Юрий Кругляк_Квантовое моделирование в квантовой химии на квантовых компьютерах_399_стр

DFT-2

Ryan Hatcher, Jorge A. Kittl, Christopher Bowen

A Method to Calculate Correlation for Density Functional Theory on a Quantum Processor

https://arxiv.org/abs/1903.05550 13.05.2019

An extension of the Variational Quantum Eigensolver (VQE) method is presented where a quantum computer generates an accurate exchange-correlation potential for a Density Functional Theory (DFT) simulation on classical hardware. The method enables efficient simulations of quantum systems by interweaving calculations on classical and quantum resources. DFT is implemented on classical hardware, which enables the efficient representation of and operation on quantum systems while being formally exact. The portion of the simulation operating on quantum hardware produces an accurate exchange-correlation potential but only requires relatively short depth quantum circuits.

DFT-3

Jun Yang, James Brown, James Daniel Whitfield

Measurement on quantum devices with applications to time-dependent density functional theory

https://arxiv.org/abs/1909.03078 06.09.2019

Quantum algorithms are touted as a way around some classically intractable problems such as the simulation of quantum mechanics. At the end of all quantum algorithms is a quantum measurement whereby classical data is extracted and utilized. In fact, any of the modern hybrid-classical approaches are essentially quantum measurements of states with short quantum circuit descriptions. Here, we compare and examine three methods of extracting the time-dependent one-particle probability density from a quantum simulation: direct Z-measurement, Bayesian phase estimation and harmonic inversion. We have tested these methods in the context of the potential inversion problem of time-dependent density functional theory. Our test results suggest that direct measurement is the preferable method. We also highlight areas where the other two methods may be useful and report on tests using Rigetti's quantum device. This study provides a starting point for imminent applications of quantum computing.

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DFT-4

Peiwei You, Jiyu Xu, Chao Lian at al

Quantum dynamics simulations: Combining path integral nuclear dynamics and real-time TDDFT

https://www.researchgate.net/publication/337371282; https://doi.org/10.1088/2516-1075/ab58fc 19.11.2019

We report a practical computational scheme for quantum electron-nuclear dynamic simulations, applicable to both finite (e.g. ozone) and periodic systems (e.g. graphene), using a combination of real-time time dependent density functional theory (rt-TDDFT) and ring polymer molecular dynamics (RPMD). This scheme could deal with quantum effects of nuclei beyond Ehrenfest dynamics in TDDFT simulations. We find that when nuclear quantum effects (NQEs) are taken into account, the atomic structure of ozone splits into normal states and cyclic states upon photoexcitation. NQEs broaden the electronic density of state and induce strong orbital couplings, leading to new nuclear trajectories and carrier dynamics different from classical simulations. We also observe a charge carrier redistribution accelerated by the quantum motions of carbon atoms in graphene, yielding an exponential decay with fast relaxation time. These developments and practices represent an advance in studying full quantum dynamics of electrons and nuclei from first-principles, towards a complete and predictive understanding of quantum interactions and dynamics in large molecules and complex materials at the microscopic level.

Новые методы и алгоритмы

NMA-1

Nicolas P. D. Sawaya, Tim Menke, Thi Ha Kyaw at al

Resource-e cient digital quantum simulation of d-level systems for photonic, vibrational, and spin-s Hamiltonians

https://arxiv.org/abs/1909.12847 27.09.2019

Simulation of quantum systems is expected to be one of the most important applications of quantum computing, with much of the theoretical work so far having focused on fermionic and spin-half systems. Here, we instead consider encodings of d-level quantum operators into multi-qubit operators, studying resource requirements for approximating operator exponentials by Trotterization. We primarily focus on spin-s and truncated bosonic operators in second quantization, observing desirable properties for approaches based on the Gray code, which to our knowledge has not been used in this context previously. After outlining a methodology for implementing an arbitrary encoding, we investigate the interplay between Hamming distances, sparsity structures, bosonic truncation, and other properties of local operators. Finally, we obtain resource counts for five common

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Hamiltonian classes used in physics and chemistry, counts that include the possibility of converting between encodings within a Trotter step. The most efficient encoding choice is heavily dependent on the application and highly sensitive to d, though clear trends are present. These operation count reductions are relevant for running algorithms on near-term quantum hardware, because the savings effectively decrease the required circuit depth. Results and procedures outlined in this work may be useful for simulating a broad class of Hamiltonians on qubit-based digital quantum computers.

NMA-2

Cristina Cîrstoiu, Zoë Holmes, Joseph Iosue et al

Variational Fast Forwarding for Quantum Simulation Beyond the Coherence Time

https://arxiv.org/abs/1910.04292v1 09.10.2019

Trotterization-based, iterative approaches to quantum simulation are restricted to simulation times less than the coherence time of the quantum computer, which limits their utility in the near term. Here, we present a hybrid quantum-classical algorithm, called Variational Fast Forwarding (VFF), for decreasing the quantum circuit depth of quantum simulations. VFF seeks an approximate diagonalization of a short-time simulation to enable longer-time simulations using a constant number of gates. Our error analysis provides two results: (1) the simulation error of VFF scales at worst linearly in the fast-forwarded simulation time, and (2) our cost function’s operational meaning as an upper bound on average-case simulation error provides a natural termination condition for VFF. We implement VFF for the Hubbard, Ising, and Heisenberg models on a simulator. Finally, we implement VFF on Rigetti’s quantum computer to show simulation beyond the coherence time.

NMA-3

Dominic W. Berry, Craig Gidney, Mario Motta, Jarrod R. McClean, Ryan Babbush

Qubitization of Arbitrary Basis Quantum Chemistry by Low Rank Factorization

https://arxiv.org/abs/1902.02134v3: FeMoco 01.10.2019

Recent work has dramatically reduced the gate complexity required to quantum simulate chemistry by using linear combinations of unitaries based methods to exploit structure in the plane wave basis Coulomb operator. Here, we show that one can achieve similar scaling even for arbitrary basis sets (which can be hundreds of times more compact than plane waves) by combining qubitized quantum walks with a low rank factorization of the Coulomb operator. We provide circuits for several variants of our algorithm (which all improve over the best scaling of prior methods) including one with e O(N3/2λ) T complexity, where N is number of orbitals and λ is the 1-norm

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of the chemistry Hamiltonian. We deploy our algorithms for simulating the FeMoco molecule (relevant to Nitrogen fixation) and obtain circuits requiring less surface code spacetime volume than prior quantum algorithms for this system, despite us using a larger and more accurate active space.

NMA-4

P. Q. Cruz, G. Catarina, R. Gautier, J. Fern´andez-Rossier

Optimizing quantum phase estimation for the simulation of Hamiltonian eigenstates

https://arxiv.org/abs/1910.06265v1 14.10.2019

We revisit quantum phase estimation algorithms for the purpose of obtaining the energy levels of many-body Hamiltonians and pay particular attention to the statistical analysis of its outputs. We introduce the estimation of the first trigonometric moment of the parent distribution associated with eigenstate inputs as a new post-processing direction. By showing how it connects with the unknown phase, we find that if used as its direct estimator it permits to match the accuracy of the standard majority rule with one qubit less, allowing for shallower algorithms. Moreover, we make evident this quantity can also be inverted to provide an unbiased estimator of the phase. We then use IBM Q hardware to carry out the digital quantum simulation of three toy models: a two-level system, a two-spin Ising model and a twosite Hubbard model at half-filling.

NMA-5

Giacomo Torlai, Guglielmo Mazzola, Giuseppe Carleo, Antonio Mezzacapo

Precise measurement of quantum observables with neural-network estimators https://arxiv.org/pdf/1910.07596: H2, LiH, BeH2

16.10.2019

The measurement precision of modern quantum simulators is intrinsically constrained by the limited set of measurements that can be e ciently implemented on hardware. This fundamental limitation is particularly severe for quantum algorithms where complex quantum observables are to be precisely evaluated. To achieve precise estimates with current methods, prohibitively large amounts of sample statistics are required in experiments. Here, we propose to reduce the measurement overhead by integrating artificial neural networks with quantum simulation platforms. We show that unsupervised learning of single-qubit data allows the trained networks to accommodate measurements of complex observables, otherwise costly using traditional post-processing techniques. The e ectiveness of this hybrid measurement protocol is demonstrated for quantum chemistry Hamiltonians using both synthetic and experimental data. Neural-network estimators attain high-precision measurements with a drastic reduction in the amount of sample statistics, without requiring additional quantum resources.

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NMA-6

Bryan O'Gorman, William J. Huggins, Eleanor G. Rieffel, K. Birgitta Whaley

Generalized swap networks for near-term quantum computing https://arxiv.org/abs/1905.05118

13.05.2019

The practical use of many types of near-term quantum computers requires accounting for their limited connectivity. One way of overcoming limited connectivity is to insert swaps in the circuit so that logical operations can be performed on physically adjacent qubits, which we refer to as solving the `routing via matchings' problem. We address the routing problem for families of quantum circuits defined by a hypergraph wherein each hyperedge corresponds to a potential gate. Our main result is that any unordered set of k-qubit gates on distinct k-qubit subsets of n logical qubits can be ordered and parallelized in O(nk−1) depth using a linear arrangement of n physical qubits; the construction is completely general and achieves optimal scaling in the

classes of problems for which our method is particularly useful. First, it applies to sets of mutually commuting gates, as in the (diagonal) phase separators of Quantum Alternating Operator Ansatz (Quantum Approximate Optimization Algorithm) circuits. For example, a single level of a QAOA circuit for Maximum Cut can be implemented in linear depth, and a single level for 3-SAT in quadratic depth. Second, it applies to sets of gates that do not commute but for which compilation efficiency is the dominant criterion in their ordering. In particular, it can be adapted to Trotterized time-evolution of fermionic Hamiltonians under the Jordan-Wigner transformation, and also to non-standard mixers in QAOA. Using our method, a single Trotter step of the electronic structure Hamiltonian in an arbitrary basis of n orbitals can be done in O(n3) depth while a Trotter step of the unitary coupled cluster singles and doubles method can be implemented in O(n2η) depth, where η is the number of electrons.

NMA-7

Róbert Izsák

Single reference coupled cluster methods for computing excitation energies in large molecules: The efficiency and accuracy of approximations https://onlinelibrary.wiley.com/doi/full/10.1002/wcms.1445; https://doi.org/10.1002/wcms.1445

12.09.2019

While methodological developments in the last decade made it possible to compute coupled cluster (CC) energies including excitations up to a perturbative triples correction for molecules containing several hundred atoms, a similar breakthrough has not yet been reported for excited state computations. Accurate CC methods for excited states are still expensive, although some promising candidates for an efficient

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and accurate excited state CC method have emerged recently. This review examines the various approximation schemes with particular emphasis on their performance for excitation energies and summarizes the best state of the art results which may pave the way for a robust excited state method applicable to molecules of hundreds of atoms. Among these, special attention will be given to exploiting the techniques of similarity transformation, perturbative approximations as well as integral decomposition, local and embedding techniques within the equation of motion CC framework.

NMA-8

Yuji Mochizuki, Koji Okuwaki, Takumi Kato, Yuichiro Minato

Reduction of Orbital Space for Molecular Orbital Calculations with Quantum Computation Simulator for Educations https://chemrxiv.org/articles/Reduction_of_Orbital_Space_for_Molecular_Orbital_C alculations_with_Quantum_Computation_Simulator_for_Educations/9863810:

H2, LiH, NaH

22.09.2019

Recently, the molecular orbital (MO) calculations with quantum computations (QCs) have attracted considerable interest. The cost of QCs highly depends on the number of qubits even on quantum simulators. Besides the frozen-core restriction for the occupied MO space, we have used the pseudo-natural orbital derived from the second-order Moller-Plesset perturbation (MP2) theory for the virtual MO space. A preliminary test on the LiH molecule (STO-3G basis) showed acceleration by a factor larger than 500 for MO-QC with the Blueqat simulator, where the required time was 72 s per solution.

NMA-9

Nikolaj Moll, Andreas Fuhrer, Peter Staar, Ivano Tavernelli

Optimizing qubit resources for quantum chemistry simulations in second quantization on a quantum computer

https://arxiv.org/abs/1510.04048: H2 11.05.2016

Quantum chemistry simulations on a quantum computer suffer from the overhead needed for encoding the fermionic problem in a bosonic system of qubits. By exploiting the block diagonality of a fermionic Hamiltonian, we show that the number of required qubits can be reduced by a factor of two or more. There is no need to go into the basis of the Hilbert space for this reduction because all operations can be performed in the operator space. The scheme is conceived as a precomputational step that would be performed on a classical computer prior to the actual quantum simulation. We apply this scheme to reduce the number of qubits necessary to simulate both the Hamiltonian of the two-site Fermi-Hubbard model and

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the hydrogen molecule. Both quantum systems can then be simulated with a twoqubit quantum computer.

NMA-10

Ammar Daskin, Teng Bian, Rongxin Xia, Sabre Kais

Context aware quantum simulation of a matrix stored in quantum memory https://arxiv.org/abs/1904.01259: H2

02.04.2019

In this paper a storage method and a context-aware circuit simulation idea are presented for the sum of block diagonal matrices. Using the design technique for a generalized circuit for the Hamiltonian dynamics through the truncated series, we generalize the idea to (0-1) matrices and discuss the generalization for the real matrices. The presented circuit requires O(n) number of quantum gates and yields the correct output with the success probability depending on the number of elements: for matrices with poly(n), the success probability is 1/poly(n). Since the operations on the circuit are controlled by the data itself, the circuit can be considered as a context aware computing gadget. In addition, it can be used in variational quantum eigensolver and in the simulation of molecular Hamiltonians.

NMA-11

Jarrod R. McClean, Fabian M. Faulstich, Qinyi Zhu, Bryan O'Gorman et al

Discontinuous Galerkin discretization for quantum simulation of chemistry https://arxiv.org/abs/1909.00028: H2, H4, H6, H8

30.08.2019

Methods for electronic structure based on Gaussian and molecular orbital discretizations offer a well established, compact representation that forms much of the foundation of correlated quantum chemistry calculations on both classical and quantum computers. Despite their ability to describe essential physics with relatively few basis functions, these representations can suffer from a quartic growth of the number of integrals. Recent results have shown that, for some quantum and classical algorithms, moving to representations with diagonal two-body operators can result in dramatically lower asymptotic costs, even if the number of functions required increases significantly. We introduce a way to interpolate between the two regimes in a systematic and controllable manner, such that the number of functions is minimized while maintaining a block diagonal structure of the two-body operator and desirable properties of an original, primitive basis. Techniques are analyzed for leveraging the structure of this new representation on quantum computers. Empirical results for hydrogen chains suggest a scaling improvement from O(N4.5) in molecular orbital representations to O(N2.6) in our representation for quantum evolution in a faulttolerant setting, and exhibit a constant factor crossover at 15 to 20 atoms. Moreover, we test these methods using modern density matrix renormalization group methods

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classically, and achieve excellent accuracy with respect to the complete basis set limit with a speedup of 1 – 2 orders of magnitude with respect to using the primitive or Gaussian basis sets alone. These results suggest our representation provides significant cost reductions while maintaining accuracy relative to molecular orbital or strictly diagonal approaches for modest-sized systems in both classical and quantum computation for correlated systems.

NMA-12

Sam McArdle, Tyson Jones, Suguru Endo, Ying Li, Simon C. Benjamin, Xiao Yuan

Variational ansatz-based quantum simulation of imaginary time evolution https://www.nature.com/articles/s41534-019-0187-2: npj Quantum Information, 5, Article number: 75 (2019): H2, LiH

05.08.2019

Imaginary time evolution is a powerful tool for studying quantum systems. While it is possible to simulate with a classical computer, the time and memory requirements generally scale exponentially with the system size. Conversely, quantum computers can efficiently simulate quantum systems, but not non-unitary imaginary time evolution. We propose a variational algorithm for simulating imaginary time evolution on a hybrid quantum computer. We use this algorithm to find the groundstate energy of many-particle systems; specifically molecular hydrogen and lithium hydride, finding the ground state with high probability. Our method can also be applied to general optimisation problems and quantum machine learning. As our algorithm is hybrid, suitable for error mitigation and can exploit shallow quantum circuits, it can be implemented with current quantum computers.

NMA-13

Jarrod R. McClean, Zhang Jiang, Nicholas C. Rubin, Ryan Babbush, Hartmut Neven

Decoding quantum errors with subspace expansions https://arxiv.org/abs/1903.05786: H2

14.03.2019

With the rapid developments in quantum hardware comes a push towards the first practical applications on these devices. While fully fault-tolerant quantum computers may still be years away, one may ask if there exist intermediate forms of error correction or mitigation that might enable practical applications before then. In this work, we consider the idea of post-processing error decoders using existing quantum codes, which are capable of mitigating errors on encoded logical qubits using classical post-processing with no complicated syndrome measurements or additional qubits beyond those used for the logical qubits. This greatly simplifies the experimental exploration of quantum codes on near-term devices, removing the need for locality of syndromes or fast feed-forward, allowing one to study performance aspects of codes on real devices. We provide a general construction equipped with a

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simple stochastic sampling scheme that does not depend explicitly on a number of terms that we extend to approximate projectors within a subspace. This theory then allows one to generalize to the correction of some logical errors in the code space, correction of some physical unencoded Hamiltonians without engineered symmetries, and corrections derived from approximate symmetries. In this work, we develop the theory of the method and demonstrate it on a simple example with the perfect [[5,1,3]] code, which exhibits a pseudo-threshold of p ≈ 0.50 under a single qubit depolarizing channel applied to all qubits. We also provide a demonstration under the application of a logical operation and performance on an unencoded hydrogen molecule, which exhibits a significant improvement over the entire range of possible errors incurred under a depolarizing channel.

NMA-14

Suguru Endo, Iori Kurata, Yuya O. Nakagawa

Calculation of the Green's function on near-term quantum computers https://arxiv.org/abs/1909.12250

26.09.2019

The Green's function plays a crucial role to study the nature of quantum many-body systems, especially strongly-correlated systems. Although the development of quantum computers in near term is expected to enable us to compute energy spectrum and energy eigenstates of such classically-intractable systems, the methods to simulate the Green's function with near-term quantum algorithms have not been proposed yet. Here, we propose two methods to calculate the Green's function of a given Hamiltonian on near-term quantum computers. The first one makes use of the variational dynamics simulation of quantum systems and computes the dynamics of the Green's function in real time directly. The second one utilizes the Lehmann representation of the Green's function and a method which calculates excited states of the Hamiltonian. Both methods require shallow quantum circuits and are compatible with near-term quantum computers. We numerically simulated the Green's function of the Hubbard model and demonstrated the validity of our proposals.

NMA-15

Jiaying Yang, Ahsan Javed Awan, Gemma Vall-Llosera

Support Vector Machines on Noisy Intermediate Scale Quantum Computers https://arxiv.org/abs/1909.11988

26.09.2019

Support vector machine (SVM) algorithms are considered essential for the implementation of automation in a radio access network. Specifically, they are critical in the prediction of the quality of user experience for video streaming based on device and network-level metrics. Quantum SVM is the quantum analogue of the classical SVM algorithm, which utilizes the properties of quantum computers to

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speed up the algorithm exponentially. In this work, we derive an optimized preprocessing unit for a quantum SVM that allows classifying any two-dimensional datasets that are linearly separable. We further provide a result readout method of the kernel matrix generation circuit to avoid quantum tomography that, in turn, reduces the quantum circuit depth. We also derive a quantum SVM system based on an optimized HHL quantum circuit with reduced circuit depth.

NMA-16

Sumsam Ullah Khan, Ahsan Javed Awan, Gemma Vall-Llosera

K-Means Clustering on Noisy Intermediate Scale Quantum Computers https://arxiv.org/abs/1909.12183

26.09.2019

Real-time clustering of big performance data generated by the telecommunication networks requires domain-specific high performance compute infrastructure to detect anomalies. In this paper, we evaluate noisy intermediate-scale quantum (NISQ) computers characterized by low decoherence times, for K-means clustering and propose three strategies to generate shorter-depth quantum circuits needed to overcome the limitation of NISQ computers. The strategies are based on exploiting; i) quantum interference, ii) negative rotations and iii) destructive interference. By comparing our implementations on IBMQX2 machine for representative data sets, we show that NISQ computers can solve the K-means clustering problem with the same level of accuracy as that of classical computers.

NMA-17

Zhang Jiang, Amir Kalev, Wojciech Mruczkiewicz, Hartmut Neven

Optimal fermion-to-qubit mapping via ternary trees with applications to reduced quantum states learning

https://arxiv.org/abs/1910.10746 23.10.2019

We introduce a fermion-to-qubit mapping defined on ternary trees, where any single

operator acting nontrivially on log3(2n+1) qubits. The mapping has a simple structure and is optimal in the sense that it is impossible to construct Pauli operators

Majorana operator on an n-mode fermionic system is mapped to a multi-qubit Pauli

in any fermion-to-qubit mapping acting nontrivially on less than log3(2n) qubits on average. We apply it to the problem of learning k-fermion reduced density matrix (RDM), a problem relevant in various quantum simulation applications. We show that using the ternary-tree mapping one can determine the elements of all k-fermion RDMs, to precision ϵ, by repeating a single quantum circuit for (2n+1)kϵ−2 times. This result is based on a method we develop here that allows one to determine the elements of all k-qubit RDMs, to precision ϵ, by repeating a single quantum circuit for 3kϵ−2 times, independent of the system size. This improves over existing schemes for determining qubit RDMs.

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