Orbital-rotated Fermi-Hubbard model as a benchmarking problem for quantum chemistry with the exact solution

Evaluating the relative performance of different quantum algorithms for quantum computers is of great significance in the research of quantum algorithms. In this study, we consider the problem of quantum chemistry, which is considered one of the important applications of quantum algorithms. While evaluating these algorithms in systems with a large number of qubits is essential to see the scalability of the algorithms, the solvable models usually used for such evaluations typically have a small number of terms compared to the molecular Hamiltonians used in quantum chemistry. The large number of terms in molecular Hamiltonians is a major bottleneck when applying quantum algorithms to quantum chemistry. Various methods are being considered to address this problem, highlighting its importance in developing quantum algorithms for quantum chemistry. Based on these points, a solvable model with a number of terms comparable to the molecular Hamiltonian is essential to evaluate the performance of such algorithms. In this paper, we propose a set of exactly solvable Hamiltonians that has a comparable order of terms with molecular Hamiltonians by applying a spin-involving orbital rotation to the one-dimensional Fermi-Hubbard Hamiltonian. We verify its similarity to the molecular Hamiltonian from some prospectives and investigate whether the difficulty of calculating the ground-state energy changes before and after orbital rotation by applying the density matrix renormalization group up to 24 sites corresponding to 48 qubits. This proposal would enable proper evaluation of the performance of quantum algorithms for quantum chemistry, serving as a guiding framework for algorithm development.


NISQ deviceQuantum chemistry
Ryota Kojima, Masahiko Kamoshita, Keita Kanno

ADAPT-QSCI: Adaptive Construction of Input State for Quantum-Selected Configuration Interaction

We present a quantum-classical hybrid algorithm for calculating the ground state and its energy of the quantum many-body Hamiltonian by proposing an adaptive construction of a quantum state for the quantum-selected configuration interaction (QSCI) method. QSCI allows us to select important electronic configurations in the system to perform CI calculation (subspace diagonalization of the Hamiltonian) by sampling measurement for a proper input quantum state on a quantum computer, but how we prepare a desirable input state has remained a challenge. We propose an adaptive construction of the input state for QSCI in which we run QSCI repeatedly to grow the input state iteratively. We numerically illustrate that our method, dubbed ADAPT-QSCI, can yield accurate ground-state energies for small molecules, including a noisy situation for eight qubits where error rates of two-qubit gates and the measurement are both as large as 1%. ADAPT-QSCI serves as a promising method to take advantage of current noisy quantum devices and pushes forward its application to quantum chemistry.


Quantum chemistryNISQ deviceJoint research
Yuya O. Nakagawa, Masahiko Kamoshita, Wataru Mizukami, Shotaro Sudo, Yu-ya Ohnishi

Bias-preserving computation with the bit-flip code

We explore the feasibility of fault-tolerant quantum computation using the bit-flip repetition code in a biased noise channel where only the bit-flip error can occur. While several logic gates can potentially produce phase-flip errors even in such a channel, we propose bias-preserving implementation of S, H, CZ, and Rz gates. We demonstrate that our scheme improves the computational precision in several tasks such as the time evolution of quantum systems and variational quantum eigensolver.


Quantum chemistryFault-tolerant quantum computer
Shoichiro Tsutsui, Keita Kanno

Classical variational optimization of PREPARE circuit for quantum phase estimation of quantum chemistry Hamiltonians

We propose a method for constructing PREPARE circuits for quantum phase estimation of a molecular Hamiltonian in quantum chemistry by using variational optimization of quantum circuits solely on classical computers. The PREPARE circuit generates a quantum state which encodes the coefficients of the terms in the Hamiltonian as probability amplitudes and plays a crucial role in the state-of-the-art efficient implementations of quantum phase estimation. We employ the automatic quantum circuit encoding algorithm [Shirakawa et al., arXiv:2112.14524] to construct PREPARE circuits, which requires classical simulations of quantum circuits of O(logN) qubits with N being the number of qubits of the Hamiltonian. The generated PREPARE circuits do not need any ancillary qubit. We demonstrate our method by investigating the number of T-gates of the obtained PREPARE circuits for quantum chemistry Hamiltonians of various molecules, which shows a constant-factor reduction compared to previous approaches that do not use ancillary qubits. Since the number of available logical qubits and T gates will be limited at the early stage of the fault-tolerant quantum computing, the proposed method is particularly of use for performing the quantum phase estimation with such limited capability.


Fault-tolerant quantum computerQuantum chemistryCondensed matter physics
Hayata Morisaki, Kosuke Mitarai, Keisuke Fujii, Yuya O. Nakagawa

Accelerated variational quantum eigensolver with joint Bell measurement

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 non-commutativity 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 joint Bell measurement VQE (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.


NISQ deviceQuantum chemistryCondensed matter physics
Chenfeng Cao, Hiroshi Yano, Yuya O. Nakagawa

Computational analysis of chemical reactions using a variational quantum eigensolver algorithm without specifying spin multiplicity

The analysis of a chemical reaction along the ground state potential energy surface in conjunction with an unknown spin state is challenging because electronic states must be separately computed several times using different spin multiplicities to find the lowest energy state. However, in principle, the ground state could be obtained with just a single calculation using a quantum computer without specifying the spin multiplicity in advance. In the present work, ground state potential energy curves for PtCO were calculated as a proof-of-concept using a variational quantum eigensolver (VQE) algorithm. This system exhibits a singlet-triplet crossover as a consequence of the interaction between Pt and CO. VQE calculations using a statevector simulator were found to converge to a singlet state in the bonding region, while a triplet state was obtained at the dissociation limit. Calculations performed using an actual quantum device provided potential energies within ±2 kcal/mol of the simulated energies after adopting error mitigation techniques. The spin multiplicities in the bonding and dissociation regions could be clearly distinguished even in the case of a small number of shots. The results of this study suggest that quantum computing can be a powerful tool for the analysis of the chemical reactions of systems for which the spin multiplicity of the ground state and variations in this parameter are not known in advance.


Quantum chemistryNISQ deviceJoint research
Soichi Shirai, Hokuto Iwakiri, Keita Kanno, Takahiro Horiba, Keita Omiya, Hirotoshi Hirai, Sho Koh

Quantum-Selected Configuration Interaction: classical diagonalization of Hamiltonians in subspaces selected by quantum computers

We propose quantum-selected configuration interaction (QSCI), a class of hybrid quantum-classical algorithms for calculating the ground- and excited-state energies of many-electron Hamiltonians on noisy quantum devices. Suppose that an approximate ground state can be prepared on a quantum computer either by variational quantum eigensolver or by some other method. Then, by sampling the state in the computational basis, which is hard for classical computation in general, one can identify the electron configurations that are important for reproducing the ground state. The Hamiltonian in the subspace spanned by those important configurations is diagonalized on classical computers to output the ground-state energy and the corresponding eigenvector. The excited-state energies can be obtained similarly. The result is robust against statistical and physical errors because the noisy quantum devices are used only to define the subspace, and the resulting ground-state energy strictly satisfies the variational principle even in the presence of such errors. The expectation values of various other operators can also be estimated for obtained eigenstates with no additional quantum cost, since the explicit eigenvectors in the subspaces are known. We verified our proposal by numerical simulations, and demonstrated it on a quantum device for an 8-qubit molecular Hamiltonian. The proposed algorithms are potentially feasible to tackle some challenging molecules by exploiting quantum devices with several tens of qubits, assisted by high-performance classical computing resources for diagonalization.


Quantum chemistryMaterial scienceNISQ device
Keita Kanno, Masaya Kohda, Ryosuke Imai, Sho Koh, Kosuke Mitarai, Wataru Mizukami, Yuya O. Nakagawa

Almost optimal measurement scheduling of molecular Hamiltonian via finite projective plane

We propose an efficient and almost optimal scheme for measuring molecular Hamiltonians in quantum chemistry on quantum computers, which requires 2N2 distinct measurements in the leading order with N being the number of molecular orbitals. It achieves the state-of-the-art by improving a previous proposal by Bonet-Monroig et al. [Phys. Rev. X 10, 031064 (2020)] which exhibits 17N2/6 scaling in the leading order. We develop a novel method based on a finite projective plane to construct sets of simultaneously-measurable operators contained in molecular Hamiltonians. Each measurement only requires a depth-O(N) circuit consisting of O(N2) one- and two-qubit gates under the Jordan-Wigner and parity mapping, assuming the linear connectivity of qubits on quantum hardwares. Because evaluating expectation values of molecular Hamiltonians is one of the major bottlenecks in the applications of quantum devices to quantum chemistry, our finding is expected to accelerate such applications.


NISQ deviceQuantum chemistry
Wataru Inoue, Koki Aoyama, Yusuke Teranishi, Keita Kanno, Yuya O. Nakagawa, Kosuke Mitarai

Clifford+T-gate Decomposition with Limited Number of T gates, its Error Analysis, and Performance of Unitary Coupled Cluster Ansatz in Pre-FTQC Era

Fault-tolerant quantum computation (FTQC) is essential to robustly implement quantum algorithms and thus to maximize advantages of quantum computers. In this context, a quantum circuit is decomposed into universal gates that can be fault-tolerantly implemented, for example, Clifford+T gates. Here, T gate is usually regarded as an essential resource for quantum computation because its action cannot be simulated efficiently on classical computers. Practically, it is highly likely that only a limited number of T gates are available in the near future due to its experimental difficulty of fault-tolerant implementation. In this paper, considering this Pre-FTQC era, we investigate Clifford+T decomposition with a limited budget of T gates and propose a new model of the error of such decomposition. More concretely, we propose an algorithm to generate the most accurate Clifford+T-gate decomposition of a given single-qubit rotation gate with a fixed number of T gates. We also propose to model the error of Clifford+T decomposition using well-known depolarizing noise by considering the average of the effects of the error. We numerically verified our model successfully explains the decomposition error for a wide variety of molecules using our decomposition algorithm. Thus, we shed light on a first-stage application of quantum computers from a practical point of view and fuel further research towards what quantum computation can achieve in the upcoming future.


Quantum chemistryNISQ deviceFault-tolerant quantum computer
Kohdai Kuroiwa, Yuya O. Nakagawa

Quantum Car-Parrinello Molecular Dynamics: A Cost-Efficient Molecular Simulation Method on Near-Term Quantum Computers

In this paper, we propose a cost-reduced method for finite-temperature molecular dynamics on a near-term quantum computer, Quantum Car-Parrinello molecular dynamics (QCPMD). One of the most promising applications of near-term quantum computers is quantum chemistry. It has been expected that simulations of molecules via molecular dynamics can be also efficiently performed on near-term quantum computers by applying a promising near-term quantum algorithm of the variational quantum eigensolver (VQE). However, this method may demand considerable computational costs to achieve a sufficient accuracy, and otherwise, statistical noise can significantly affect the results. To resolve these problems, we invent an efficient method for molecular time evolution inspired by Car-Parrinello method. In our method, parameters characterizing the quantum state evolve based on equations of motion instead of being optimized. Furthermore, by considering Langevin dynamics, we can make use of the intrinsic statistical noise. As an application of QCPMD, we propose an efficient method for vibrational frequency analysis of molecules in which we can use the results of the molecular dynamics calculated by QCPMD. Numerical experiments show that our method can precisely simulate the Langevin dynamics at the equilibrium state, and we can successfully predict a given molecule's eigen frequencies. Furthermore, in the numerical simulation, our method achieves a substantial cost reduction compared with molecular dynamics using the VQE. Our method achieves an efficient computation without using widely employed method of the VQE. In this sense, we open up a new possibility of molecular dynamics on near-term quantum computers. We expect our results inspire further invention of efficient near-term quantum algorithms for simulation of molecules.


NISQ deviceJoint researchQuantum chemistry
Kohdai Kuroiwa, Takahiro Ohkuma, Hirokazu Sato, Ryosuke Imai