Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2610.00521

First-quantized quantum simulation with non-local potentials by matrix-product-state encoding

Kazuki Tsuoka, Taichi Kosugi, Masari Watanabe, Hirofumi Nishi, Yu-ichiro Matsushita

quant-phSubmitted Sep 30, 2026• Announced Oct 2, 2026• 0 citations
arXivHTMLPDF

Abstract

Quantum simulations based on first-quantized approaches can offer lower space and gate complexities than second-quantized approaches. The gate requirements for these simulations can be further reduced by employing pseudopotentials, a technique widely used in classical quantum chemistry calculations to decrease the number of simulated electrons. However, implementing pseudopotentials in quantum circuits remains challenging due to their non-local nature. In this study, we proposed a method to efficiently implement non-local potentials using matrix product state (MPS) encoding. MPS encoding allows us to efficiently implement a Trotter step operator with a circuit depth of $O(\mathrm{deg}(V_{\mathrm{loc}}) n^{\mathrm{deg}(V_{\mathrm{loc}})} + 8Lnn_{\mathrm{atom}}χ^2 + 2n^2)$, where $\mathrm{deg}(V_{\mathrm{loc}})$ is the degree of the polynomial local potential, $n_{\mathrm{atom}}$ is the number of atoms in the system, $L$ is the maximum number of projector functions across all atoms, and $n$ is the number of qubits used to represent the binary-encoded wave function in one-dimensional space. We demonstrated the effectiveness of this approach by applying it to a model of the ionization of a one-dimensional hydrogen atom driven by an ultrashort laser pulse. Our results showed that MPS encoding significantly reduces the required circuit depth compared to a direct unitary decomposition, providing a pathway for practical first-quantized simulations.

Related papers

Approximate encoding of quantum states using shallow circuits

Matan Ben-Dov, David Shnaiderov, Adi Makmal +1 more

Abstract Quantum algorithms and simulations often require the preparation of complex states through sequences of 2-qubit gates. For a generic quantum state, the number of required gates grows exponentially with the number of qubits, becoming unfeasible on near-term quantum devices. Here, we aim at creating an approximate encoding of the target state using a limited number of gates. As a first step, we consider a quantum state that is efficiently represented classically, such as a one-dimensional matrix product state. Using tensor network techniques, we develop and implement an efficient optimization algorithm that approaches the optimal implementation, requiring a polynomial number of iterations. We, next, consider the implementation of the proposed optimization algorithm directly on a quantum computer and overcome inherent barren plateaus by employing a local cost function. Our work offers a universal method to prepare target states using local gates and represents a significant improvement over known strategies.

arXiv:2207.00028•npj Quantum Information•Jul 2, 2024•26 citations

Complexity of Implementing Trotter Steps

Guang Hao Low, Yuan Su, Yu Tong +1 more

Quantum dynamics can be simulated on a quantum computer by exponentiating elementary terms from the Hamiltonian in a sequential manner.However, such an implementation of Trotter steps has gate complexity depending on the total Hamiltonian term number, comparing unfavorably to algorithms using more advanced techniques.We develop methods to perform faster Trotter steps with complexity sublinear in the number of terms.We achieve this for a class of Hamiltonians whose interaction strength decays with distance according to power law.Our methods include one based on a recursive block encoding and one based on an average-cost simulation, overcoming the normalization-factor barrier of these advanced quantum simulation techniques.We also realize faster Trotter steps when certain blocks of Hamiltonian coefficients have low rank.Combining with a tighter error analysis, we show that it suffices to use η 1/3 n 1/3 + n 2/3 /η 2/3 n 1+o(1) gates to simulate uniform electron gas with n spin orbitals and η electrons in second quantization in real space, asymptotically improving over the best previous work.We obtain an analogous result when the external potential of nuclei is introduced under the Born-Oppenheimer approximation.We prove a circuit lower bound when the Hamiltonian coefficients take a continuum range of values, showing that generic n-qubit two-local Hamiltonians with commuting terms require at least (n 2 ) gates to evolve with accuracy = (1/poly(n)) for time t = ( ).Our proof is based on a gateefficient reduction from the approximate synthesis of diagonal unitaries within the Hamming weight-2 subspace, which may be of independent interest.Our result thus suggests the use of Hamiltonian structural properties as both necessary and sufficient to implement Trotter steps with lower gate complexity.

arXiv:2211.09133•PRX Quantum•May 9, 2023•29 citations

Quantum Circuit Matrix Product State Ansatz for Large-Scale Simulations of Molecules

Yi Fan, Jie Liu, Zhenyu Li +1 more

As demonstrated in the density matrix renormalization group (DMRG) method, approximating many-body wave function of electrons using a matrix product state (MPS) is a promising way to solve electronic structure problems. The expressibility of an MPS is determined by the size of the matrices or, in other words, the bond dimension, which unfortunately may be required to be very large in quantum chemistry simulations. In this study, we propose to calculate the ground state energies of molecular systems by variationally optimizing quantum circuit MPS (QCMPS) with a relatively small number of qubits. It is demonstrated that with carefully chosen circuit structure and orbital localization scheme, QCMPS can reach a similar accuracy as that achieved in DMRG with an exponentially large bond dimension. QCMPS simulation of a linear hydrogen molecular chain with 50 orbitals can reach the chemical accuracy using only 6 qubits at a moderate circuit depth. These results suggest that QCMPS is a promising wave function ansatz in the variational quantum eigensolver algorithm for molecular systems.

arXiv:2301.06376•Journal of Chemical Theory and Computation•Jul 28, 2023•13 citations

Faster quantum simulation of quantum chemistry with tailored algorithms and Hamiltonians

Ian Kivlichan

With quantum computers on the horizon, it is essential to understand what problems they can be used to solve. One potential application is simulating quantum systems, in particular, systems with many interacting electrons. However, there is a disparity between the computational resources required to study these problems and the capabilities of modern experimental quantum devices. In this thesis, we make progress toward closing this gap by reducing the cost of quantum simulation of quantum chemistry on a number of fronts. In the first half, we develop three improved quantum simulation algorithms by approximating time evolution under the Hamiltonian using a truncated Taylor series rather than the Trotter-Suzuki decomposition. We do this first for the molecular electronic structure Hamiltonian in two representations, obtaining asymptotic improvements in terms of both the number of orbitals and the required precision. We also apply the truncated Taylor series paradigm to quantum simulation in real space, where we consider simulation of general wave functions discretized on a uniform grid. In the second half, by working with Hamiltonians tailored specifically to quantum algorithms, we demonstrate that Trotter steps of the electronic structure Hamiltonian can be simulated in exactly N depth and with only linear connectivity, where N is the number of spin-orbitals. This resolves the problem of non-locality in fermion-to-qubit mappings, and has wide-ranging applications for variational and phase estimation-based quantum simulation of quantum chemistry. We further study this algorithm on an error-corrected quantum computer using the surface code. Benchmarking it alongside other product formula approaches, we show that quantum simulation of scientifically important instances of the electronic structure problem that are classically intractable can be performed with only a few hundred thousand physical qubits and fewer than one billion costly fault-tolerant gates. Finally, we study the effect of using randomized Hamiltonians in iterative phase estimation, and how this allows us to reduce the costs of simulation as well as to exploit knowledge of the ground state in how we design our simulation algorithm.

•Jan 8, 2020•0 citations

Quantum Encoding of Structured Data with Matrix Product States

Josh Green, Jingbo B. Wang

The amplitude encoding of an arbitrary $n$-qubit state vector requires $Ω(2^n)$ gate operations, owing to the exponential dimension of the Hilbert space. We can, however, form dimensionality-reduced representations of quantum states using matrix product states (MPS). In this article, we illustrate that MPS techniques enable the preparation of quantum states representative of functions with complexity up to low-degree piecewise polynomials via shallow-depth quantum circuits with accuracy exceeding 99.99\%. We extend these results to the approximate amplitude encoding of pixel values. We showcase this approach by efficiently preparing a $128\times 128$ ChestMNIST medical image (https://medmnist.com/) on 14 qubits with fidelity exceeding 99.2\% on a circuit with a total depth of just 425 single-qubit rotation and CNOT gates.

arXiv:2502.16464•Feb 23, 2025•0 citations

Quantum simulation of realistic materials in first quantization using non-local pseudopotentials

Dominic William Berry, Nicholas C. Rubin, Ahmed O. Elnabawy +5 more

This paper improves and demonstrates the usefulness of the first quantized plane-wave algorithms for the quantum simulation of electronic structure. We describe our quantum algorithm for first quantized simulation that accurately includes pseudopotentials. We focus on the Goedecker-Tetter-Hutter pseudopotential, and despite its complicated form, we block encode the associated operator without significantly increasing the overall cost of quantum simulation. This is surprising since simulating the nuclear potential is much simpler without pseudopotentials, yet is still the bottleneck. We also generalize prior methods to enable the simulation of materials with non-cubic unit cells, which requires nontrivial modifications. Finally, we combine these techniques to estimate block-encoding costs for commercially relevant instances of heterogeneous catalysis (e.g. carbon monoxide adsorption) and compare to the quantum resources needed to simulate materials in second quantization. We conclude that for computational cells with many particles, first quantization often requires meaningfully less spacetime volume.

arXiv:2312.07654•npj Quantum Information•Dec 19, 2024•18 citations
Quantum simulations based on first-quantized approaches can offer lower space and gate complexities than second-quantized approaches. The gate requirements for these simulations can be further reduced by employing pseudopotentials, a technique widely used in classical quantum chemistry calculations to decrease the number of simulated electrons. However, implementing pseudopotentials in quantum circuits remains challenging due to their non-local nature. In this study, we proposed a method to efficiently implement non-local potentials using matrix product state (MPS) encoding. MPS encoding allows us to efficiently implement a Trotter step operator with a circuit depth of O(deg(Vloc)ndeg(Vloc)+8Lnnatomχ2+2n2)O(\mathrm{deg}(V_{\mathrm{loc}}) n^{\mathrm{deg}(V_{\mathrm{loc}})} + 8Lnn_{\mathrm{atom}}χ^2 + 2n^2)O(deg(Vloc​)ndeg(Vloc​)+8Lnnatom​χ2+2n2), where deg(Vloc)\mathrm{deg}(V_{\mathrm{loc}})deg(Vloc​) is the degree of the polynomial local potential, natomn_{\mathrm{atom}}natom​ is the number of atoms in the system, LLL is the maximum number of projector functions across all atoms, and nnn is the number of qubits used to represent the binary-encoded wave function in one-dimensional space. We demonstrated the effectiveness of this approach by applying it to a model of the ionization of a one-dimensional hydrogen atom driven by an ultrashort laser pulse. Our results showed that MPS encoding significantly reduces the required circuit depth compared to a direct unitary decomposition, providing a pathway for practical first-quantized simulations.
Abstract Quantum algorithms and simulations often require the preparation of complex states through sequences of 2-qubit gates. For a generic quantum state, the number of required gates grows exponentially with the number of qubits, becoming unfeasible on near-term quantum devices. Here, we aim at creating an approximate encoding of the target state using a limited number of gates. As a first step, we consider a quantum state that is efficiently represented classically, such as a one-dimensional matrix product state. Using tensor network techniques, we develop and implement an efficient optimization algorithm that approaches the optimal implementation, requiring a polynomial number of iterations. We, next, consider the implementation of the proposed optimization algorithm directly on a quantum computer and overcome inherent barren plateaus by employing a local cost function. Our work offers a universal method to prepare target states using local gates and represents a significant improvement over known strategies.
Quantum dynamics can be simulated on a quantum computer by exponentiating elementary terms from the Hamiltonian in a sequential manner.However, such an implementation of Trotter steps has gate complexity depending on the total Hamiltonian term number, comparing unfavorably to algorithms using more advanced techniques.We develop methods to perform faster Trotter steps with complexity sublinear in the number of terms.We achieve this for a class of Hamiltonians whose interaction strength decays with distance according to power law.Our methods include one based on a recursive block encoding and one based on an average-cost simulation, overcoming the normalization-factor barrier of these advanced quantum simulation techniques.We also realize faster Trotter steps when certain blocks of Hamiltonian coefficients have low rank.Combining with a tighter error analysis, we show that it suffices to use η 1/3 n 1/3 + n 2/3 /η 2/3 n 1+o(1) gates to simulate uniform electron gas with n spin orbitals and η electrons in second quantization in real space, asymptotically improving over the best previous work.We obtain an analogous result when the external potential of nuclei is introduced under the Born-Oppenheimer approximation.We prove a circuit lower bound when the Hamiltonian coefficients take a continuum range of values, showing that generic n-qubit two-local Hamiltonians with commuting terms require at least (n 2 ) gates to evolve with accuracy = (1/poly(n)) for time t = ( ).Our proof is based on a gateefficient reduction from the approximate synthesis of diagonal unitaries within the Hamming weight-2 subspace, which may be of independent interest.Our result thus suggests the use of Hamiltonian structural properties as both necessary and sufficient to implement Trotter steps with lower gate complexity.
As demonstrated in the density matrix renormalization group (DMRG) method, approximating many-body wave function of electrons using a matrix product state (MPS) is a promising way to solve electronic structure problems. The expressibility of an MPS is determined by the size of the matrices or, in other words, the bond dimension, which unfortunately may be required to be very large in quantum chemistry simulations. In this study, we propose to calculate the ground state energies of molecular systems by variationally optimizing quantum circuit MPS (QCMPS) with a relatively small number of qubits. It is demonstrated that with carefully chosen circuit structure and orbital localization scheme, QCMPS can reach a similar accuracy as that achieved in DMRG with an exponentially large bond dimension. QCMPS simulation of a linear hydrogen molecular chain with 50 orbitals can reach the chemical accuracy using only 6 qubits at a moderate circuit depth. These results suggest that QCMPS is a promising wave function ansatz in the variational quantum eigensolver algorithm for molecular systems.
With quantum computers on the horizon, it is essential to understand what problems they can be used to solve. One potential application is simulating quantum systems, in particular, systems with many interacting electrons. However, there is a disparity between the computational resources required to study these problems and the capabilities of modern experimental quantum devices. In this thesis, we make progress toward closing this gap by reducing the cost of quantum simulation of quantum chemistry on a number of fronts. In the first half, we develop three improved quantum simulation algorithms by approximating time evolution under the Hamiltonian using a truncated Taylor series rather than the Trotter-Suzuki decomposition. We do this first for the molecular electronic structure Hamiltonian in two representations, obtaining asymptotic improvements in terms of both the number of orbitals and the required precision. We also apply the truncated Taylor series paradigm to quantum simulation in real space, where we consider simulation of general wave functions discretized on a uniform grid. In the second half, by working with Hamiltonians tailored specifically to quantum algorithms, we demonstrate that Trotter steps of the electronic structure Hamiltonian can be simulated in exactly N depth and with only linear connectivity, where N is the number of spin-orbitals. This resolves the problem of non-locality in fermion-to-qubit mappings, and has wide-ranging applications for variational and phase estimation-based quantum simulation of quantum chemistry. We further study this algorithm on an error-corrected quantum computer using the surface code. Benchmarking it alongside other product formula approaches, we show that quantum simulation of scientifically important instances of the electronic structure problem that are classically intractable can be performed with only a few hundred thousand physical qubits and fewer than one billion costly fault-tolerant gates. Finally, we study the effect of using randomized Hamiltonians in iterative phase estimation, and how this allows us to reduce the costs of simulation as well as to exploit knowledge of the ground state in how we design our simulation algorithm.
The amplitude encoding of an arbitrary nnn-qubit state vector requires Ω(2n)Ω(2^n)Ω(2n) gate operations, owing to the exponential dimension of the Hilbert space. We can, however, form dimensionality-reduced representations of quantum states using matrix product states (MPS). In this article, we illustrate that MPS techniques enable the preparation of quantum states representative of functions with complexity up to low-degree piecewise polynomials via shallow-depth quantum circuits with accuracy exceeding 99.99\%. We extend these results to the approximate amplitude encoding of pixel values. We showcase this approach by efficiently preparing a 128×128128\times 128128×128 ChestMNIST medical image (https://medmnist.com/) on 14 qubits with fidelity exceeding 99.2\% on a circuit with a total depth of just 425 single-qubit rotation and CNOT gates.
This paper improves and demonstrates the usefulness of the first quantized plane-wave algorithms for the quantum simulation of electronic structure. We describe our quantum algorithm for first quantized simulation that accurately includes pseudopotentials. We focus on the Goedecker-Tetter-Hutter pseudopotential, and despite its complicated form, we block encode the associated operator without significantly increasing the overall cost of quantum simulation. This is surprising since simulating the nuclear potential is much simpler without pseudopotentials, yet is still the bottleneck. We also generalize prior methods to enable the simulation of materials with non-cubic unit cells, which requires nontrivial modifications. Finally, we combine these techniques to estimate block-encoding costs for commercially relevant instances of heterogeneous catalysis (e.g. carbon monoxide adsorption) and compare to the quantum resources needed to simulate materials in second quantization. We conclude that for computational cells with many particles, first quantization often requires meaningfully less spacetime volume.