Quantum computers still suffer from errors that keep them in the category of Noisy Intermediate-Scale Quantum (NISQ) computers. Among the various tools capable of circumventing some of the problems of NISQ computers, there is the Variational Quantum Algorithm (VQA). This algorithm can be useful in almost all sub-areas of quantum computing, such as error correction, compilation, combinatorial optimization, and dynamic simulations. These algorithms involve optimization processes in parametric rotation gates and can make use of well-founded optimizers in classical machine learning in computing. Some studies in the formalism of quantum mechanics lead us to possible experiments on a quantum computer, but due to its sensitivity, these experiments require daily calibration. When dealing with an external environment, the system behaves as an open system, governed by a unitary operator in the global state space and described by a completely positive trace-preserving map (CPTP) in the local state space. We call these processes quantum channels, which result in an evolved state over time. Through the formalism of the representation in terms of Kraus operators, we obtain the evolved states through analytical calculations and compare the coherence of this state with the coherence of the evolved state obtained through the protocol of simulating noisy quantum channels proposed in [M. S. Zanetti, D. F. Pinto, M. L. W. Basso, and J. Maziero, Simulating noisy quantum channels via quantum state preparation algorithms, J. Phys. B: At. Mol. Opt. Phys. 56, 115501 (2023)]. During a step of this protocol, state preparations are performed using functions from the Qiskit library. The present work performs state preparations using VQA. We simulate channels associated with bit-flip, phase-flip, bit-phase-flip, depolarizing, amplitude damping, phase damping, generalized amplitude damping, Lorentz, and Heisenberg-Weyl dephasing. The results of simulations with VQA are satisfactory, with the coherence of the simulated state approaching the analytical value. This demonstrates the usefulness of VQA in state preparation for this problem.
Quantum computers still suffer from errors that keep them in the category of Noisy Intermediate-Scale Quantum (NISQ) computers. Among the various tools capable of circumventing some of the problems of NISQ computers, there is the Variational Quantum Algorithm (VQA). This algorithm can be useful in almost all sub-areas of quantum computing, such as error correction, compilation, combinatorial optimization, and dynamic simulations. These algorithms involve optimization processes in parametric rotation gates and can make use of well-founded optimizers in classical machine learning in computing. Some studies in the formalism of quantum mechanics lead us to possible experiments on a quantum computer, but due to its sensitivity, these experiments require daily calibration. When dealing with an external environment, the system behaves as an open system, governed by a unitary operator in the global state space and described by a completely positive trace-preserving map (CPTP) in the local state space. We call these processes quantum channels, which result in an evolved state over time. Through the formalism of the representation in terms of Kraus operators, we obtain the evolved states through analytical calculations and compare the coherence of this state with the coherence of the evolved state obtained through the protocol of simulating noisy quantum channels proposed in [M. S. Zanetti, D. F. Pinto, M. L. W. Basso, and J. Maziero, Simulating noisy quantum channels via quantum state preparation algorithms, J. Phys. B: At. Mol. Opt. Phys. 56, 115501 (2023)]. During a step of this protocol, state preparations are performed using functions from the Qiskit library. The present work performs state preparations using VQA. We simulate channels associated with bit-flip, phase-flip, bit-phase-flip, depolarizing, amplitude damping, phase damping, generalized amplitude damping, Lorentz, and Heisenberg-Weyl dephasing. The results of simulations with VQA are satisfactory, with the coherence of the simulated state approaching the analytical value. This demonstrates the usefulness of VQA in state preparation for this problem.
This work presents a new protocol for simulating quantum systems based on quantum state preparation, with various applications such as the simulation of quantum channels and the implementation of general measurements. In general, these applications are modeled by quantum operations and involve completely positive trace-preserving (CPTP) maps. These maps correspond to sets of operators known as Kraus operators, when applied to the simulation of quantum channels, and general measurement operators, when applied to the implementation of non-selective measurements. In the literature, especially when the experimental platform is circuit-based quantum computing, these operators are employed to adjust the parameters of unitary matrices, which are the ones that actually operate in the experiment. To do this, it is necessary to solve a system of nonlinear equations numerically. The new protocol avoids these complications by applying the operators directly over the analytical description of the quantum systems to be simulated. Through the use of isometries and purifications, it models the quantum operations of interest to the experimentalist, whether they are quantum channels or general measurements, enabling the use of software libraries for symbolic algebra to perform exact and automated manipulation of the state vector, with complex amplitudes, to be prepared for simulation. The algebraic manipulation of the state vector provides greater flexibility in experimental planning, as it can be used on any quantum simulation platform. More specifically, it is not restricted to circuit-based quantum computing. Finally, the use of the new protocol is illustrated in the simulation of one-qubit channels, qudits, Lorentz transformations, mixed-state simulations, POVM implementation, and the certification of quantum contextuality via POVMs.
The preparation of an equilibrium thermal state of a quantum many-body system on noisy intermediate-scale quantum (NISQ) devices is an important task in order to extend the range of applications of quantum computation. Faithful Gibbs state preparation would pave the way to investigate protocols such as thermalization and out-of-equilibrium thermodynamics and provide useful resources for quantum algorithms, where sampling from Gibbs states constitutes a key subroutine. We propose a variational quantum algorithm (VQA) to prepare Gibbs states of a quantum many-body system. The novelty of our VQA consists of implementing a parameterized quantum circuit acting on two distinct, yet connected (via cnot gates), quantum registers. The VQA evaluates the Helmholtz free energy, where the von Neumann entropy is obtained via postprocessing of computational-basis measurements on one register, while the Gibbs state is prepared on the other register via a unitary rotation in the energy basis. Finally, we benchmark our VQA by preparing Gibbs states of the transverse-field Ising and Heisenberg XXZ models and achieve remarkably high fidelities across a broad range of temperatures in state-vector simulations. We also assess the performance of the VQA on IBM quantum computers, showcasing its feasibility on current NISQ devices.
Traditional quantum algorithms with guaranteed performance generally require fully coherent quantum computers to operate, making error-correction a necessity. In contrast, variational quantum algorithms seek to remove this requirement by formulating the computation as the approximate optimization of a functional. This optimization is carried out in the space of heuristic functions defined by a parameterized (variational) quantum circuit. Taking advantage of the flexibility in the definition of these heuristics, we can expand the operability of variational algorithms within the regime of coherence times of noisy intermediate-scale quantum devices, avoiding the necessity of error-correction. In this dissertation, we present the application of the variational quantum computing approach to problems in quantum simulation, quantum state preparation, quantum error-correction, and generative modeling.\nIn the first part, we investigate the use of the variational quantum eigensolver (VQE) for simulating the ground state of fermionic systems. We start by studying the implementation of VQE using the unitary coupled cluster (UCC) ansatz and propose strategies to reduce its cost. We use these insights to carry out some of the first experimental quantum computations of molecular energies using simplified versions of this ansatz. We also propose an extension of UCC to study problems in condensed matter physics, along with a new low-depth circuit ansatz for preparing non-gaussian fermionic states on quantum computers. We show the potential of our approach to describe strongly correlated fermionic systems.\nIn the second part, we develop three new variational quantum algorithms for problems in quantum computing and machine learning, namely: 1) the quantum autoencoder, to compress ensembles of quantum states, 2) the variational quantum error corrector, to find device-tailored quantum encoding and recovery circuits for quantum error correction, and 3) the variational quantum generator, to generate classical probability distributions. These techniques offer efficient ways to design new quantum circuits for state preparation, to find more effective error-correcting codes and to perform generative modeling with quantum computers. Our work provides insights into the design of variational quantum algorithms and establishes practical guidelines to implement these methods on near-term quantum computers, as well as some future research directions for this field.
Variational quantum algorithms (VQAs) have established themselves as a central computational paradigm in the Noisy Intermediate-Scale Quantum (NISQ) era. By coupling parameterized quantum circuits (PQCs) with classical optimization, they operate effectively under strict hardware limitations. However, as quantum architectures transition toward early fault-tolerant (EFT) and ultimate fault-tolerant (FT) regimes, the foundational principles and long-term viability of VQAs require systematic reassessment. This review offers an insightful analysis of VQAs and their progression toward the fault-tolerant regime. We deconstruct the core algorithmic framework by examining ansatz design and classical optimization strategies, including cost function formulation, gradient computation, and optimizer selection. Concurrently, we evaluate critical training bottlenecks, notably barren plateaus (BPs), alongside established mitigation strategies. The discussion then explores the EFT phase, detailing how the integration of quantum error mitigation and partial error correction can sustain algorithmic performance. Addressing the FT phase, we analyze the inherent challenges confronting current hybrid VQA models. Furthermore, we synthesize recent VQA applications across diverse domains, including many-body physics, quantum chemistry, machine learning, and mathematical optimization. Ultimately, this review outlines a theoretical roadmap for adapting quantum algorithms to future hardware generations, elucidating how variational principles can be systematically refined to maintain their relevance and efficiency within an error-corrected computational environment.
Variational hybrid quantum-classical algorithms (VHQCAs) are near-term algorithms that leverage classical optimization to minimize a cost function, which is efficiently evaluated on a quantum computer. Recently VHQCAs have been proposed for quantum compiling, where a target unitary U is compiled into a short-depth gate sequence V. In this work, we report on a surprising form of noise resilience for these algorithms. Namely, we find one often learns the correct gate sequence V (i.e. the correct variational parameters) despite various sources of incoherent noise acting during the cost-evaluation circuit. Our main results are rigorous theorems stating that the optimal variational parameters are unaffected by a broad class of noise models, such as measurement noise, gate noise, and Pauli channel noise. Furthermore, our numerical implementations on IBM's noisy simulator demonstrate resilience when compiling the quantum Fourier transform, Toffoli gate, and W-state preparation. Hence, variational quantum compiling, due to its robustness, could be practically useful for noisy intermediate-scale quantum devices. Finally, we speculate that this noise resilience may be a general phenomenon that applies to other VHQCAs such as the variational quantum eigensolver.
In recent years, variational quantum algorithms have gained significant attention due to their adaptability and efficiency on near-term quantum hardware. They have shown potential in a variety of tasks, including linear algebra, search problems, Gibbs, and ground state preparation. Nevertheless, the presence of noise in current day quantum hardware severely limits their performance. In this work, we introduce dissipative variational quantum algorithms (D-VQAs) by incorporating dissipative operations, such as qubit RESET and stochastic gates, as an intrinsic part of a variational quantum circuit. We argue that such dissipative variational algorithms possess some natural resilience to dissipative noise. We demonstrate how such algorithms can prepare Gibbs states over a wide range of quantum many-body Hamiltonians and temperatures, while significantly reducing errors due to both coherent and noncoherent noise. An additional advantage of our approach is that no ancilla qubits are need. Our results highlight the potential of D-VQAs to enhance the robustness and accuracy of variational quantum computations on noisy intermediate-scale quantum (NISQ) devices.