arXiv subjects

Artur F. Izmaylov

Publications and source records attributed to Artur F. Izmaylov.

At least 55 records · Page 3Linked to original sources

arXiv2022

Deterministic improvements of quantum measurements with grouping of compatible operators, non-local transformations, and covariance estimates

Obtaining the expectation value of an observable on a quantum computer is a crucial step in the variational quantum algorithms. For complicated observables such as molecular electronic Hamiltonians, a common strategy is to present the observable as a linear combination of measurable fragments. The main problem of this approach is a large number of measurements required for accurate sampling of the observable's expectation value. We consider several partitioning schemes based on grouping of commuting multi-qubit Pauli products with the goal of minimizing the number of measurements. Three main directions are explored: 1) grouping commuting operators using the greedy approach, 2) involving non-local unitary transformations for measuring, and 3) taking advantage of compatibility of some Pauli products with several measurable groups. The last direction gives rise to a general framework that not only provides improvements over previous methods but also connects measurement grouping approaches with recent advances in techniques of shadow tomography. Following this direction, we develop two new measurement schemes that achieve a severalfold reduction in the number of measurements for a set of model molecules compared to previous state-of-the-art methods.

quant-ph↗

arXiv2022

A Quantum Computing View on Unitary Coupled Cluster Theory

We present a review of the Unitary Coupled Cluster (UCC) ansatz and related ansätze which are used to variationally solve the electronic structure problem on quantum computers. A brief history of coupled cluster (CC) methods is provided, followed by a broad discussion of the formulation of CC theory. This includes touching on the merits and difficulties of the method and several variants, UCC among them, in the classical context, to motivate their applications on quantum computers. In the core of the text, the UCC ansatz and its implementation on a quantum computer are discussed at length, in addition to a discussion on several derived and related ansätze specific to quantum computing. The review concludes with a unified perspective on the discussed ansätze, attempting to bring them under a common framework, as well as with a reflection upon open problems within the field.

quant-ph↗

arXiv2022

Controlling energy conservation in quantum dynamics with independently moving basis functions: Application to Multi-Configuration Ehrenfest

Application of the time-dependent variational principle to a linear combination of frozen-width Gaussians describing the nuclear wavefunction provides a formalism where the total energy is conserved. The computational downside of this formalism is that trajectories of individual Gaussians are solutions of a coupled system of differential equations, limiting implementation to serial propagation algorithms. To allow for parallelization and acceleration of the computation, independent trajectories based on simplified equations of motion were suggested. Unfortunately, within practical realizations involving finite Gaussian bases, this simplification leads to breaking the energy conservation. We offer a solution for this problem by using Lagrange multipliers to ensure the energy and norm conservation regardless of basis function trajectories or basis completeness. We illustrate our approach within the Multi-Configuration Ehrenfest method considering a linear vibronic coupling model.

physics.chem-ph↗

arXiv2021

Analytic gradients in variational quantum algorithms: Algebraic extensions of the parameter-shift rule to general unitary transformations

Optimization of unitary transformations in Variational Quantum Algorithms benefits highly from efficient evaluation of cost function gradients with respect to amplitudes of unitary generators. We propose several extensions of the parametric-shift-rule to formulating these gradients as linear combinations of expectation values for generators with general eigen-spectrum (i.e. with more than two eigenvalues). Our approaches are exact and do not use any auxiliary qubits, instead they rely on a generator eigen-spectrum analysis. Two main directions in the parametric-shift-rule extensions are 1) polynomial expansion of the exponential unitary operator based on a limited number of different eigenvalues in the generator and 2) decomposition of the generator as a linear combination of low-eigenvalue operators (e.g. operators with only 2 or 3 eigenvalues). These techniques have a range of scalings for the number of needed expectation values with the number of generator eigenvalues from quadratic (for polynomial expansion) to linear and even $\log_2$ (for generator decompositions). This allowed us to propose efficient differentiation schemes superior to previous approaches for commonly used 2-qubit transformations (e.g. match-gates, transmon and fSim gates) and $\hat S^2$-conserving fermionic operators for the variational quantum eigensolver.

quant-ph↗

arXiv2021

Cartan sub-algebra approach to efficient measurements of quantum observables

An arbitrary operator corresponding to a physical observable cannot be measured in a single measurement on currently available quantum hardware. To obtain the expectation value of the observable, one needs to partition its operator to measurable fragments. However, the observable and its fragments generally do not share any eigenstates, and thus the number of measurements needed to obtain the expectation value of the observable can grow rapidly even when the wavefunction prepared is close to an eigenstate of the observable. We provide a unified Lie algebraic framework for developing efficient measurement schemes for quantum observables, it is based on two elements: 1) embedding the observable operator in a Lie algebra and 2) transforming Lie algebra elements into those of a Cartan sub-algebra (CSA) using unitary operators. The CSA plays the central role because all its elements are mutually commutative and thus can be measured simultaneously. We illustrate the framework on measuring expectation values of Hamiltonians appearing in the Variational Quantum Eigensolver approach to quantum chemistry. The CSA approach puts many recently proposed methods for the measurement optimization within a single framework, and allows one not only to reduce the number of measurable fragments but also the total number of measurements.

quant-ph↗

arXiv2021

How to define quantum mean-field solvable Hamiltonians using Lie algebras

Necessary and sufficient conditions for quantum Hamiltonians to be exactly solvable within mean-field theories have not been formulated so far. To resolve this problem, first, we define what mean-field theory is, independently of a Hamiltonian realization in a particular set of operators. Second, using a Lie-algebraic framework we formulate a criterion for a Hamiltonian to be mean-field solvable. The criterion is applicable for both distinguishable and indistinguishable particle cases. For the electronic Hamiltonians, our approach reveals the existence of mean-field solvable Hamiltonians of higher fermionic operator powers than quadratic. Some of the mean-field solvable Hamiltonians require different sets of quasi-particle rotations for different eigenstates, which reflects a more complicated structure of such Hamiltonians.

quant-ph↗

arXiv2021

Computational approaches to efficient generation of the stationary state for incoherent light excitation

Light harvesting processes are often computationally studied from a time-dependent viewpoint, in line with ultrafast coherent spectroscopy experiments. Yet, natural processes take place in the presence of incoherent light, which induces a stationary state. Such stationary states can be described using the eigenbasis of the molecular Hamiltonian, but for realistic systems a full diagonalization is prohibitively expensive. We propose three efficient computational approaches to obtaining the stationary state that circumvent system Hamiltonian diagonalization. The connection between the incoherent perturbations, decoherence, and Kraus operators is established.

physics.chem-ph↗

arXiv2021

Tequila: A platform for rapid development of quantum algorithms

Variational quantum algorithms are currently the most promising class of algorithms for deployment on near-term quantum computers. In contrast to classical algorithms, there are almost no standardized methods in quantum algorithmic development yet, and the field continues to evolve rapidly. As in classical computing, heuristics play a crucial role in the development of new quantum algorithms, resulting in high demand for flexible and reliable ways to implement, test, and share new ideas. Inspired by this demand, we introduce tequila, a development package for quantum algorithms in python, designed for fast and flexible implementation, prototyping, and deployment of novel quantum algorithms in electronic structure and other fields. Tequila operates with abstract expectation values which can be combined, transformed, differentiated, and optimized. On evaluation, the abstract data structures are compiled to run on state-of-the-art quantum simulators or interfaces.

quant-ph↗

arXiv2020

Unitary transformation of the electronic Hamiltonian with an exact quadratic truncation of the Baker-Campbell-Hausdorff expansion

Application of current and near-term quantum hardware to the electronic structure problem is highly limited by qubit counts, coherence times, and gate fidelities. To address these restrictions within the variational quantum eigensolver (VQE) framework, many recent contributions have suggested dressing the electronic Hamiltonian to include a part of electron correlation, leaving the rest to be accounted by VQE state preparation. We present a new dressing scheme that combines preservation of the Hamiltonian hermiticity and an exact quadratic truncation of the Baker-Campbell-Hausdorff expansion. The new transformation is constructed as the exponent of an involutory linear combination (ILC) of anti-commuting Pauli products. It incorporates important strong correlation effects in the dressed Hamiltonian and can be viewed as a classical preprocessing step alleviating the resource requirements of the subsequent VQE application. The assessment of the new computational scheme for electronic structure of the LiH, H$_2$O, and N$_2$ molecules shows significant increase in efficiency compared to conventional qubit coupled cluster dressings.

quant-ph↗

arXiv2020

A posteriori corrections to the Iterative Qubit Coupled Cluster method to minimize the use of quantum resources in large-scale calculations

The iterative qubit coupled cluster (iQCC) method is a systematic variational approach to solve the electronic structure problem on universal quantum computers. It is able to use arbitrarily shallow quantum circuits at expense of iterative canonical transformation of the Hamiltonian and rebuilding a circuit. Here we present a variety of a posteriori corrections to the iQCC energies to reduce the number of iterations to achieve the desired accuracy. Our energy corrections are based on a low-order perturbation theory series that can be efficiently evaluated on a classical computer. Moreover, capturing a part of the total energy perturbatively, allows us to formulate the qubit active-space concept, in which only a subset of all qubits is treated variationally. As a result, further reduction of quantum resource requirements is achieved. We demonstrate the utility and efficiency of our approach numerically on the examples of 10-qubit N$_2$ molecule dissociation, the 24-qubit H$_2$O symmetric stretch, and 56-qubit singlet-triplet gap calculations for the technologically important complex, tris-(2-phenylpyridine)iridium(III), Ir(ppy)$_3$.

quant-ph↗

arXiv2020

On the order problem in construction of unitary operators for the Variational Quantum Eigensolver

One of the main challenges in the Variational Quantum Eigensolver (VQE) framework is construction of the unitary transformation. The dimensionality of the space for unitary rotations of $N$ qubits is $4^N-1$, which makes the choice of a polynomial subset of generators exponentially difficult process. Moreover, due to non-commutativity of generators, the order in which they are used strongly affects results. Choosing the optimal order in a particular subset of generators requires testing the factorial number of combinations. We propose an approach based on the Lie algebra - Lie group connection and corresponding closure relations that systematically eliminates the order problem.

quant-ph↗

arXiv2020

Measurement Optimization in the Variational Quantum Eigensolver Using a Minimum Clique Cover

Solving the electronic structure problem using the Variational Quantum Eigensolver (VQE) technique involves measurement of the Hamiltonian expectation value. Current hardware can perform only projective single-qubit measurements, and thus, the Hamiltonian expectation value is obtained by measuring parts of the Hamiltonian rather than the full Hamiltonian. This restriction makes the measurement process inefficient because the number of terms in the Hamiltonian grows as $O(N^4)$ with the size of the system, $N$. To optimize VQE measurement one can try to group as many Hamiltonian terms as possible for their simultaneous measurement. Single-qubit measurements allow one to group only the terms that commute within corresponding single-qubit subspaces or qubit-wise commuting. We found that qubit-wise commutativity between the Hamiltonian terms can be expressed as a graph and the problem of the optimal grouping is equivalent of finding a minimum clique cover (MCC) for the Hamiltonian graph. The MCC problem is NP-hard but there exist several polynomial heuristic algorithms to solve it approximately. Several of these heuristics were tested in this work for a set of molecular electronic Hamiltonians. On average, grouping qubit-wise commuting terms reduced the number of operators to measure three times compared to the total number of terms in the considered Hamiltonians.

quant-ph↗

arXiv2020

Measuring all compatible operators in one series of a single-qubit measurements using unitary transformations

The Variational Quantum Eigensolver approach to the electronic structure problem on a quantum computer involves measurement of the Hamiltonian expectation value. Formally, quantum mechanics allows one to measure all mutually commuting or compatible operators simultaneously. Unfortunately, the current hardware permits measuring only a much more limited subset of operators that share a common tensor product eigen-basis. We introduce unitary transformations that transform any fully commuting group of operators to a group that can be measured on current hardware. These unitary operations can be encoded as a sequence of Clifford gates and let us not only measure much larger groups of terms but also to obtain these groups efficiently on a classical computer. The problem of finding the minimum number of fully commuting groups of terms covering the whole Hamiltonian is found to be equivalent to the minimum clique cover problem for a graph representing Hamiltonian terms as vertices and commutativity between them as edges. Tested on few molecular electronic Hamiltonians with up to 50 thousand terms, the introduced technique allows for the reduction of the number of separately measurable operator groups down to few hundreds, thus achieving up to 2 orders of magnitude reduction. It was also shown that this gain scales at least linearly with the number of qubits.

quant-ph↗

arXiv2019

Iterative Qubit Coupled Cluster approach with efficient screening of generators

An iterative version of the qubit coupled cluster (QCC) method [I.G. Ryabinkin et al., J. Chem. Theory Comput. 14, 6317 (2019)] is proposed. The new method seeks to find ground electronic energies of molecules on noisy intermediate-scale quantum (NISQ) devices. Each iteration involves a canonical transformation of the Hamiltonian and employs constant-size quantum circuits at the expense of increasing the Hamiltonian size. We numerically studied the convergence of the method on ground-state calculations for LiH, H$_2$O, and N$_2$ molecules and found that the exact ground-state energies can be systematically approached only if the generators of the QCC ansatz are sampled from a specific set of operators. We report an algorithm for constructing this set that scales linearly with the size of a Hamiltonian.

quant-ph↗

arXiv2019

Unitary partitioning approach to the measurement problem in the Variational Quantum Eigensolver method

To obtain estimates of electronic energies, the Variational Quantum Eigensolver (VQE) technique performs separate measurements for multiple parts of the system Hamiltonian. Current quantum hardware is restricted to projective single-qubit measurements, and thus, only parts of the Hamiltonian which form mutually qubit-wise commuting groups can be measured simultaneously. The number of such groups in the electronic structure Hamiltonians grows as $N^4$, where $N$ is the number of qubits, and thus puts serious restrictions on the size of the systems that can be studied. Using a partitioning of the system Hamiltonian as a linear combination of unitary operators we found a circuit formulation of the VQE algorithm that allows one to measure a group of fully anti-commuting terms of the Hamiltonian in a single series of single-qubit measurements. Numerical comparison of the unitary partitioning to previously used grouping of Hamiltonian terms based on their qubit-wise commutativity shows an $N$-fold reduction in the number of measurable groups.

quant-ph↗

arXiv2019

Exact and approximate symmetry projectors for the electronic structure problem on a quantum computer

Solving the electronic structure problem on a universal-gate quantum computer within the variational quantum eigensolver (VQE) methodology requires constraining the search procedure to a subspace defined by relevant physical symmetries. Ignoring symmetries results in convergence to the lowest eigenstate of the Fock space for the second quantized electronic Hamiltonian. Moreover, this eigenstate can be symmetry broken due to limitations of the wavefunction ansatz. To address this VQE problem, we introduce and assess methods of exact and approximate projection operators to irreducible eigen-subspaces of available physical symmetries. Feasibility of symmetry projection operators in the VQE framework is discussed, and their efficiency is compared with symmetry constraint optimization procedures. Generally, projectors introduce higher numbers of terms for VQE measurement compared to the constraint approach. On the other hand, the projection formalism improves accuracy of the variational wavefunction ansatz without introducing additional unitary transformations, which is beneficial for reducing depths of quantum circuits.

quant-ph↗

arXiv2019

Revising measurement process in the variational quantum eigensolver: Is it possible to reduce the number of separately measured operators?

Current implementations of the Variational Quantum Eigensolver (VQE) technique for solving the electronic structure problem involve splitting the system qubit Hamiltonian into parts whose elements commute within their single qubit subspaces. The number of such parts rapidly grows with the size of the molecule, this increases the uncertainty in the measurement of the energy expectation value because elements from different parts need to be measured independently. To address this problem we introduce a more efficient partitioning of the qubit Hamiltonian using fewer parts that need to be measured separately. The new partitioning scheme is based on two ideas: 1) grouping terms into parts whose eigenstates have a single-qubit product structure, and 2) devising multi-qubit unitary transformations for the Hamiltonian or its parts to produce less entangled operators. The first condition allows the new parts to be measured in the number of involved qubit consequential one-particle measurements. Advantages of the new partitioning scheme resulting in several-fold reduction of separately measured terms are illustrated on the H2 and LiH problems.

quant-ph↗

arXiv2019

On construction of projection operators

The problem of construction of projection operators on eigen-subspaces of symmetry operators is considered. This problem arises in many approximate methods for solving time-independent and time-dependent quantum problems, and its solution ensures proper physical symmetries in development of approximate methods. The projector form is sought as a function of symmetry operators and their eigenvalues characterizing the eigen-subspace of interest. This form is obtained in two steps: 1) identification of algebraic structures within a set of symmetry operators (e.g. groups and Lie algebras), and 2) construction of the projection operators for individual symmetry operators. The first step is crucial for efficient projection operator construction because it allows for using information on irreducible representations of the present algebraic structure. The discussed approaches have promise to stimulate further developments of variational approaches for electronic structure of strongly correlated systems and in quantum computing.

quant-ph↗