness of the measurement outcomes by reusing the ground-state energy estimation data to perform high-fidelity reduced state tomography.
Variational quantum algorithms stand as the most promising approaches towards practical applications of near-term quantum computers. However, these methodologies usually require a large number of measurements, which represents an important roadblock for future real-world use cases. We introduce a novel approach to tackle this problem: a variational measurement scheme. We present an algorithm that optimises informationally complete POVMs on the fly in order to minimise the statistical fluctuations in the estimation of relevant cost functions. We use it in combination with the Variational Quantum Eigensolver to calculate ground-state energies of molecular Hamiltonians in numerical simulations and show that it is competitive with state-of-the-art measurement reduction approaches. We also highlight the potential of the informational completeness of the measurement outcomes by reusing the ground-state energy estimation data to perform high-fidelity reduced state tomography.
Many prominent quantum computing algorithms with applications in fields such as chemistry and materials science require a large number of measurements, which represents an important roadblock for future real-world use cases.We introduce a novel approach to tackle this problem through an adaptive measurement scheme.We present an algorithm that optimizes informationally complete positive operator-valued measurements (POVMs) on the fly in order to minimize the statistical fluctuations in the estimation of relevant cost functions.We show its advantage by improving the efficiency of the variational quantum eigensolver in calculating ground-state energies of molecular Hamiltonians with extensive numerical simulations.Our results indicate that the proposed method is competitive with state-of-the-art measurement-reduction approaches in terms of efficiency.In addition, the informational completeness of the approach offers a crucial advantage, as the measurement data can be reused to infer other quantities of interest.We demonstrate the feasibility of this prospect by reusing ground-state energy-estimation data to perform high-fidelity reduced state tomography.
Adaptive derivative-assembled problem-tailored ansatz variational quantum eigensolver (ADAPT-VQE) stands out as a robust algorithm for constructing compact ansätze for molecular simulation. It enables to significantly reduce the circuit depth with respect to other methods, such as unitary coupled cluster singles and doubles, while achieving higher accuracy and not suffering from so-called barren plateaus that hinder the variational optimization of many hardware-efficient ansätze. In its standard implementation, however, it introduces a considerable measurement overhead in the form of gradient evaluations through estimations of many commutator operators. In this work, we mitigate this measurement overhead by exploiting a recently introduced method for energy evaluation relying on adaptive informationally complete generalized measurements (AIMs). Besides offering an efficient way to measure the energy itself, informationally complete (IC) measurement data can be reused to estimate all the commutators of the operators in the operator pool of ADAPT-VQE, using only classically efficient postprocessing. The framework is generic for any IC positive operator-valued measures (POVMs) implementation, and in this work, we demonstrate it’s effectiveness with dilation POVMs. We present the AIM-ADAPT-VQE scheme in detail, and investigate its performance with H 4 , N 2 , and 1,3,5,7-octatetraene Hamiltonians and several operator pools. Our numerical simulations indicate that the measurement data obtained to evaluate the energy can be reused to implement ADAPT-VQE with no additional measurement overhead for the systems considered here. In addition, we show that, if the energy is measured within chemical precision, the count in the resulting circuits is close to the ideal one. With scarce measurement data, AIM-ADAPT-VQE still converges to the ground state with high probability, albeit with an increased circuit depth in some cases.
We present adaptive measurement techniques tailored for variational quantum algorithms on near-term small and noisy devices. In particular, we generalise earlier "learning to measure" strategies in two ways. First, by considering a class of adaptive positive operator valued measures (POVMs) that can be simulated with simple projective measurements without ancillary qubits, we decrease the amount of required qubits and two-qubit gates. Second, by introducing a method based on Quantum Detector Tomography to mitigate the effect of noise, we are able to optimise the POVMs as well as to infer expectation values reliably in the currently available noisy quantum devices. Our numerical simulations clearly indicate that the presented strategies can significantly reduce the number of needed shots to achieve chemical accuracy in variational quantum eigensolvers, thus helping to solve one of the bottlenecks of near-term quantum computing.
The variational quantum eigensolver (VQE) stands as a prominent quantum-classical hybrid algorithm for near-term quantum computers to obtain the ground states of molecular Hamiltonians in quantum chemistry. However, due to the noncommutativity of the Pauli operators in the Hamiltonian, the number of measurements required on quantum computers increases significantly as the system size grows, which may hinder practical applications of VQE. In this work, we present a protocol termed (JBM-VQE) to reduce the number of measurements and speed up the VQE algorithm. Our method employs joint Bell measurements, enabling the simultaneous measurement of the absolute values of all expectation values of Pauli operators present in the Hamiltonian. In the course of the optimization, JBM-VQE estimates the absolute values of the expectation values of the Pauli operators for each iteration by the joint Bell measurement, while the signs of them are measured less frequently by the conventional method to measure the expectation values. Our approach is based on the empirical observation that the signs do not often change during optimization. We illustrate the speed-up of JBM-VQE compared to conventional VQE by numerical simulations for finding the ground states of molecular Hamiltonians of small molecules, and the speed-up of JBM-VQE at the early stage of the optimization becomes increasingly pronounced in larger systems. Our approach based on the joint Bell measurement is not limited to VQE and can be utilized in various quantum algorithms whose cost functions are expectation values of many Pauli operators. Published by the American Physical Society 2024
Quantum computing has become a potential way to solve complex problems that conventional computers can’t handle, especially those involving strongly correlated quantum systems. Classical simulation of these systems becomes exponentially more challenging as the number of interacting particles rises, resulting in significant computational and memory limitations. As a result, classical computational approaches are unable to handle many significant physical and chemical systems, including molecular structures, biological molecules, and quantum materials. Quantum computers operate using the same quantum mechanical principles as these systems. Consequently, they provide a natural framework for exploring large Hilbert spaces and capturing complex quantum correlations. This thesis begins by discussing the fundamental principles of quantum mechanics that are applicable to computation, such as qubits, superposition, and entanglement. The construction of quantum circuits is discussed after a discussion of basic quantum gates and their matrix representations. These ideas are the foundation of quantum algorithms and offer the essential framework for comprehending the modeling and manipulation of complicated quantum states using quantum hardware. Building on this foundation, the thesis focuses on Variational Quantum Algorithms (VQAs), which offer a feasible hybrid quantum-classical approach to optimizing near-term quantum devices. In VQAs, a parameterized quantum circuit is used to approximate the ground-state energy of the quantum system. A classical optimizer then iteratively updates the variational parameters to minimize a cost function, usually the expectation value of the system’s Hamiltonian. A thorough analysis of the function of gradient-based optimization is conducted, encompassing the parameter-shift rule that is employed to calculate precise energy derivatives. The discussion is expanded to include major applications such as the Quantum Approximate Optimization Algorithm and developing approaches to quantum machine learning, emphasizing the importance of efficient and accurate gradient evaluation. A major challenge in the real-world application of VQAs is the high resource cost of gradient estimation using conventional direct measurement techniques. These techniques require repeated projective measurements and several executions of parameter-shifted quantum circuits, which lead to significant measurement overhead and deeper circuits. Quantum Non-Demolition Measurement (QNDM) is examined in this thesis as a potential framework for gradient evaluation to get around this limitation. Based on recent theoretical advances, QNDM introduces an ancillary detector qubit that coherently stores gradient information within its phase. Derivatives can then be estimated indirectly through detector measurements. This allows gradient information to be extracted without destroying the quantum state and with a greatly reduced number of measurements. The efficiency of QNDM is evaluated by extending the study to systems with bigger Hamiltonians and greater molecular complexity, as well as by comparing it in detail with direct measurement techniques utilizing benchmark molecular systems like LiH, Li2, and H2. Although both methods successfully converge to the ground-state energy, as shown by numerical results, QNDM consistently uses fewer quantum resources in terms of logical gate operations, measurement shots, and circuit executions. Additionally, as system size grows, QNDM’s resource advantage becomes more noticeable, suggesting that it has favorable scalability for bigger molecular systems. This thesis concludes that, while both direct measurement and quantum non-demolition measurement approaches allow for precise variational optimization, QNDM gives a significant decrease in quantum resources. These results establish QNDM as an effective and scalable method for developing quantum simulations of realistic physical and chemical systems and for executing variational quantum algorithms on near-term quantum hardware.
Quantum state tomography (QST) is a fundamental technique in quantum information science, aiming to reconstruct the density matrix of an unknown quantum state from experimental measurements. However, traditional QST methods often suffer from significant computational demands and are highly susceptible to measurement noise, hindering their practical application, particularly with near-term quantum computers. This work proposes a novel approach to QST utilizing variational quantum algorithms (VQAs). VQAs iteratively optimize a parameterized quantum circuit to minimize a cost function related to the fidelity between the desired state and the state predicted by the circuit. This approach dramatically reduces the number of required measurements and offers the potential for achieving high-fidelity state reconstruction with lower quantum resource requirements. We detail the theoretical framework, outlining the core algorithms and mathematical formulations. The key advantage lies in the ability to learn the state directly, mitigating the limitations of traditional methods. We explore the optimization landscape and discuss strategies for improving convergence and robustness. The results demonstrate a promising path toward scalable and robust QST on quantum devices.