Significant advances have been made in the study of quantum advantage both in theory and experiment, although these have mostly been limited to artificial setups. In this work, we extend the scope to address quantum advantage in tasks relevant to chemistry and physics. Specifically, we consider the unitary cluster Jastrow (UCJ) ansatz-a variant of the unitary coupled cluster ansatz, which is widely used to solve the electronic structure problem on quantum computers-to show that sampling from the output distributions of quantum circuits implementing the UCJ ansatz is likely to be classically hard. More specifically, we show that there exist UCJ circuits for which classical simulation of sampling cannot be performed in polynomial time, under a reasonable complexity-theoretical assumption that the polynomial hierarchy does not collapse. Our main contribution is to show that a class of UCJ circuits can be used to perform arbitrary instantaneous quantum polynomial-time (IQP) computations, which are already known to be classically hard to simulate under the same complexity assumption. As a side result, we also show that UCJ equipped with post-selection can generate the class post-BQP. Our demonstration, worst-case nonsimulatability of UCJ, would potentially imply quantum advantage in quantum algorithms for chemistry and physics using unitary coupled cluster type ansatzes, such as the variational quantum eigensolver and quantum-selected configuration interaction.

Quantum-selected configuration interaction (QSCI) is a novel quantum-classical hybrid algorithm for quantum chemistry calculations. This method identifies electron configurations having large weights for the target state using quantum devices and allows CI calculations to be performed with the selected configurations on classical computers. In principle, the QSCI algorithm can take advantage of the ability to handle large configuration spaces while reducing the negative effects of noise on the calculated values. At present, QSCI calculations are limited by qubit noise during the input state preparation and measurement process, restricting them to small active spaces. These limitations make it difficult to perform calculations with quantitative accuracy. The present study demonstrates a computational scheme based on multireference perturbation theory calculations on a classical computer, using the QSCI wavefunction as a reference. This method was applied to ground and excited state calculations for two typical aromatic molecules, naphthalene and tetracene. The incorporation of the perturbation treatment was found to provide improved accuracy. Extension of the reference space based on the QSCI-selected configurations as a means of further improvement was also investigated.

We present a comprehensive study of the commute time kernel method via the effective resistance framework analyzing the quantum complexity of the originally classical approach. Our study reveals that while there is a trade-off between accuracy and computational complexity, significant improvements can be achieved in terms of runtime efficiency without substantially compromising on precision. Our investigation highlights a notable quantum speedup in calculating the kernel, which offers a quadratic improvement in time complexity over classical approaches in certain instances. In addition, we introduce methodical improvements over the original work on the commute time kernel and provide empirical evidence suggesting the potential reduction of kernel queries without significant impact on result accuracy. Benchmarking our method on several chemistry-based datasets: AIDS, NCL1, PTC−MR, MUTAG, PROTEINS - data points previously unexplored in existing literature, shows that while not always the most accurate, it excels in time efficiency. This makes it a compelling alternative for applications where computational speed is crucial. Our results highlight the balance between accuracy, computational complexity, and speedup offered by quantum computing, promoting further research into efficient algorithms for kernel methods and their applications in chemistry-based datasets.

Expectation value estimation is ubiquitous in quantum algorithms. The expectation value of a Hamiltonian, which is essential in various practical applications, is often estimated by measuring a large number of Pauli strings on quantum computers and performing classical post-processing. In the case of n-qubit molecular Hamiltonians in quantum chemistry calculations, it is necessary to evaluate O(n⁴) Pauli strings, requiring a large number of measurements for accurate estimation. To reduce the measurement cost, we assess an existing idea that uses two copies of an n-qubit quantum state of interest and coherently measures them in the Bell basis, which enables the simultaneous estimation of the absolute values of expectation values of all the n-qubit Pauli strings. We numerically investigate the efficiency of energy estimation for molecular Hamiltonians of up to 12 qubits. The results show that, when the target precision is no smaller than tens of milli-Hartree, this method requires fewer measurements than conventional sampling methods. This suggests that the method may be useful for many applications that rely on expectation value estimation of Hamiltonians and other observables as well when moderate precision is sufficient.

Quantum-selected configuration interaction (QSCI) utilizes an input quantum state on a quantum device to select important bases (electron configurations in quantum chemistry) which define a subspace where we diagonalize a target Hamiltonian, i.e., perform selected configuration interaction, on classical computers. Previous proposals for preparing a good input state, which is crucial for the quality of QSCI, based on optimization of quantum circuits may suffer from optimization difficulty and require many runs of the quantum device. Here we propose using a time-evolved state by the target Hamiltonian (for some initial state) as an input of QSCI. Our proposal is based on the intuition that the time evolution by the Hamiltonian creates electron excitations of various orders when applied to the initial state. We numerically investigate the accuracy of the energy obtained by the proposed method for quantum chemistry Hamiltonians describing electronic states of small molecules. Numerical results reveal that our method can yield sufficiently accurate ground-state energies for the investigated molecules. Our proposal provides a systematic and optimization-free method to prepare the input state of QSCI and could contribute to practical applications of quantum computers to quantum chemistry calculations.

In this review, we give an overview of the proposed applications in the early-FTQC (EFTQC) era. Starting from the error correction architecture for EFTQC device, we first review the recently developed space-time efficient analogue rotation (STAR) architecture, which is a partially fault-tolerant error correction architecture. Then, we review the requirements of an EFTQC algorithm. In particular, the class of ground state energy estimation (GSEE) algorithm known as the statistical phase estimation algorithm (SPE) is studied. We especially cast our attention on two SPE-type algorithms, the step-function filter-based variant by Lin and Tong (LT22) and Gaussian Filter. Based on the latter, we introduce the Gaussian Fitting algorithm, which uses an alternative post-processing procedure compared to the previous study. Finally, we systematically simulate the aforementioned algorithms and Variational Quantum Eigensolver (VQE) using the 1-uCJ ansatz with different shot counts. Most importantly, we perform noisy simulations based on the STAR architecture. We find that for estimating the ground state energy of the 4-qubit H2 Hamiltonian in the STO-3G basis, SPE becomes more advantageous over VQE when the physical error rate is sufficiently low.

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.

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.

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.

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.