arXiv subjects

Nick S. Blunt

Publications and source records attributed to Nick S. Blunt.

At least 19 recordsLinked to original sources

arXiv2026

More efficient Clifford+T synthesis for small-angle rotations and application to Trotterization

Clifford+T synthesis of rotation gates is an important routine in fault-tolerant quantum compilation. While Clifford+T synthesis is scalable, it has a high overhead of tens of T gates per rotation in practice, translating to high resource estimates for many fault-tolerant algorithms. However, these well-known results, including those using probabilistic mixtures [Quantum 7, 1208 (2023)], are independent of the rotation angle $θ$, requiring $O(\log 1/δ)$ T gates. We show that it is possible to do much better for small angles, reducing the T cost to $\tilde{O}(θ^2/δ)$, and returning to existing $O(\log1/δ)$ results in the worst case. This is particularly important since many algorithms, such as Trotterization, are dominated by small-angle rotations. Further, we perform a detailed theoretical and numerical study of quasi-probabilities, which can further reduce the total T cost of large circuits by orders of magnitude with only a small overhead in sample complexity. We also develop a scheme based on quasi-probability mixtures of Clifford+T fallback channels. We derive new $θ$-dependent formulas that can be used for resource estimation of fault-tolerant quantum algorithms. As an application of our results, we show that the gate cost of Trotterization circuits compiled to a Clifford+T gate set is constant in the small Trotter step size limit, and can be reduced by orders of magnitude even for large step sizes. The cost of fault-tolerant Trotterization for a variety of applications should be re-examined in light of these results. Our work dispels the widely-stated claim that Clifford+T rotation synthesis has a high cost independent of $θ$, and further develops a scalable quasi-probability method for rotation synthesis. We also expect our results to bring forward useful early fault-tolerant quantum computing by reducing required magic state resources.

quant-ph↗

arXiv2026

Quantum simulation of nanographenes and Trotter error cancellation

Fault-tolerant quantum computing is a promising tool for simulating molecules and materials, but frequently-considered applications require substantial resources, and the gap between hardware capabilities and requirements remains significant. We propose quantum simulation of nanographene $π$-systems as relevant and scalable problems to span the gap between early and large-scale fault-tolerant quantum computing. We examine the efficiency of Trotterized quantum simulation, present a detailed analysis of worst-case, average-case and energy eigenvalue Trotter errors, and show that these Trotter error estimates vary by orders of magnitude. Trotter eigenvalue errors are obtained from a novel tensor-network-based approach which allows spectral analysis of product formulas for systems beyond brute-force calculation. Notably, we observe a Trotter error cancellation phenomenon whereby the Trotter error for energy differences between low-lying eigenstates is significantly smaller than the Trotter error for absolute energies, resulting in approximately an order of magnitude circuit depth reduction for quantum phase estimation calculation of energy gaps. This is a significant result because for most chemical applications, only energy differences are of practical relevance. We estimate that calculation of energy gaps to chemical accuracy between the ground- and excited-states within the Pariser--Parr--Pople model for large 2D nanographenes (up to 140 spin orbitals) requires circuits with $< 3.2 \times 10^7$ Toffoli gates. This work shows that considering details of chemically-relevant applications and exploiting error cancellation can lead to substantial reductions in resource requirements.

quant-ph↗

arXiv2025

A Monte Carlo approach to bound Trotter error

Trotter product formulas are a natural and powerful approach to perform quantum simulation. However, the error analysis of product formulas is challenging, and their cost is often overestimated. It is established that Trotter error can be bounded in terms of spectral norms of nested commutators of the Hamiltonian partitions [Childs et al., Phys. Rev. X 11, 011020], but evaluating these expressions is challenging, often achieved by repeated application of the triangle inequality, significantly loosening the bound. Here, we show that the spectral norm of an operator can be upper bounded by the spectral norm of an equivalent sign-problem-free operator, which can be calculated efficiently to large system sizes using projector Monte Carlo simulation. For a range of Hamiltonians and considering second-order formulas, we demonstrate that this Monte Carlo-based bound is often extremely tight, and even exact in some instances. For the uniform electron gas we reduce the cost of performing Trotterization from the literature by an order of magnitude. For the Pariser-Parr-Pople model for linear acene molecules, which has $\mathcal{O}(N^2)$ long-range interaction terms, we show that it suffices to use $\mathcal{O}(N^{0.57})$ Trotter steps and circuit depth $\mathcal{O}(N^{1.57})$ to implement Hamiltonian simulation. We hope that this approach will lead to a better understanding of the potential accuracy of Trotterization in a range of important applications.

quant-ph↗

arXiv2025

Fault-tolerant quantum simulation of generalized Hubbard models

Quantum simulations of strongly interacting fermionic systems, such as those described by the Hubbard model, are promising candidates for useful early fault-tolerant quantum computing applications. This paper presents Tile Trotterization, a generalization of plaquette Trotterization (PLAQ), which uses a set of tiles to construct Trotter decompositions of arbitrary lattice Hubbard models. The Tile Trotterization scheme also enables the simulation of more complex models, including the extended Hubbard model. We improve previous Hubbard model commutator bounds, further provide tight commutator bounds for periodic extended Hubbard models, and demonstrate the use of tensor network methods for this task. We consider applications of Tile Trotterization to simulate hexagonal lattice Hubbard models and compare the resource requirements of Tile Trotterization for performing quantum phase estimation to a qubitization-based approach, demonstrating that Tile Trotterization scales more efficiently with system size. These advancements significantly broaden the potential applications of early fault-tolerant quantum computers to models of practical interest in materials research and organic chemistry.

quant-ph↗

arXiv2025

The Electronic Structure of the Hydrogen Molecule: A Tutorial Exercise in Classical and Quantum Computation

In this educational paper, we will discuss calculations on the hydrogen molecule both on classical and quantum computers. In the former case, we will discuss the calculation of molecular integrals that can then be used to calculate potential energy curves at the Hartree--Fock level and to correct them by obtaining the exact results for all states in the minimal basis. Some aspects of spin-symmetry will also be discussed. In the case of quantum computing, we will start out from the second-quantized Hamiltonian and qubit mappings. Using quantum phase estimation, we then provide the circuits for two different algorithms: Trotteization and qubitization. Finally, the significance of quantum error correction will be briefly discussed.

physics.chem-ph↗

arXiv2024

A quantum computing approach to fixed-node Monte Carlo using classical shadows

Quantum Monte Carlo (QMC) methods are powerful approaches for solving electronic structure problems. Although they often provide high-accuracy solutions, the precision of most QMC methods is ultimately limited by a trial wave function that must be used. Recently, an approach has been demonstrated to allow the use of trial wave functions prepared on a quantum computer [Nature 603, 416 (2022)] in the auxiliary-field QMC (AFQMC) method, using classical shadows to estimate the required overlaps. However, this approach has an exponential post-processing step to construct these overlaps, when performing classical shadows obtained using random Clifford circuits. Here, we study an approach to avoid this exponentially scaling step by using a fixed-node Monte Carlo method, based on full configuration interaction quantum Monte Carlo (FCIQMC). This method is applied to the local unitary cluster Jastrow (LUCJ) ansatz. We consider H$_4$, ferrocene and benzene molecules using up to 12 qubits as examples. Circuits are compiled to native gates for typical near-term architectures, and we assess the impact of circuit-level depolarizing noise on the method. We also provide a comparison of AFQMC and fixed-node approaches, demonstrating that AFQMC is more robust to errors, although extrapolations of the fixed-node energy reduce this discrepancy. Although the method can be used to reach chemical accuracy, the sampling cost to achieve this is high even for small active spaces, suggesting caution for the prospect of outperforming conventional QMC approaches.

quant-ph↗

arXiv2024

Demonstrating real-time and low-latency quantum error correction with superconducting qubits

Quantum error correction (QEC) will be essential to achieve the accuracy needed for quantum computers to realise their full potential. The field has seen promising progress with demonstrations of early QEC and real-time decoded experiments. As quantum computers advance towards demonstrating a universal fault-tolerant logical gate set, implementing scalable and low-latency real-time decoding will be crucial to prevent the backlog problem, avoiding an exponential slowdown and maintaining a fast logical clock rate. Here, we demonstrate low-latency feedback with a scalable FPGA decoder integrated into the control system of a superconducting quantum processor. We perform an 8-qubit stability experiment with up to $25$ decoding rounds and a mean decoding time per round below $1$ $μs$, showing that we avoid the backlog problem even on superconducting hardware with the strictest speed requirements. We observe logical error suppression as the number of decoding rounds is increased. We also implement and time a fast-feedback experiment demonstrating a decoding response time of $9.6$ $μs$ for a total of $9$ measurement rounds. The decoder throughput and latency developed in this work, combined with continued device improvements, unlock the next generation of experiments that go beyond purely keeping logical qubits alive and into demonstrating building blocks of fault-tolerant computation, such as lattice surgery and magic state teleportation.

quant-ph↗

arXiv2024

Compilation of a simple chemistry application to quantum error correction primitives

A number of exciting recent results have been seen in the field of quantum error correction. These include initial demonstrations of error correction on current quantum hardware, and resource estimates which improve understanding of the requirements to run large-scale quantum algorithms for real-world applications. In this work, we bridge the gap between these two developments by performing careful estimation of the resources required to fault-tolerantly perform quantum phase estimation (QPE) on a minimal chemical example. Specifically, we describe a detailed compilation of the QPE circuit to lattice surgery operations for the rotated surface code, for a hydrogen molecule in a minimal basis set. We describe a number of optimisations at both the algorithmic and error correction levels. We find that implementing even a simple chemistry circuit requires 1,000 qubits and 2,300 quantum error correction rounds, emphasising the need for improved error correction techniques specifically targeting the early fault-tolerant regime.

quant-ph↗

arXiv2023

Statistical phase estimation and error mitigation on a superconducting quantum processor

Quantum phase estimation (QPE) is a key quantum algorithm, which has been widely studied as a method to perform chemistry and solid-state calculations on future fault-tolerant quantum computers. Recently, several authors have proposed statistical alternatives to QPE that have benefits on early fault-tolerant devices, including shorter circuits and better suitability for error mitigation techniques. However, practical implementations of the algorithm on real quantum processors are lacking. In this paper we practically implement statistical phase estimation on Rigetti's superconducting processors. We specifically use the method of Lin and Tong [PRX Quantum 3, 010318 (2022)] using the improved Fourier approximation of Wan et al. [PRL 129, 030503 (2022)], and applying a variational compilation technique to reduce circuit depth. We then incorporate error mitigation strategies including zero-noise extrapolation and readout error mitigation with bit-flip averaging. We propose a simple method to estimate energies from the statistical phase estimation data, which is found to improve the accuracy in final energy estimates by one to two orders of magnitude with respect to prior theoretical bounds, reducing the cost to perform accurate phase estimation calculations. We apply these methods to chemistry problems for active spaces up to 4 electrons in 4 orbitals, including the application of a quantum embedding method, and use them to correctly estimate energies within chemical precision. Our work demonstrates that statistical phase estimation has a natural resilience to noise, particularly after mitigating coherent errors, and can achieve far higher accuracy than suggested by previous analysis, demonstrating its potential as a valuable quantum algorithm for early fault-tolerant devices.

quant-ph↗

arXiv2023

Measuring Electron Correlation. The Impact of Symmetry and Orbital Transformations

In this perspective, the various measures of electron correlation used in wavefunction theory, density functional theory and quantum information theory are briefly reviewed. We then focus on a more traditional metric based on dominant weights in the full configuration solution and discuss its behaviour with respect to the choice of the $N$-electron and the one-electron basis. The impact of symmetry is discussed and we emphasize that the distinction between determinants, configuration state functions and configurations as reference functions is useful because the latter incorporate spin-coupling into the reference and should thus reduce the complexity of the wavefunction expansion. The corresponding notions of single determinant, single spin-coupling and single configuration wavefunctions are discussed and the effect of orbital rotations on the multireference character is reviewed by analysing a simple model system. In molecular systems, the extent of correlation effects should be limited by finite system size and in most cases the appropriate choices of one-electron and $N$-electron bases should be able to incorporate these into a low-complexity reference function, often a single configurational one.

physics.chem-ph↗

arXiv2023

A perspective on the current state-of-the-art of quantum computing for drug discovery applications

Computational chemistry is an essential tool in the pharmaceutical industry. Quantum computing is a fast evolving technology that promises to completely shift the computational capabilities in many areas of chemical research by bringing into reach currently impossible calculations. This perspective illustrates the near-future applicability of quantum computation to pharmaceutical problems. We briefly summarize and compare the scaling properties of state-of-the-art quantum algorithms, and provide novel estimates of the quantum computational cost of simulating progressively larger embedding regions of a pharmaceutically relevant covalent protein-drug complex involving the drug Ibrutinib. Carrying out these calculations requires an error-corrected quantum architecture, that we describe. Our estimates showcase that recent developments on quantum algorithms have dramatically reduced the quantum resources needed to run fully quantum calculations in active spaces of around 50 orbitals and electrons, from estimated over 1000 years using the Trotterisation approach to just a few days with sparse qubitisation, painting a picture of fast and exciting progress in this nascent field.

physics.chem-ph↗

arXiv2022

Quantum Computing in Pharma: A Multilayer Embedding Approach for Near Future Applications

Quantum computers are special purpose machines that are expected to be particularly useful in simulating strongly correlated chemical systems. The quantum computer excels at treating a moderate number of orbitals within an active space in a fully quantum mechanical manner. We present a quantum phase estimation calculation on F$_2$ in a (2,2) active space on Rigetti's Aspen-11 QPU. While this is a promising start, it also underlines the need for carefully selecting the orbital spaces treated by the quantum computer. In this work, a scheme for selecting such an active space automatically is described and simulated results obtained using both the quantum phase estimation (QPE) and variational quantum eigensolver (VQE) algorithms are presented and combined with a subtractive method to enable accurate description of the environment. The active occupied space is selected from orbitals localized on the chemically relevant fragment of the molecule, while the corresponding virtual space is chosen based on the magnitude of interactions with the occupied space calculated from perturbation theory. This protocol is then applied to two chemical systems of pharmaceutical relevance: the enzyme [Fe] hydrogenase and the photosenzitizer temoporfin. While the sizes of the active spaces currently amenable to a quantum computational treatment are not enough to demonstrate quantum advantage, the procedure outlined here is applicable to any active space size, including those that are outside the reach of classical computation.

physics.chem-ph↗

arXiv2021

Fixed and partial-node approximations in Slater determinant space for molecules

We present a study of fixed and partial-node approximations in Slater determinant basis sets, using full configuration interaction quantum Monte Carlo (FCIQMC) to perform sampling. Walker annihilation in the FCIQMC method allows partial-node simulations to be performed, relaxing the nodal constraint to converge to the FCI solution. This is applied to ab initio molecular systems, using symmetry-projected Jastrow mean-field wave functions for complete active space (CAS) problems. Convergence and the sign problem within the partial-node approximation are studied, which is shown to eventually be limited in its use due to the large walker populations required. However the fixed-node approximation results in an accurate and practical method. We apply these approaches to various molecular systems and active spaces, including ferrocene and acenes. This also provides a test of symmetry-projected Jastrow mean-field wave functions in variational Monte Carlo (VMC) for a new set of problems. For trans-polyacetylene molecules and acenes we find that the time to perform a constant number of fixed-node FCIQMC iterations scales as O(N^{1.44}) and O(N^{1.75}) respectively, resulting in an efficient method for CAS-based problems that can be applied accurately to large active spaces.

physics.chem-ph↗

arXiv2020

Efficient multireference perturbation theory without high-order reduced density matrices

We present a stochastic approach to perform strongly contracted n-electron valence state perturbation theory (SC-NEVPT), which only requires one- and two-body reduced density matrices, without introducing approximations. We use this method to perform SC-NEVPT2 for CASSCF wave functions obtained from selected configuration interaction, although the approach is applicable to a larger class of wave functions, including those from variational Monte Carlo (VMC). The accuracy of this approach is demonstrated for small test systems, and the scaling is investigated with the number of virtual orbitals and the molecule size. We also find the SC-NEVPT2 energy to be relatively insensitive to the quality of the reference wave function. Finally, the method is applied to the Fe(II)-Porphyrin system with a (32e, 29o) active space, and to the isomerization of Cu2O2 in a (28e, 32o) active space.

physics.chem-ph↗

arXiv2020

Recent developments in the PySCF program package

PYSCF is a Python-based general-purpose electronic structure platform that both supports first-principles simulations of molecules and solids, as well as accelerates the development of new methodology and complex computational workflows. The present paper explains the design and philosophy behind PYSCF that enables it to meet these twin objectives. With several case studies, we show how users can easily implement their own methods using PYSCF as a development environment. We then summarize the capabilities of PYSCF for molecular and solid-state simulations. Finally, we describe the growing ecosystem of projects that use PYSCF across the domains of quantum chemistry, materials science, machine learning and quantum information science.

physics.chem-ph↗

arXiv2020

NECI: N-Electron Configuration Interaction with emphasis on state-of-the-art stochastic methods

We present NECI, a state-of-the-art implementation of the Full Configuration Interaction Quantum Monte Carlo algorithm, a method based on a stochastic application of the Hamiltonian matrix on a sparse sampling of the wave function. The program utilizes a very powerful parallelization and scales efficiently to more than 24000 CPU cores. In this paper, we describe the core functionalities of NECI and recent developments. This includes the capabilities to calculate ground and excited state energies, properties via the one- and two-body reduced density matrices, as well as spectral and Green's functions for ab initio and model systems. A number of enhancements of the bare FCIQMC algorithm are available within NECI, allowing to use a partially deterministic formulation of the algorithm, working in a spin-adapted basis or supporting transcorrelated Hamiltonians. NECI supports the FCIDUMP file format for integrals, supplying a convenient interface to numerous quantum chemistry programs and it is licensed under GPL-3.0.

physics.comp-ph↗

arXiv2019

Multireference configuration interaction and perturbation theory without reduced density matrices

The computationally expensive evaluation and storage of high-rank reduced density matrices (RDMs) has been the bottleneck in the calculation of dynamic correlation for multireference wave functions in large active spaces. We present a stochastic formulation of multireference configuration interaction (MRCI) and perturbation theory (MRPT) that avoids the need for these expensive RDMs. The algorithm presented here is flexible enough to incorporate a wide variety of active space reference wave functions, including selected configuration interaction, matrix product states, and symmetry-projected Jastrow mean field wave functions. It enjoys the usual attractive features of Monte Carlo methods, such as embarrassing parallelizability and low memory costs. We find that the stochastic algorithm is already competitive with the deterministic algorithm for small active spaces, containing as few as 14 orbitals. We illustrate the utility of our stochastic formulation using benchmark applications.

physics.chem-ph↗

arXiv2019

A hybrid approach to extending selected configuration interaction and full configuration interaction quantum Monte Carlo

We present an approach to combining selected configuration interaction (SCI) and initiator full configuration interaction quantum Monte Carlo (i-FCIQMC). In the current i-FCIQMC scheme, the space of initiators is chosen dynamically by a population threshold. Here, we instead choose initiators as the selected space (V) from a prior SCI calculation, allowing substantially larger initiator spaces for a given walker population. While SCI+PT2 adds a perturbative correction in the first-order interacting space (FOIS) beyond V, the approach presented here allows a variational calculation in the same space, and a perturbative correction in the second-order interacting space. The use of a fixed initiator space reintroduces population plateaus into FCIQMC, but it is shown that the plateau height is typically only a small multiple of the size of V. Thus, for a comparable fundamental memory cost to SCI+PT2, a substantially larger space can be sampled. The resulting method can be seen as a complementary approach to SCI+PT2, which is more accurate but slower for a common selected/initiator space. More generally, our results show that approaches exist to significantly improve initiator energies in i-FCIQMC, while still ameliorating the fermion sign problem relative to the original FCIQMC method.

physics.chem-ph↗