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:2404.10050

The Cost of Entanglement Renormalization on a Fault-Tolerant Quantum Computer

Joshua Job, Isaac H. Kim, Johnston, Eric, Steve Adachi

Published Apr 15, 2024• 0 citations
arXivHTMLPDF

Abstract

We perform a detailed resource estimate for the prospect of using deep entanglement renormalization ansatz (DMERA) on a fault-tolerant quantum computer, focusing on the regime in which the target system is large. For probing a relatively large system size ($64\times 64$), we observe up to an order of magnitude reduction in the number of qubits, compared to the approaches based on quantum phase estimation (QPE). We discuss two complementary strategies to measure the energy. The first approach is based on a random sampling of the local terms of the Hamiltonian, requiring $\mathcal{O}(1/ε^2)$ invocations of quantum circuits, each of which have depth of at most $\mathcal{O}(\log N)$, where $ε$ is the relative precision in the energy and $N$ is the system size. The second approach is based on a coherent estimation of the expectation value of observables averaged over space, which achieves the Heisenberg scaling while incurring only a logarithmic cost in the system size. For estimating the energy per site of $ε$, $\mathcal{O}\left(\frac{\log N}ε \right)$ $T$ gates and $\mathcal{O}\left(\log N \right)$ qubits suffice. The constant factor of the leading contribution is shown to be determined by the depth of the DMERA circuit, the gates used in the ansatz, and the periodicity of the circuit. We also derive tight bounds on the variance of the energy gradient, assuming the gates are random Pauli rotations.

DOI
10.48550/arxiv.2404.10050

Related papers

Assessment of quantum phase estimation protocols for early fault-tolerant quantum computers

Jacob S. Nelson, Andrew Baczewski

We compare several quantum phase estimation (QPE) protocols intended for early fault-tolerant (EFT) quantum computers in the context of models of their implementations on a surface code architecture. We estimate the logical and physical resources required to use these protocols to calculate the ground-state energy of molecular hydrogen in a minimal basis with error below ${10}^{\ensuremath{-}3}$ atomic units, in the presence of depolarizing logical errors. Accounting for the overhead of rotation synthesis and magic state distillation, we find that the total $T$-gate counts do not vary significantly among the EFT QPE protocols at fixed state overlap. In addition to reducing the number of ancilla qubits and circuit depth, the noise robustness of the EFT protocols can be leveraged to reduce resource requirements below those of textbook QPE, realizing an approximately 300-fold reduction in aggregate computational volume in some cases. We also compare resource requirements for QPE on Trotterized time evolution and qubitized walk operators, for this same minimal exemplar. We estimate the costs of these two approaches to be comparable, with the Trotter approach requiring slightly fewer physical qubits and the qubitized approach requiring slightly fewer code cycles. Even so, our estimates are well beyond the scale of existing early fault-tolerance demonstrations.

Physical Review A•Oct 16, 2024•7 citations

Variational quantum simulation of the critical Ising model with symmetry averaging

Troy J. Sewell, Ning Bao, Stephen P. Jordan

Here we investigate the use of deep multiscale entanglement renormalization ansatz (DMERA) circuits as a variational ansatz. We use the exactly solvable one-dimensional critical transverse-field Ising model as a test bed. Numerically exact simulation of the quantum circuit ansatz can in this case be carried out to hundreds of qubits by exploiting efficient classical algorithms for simulating matchgate circuits. We find that, for this system, the DMERA strongly outperforms a standard quantum approximate optimization algorithm (QAOA)--style ansatz, and that a major source of systematic error in correlation functions approximated using the DMERA is the breaking of the translational and Kramers-Wannier symmetries of the transverse-field Ising model. We are able to reduce this error by up to four orders of magnitude by symmetry averaging, without incurring additional cost in qubits or circuit depth. We propose that this technique for mitigating systematic error could be applied to noisy intermediate-scale quantum (NISQ) simulations of physical systems with other symmetries.

arXiv:2210.15053•Physical Review A•Apr 26, 2023•3 citations

Heisenberg-Limited Ground-State Energy Estimation for Early Fault-Tolerant Quantum Computers

Lin Lin, Yu Tong

Under suitable assumptions, the quantum-phase-estimation (QPE) algorithm is able to achieve Heisenberg-limited precision scaling in estimating the ground-state energy.However, QPE requires a large number of ancilla qubits and a large circuit depth, as well as the ability to perform inverse quantum Fourier transform, making it expensive to implement on an early fault-tolerant quantum computer.We propose an alternative method to estimate the ground-state energy of a Hamiltonian with Heisenberglimited precision scaling, which employs a simple quantum circuit with one ancilla qubit, and a classical postprocessing procedure.Besides the ground-state energy, our algorithm also produces an approximate cumulative distribution function of the spectral measure, which can be used to compute other spectral properties of the Hamiltonian.

arXiv:2102.11340•PRX Quantum•Feb 2, 2022•176 citations

An assessment of quantum phase estimation protocols for early fault-tolerant quantum computers

Jacob S. Nelson, Andrew Baczewski

We compare several quantum phase estimation (QPE) protocols intended for early fault-tolerant quantum computers (EFTQCs) in the context of models of their implementations on a surface code architecture. We estimate the logical and physical resources required to use these protocols to calculate the ground state energy of molecular hydrogen in a minimal basis with error below $10^{-3}$ atomic units in the presence of depolarizing logical errors. Accounting for the overhead of rotation synthesis and magic state distillation, we find that the total $T$-gate counts do not vary significantly among the EFT QPE protocols at fixed state overlap. In addition to reducing the number of ancilla qubits and circuit depth, the noise robustness of the EFT protocols can be leveraged to reduce resource requirements below those of textbook QPE, realizing approximately a 300-fold reduction in computational volume in some cases. Even so, our estimates are well beyond the scale of existing early fault-tolerance demonstrations.

arXiv:2403.00077•Feb 29, 2024•0 citations

Fault-tolerant quantum computation of molecular observables

Mark Steudtner, Sam Morley-Short, William Pol +8 more

Over the past three decades significant reductions have been made to the cost of estimating ground-state energies of molecular Hamiltonians with quantum computers. However, comparatively little attention has been paid to estimating the expectation values of other observables with respect to said ground states, which is important for many industrial applications. In this work we present a novel expectation value estimation (EVE) quantum algorithm which can be applied to estimate the expectation values of arbitrary observables with respect to any of the system's eigenstates. In particular, we consider two variants of EVE: std-EVE, based on standard quantum phase estimation, and QSP-EVE, which utilizes quantum signal processing (QSP) techniques. We provide rigorous error analysis for both both variants and minimize the number of individual phase factors for QSPEVE. These error analyses enable us to produce constant-factor quantum resource estimates for both std-EVE and QSP-EVE across a variety of molecular systems and observables. For the systems considered, we show that QSP-EVE reduces (Toffoli) gate counts by up to three orders of magnitude and reduces qubit width by up to 25% compared to std-EVE. While estimated resource counts remain far too high for the first generations of fault-tolerant quantum computers, our estimates mark a first of their kind for both the application of expectation value estimation and modern QSP-based techniques.

arXiv:2303.14118•Quantum•Nov 6, 2023•16 citations

Early Fault-Tolerant Quantum Algorithms in Practice: Application to Ground-State Energy Estimation

Oriel Kiss, Utkarsh Azad, Borja Requena +3 more

We investigate the feasibility of early fault-tolerant quantum algorithms focusing on ground-state energy estimation problems. In particular, we examine the computation of the cumulative distribution function (CDF) of the spectral measure of a Hamiltonian and the identification of its discontinuities. Scaling these methods to larger system sizes reveals three key challenges: the smoothness of the CDF for large supports, the lack of tight lower bounds on the overlap with the true ground state, and the difficulty of preparing high-quality initial states. To address these challenges, we propose a signal processing approach to find these estimates automatically, in the regime where the quality of the initial state is unknown. Rather than aiming for exact ground-state energy, we advocate for improving classical estimates by targeting the low-energy support of the initial state. Additionally, we provide quantitative resource estimates, demonstrating a constant factor improvement in the number of samples required to detect a specified change in CDF. Our numerical experiments, conducted on a 26-qubit fully connected Heisenberg model, leverage a truncated density-matrix renormalization group (DMRG) initial state with a low bond dimension. The results show that the predictions from the quantum algorithm align closely with the DMRG-converged energies at larger bond dimensions while requiring several orders of magnitude fewer samples than theoretical estimates suggest. These findings underscore that CDF-based quantum algorithms are a practical and resource-efficient alternative to quantum phase estimation, particularly in resource-constrained scenarios.

arXiv:2405.03754•Quantum•Apr 1, 2025•11 citations
We perform a detailed resource estimate for the prospect of using deep entanglement renormalization ansatz (DMERA) on a fault-tolerant quantum computer, focusing on the regime in which the target system is large. For probing a relatively large system size (64×6464\times 6464×64), we observe up to an order of magnitude reduction in the number of qubits, compared to the approaches based on quantum phase estimation (QPE). We discuss two complementary strategies to measure the energy. The first approach is based on a random sampling of the local terms of the Hamiltonian, requiring O(1/ε2)\mathcal{O}(1/ε^2)O(1/ε2) invocations of quantum circuits, each of which have depth of at most O(log⁡N)\mathcal{O}(\log N)O(logN), where εεε is the relative precision in the energy and NNN is the system size. The second approach is based on a coherent estimation of the expectation value of observables averaged over space, which achieves the Heisenberg scaling while incurring only a logarithmic cost in the system size. For estimating the energy per site of εεε, O(log⁡Nε)\mathcal{O}\left(\frac{\log N}ε \right)O(εlogN​) TTT gates and O(log⁡N)\mathcal{O}\left(\log N \right)O(logN) qubits suffice. The constant factor of the leading contribution is shown to be determined by the depth of the DMERA circuit, the gates used in the ansatz, and the periodicity of the circuit. We also derive tight bounds on the variance of the energy gradient, assuming the gates are random Pauli rotations.
We compare several quantum phase estimation (QPE) protocols intended for early fault-tolerant (EFT) quantum computers in the context of models of their implementations on a surface code architecture. We estimate the logical and physical resources required to use these protocols to calculate the ground-state energy of molecular hydrogen in a minimal basis with error below ${10}^{\ensuremath{-}3}$ atomic units, in the presence of depolarizing logical errors. Accounting for the overhead of rotation synthesis and magic state distillation, we find that the total TTT-gate counts do not vary significantly among the EFT QPE protocols at fixed state overlap. In addition to reducing the number of ancilla qubits and circuit depth, the noise robustness of the EFT protocols can be leveraged to reduce resource requirements below those of textbook QPE, realizing an approximately 300-fold reduction in aggregate computational volume in some cases. We also compare resource requirements for QPE on Trotterized time evolution and qubitized walk operators, for this same minimal exemplar. We estimate the costs of these two approaches to be comparable, with the Trotter approach requiring slightly fewer physical qubits and the qubitized approach requiring slightly fewer code cycles. Even so, our estimates are well beyond the scale of existing early fault-tolerance demonstrations.
Here we investigate the use of deep multiscale entanglement renormalization ansatz (DMERA) circuits as a variational ansatz. We use the exactly solvable one-dimensional critical transverse-field Ising model as a test bed. Numerically exact simulation of the quantum circuit ansatz can in this case be carried out to hundreds of qubits by exploiting efficient classical algorithms for simulating matchgate circuits. We find that, for this system, the DMERA strongly outperforms a standard quantum approximate optimization algorithm (QAOA)--style ansatz, and that a major source of systematic error in correlation functions approximated using the DMERA is the breaking of the translational and Kramers-Wannier symmetries of the transverse-field Ising model. We are able to reduce this error by up to four orders of magnitude by symmetry averaging, without incurring additional cost in qubits or circuit depth. We propose that this technique for mitigating systematic error could be applied to noisy intermediate-scale quantum (NISQ) simulations of physical systems with other symmetries.
Under suitable assumptions, the quantum-phase-estimation (QPE) algorithm is able to achieve Heisenberg-limited precision scaling in estimating the ground-state energy.However, QPE requires a large number of ancilla qubits and a large circuit depth, as well as the ability to perform inverse quantum Fourier transform, making it expensive to implement on an early fault-tolerant quantum computer.We propose an alternative method to estimate the ground-state energy of a Hamiltonian with Heisenberglimited precision scaling, which employs a simple quantum circuit with one ancilla qubit, and a classical postprocessing procedure.Besides the ground-state energy, our algorithm also produces an approximate cumulative distribution function of the spectral measure, which can be used to compute other spectral properties of the Hamiltonian.
We compare several quantum phase estimation (QPE) protocols intended for early fault-tolerant quantum computers (EFTQCs) in the context of models of their implementations on a surface code architecture. We estimate the logical and physical resources required to use these protocols to calculate the ground state energy of molecular hydrogen in a minimal basis with error below 10−310^{-3}10−3 atomic units in the presence of depolarizing logical errors. Accounting for the overhead of rotation synthesis and magic state distillation, we find that the total TTT-gate counts do not vary significantly among the EFT QPE protocols at fixed state overlap. In addition to reducing the number of ancilla qubits and circuit depth, the noise robustness of the EFT protocols can be leveraged to reduce resource requirements below those of textbook QPE, realizing approximately a 300-fold reduction in computational volume in some cases. Even so, our estimates are well beyond the scale of existing early fault-tolerance demonstrations.
Over the past three decades significant reductions have been made to the cost of estimating ground-state energies of molecular Hamiltonians with quantum computers. However, comparatively little attention has been paid to estimating the expectation values of other observables with respect to said ground states, which is important for many industrial applications. In this work we present a novel expectation value estimation (EVE) quantum algorithm which can be applied to estimate the expectation values of arbitrary observables with respect to any of the system's eigenstates. In particular, we consider two variants of EVE: std-EVE, based on standard quantum phase estimation, and QSP-EVE, which utilizes quantum signal processing (QSP) techniques. We provide rigorous error analysis for both both variants and minimize the number of individual phase factors for QSPEVE. These error analyses enable us to produce constant-factor quantum resource estimates for both std-EVE and QSP-EVE across a variety of molecular systems and observables. For the systems considered, we show that QSP-EVE reduces (Toffoli) gate counts by up to three orders of magnitude and reduces qubit width by up to 25% compared to std-EVE. While estimated resource counts remain far too high for the first generations of fault-tolerant quantum computers, our estimates mark a first of their kind for both the application of expectation value estimation and modern QSP-based techniques.
We investigate the feasibility of early fault-tolerant quantum algorithms focusing on ground-state energy estimation problems. In particular, we examine the computation of the cumulative distribution function (CDF) of the spectral measure of a Hamiltonian and the identification of its discontinuities. Scaling these methods to larger system sizes reveals three key challenges: the smoothness of the CDF for large supports, the lack of tight lower bounds on the overlap with the true ground state, and the difficulty of preparing high-quality initial states. To address these challenges, we propose a signal processing approach to find these estimates automatically, in the regime where the quality of the initial state is unknown. Rather than aiming for exact ground-state energy, we advocate for improving classical estimates by targeting the low-energy support of the initial state. Additionally, we provide quantitative resource estimates, demonstrating a constant factor improvement in the number of samples required to detect a specified change in CDF. Our numerical experiments, conducted on a 26-qubit fully connected Heisenberg model, leverage a truncated density-matrix renormalization group (DMRG) initial state with a low bond dimension. The results show that the predictions from the quantum algorithm align closely with the DMRG-converged energies at larger bond dimensions while requiring several orders of magnitude fewer samples than theoretical estimates suggest. These findings underscore that CDF-based quantum algorithms are a practical and resource-efficient alternative to quantum phase estimation, particularly in resource-constrained scenarios.